The Probability Function
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Expectation, Variance, and Standard Deviation
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Linear Transformation
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Expectation and Variance of a Sum of Random Variables
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Special Discrete Probability Distributions - Binomial Probability
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Special Discrete Probability Distributions - Geometric Probability
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Special Discrete Probability Distributions - Uniform Probability
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Special Discrete Probability Distributions - Hypergeometric Probability
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Special Discrete Probability Distributions - Negative Binomial Distribution
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Poisson Approximation of the Binomial Probability Distribution
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The Discrete Random Variable - Summary Questions
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Special Discrete Probability Distributions - Poisson Probability
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[{"Name":"The Probability Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"4m 36s","ChapterTopicVideoID":12479,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.170 ","End":"00:03.240","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:03.240 ","End":"00:07.125","Text":"discrete random variables and the probability function."},{"Start":"00:07.125 ","End":"00:12.480","Text":"Now, what\u0027s a discrete random variable?"},{"Start":"00:12.480 ","End":"00:18.465","Text":"Well, it\u0027s a variable that can receive individual values with various probabilities."},{"Start":"00:18.465 ","End":"00:23.370","Text":"For example, let\u0027s define X as"},{"Start":"00:23.370 ","End":"00:29.830","Text":"a random variable having values when you throw a die."},{"Start":"00:29.830 ","End":"00:32.480","Text":"Well, what are the values when you throw a die?"},{"Start":"00:32.480 ","End":"00:34.655","Text":"Can be either 1 or 2,"},{"Start":"00:34.655 ","End":"00:37.410","Text":"or 3, or 4 or 5,"},{"Start":"00:37.410 ","End":"00:44.195","Text":"or 6, and what are the probabilities for each one of the values?"},{"Start":"00:44.195 ","End":"00:46.115","Text":"Well, if it\u0027s a fair die,"},{"Start":"00:46.115 ","End":"00:53.615","Text":"then the probability of X for each X will equal to 1 over 6."},{"Start":"00:53.615 ","End":"00:58.520","Text":"Another example, let\u0027s say y is when you toss a coin."},{"Start":"00:58.520 ","End":"01:02.105","Text":"Now, what is the values of tossing the coin?"},{"Start":"01:02.105 ","End":"01:05.210","Text":"Well, it\u0027s either heads or tails."},{"Start":"01:05.210 ","End":"01:12.315","Text":"What\u0027s the probability of each one of the values when you toss the coin?"},{"Start":"01:12.315 ","End":"01:14.685","Text":"Well, it\u0027s 1/2."},{"Start":"01:14.685 ","End":"01:18.740","Text":"You can have 1/2 for heads and 1/2 for tails if it\u0027s a fair coin."},{"Start":"01:18.740 ","End":"01:25.385","Text":"Now, let\u0027s take a look at an example for a roulette wheel."},{"Start":"01:25.385 ","End":"01:32.025","Text":"Now, let\u0027s assume that this is our roulette wheel and let\u0027s assume or let\u0027s define"},{"Start":"01:32.025 ","End":"01:40.355","Text":"X as the profit that one has when one turns the roulette wheel."},{"Start":"01:40.355 ","End":"01:47.750","Text":"Excellent. What are the values that X can have when he tosses a roulette wheel?"},{"Start":"01:47.750 ","End":"01:50.975","Text":"Well, he can win 10,"},{"Start":"01:50.975 ","End":"01:55.710","Text":"or he can win 20, or he can win 30."},{"Start":"01:55.710 ","End":"02:00.875","Text":"What are the probabilities for each one of the values?"},{"Start":"02:00.875 ","End":"02:04.100","Text":"Well, if X equals 10,"},{"Start":"02:04.100 ","End":"02:09.770","Text":"the probability is quarter 0.25. Why is that?"},{"Start":"02:09.770 ","End":"02:15.805","Text":"Because that\u0027s the area of the circle that\u0027s associated with the number 10."},{"Start":"02:15.805 ","End":"02:19.880","Text":"What\u0027s the probability of X equaling 20?"},{"Start":"02:19.880 ","End":"02:22.820","Text":"Well, again, that\u0027s a quarter."},{"Start":"02:22.820 ","End":"02:26.015","Text":"That\u0027s just like 10."},{"Start":"02:26.015 ","End":"02:30.020","Text":"This is the area associated with the number 20."},{"Start":"02:30.020 ","End":"02:34.085","Text":"What\u0027s the probability of x equaling 30?"},{"Start":"02:34.085 ","End":"02:37.670","Text":"Well, here it\u0027s 1/2. Why is that?"},{"Start":"02:37.670 ","End":"02:41.120","Text":"Because this area right here,"},{"Start":"02:41.120 ","End":"02:45.920","Text":"half of the circle, half of the roulette wheel is associated with the profits of 30."},{"Start":"02:45.920 ","End":"02:51.319","Text":"When we want to write the probability function,"},{"Start":"02:51.319 ","End":"02:53.615","Text":"we do it in a table form."},{"Start":"02:53.615 ","End":"02:56.730","Text":"This is how we do it."},{"Start":"02:58.250 ","End":"03:09.455","Text":"Here we write down the values of X. X can have a value of 10, 20, or 30."},{"Start":"03:09.455 ","End":"03:12.005","Text":"Now, what\u0027s the probability of X?"},{"Start":"03:12.005 ","End":"03:14.420","Text":"Well, we\u0027ve calculated that right here."},{"Start":"03:14.420 ","End":"03:18.810","Text":"That\u0027s 25 percent if X equals 10,"},{"Start":"03:18.810 ","End":"03:22.550","Text":"25 percent if X equals 20,"},{"Start":"03:22.550 ","End":"03:26.060","Text":"and 50 percent if X equals 30."},{"Start":"03:26.060 ","End":"03:28.940","Text":"Now the last thing that you have to do is make"},{"Start":"03:28.940 ","End":"03:35.375","Text":"sure that the sum of all the probabilities have to add up to 1."},{"Start":"03:35.375 ","End":"03:38.164","Text":"If they don\u0027t, you\u0027ve made a mistake."},{"Start":"03:38.164 ","End":"03:39.980","Text":"Let\u0027s take a look."},{"Start":"03:39.980 ","End":"03:42.065","Text":"There\u0027s 50, 75, 1."},{"Start":"03:42.065 ","End":"03:44.330","Text":"That\u0027s excellent."},{"Start":"03:44.330 ","End":"03:47.045","Text":"Let\u0027s just recap."},{"Start":"03:47.045 ","End":"03:48.935","Text":"First of all,"},{"Start":"03:48.935 ","End":"03:53.310","Text":"you have to identify the variable."},{"Start":"03:54.800 ","End":"04:01.530","Text":"Secondly, you have to see what its values are."},{"Start":"04:03.250 ","End":"04:08.150","Text":"The third step is to calculate"},{"Start":"04:08.150 ","End":"04:16.185","Text":"the probabilities for each one of the values and 4,"},{"Start":"04:16.185 ","End":"04:18.355","Text":"let\u0027s not forget our check."},{"Start":"04:18.355 ","End":"04:24.370","Text":"You have to check that the sum equals 1."},{"Start":"04:24.700 ","End":"04:27.380","Text":"Now, in this chapter,"},{"Start":"04:27.380 ","End":"04:31.175","Text":"you\u0027ll be given a whole bunch of problems,"},{"Start":"04:31.175 ","End":"04:37.140","Text":"please solve them by yourselves before you see the solution videos."}],"ID":12958},{"Watched":false,"Name":"Exercise 1","Duration":"4m 9s","ChapterTopicVideoID":12480,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this question we\u0027re given that the distribution of cars in"},{"Start":"00:03.600 ","End":"00:08.385","Text":"a given community is as follows: 50 families have no cars,"},{"Start":"00:08.385 ","End":"00:10.335","Text":"70 families have 1 car,"},{"Start":"00:10.335 ","End":"00:12.255","Text":"60 families have 2 cars,"},{"Start":"00:12.255 ","End":"00:14.625","Text":"and 20 families have 3 cars,"},{"Start":"00:14.625 ","End":"00:18.285","Text":"and a family is randomly selected from the community."},{"Start":"00:18.285 ","End":"00:23.190","Text":"We define X as the number of cars owned by the selected family."},{"Start":"00:23.190 ","End":"00:27.210","Text":"We\u0027re asked to construct the probability function of X."},{"Start":"00:27.210 ","End":"00:29.880","Text":"Well, the first thing that we need to know"},{"Start":"00:29.880 ","End":"00:33.870","Text":"is what\u0027s the number of families in the community?"},{"Start":"00:33.870 ","End":"00:36.690","Text":"What\u0027s the total number of families in the community?"},{"Start":"00:36.690 ","End":"00:40.305","Text":"Well, let\u0027s just sum up these guys right here."},{"Start":"00:40.305 ","End":"00:45.290","Text":"50 plus 70 plus 60 plus 20."},{"Start":"00:45.290 ","End":"00:50.390","Text":"Now, n is the total number of families in the community."},{"Start":"00:50.390 ","End":"00:57.440","Text":"Again, that\u0027s 50 plus 70 plus 60 plus 20,"},{"Start":"00:57.440 ","End":"01:00.149","Text":"and that equals to 200."},{"Start":"01:00.190 ","End":"01:05.660","Text":"Now the next thing that we need to figure out are what are the values of X?"},{"Start":"01:05.660 ","End":"01:10.190","Text":"Now remember we defined X as the number of cars owned by the selected family."},{"Start":"01:10.190 ","End":"01:12.455","Text":"Well, we\u0027re given that here."},{"Start":"01:12.455 ","End":"01:14.985","Text":"No car, 1 car,"},{"Start":"01:14.985 ","End":"01:16.810","Text":"2 cars, and 3 cars."},{"Start":"01:16.810 ","End":"01:20.885","Text":"That means that the values that X can have is 0,"},{"Start":"01:20.885 ","End":"01:25.695","Text":"1, 2, and 3."},{"Start":"01:25.695 ","End":"01:29.465","Text":"Now, what we\u0027re asked,"},{"Start":"01:29.465 ","End":"01:34.010","Text":"we\u0027re asked to construct a probability function of X."},{"Start":"01:34.010 ","End":"01:36.590","Text":"That means that we want to know what\u0027s"},{"Start":"01:36.590 ","End":"01:41.000","Text":"the probability of X when it equals 0, for example."},{"Start":"01:41.000 ","End":"01:44.060","Text":"Now, in order to do that,"},{"Start":"01:44.060 ","End":"01:48.422","Text":"we have to take the proportion of families that have no car"},{"Start":"01:48.422 ","End":"01:56.975","Text":"from the total number of families that are under community."},{"Start":"01:56.975 ","End":"01:59.540","Text":"In this case where X equals 0,"},{"Start":"01:59.540 ","End":"02:02.765","Text":"that\u0027s 50 over 200,"},{"Start":"02:02.765 ","End":"02:08.600","Text":"and that equals to 0.25."},{"Start":"02:08.600 ","End":"02:11.150","Text":"The probability of X equals 1,"},{"Start":"02:11.150 ","End":"02:14.735","Text":"that\u0027s the probability of a family having 1 car."},{"Start":"02:14.735 ","End":"02:17.030","Text":"Well, that\u0027s the proportion of families that have"},{"Start":"02:17.030 ","End":"02:19.355","Text":"1 car from the total number of families,"},{"Start":"02:19.355 ","End":"02:26.425","Text":"that\u0027s 70 over 200, and that\u0027s 0.35."},{"Start":"02:26.425 ","End":"02:32.565","Text":"Now, let\u0027s do that for the probability of X equals 2,"},{"Start":"02:32.565 ","End":"02:36.465","Text":"and that\u0027s 60 over 200,"},{"Start":"02:36.465 ","End":"02:38.715","Text":"and that\u0027s 0.3,"},{"Start":"02:38.715 ","End":"02:43.190","Text":"and the probability of X equals 3, well,"},{"Start":"02:43.190 ","End":"02:46.230","Text":"that\u0027s 20 over 200,"},{"Start":"02:46.230 ","End":"02:49.445","Text":"and that equals to 0.1."},{"Start":"02:49.445 ","End":"02:54.740","Text":"Now, let\u0027s construct our probability function."},{"Start":"02:54.740 ","End":"02:58.325","Text":"Let\u0027s draw this in a table form."},{"Start":"02:58.325 ","End":"03:01.430","Text":"Let\u0027s do that here."},{"Start":"03:01.430 ","End":"03:05.635","Text":"X and P of x."},{"Start":"03:05.635 ","End":"03:11.369","Text":"Now we said that the values of X are 0, 1, 2."},{"Start":"03:11.369 ","End":"03:19.610","Text":"We\u0027ve already figured out the probabilities of each 1 of the values."},{"Start":"03:19.610 ","End":"03:24.145","Text":"That\u0027s right here, that\u0027s 0.25, right here."},{"Start":"03:24.145 ","End":"03:28.710","Text":"0.35 for X equals 1."},{"Start":"03:28.710 ","End":"03:32.235","Text":"For X equals 2, that\u0027s 0.3,"},{"Start":"03:32.235 ","End":"03:36.065","Text":"and for X equals 3, that\u0027s 0.1."},{"Start":"03:36.065 ","End":"03:41.075","Text":"Now, this is a probability function right here."},{"Start":"03:41.075 ","End":"03:45.170","Text":"The only thing that we have to check is that,"},{"Start":"03:45.170 ","End":"03:48.425","Text":"the sum of the probabilities equals 1."},{"Start":"03:48.425 ","End":"03:51.800","Text":"Well, that\u0027s 0.1 plus 0.3."},{"Start":"03:51.800 ","End":"03:53.495","Text":"That\u0027s your 0.4."},{"Start":"03:53.495 ","End":"03:56.880","Text":"0.4 plus 0.35,"},{"Start":"03:56.880 ","End":"04:01.155","Text":"that\u0027s 0.75 plus 0.25, that equals what?"},{"Start":"04:01.155 ","End":"04:08.970","Text":"We\u0027re okay, and here is our probability function for this question."}],"ID":12959},{"Watched":false,"Name":"Exercise 2","Duration":"5m ","ChapterTopicVideoID":12481,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this question, we\u0027ll be dealing with codes."},{"Start":"00:02.910 ","End":"00:07.350","Text":"Now, a two-letter code is composed from the letters A, B,"},{"Start":"00:07.350 ","End":"00:10.260","Text":"and C and we\u0027re asked how many codes can be"},{"Start":"00:10.260 ","End":"00:15.270","Text":"composed and we\u0027re also asked to list all the possible codes."},{"Start":"00:15.270 ","End":"00:19.170","Text":"Let\u0027s get added. In Section a,"},{"Start":"00:19.170 ","End":"00:21.450","Text":"how many codes can be composed?"},{"Start":"00:21.450 ","End":"00:25.470","Text":"Well, let\u0027s define n_1 as"},{"Start":"00:25.470 ","End":"00:31.785","Text":"the possible number of letters that we can put in the first place. Well, that\u0027s 3."},{"Start":"00:31.785 ","End":"00:33.950","Text":"You can put it in the first place,"},{"Start":"00:33.950 ","End":"00:39.270","Text":"either A or B or C. The same goes for the second place."},{"Start":"00:39.270 ","End":"00:43.865","Text":"Let\u0027s define n_2 as a possible number of letters that we can put in the second place."},{"Start":"00:43.865 ","End":"00:46.080","Text":"That\u0027s also 3."},{"Start":"00:46.520 ","End":"00:57.665","Text":"The possible number of letters that the code can have is n_1 times n_2 equals 3 times 3,"},{"Start":"00:57.665 ","End":"00:59.800","Text":"that equals to 9."},{"Start":"00:59.800 ","End":"01:02.005","Text":"In Section b,"},{"Start":"01:02.005 ","End":"01:04.760","Text":"we\u0027re asked to list all the possible codes."},{"Start":"01:04.760 ","End":"01:08.390","Text":"Well, in essence, what we\u0027re trying to do is to map out"},{"Start":"01:08.390 ","End":"01:13.280","Text":"the sample space let\u0027s list out all the possible codes."},{"Start":"01:13.280 ","End":"01:20.075","Text":"But we have AA, AB, and AC."},{"Start":"01:20.075 ","End":"01:23.125","Text":"We also have BA,"},{"Start":"01:23.125 ","End":"01:27.180","Text":"BB, and BC."},{"Start":"01:27.180 ","End":"01:30.165","Text":"We also have CA,"},{"Start":"01:30.165 ","End":"01:33.745","Text":"CB, and CC."},{"Start":"01:33.745 ","End":"01:38.495","Text":"Now, all of this is the sample space."},{"Start":"01:38.495 ","End":"01:41.630","Text":"In Section c,"},{"Start":"01:41.630 ","End":"01:46.285","Text":"we define X as the number of times that B appears in the code."},{"Start":"01:46.285 ","End":"01:50.440","Text":"We\u0027re asked to construct a probability function of X."},{"Start":"01:50.440 ","End":"01:53.230","Text":"Well before we can do that,"},{"Start":"01:53.230 ","End":"01:56.620","Text":"let\u0027s first go over the definition of X."},{"Start":"01:56.620 ","End":"02:01.430","Text":"X is the number of times that B appears in the corner."},{"Start":"02:01.430 ","End":"02:02.510","Text":"This is very important."},{"Start":"02:02.510 ","End":"02:06.170","Text":"We have to understand what our variable"},{"Start":"02:06.170 ","End":"02:10.340","Text":"is before we can even construct a probability function."},{"Start":"02:10.340 ","End":"02:12.575","Text":"Well, let\u0027s see."},{"Start":"02:12.575 ","End":"02:14.660","Text":"From Question b,"},{"Start":"02:14.660 ","End":"02:16.435","Text":"of Section b above,"},{"Start":"02:16.435 ","End":"02:19.590","Text":"this is our sample space."},{"Start":"02:19.590 ","End":"02:21.140","Text":"What we\u0027re asked to do,"},{"Start":"02:21.140 ","End":"02:23.030","Text":"we\u0027re asked to find out, first of all,"},{"Start":"02:23.030 ","End":"02:26.780","Text":"what\u0027s the number of times that B appears in the code?"},{"Start":"02:26.780 ","End":"02:30.660","Text":"Well, let\u0027s see here,0,"},{"Start":"02:30.770 ","End":"02:34.620","Text":"this is a number of times the B appears in this code."},{"Start":"02:34.620 ","End":"02:36.855","Text":"Here, B appears once,"},{"Start":"02:36.855 ","End":"02:38.925","Text":"here it appears 0,"},{"Start":"02:38.925 ","End":"02:41.815","Text":"here it appears once, here,"},{"Start":"02:41.815 ","End":"02:44.584","Text":"it appears twice, once,"},{"Start":"02:44.584 ","End":"02:48.245","Text":"0, once, and 0."},{"Start":"02:48.245 ","End":"02:55.515","Text":"This is the number of times that B appears in the code."},{"Start":"02:55.515 ","End":"03:01.380","Text":"Now we can construct the probability function of X."},{"Start":"03:01.380 ","End":"03:05.460","Text":"Now, how many values can x have?"},{"Start":"03:05.460 ","End":"03:07.170","Text":"Well, we can see from here."},{"Start":"03:07.170 ","End":"03:11.800","Text":"It can have either 0 or 1 or 2."},{"Start":"03:12.440 ","End":"03:17.314","Text":"Let\u0027s construct our probability function."},{"Start":"03:17.314 ","End":"03:22.140","Text":"We have here the values of X, that\u0027s 0,1,"},{"Start":"03:22.140 ","End":"03:29.125","Text":"and 2 and probabilities of X."},{"Start":"03:29.125 ","End":"03:35.990","Text":"Now, what\u0027s the probability of X being equal to 0?"},{"Start":"03:35.990 ","End":"03:42.965","Text":"Well, all we have to do is find the relative frequency of X when it equals to 0."},{"Start":"03:42.965 ","End":"03:45.800","Text":"How many times does 0 appear here?"},{"Start":"03:45.800 ","End":"03:48.520","Text":"Well, that\u0027s once, twice,"},{"Start":"03:48.520 ","End":"03:50.775","Text":"3, and 4 times."},{"Start":"03:50.775 ","End":"03:52.995","Text":"That\u0027s 4 out of what?"},{"Start":"03:52.995 ","End":"03:58.180","Text":"Well, we have 9 possible values for the code."},{"Start":"03:58.250 ","End":"04:02.685","Text":"The probability of X equal 1."},{"Start":"04:02.685 ","End":"04:05.400","Text":"Well, how many times does X equal?"},{"Start":"04:05.400 ","End":"04:09.390","Text":"1, 2, 3, and 4."},{"Start":"04:09.390 ","End":"04:13.080","Text":"Again, that\u0027s 4/9."},{"Start":"04:13.080 ","End":"04:16.730","Text":"What\u0027s the probability of X equaling 2?"},{"Start":"04:16.730 ","End":"04:20.755","Text":"That\u0027s only once, That\u0027s 1/9."},{"Start":"04:20.755 ","End":"04:24.965","Text":"Let\u0027s put in these probabilities back into the table."},{"Start":"04:24.965 ","End":"04:29.125","Text":"The probability of X equals 0, that\u0027s 4/9."},{"Start":"04:29.125 ","End":"04:32.220","Text":"The probability of X equals 1,"},{"Start":"04:32.220 ","End":"04:39.215","Text":"that\u0027s 4/9 and the probability of X equaling 2, that\u0027s 1/9."},{"Start":"04:39.215 ","End":"04:46.490","Text":"All we have to do now is to make sure that the sum of the probabilities equals 1."},{"Start":"04:46.490 ","End":"04:49.700","Text":"Well that\u0027s 4 plus 4 plus 1 over 9."},{"Start":"04:49.700 ","End":"04:54.320","Text":"That means it\u0027s 9/9 equals 1, so we\u0027re okay."},{"Start":"04:54.320 ","End":"05:00.510","Text":"This is our probability function."}],"ID":12960},{"Watched":false,"Name":"Exercise 3","Duration":"6m 1s","ChapterTopicVideoID":12482,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this question, we\u0027ll be talking about the chances of a student"},{"Start":"00:03.060 ","End":"00:06.465","Text":"passing an economics and a statistics exam."},{"Start":"00:06.465 ","End":"00:11.580","Text":"Now, a student takes semester exams in economics and statistics."},{"Start":"00:11.580 ","End":"00:14.310","Text":"The chances of passing the economics exam are"},{"Start":"00:14.310 ","End":"00:19.230","Text":"80 percent and the chances of passing the statistics exam are 90 percent."},{"Start":"00:19.230 ","End":"00:23.595","Text":"The chances of passing both exams are 75 percent."},{"Start":"00:23.595 ","End":"00:27.540","Text":"Now, let X be the number of exams that the student passes."},{"Start":"00:27.540 ","End":"00:31.680","Text":"We have to construct the probability function of X."},{"Start":"00:31.680 ","End":"00:36.075","Text":"In order to do that, let\u0027s define our variables first."},{"Start":"00:36.075 ","End":"00:42.880","Text":"Let A be the event that the student passes the economics exam,"},{"Start":"00:44.210 ","End":"00:53.045","Text":"and B is the event that the student passes the statistics exam."},{"Start":"00:53.045 ","End":"00:56.210","Text":"Now the probability of A is given to us,"},{"Start":"00:56.210 ","End":"00:57.905","Text":"that\u0027s 80 percent,"},{"Start":"00:57.905 ","End":"01:03.065","Text":"and the probability of B is 90 percent."},{"Start":"01:03.065 ","End":"01:12.080","Text":"The probability of passing both the economics and the statistics exam is 75 percent."},{"Start":"01:12.080 ","End":"01:13.775","Text":"Now, we see that here,"},{"Start":"01:13.775 ","End":"01:15.080","Text":"that\u0027s given to us."},{"Start":"01:15.080 ","End":"01:17.495","Text":"That\u0027s 80 percent for economics,"},{"Start":"01:17.495 ","End":"01:19.715","Text":"90 percent for statistics,"},{"Start":"01:19.715 ","End":"01:23.730","Text":"and 75 percent for both."},{"Start":"01:24.050 ","End":"01:29.390","Text":"Now, there are several ways that we can present the data."},{"Start":"01:29.390 ","End":"01:33.290","Text":"Well, we can do it through a probability matrix,"},{"Start":"01:33.290 ","End":"01:36.635","Text":"or we can do this through Venn diagrams."},{"Start":"01:36.635 ","End":"01:41.704","Text":"Now, I want to show you how we do this through a Venn diagram."},{"Start":"01:41.704 ","End":"01:45.470","Text":"Let\u0027s draw this out, and here it is."},{"Start":"01:45.470 ","End":"01:49.640","Text":"We can see the probabilities right on the Venn diagram."},{"Start":"01:49.640 ","End":"01:54.020","Text":"Let\u0027s start with the probability of A intersect B."},{"Start":"01:54.020 ","End":"01:56.285","Text":"Well, that\u0027s 75 percent."},{"Start":"01:56.285 ","End":"02:02.610","Text":"We can see that here in the intersect between A and B."},{"Start":"02:02.610 ","End":"02:07.240","Text":"Now, let\u0027s look at the probability of A, that\u0027s 80 percent."},{"Start":"02:07.240 ","End":"02:12.160","Text":"Now, 75 percent is in the intersection,"},{"Start":"02:12.160 ","End":"02:14.425","Text":"in order to make it up to 80,"},{"Start":"02:14.425 ","End":"02:18.400","Text":"that means that only 5 percent is outside of the intersect,"},{"Start":"02:18.400 ","End":"02:22.215","Text":"in order for the probability of all of it to be 80 percent."},{"Start":"02:22.215 ","End":"02:25.500","Text":"This 5 percent just represents the people who passed"},{"Start":"02:25.500 ","End":"02:30.235","Text":"the economics exam but did not pass the statistics exam."},{"Start":"02:30.235 ","End":"02:34.870","Text":"Likewise, the probability of B is 90 percent."},{"Start":"02:34.870 ","End":"02:39.730","Text":"Now, we know that 75 percent of the 90 percent is in the intersect."},{"Start":"02:39.730 ","End":"02:46.980","Text":"That means that 15 percent has to be outside of the intersect and all that means is"},{"Start":"02:46.980 ","End":"02:49.790","Text":"that 15 percent of the students have passed"},{"Start":"02:49.790 ","End":"02:54.485","Text":"the statistics exam but did not pass the economics exam."},{"Start":"02:54.485 ","End":"03:02.845","Text":"Now, if we look at all the probabilities within the events A,"},{"Start":"03:02.845 ","End":"03:04.384","Text":"B, and the intersect,"},{"Start":"03:04.384 ","End":"03:08.210","Text":"we see that we\u0027re missing about 5 percent, that\u0027s here."},{"Start":"03:08.210 ","End":"03:13.400","Text":"What does that mean? That means that this is a probability of"},{"Start":"03:13.400 ","End":"03:20.345","Text":"the students that did not pass either the economics or the statistics exam."},{"Start":"03:20.345 ","End":"03:22.955","Text":"Now, having said that,"},{"Start":"03:22.955 ","End":"03:26.600","Text":"let\u0027s get back to our question. What do we want to do?"},{"Start":"03:26.600 ","End":"03:31.085","Text":"We want to construct the probability function of X,"},{"Start":"03:31.085 ","End":"03:33.005","Text":"where X is what?"},{"Start":"03:33.005 ","End":"03:37.200","Text":"The number of exams that the student passes."},{"Start":"03:37.580 ","End":"03:42.140","Text":"How many exams can the student pass?"},{"Start":"03:42.140 ","End":"03:45.615","Text":"Well, he can pass either 0 exams,"},{"Start":"03:45.615 ","End":"03:49.385","Text":"where he doesn\u0027t pass either the statistics or the economics exam."},{"Start":"03:49.385 ","End":"03:51.440","Text":"He can pass 1 exam,"},{"Start":"03:51.440 ","End":"03:53.090","Text":"and it doesn\u0027t matter which right now,"},{"Start":"03:53.090 ","End":"03:55.535","Text":"it could be the economics or the statistics exam,"},{"Start":"03:55.535 ","End":"03:59.050","Text":"or he can pass both exams."},{"Start":"04:03.770 ","End":"04:10.330","Text":"What\u0027s the probability of X being equal to 0?"},{"Start":"04:10.330 ","End":"04:16.670","Text":"Well, we have that right here. That\u0027s 0.05."},{"Start":"04:16.670 ","End":"04:21.230","Text":"What\u0027s the probability of X being equal to 1?"},{"Start":"04:21.230 ","End":"04:24.695","Text":"That means that we\u0027re looking at the probability of"},{"Start":"04:24.695 ","End":"04:29.510","Text":"a student passing the economics and not passing the statistics,"},{"Start":"04:29.510 ","End":"04:34.685","Text":"that\u0027s 0.05 plus,"},{"Start":"04:34.685 ","End":"04:39.230","Text":"or the probability that a student"},{"Start":"04:39.230 ","End":"04:45.780","Text":"passes the statistics but doesn\u0027t pass the economics, that\u0027s 0.15."},{"Start":"04:45.780 ","End":"04:49.520","Text":"That comes out to 0.2."},{"Start":"04:49.810 ","End":"04:55.810","Text":"Then what\u0027s left is the probability of X passing both exams."},{"Start":"04:55.810 ","End":"04:59.875","Text":"Well, that\u0027s right here in the intersect, that\u0027s 0.75."},{"Start":"04:59.875 ","End":"05:05.110","Text":"Now we have all the information that we need to construct our probability function."},{"Start":"05:05.110 ","End":"05:11.875","Text":"Let\u0027s do that. We have X and we have the probability of X."},{"Start":"05:11.875 ","End":"05:14.710","Text":"Now, X can be either 0, or 1, or 2."},{"Start":"05:14.710 ","End":"05:19.840","Text":"We said that. We\u0027ve already calculated the probability of"},{"Start":"05:19.840 ","End":"05:26.225","Text":"X where X equals 0, that\u0027s 0.05."},{"Start":"05:26.225 ","End":"05:32.100","Text":"The probability of X being equal to 1, well that\u0027s 0.2."},{"Start":"05:32.110 ","End":"05:38.700","Text":"The probability of X being equal to 2, that\u0027s 75 percent."},{"Start":"05:38.700 ","End":"05:42.770","Text":"Excellent. All we have to do right now is"},{"Start":"05:42.770 ","End":"05:47.180","Text":"to make sure that the sum of our probabilities add up to 1."},{"Start":"05:47.180 ","End":"05:51.060","Text":"That\u0027s 75 percent plus 20 percent,"},{"Start":"05:51.060 ","End":"05:52.440","Text":"that\u0027s 95 percent,"},{"Start":"05:52.440 ","End":"05:54.735","Text":"plus 5 percent, that\u0027s 100 percent."},{"Start":"05:54.735 ","End":"06:01.650","Text":"We\u0027re good. This then is the probability function of X."}],"ID":12961},{"Watched":false,"Name":"Exercise 4","Duration":"5m 3s","ChapterTopicVideoID":12483,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this question, we\u0027ll be talking about"},{"Start":"00:02.820 ","End":"00:06.104","Text":"the chances of a person winning a series of games."},{"Start":"00:06.104 ","End":"00:11.130","Text":"Now the chances of winning 1 specific game, are 30 percent."},{"Start":"00:11.130 ","End":"00:14.496","Text":"A person plays the game until he wins."},{"Start":"00:14.496 ","End":"00:15.945","Text":"When he wins he quits."},{"Start":"00:15.945 ","End":"00:19.935","Text":"But in any case, he doesn\u0027t play the game more than 4 times."},{"Start":"00:19.935 ","End":"00:23.040","Text":"We define x as the number of times he plays"},{"Start":"00:23.040 ","End":"00:27.795","Text":"the game and we\u0027re asked to construct the probability function for x."},{"Start":"00:27.795 ","End":"00:31.035","Text":"Now before we can do that,"},{"Start":"00:31.035 ","End":"00:34.075","Text":"we want to describe the data."},{"Start":"00:34.075 ","End":"00:38.285","Text":"Now the best way to describe the data is by probability tree."},{"Start":"00:38.285 ","End":"00:45.275","Text":"Why is that? Well, that\u0027s because the data or the story is presented to us in stages."},{"Start":"00:45.275 ","End":"00:47.595","Text":"He plays the first game,"},{"Start":"00:47.595 ","End":"00:49.850","Text":"if he wins, he quits, if he doesn\u0027t,"},{"Start":"00:49.850 ","End":"00:53.270","Text":"he goes on to the next stage which is the next game, and again,"},{"Start":"00:53.270 ","End":"00:57.065","Text":"he plays that if he wins he quits and so on and so forth."},{"Start":"00:57.065 ","End":"00:59.300","Text":"Now if that\u0027s the case,"},{"Start":"00:59.300 ","End":"01:02.105","Text":"let\u0027s construct our probability tree."},{"Start":"01:02.105 ","End":"01:04.505","Text":"Here we see the first stage."},{"Start":"01:04.505 ","End":"01:06.740","Text":"If he wins, he quits,"},{"Start":"01:06.740 ","End":"01:08.390","Text":"and if he doesn\u0027t,"},{"Start":"01:08.390 ","End":"01:11.130","Text":"then he plays another game."},{"Start":"01:11.560 ","End":"01:13.580","Text":"Here\u0027s the other game."},{"Start":"01:13.580 ","End":"01:15.560","Text":"The second game, if he wins again,"},{"Start":"01:15.560 ","End":"01:18.500","Text":"he quits and if he doesn\u0027t win,"},{"Start":"01:18.500 ","End":"01:21.335","Text":"then he goes onto play the third game."},{"Start":"01:21.335 ","End":"01:23.155","Text":"If he wins he quits,"},{"Start":"01:23.155 ","End":"01:26.818","Text":"if he loses he goes on to play the fourth game,"},{"Start":"01:26.818 ","End":"01:31.760","Text":"and at this stage he either wins or loses, but that\u0027s it."},{"Start":"01:31.760 ","End":"01:34.646","Text":"He doesn\u0027t play anymore."},{"Start":"01:34.646 ","End":"01:36.515","Text":"So if that\u0027s the case,"},{"Start":"01:36.515 ","End":"01:46.265","Text":"let\u0027s see what\u0027s the probability of x being 1 or equaling 1."},{"Start":"01:46.265 ","End":"01:52.525","Text":"What\u0027s that, that\u0027s basically the probability of a person winning at the first game."},{"Start":"01:52.525 ","End":"01:56.115","Text":"Well, that\u0027s 0.3, that\u0027s right here."},{"Start":"01:56.115 ","End":"01:59.360","Text":"He wins. Now what\u0027s"},{"Start":"01:59.360 ","End":"02:04.550","Text":"the probability of a person winning on the second try or the second game?"},{"Start":"02:04.550 ","End":"02:09.470","Text":"Well, that\u0027s the probability of the person losing in the first game,"},{"Start":"02:09.470 ","End":"02:13.235","Text":"and winning in the second game that\u0027s 0.7,"},{"Start":"02:13.235 ","End":"02:14.780","Text":"losing in the first game,"},{"Start":"02:14.780 ","End":"02:18.680","Text":"times winning on the second that\u0027s 0.3."},{"Start":"02:18.680 ","End":"02:22.052","Text":"The probability of x equals 3,"},{"Start":"02:22.052 ","End":"02:24.740","Text":"that means he loses in the first game,"},{"Start":"02:24.740 ","End":"02:26.885","Text":"loses in the second game,"},{"Start":"02:26.885 ","End":"02:28.820","Text":"and in the third game he wins."},{"Start":"02:28.820 ","End":"02:36.365","Text":"That equals to 0.7 times 0.7 times 0.3."},{"Start":"02:36.365 ","End":"02:41.045","Text":"Loses, loses, and then he wins."},{"Start":"02:41.045 ","End":"02:48.020","Text":"Now what\u0027s the probability of x equals 4 if the person place 4 games."},{"Start":"02:48.020 ","End":"02:50.555","Text":"Well, he loses in the first,"},{"Start":"02:50.555 ","End":"02:52.040","Text":"loses in the second,"},{"Start":"02:52.040 ","End":"02:55.387","Text":"loses in the third,"},{"Start":"02:55.387 ","End":"03:02.180","Text":"so we have 0.7 times 0.7 times 0.7,"},{"Start":"03:02.180 ","End":"03:04.520","Text":"and then he either wins or loses."},{"Start":"03:04.520 ","End":"03:09.780","Text":"So that means that you either multiply it by the probability of winning,"},{"Start":"03:10.310 ","End":"03:15.275","Text":"or you multiply it by the probability of him losing that\u0027s"},{"Start":"03:15.275 ","End":"03:21.875","Text":"0.7 times 0.7 times 0.7 times 0.7."},{"Start":"03:21.875 ","End":"03:24.470","Text":"This is the part where he wins,"},{"Start":"03:24.470 ","End":"03:27.410","Text":"the probability of him winning in the fourth game,"},{"Start":"03:27.410 ","End":"03:30.830","Text":"and this is the probability of him losing in the fourth game."},{"Start":"03:30.830 ","End":"03:35.650","Text":"Now let\u0027s calculate the probabilities."},{"Start":"03:35.650 ","End":"03:38.805","Text":"For x equals 1,"},{"Start":"03:38.805 ","End":"03:42.160","Text":"obviously that equals 0.3."},{"Start":"03:43.400 ","End":"03:47.430","Text":"For x equals 2,"},{"Start":"03:47.430 ","End":"03:51.700","Text":"well, that equals to 0.21."},{"Start":"03:52.700 ","End":"03:59.265","Text":"For x equals 3, well,"},{"Start":"03:59.265 ","End":"04:05.435","Text":"that equals to 0.147,"},{"Start":"04:05.435 ","End":"04:07.475","Text":"and for x equals 4,"},{"Start":"04:07.475 ","End":"04:11.495","Text":"well, that equals to 0.343."},{"Start":"04:11.495 ","End":"04:17.315","Text":"Now let\u0027s represent that better in a table."},{"Start":"04:17.315 ","End":"04:21.544","Text":"So again, we have x and we have the probability of x,"},{"Start":"04:21.544 ","End":"04:23.840","Text":"now x can either be 1,"},{"Start":"04:23.840 ","End":"04:27.215","Text":"or 2, or 3, or 4."},{"Start":"04:27.215 ","End":"04:32.180","Text":"Now the probability of x being 1,"},{"Start":"04:32.180 ","End":"04:35.160","Text":"0.3, when x equals 2,"},{"Start":"04:35.160 ","End":"04:38.400","Text":"that\u0027s 0.21, when x equals 3,"},{"Start":"04:38.400 ","End":"04:42.720","Text":"that\u0027s 0.147, and when x equals 4,"},{"Start":"04:42.720 ","End":"04:47.255","Text":"that equals to 0.343."},{"Start":"04:47.255 ","End":"04:50.825","Text":"Now all we need to do right now,"},{"Start":"04:50.825 ","End":"04:56.825","Text":"is to check that the probabilities equal 1 and they do, trust me."},{"Start":"04:56.825 ","End":"05:03.450","Text":"This is our probability function for x."}],"ID":12962},{"Watched":false,"Name":"Exercise 5","Duration":"9m 31s","ChapterTopicVideoID":12484,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"Project deals with the success of projects."},{"Start":"00:03.645 ","End":"00:08.475","Text":"Now, a project management company manages 3 projects simultaneously."},{"Start":"00:08.475 ","End":"00:11.655","Text":"The chances of project A succeeding are 0.7,"},{"Start":"00:11.655 ","End":"00:14.610","Text":"the chances of project B succeeding are 0.8,"},{"Start":"00:14.610 ","End":"00:17.895","Text":"and the chances of project C succeeding are 0.9."},{"Start":"00:17.895 ","End":"00:21.870","Text":"The success of each project is independent of other projects."},{"Start":"00:21.870 ","End":"00:24.180","Text":"We define X as the number of"},{"Start":"00:24.180 ","End":"00:30.240","Text":"successful projects and we\u0027re asked to construct the probability function of X."},{"Start":"00:30.240 ","End":"00:34.470","Text":"Now, the first thing that we need to do is to describe the data,"},{"Start":"00:34.470 ","End":"00:39.435","Text":"and the best way that we can do this is through a probability tree."},{"Start":"00:39.435 ","End":"00:42.190","Text":"We can describe the data in"},{"Start":"00:42.190 ","End":"00:46.505","Text":"many other techniques such as the probability matrix and so on and so forth,"},{"Start":"00:46.505 ","End":"00:48.380","Text":"but as we said,"},{"Start":"00:48.380 ","End":"00:51.530","Text":"the best way we can do this is for a probability tree,"},{"Start":"00:51.530 ","End":"00:53.970","Text":"and let\u0027s draw this out right now,"},{"Start":"00:53.970 ","End":"00:55.950","Text":"and here it is."},{"Start":"00:55.950 ","End":"01:00.800","Text":"The first thing we need to know is that since Project A,"},{"Start":"01:00.800 ","End":"01:03.980","Text":"B, and C are independent of each other,"},{"Start":"01:03.980 ","End":"01:11.735","Text":"then we can write out the various levels of the probability tree in various ways."},{"Start":"01:11.735 ","End":"01:17.150","Text":"For example, we can put here in the first level project C and here project A,"},{"Start":"01:17.150 ","End":"01:18.665","Text":"it really doesn\u0027t matter,"},{"Start":"01:18.665 ","End":"01:21.685","Text":"but at least we need to be consistent,"},{"Start":"01:21.685 ","End":"01:26.150","Text":"so let\u0027s make the first level of the tree project A,"},{"Start":"01:26.150 ","End":"01:28.280","Text":"the second level project B,"},{"Start":"01:28.280 ","End":"01:30.125","Text":"and the third level,"},{"Start":"01:30.125 ","End":"01:36.650","Text":"project C. The first thing that we need to do is write the probabilities."},{"Start":"01:36.650 ","End":"01:38.545","Text":"Well, in project A,"},{"Start":"01:38.545 ","End":"01:44.255","Text":"we see that the probability of success is 0.7."},{"Start":"01:44.255 ","End":"01:51.230","Text":"If that\u0027s the case, let\u0027s write down 0.7 every time we see a plus,"},{"Start":"01:51.230 ","End":"01:54.392","Text":"that means that the project is successful,"},{"Start":"01:54.392 ","End":"01:58.040","Text":"so this is the branch where project A is"},{"Start":"01:58.040 ","End":"02:01.925","Text":"successful and this is a branch we\u0027re project A is not successful,"},{"Start":"02:01.925 ","End":"02:04.060","Text":"that\u0027s because there\u0027s a minus here."},{"Start":"02:04.060 ","End":"02:09.640","Text":"The probability of A being not successful is 0.3."},{"Start":"02:10.400 ","End":"02:16.070","Text":"The probability of B succeeding is 0.8."},{"Start":"02:16.070 ","End":"02:18.245","Text":"Let\u0027s write that down."},{"Start":"02:18.245 ","End":"02:22.700","Text":"Here that\u0027s the branch of B being successful, that\u0027s 0.8,"},{"Start":"02:22.700 ","End":"02:26.000","Text":"so every time we see a plus at this level,"},{"Start":"02:26.000 ","End":"02:29.420","Text":"we\u0027ll write down 0.8, and here we have 0.8."},{"Start":"02:29.420 ","End":"02:32.855","Text":"That means that this is 0.2, that\u0027s the minus."},{"Start":"02:32.855 ","End":"02:34.580","Text":"Here is again that\u0027s a minus,"},{"Start":"02:34.580 ","End":"02:36.695","Text":"so that\u0027s 0.2,"},{"Start":"02:36.695 ","End":"02:42.440","Text":"and let\u0027s go down to level C. The success of project C,"},{"Start":"02:42.440 ","End":"02:47.430","Text":"the probability of project C succeeding is 0.9."},{"Start":"02:47.590 ","End":"02:51.560","Text":"Every time we see a plus at this level,"},{"Start":"02:51.560 ","End":"02:54.270","Text":"we\u0027ll write 0.9, so that\u0027s here,"},{"Start":"02:54.270 ","End":"02:57.103","Text":"that\u0027s 0.9, that\u0027s 0.9,"},{"Start":"02:57.103 ","End":"03:01.555","Text":"0.9, and 0.9."},{"Start":"03:01.555 ","End":"03:05.855","Text":"That means that the probability of not succeeding 0.1,"},{"Start":"03:05.855 ","End":"03:08.810","Text":"and we\u0027ll write that every time we see a minus,"},{"Start":"03:08.810 ","End":"03:09.920","Text":"so here\u0027s the minus,"},{"Start":"03:09.920 ","End":"03:14.270","Text":"that\u0027s 0.1, 0.1,"},{"Start":"03:14.270 ","End":"03:16.475","Text":"0.1, and 0.1."},{"Start":"03:16.475 ","End":"03:23.895","Text":"Great. The next step is to define X."},{"Start":"03:23.895 ","End":"03:27.005","Text":"We defined X as what?"},{"Start":"03:27.005 ","End":"03:32.120","Text":"As the number of successful projects. What does that mean?"},{"Start":"03:32.120 ","End":"03:37.100","Text":"That means that every time we see a plus at each level,"},{"Start":"03:37.100 ","End":"03:38.675","Text":"then we have to count it."},{"Start":"03:38.675 ","End":"03:43.295","Text":"For example, if we go to this branch right here,"},{"Start":"03:43.295 ","End":"03:46.695","Text":"we see that project A is successful,"},{"Start":"03:46.695 ","End":"03:48.705","Text":"project B is successful,"},{"Start":"03:48.705 ","End":"03:51.239","Text":"and project C is successful."},{"Start":"03:51.239 ","End":"03:54.810","Text":"That means that here, X equals 3."},{"Start":"03:54.810 ","End":"03:57.210","Text":"Now, if we go here,"},{"Start":"03:57.210 ","End":"03:59.085","Text":"project A is successful,"},{"Start":"03:59.085 ","End":"04:01.470","Text":"project B is successful and project C,"},{"Start":"04:01.470 ","End":"04:05.320","Text":"not successful, that means that x equals 2."},{"Start":"04:05.320 ","End":"04:08.705","Text":"Here we have plus minus plus,"},{"Start":"04:08.705 ","End":"04:10.950","Text":"that\u0027s X equals 2."},{"Start":"04:10.950 ","End":"04:12.780","Text":"We\u0027re counting pluses,"},{"Start":"04:12.780 ","End":"04:16.125","Text":"and here we have plus, minus, minus."},{"Start":"04:16.125 ","End":"04:19.350","Text":"That means here X equals 1."},{"Start":"04:19.350 ","End":"04:22.795","Text":"Let\u0027s go to the other branch here."},{"Start":"04:22.795 ","End":"04:25.100","Text":"That\u0027s minus plus, plus."},{"Start":"04:25.100 ","End":"04:28.455","Text":"That\u0027s X equals 2, 2 pluses."},{"Start":"04:28.455 ","End":"04:32.455","Text":"Minus plus minus, that\u0027s X equals 1."},{"Start":"04:32.455 ","End":"04:37.360","Text":"That\u0027s minus minus plus that\u0027s X equals 1."},{"Start":"04:37.360 ","End":"04:42.650","Text":"That\u0027s minus minus minus that means that X equals 0."},{"Start":"04:43.900 ","End":"04:49.885","Text":"Now, we have basically all the values that X can have,"},{"Start":"04:49.885 ","End":"04:51.850","Text":"so let\u0027s write it out here."},{"Start":"04:51.850 ","End":"04:55.870","Text":"X can have the values of 0,"},{"Start":"04:55.870 ","End":"05:00.305","Text":"1, 2, and 3."},{"Start":"05:00.305 ","End":"05:04.850","Text":"Excellent. Let\u0027s now try"},{"Start":"05:04.850 ","End":"05:09.200","Text":"to figure out what the probabilities are for each one of these values."},{"Start":"05:09.200 ","End":"05:15.860","Text":"The probability of X being equal to 0, what does that equal?"},{"Start":"05:15.860 ","End":"05:19.760","Text":"That equals to this branch right here."},{"Start":"05:19.760 ","End":"05:26.135","Text":"That\u0027s 0.3, project A not being successful times"},{"Start":"05:26.135 ","End":"05:35.570","Text":"0.2 that\u0027s the probability of B not being successful times 0.1,"},{"Start":"05:35.570 ","End":"05:43.895","Text":"that\u0027s the probability of C not being successful, and that\u0027s 0.006."},{"Start":"05:43.895 ","End":"05:49.805","Text":"What\u0027s the probability of X equaling 1?"},{"Start":"05:49.805 ","End":"05:55.880","Text":"That means, what\u0027s the probability of at least 1 project being successful?"},{"Start":"05:55.880 ","End":"05:58.100","Text":"Well, let\u0027s see what we have here."},{"Start":"05:58.100 ","End":"06:01.190","Text":"Well, we have X equals 1 here,"},{"Start":"06:01.190 ","End":"06:04.710","Text":"here, here, and here."},{"Start":"06:04.750 ","End":"06:09.515","Text":"Let\u0027s write out the probabilities for each 1 of these branches."},{"Start":"06:09.515 ","End":"06:10.760","Text":"This branch right here,"},{"Start":"06:10.760 ","End":"06:15.200","Text":"it\u0027s 0.3 times 0.2 times"},{"Start":"06:15.200 ","End":"06:24.510","Text":"0.9 plus where else"},{"Start":"06:24.510 ","End":"06:25.680","Text":"do we have X equals 1?"},{"Start":"06:25.680 ","End":"06:34.775","Text":"Right here. 0.3 times 0.8 times 0.1."},{"Start":"06:34.775 ","End":"06:35.900","Text":"That\u0027s this guy right here,"},{"Start":"06:35.900 ","End":"06:38.210","Text":"that\u0027s this branch right here."},{"Start":"06:38.210 ","End":"06:42.945","Text":"Where else do we have X equals 1 plus? Right here."},{"Start":"06:42.945 ","End":"06:49.955","Text":"0.7 times 0.2 times 0.1"},{"Start":"06:49.955 ","End":"06:58.080","Text":"times 0.1 and that equals to 0.092."},{"Start":"07:01.250 ","End":"07:04.685","Text":"We have this for X equals 0."},{"Start":"07:04.685 ","End":"07:08.825","Text":"Here we have the probability for X equals 1."},{"Start":"07:08.825 ","End":"07:14.465","Text":"What\u0027s the probability of X equaling 2?"},{"Start":"07:14.465 ","End":"07:18.540","Text":"Well, again, that\u0027s where X equals 2,"},{"Start":"07:18.540 ","End":"07:23.005","Text":"that\u0027s here, here, and here."},{"Start":"07:23.005 ","End":"07:25.985","Text":"Let\u0027s calculate that."},{"Start":"07:25.985 ","End":"07:29.870","Text":"That\u0027s 0.3 times 0.8 times"},{"Start":"07:29.870 ","End":"07:35.870","Text":"0.90"},{"Start":"07:36.080 ","End":"07:41.790","Text":"plus this guy right here,"},{"Start":"07:42.250 ","End":"07:52.840","Text":"0.7 times 0.2 times 0.9 plus this branch right here."},{"Start":"07:52.970 ","End":"07:59.640","Text":"It\u0027s plus 0.7 times 0.8 times"},{"Start":"07:59.640 ","End":"08:09.010","Text":"0.1 and all this equals 0.498."},{"Start":"08:12.590 ","End":"08:17.900","Text":"What\u0027s the probability of X being equal to 3?"},{"Start":"08:17.900 ","End":"08:24.124","Text":"Well, that\u0027s this guy right here,"},{"Start":"08:24.124 ","End":"08:29.435","Text":"so it\u0027s 0.7 times 0.8"},{"Start":"08:29.435 ","End":"08:36.390","Text":"times 0.9 and that equals to 0.504."},{"Start":"08:36.890 ","End":"08:39.140","Text":"Now, if that\u0027s the case,"},{"Start":"08:39.140 ","End":"08:42.665","Text":"let\u0027s write out the probability function."},{"Start":"08:42.665 ","End":"08:47.100","Text":"That\u0027s X. That\u0027s the probability of X."},{"Start":"08:47.200 ","End":"08:51.860","Text":"Now X can be either 0, 1, 2,"},{"Start":"08:51.860 ","End":"08:58.940","Text":"or 3, so if X equals 0, we have 0.006."},{"Start":"08:58.940 ","End":"09:04.070","Text":"If X equals 1, we have 0.092."},{"Start":"09:04.070 ","End":"09:05.900","Text":"If X equals 2,"},{"Start":"09:05.900 ","End":"09:15.465","Text":"we have 0.498 and if X equals 3, we have 0.504."},{"Start":"09:15.465 ","End":"09:20.375","Text":"The only thing that needs to be done is to make sure"},{"Start":"09:20.375 ","End":"09:25.740","Text":"that the probabilities add up to 1 and they do."},{"Start":"09:25.900 ","End":"09:31.920","Text":"This is the probability function of X."}],"ID":12963},{"Watched":false,"Name":"Exercise 6","Duration":"2m 58s","ChapterTopicVideoID":12485,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"The following is a probability function of a given random variable."},{"Start":"00:03.930 ","End":"00:10.755","Text":"We have the probability of X equaling k equals k over A, where what?"},{"Start":"00:10.755 ","End":"00:13.380","Text":"Where k equals 1, 2, 3,"},{"Start":"00:13.380 ","End":"00:18.615","Text":"or 4 and we\u0027re asked to find the value of k. Now,"},{"Start":"00:18.615 ","End":"00:20.865","Text":"in order to do that, first,"},{"Start":"00:20.865 ","End":"00:25.365","Text":"let\u0027s write down the probability function with the variable."},{"Start":"00:25.365 ","End":"00:31.700","Text":"We have x and we have the probability of x."},{"Start":"00:31.700 ","End":"00:34.040","Text":"Now x can take on the values 1,"},{"Start":"00:34.040 ","End":"00:36.305","Text":"2, 3, and 4. Right here."},{"Start":"00:36.305 ","End":"00:39.890","Text":"It\u0027s 1, 2, 3, and 4."},{"Start":"00:39.890 ","End":"00:45.500","Text":"Now what\u0027s the probability of x given that x equals 1?"},{"Start":"00:45.500 ","End":"00:52.200","Text":"Well, that\u0027s 1 over A and where x equals 2."},{"Start":"00:52.340 ","End":"00:55.665","Text":"That\u0027s 2 over A,"},{"Start":"00:55.665 ","End":"00:58.305","Text":"and where x equals 3,"},{"Start":"00:58.305 ","End":"01:00.300","Text":"that\u0027s 3 over A,"},{"Start":"01:00.300 ","End":"01:03.255","Text":"and 4 that\u0027ll be 4 over A."},{"Start":"01:03.255 ","End":"01:09.965","Text":"Now, we know that the sum of all the probabilities have to add up to 1."},{"Start":"01:09.965 ","End":"01:13.080","Text":"Let\u0027s write this down;"},{"Start":"01:13.080 ","End":"01:23.000","Text":"1 over A plus 2 over A plus 3 over A plus 4 over A must equal to 1."},{"Start":"01:23.000 ","End":"01:26.735","Text":"Now let\u0027s multiply both sides by A."},{"Start":"01:26.735 ","End":"01:32.930","Text":"We have 1 plus 2 plus 3 plus 4 equals A."},{"Start":"01:32.930 ","End":"01:37.290","Text":"That means that A equals 10."},{"Start":"01:37.700 ","End":"01:41.810","Text":"Now, let\u0027s write this out."},{"Start":"01:41.810 ","End":"01:49.690","Text":"Write the probability function of x now inserting A as 10."},{"Start":"01:49.690 ","End":"01:56.760","Text":"That\u0027ll be x, that\u0027ll be the probability of x as 1,"},{"Start":"01:56.760 ","End":"02:00.120","Text":"2, 3, and 4."},{"Start":"02:00.120 ","End":"02:03.360","Text":"That\u0027ll be 1 over 10,"},{"Start":"02:03.360 ","End":"02:06.945","Text":"2 over 10, 3 over 10,"},{"Start":"02:06.945 ","End":"02:09.070","Text":"and 4 over 10."},{"Start":"02:09.070 ","End":"02:11.135","Text":"Now here we have this."},{"Start":"02:11.135 ","End":"02:15.405","Text":"We have the probability function as a table"},{"Start":"02:15.405 ","End":"02:20.809","Text":"and we can also write the probability function as an equation."},{"Start":"02:20.809 ","End":"02:25.110","Text":"Here we go, the probability of x being equal to k,"},{"Start":"02:25.110 ","End":"02:28.665","Text":"where k equals 1, 2,"},{"Start":"02:28.665 ","End":"02:34.095","Text":"and 3, and 4, and that equals to k over 10."},{"Start":"02:34.095 ","End":"02:44.045","Text":"Now, as we said we can write the probability function as an equation or as a table."},{"Start":"02:44.045 ","End":"02:47.210","Text":"They\u0027re equivalent, they\u0027re the same."},{"Start":"02:47.210 ","End":"02:51.275","Text":"But getting back to our question,"},{"Start":"02:51.275 ","End":"02:55.160","Text":"we\u0027re asked to find the value of A and this is what we did."},{"Start":"02:55.160 ","End":"02:58.260","Text":"Here we go, A equals 10."}],"ID":12964},{"Watched":false,"Name":"Exercise 7","Duration":"3m 57s","ChapterTopicVideoID":12486,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"This problem deals with kindergarten,"},{"Start":"00:02.580 ","End":"00:04.755","Text":"and children, and plays."},{"Start":"00:04.755 ","End":"00:07.860","Text":"Now, a kindergarten has 8 children,"},{"Start":"00:07.860 ","End":"00:10.035","Text":"5 boys and 3 girls."},{"Start":"00:10.035 ","End":"00:13.845","Text":"3 children are randomly selected to take part in a play."},{"Start":"00:13.845 ","End":"00:18.855","Text":"We define X as the number of boys selected to take part in the play."},{"Start":"00:18.855 ","End":"00:23.220","Text":"We\u0027re asked to construct the probability function of X."},{"Start":"00:23.220 ","End":"00:27.390","Text":"Now, there are many ways to solve this problem,"},{"Start":"00:27.390 ","End":"00:32.380","Text":"but let\u0027s use what we know from combinatorics."},{"Start":"00:33.530 ","End":"00:38.760","Text":"We know that we have X."},{"Start":"00:38.760 ","End":"00:42.970","Text":"What\u0027s that? That\u0027s the number of boys selected."},{"Start":"00:43.120 ","End":"00:46.310","Text":"We need the probability of X."},{"Start":"00:46.310 ","End":"00:48.350","Text":"Now how many boys can be selected?"},{"Start":"00:48.350 ","End":"00:53.300","Text":"Well, only 3 people or 3 children are selected for the play."},{"Start":"00:53.300 ","End":"00:58.025","Text":"The number of boys that are selected to be either 0,"},{"Start":"00:58.025 ","End":"01:02.555","Text":"or 1, or 2, or 3."},{"Start":"01:02.555 ","End":"01:05.225","Text":"Now, as we said,"},{"Start":"01:05.225 ","End":"01:08.190","Text":"what\u0027s our sample space?"},{"Start":"01:09.200 ","End":"01:19.900","Text":"The Sample space is basically the fact that we need to take 8 and 3."},{"Start":"01:19.900 ","End":"01:25.215","Text":"We have 8 children and we need to take 3 of them."},{"Start":"01:25.215 ","End":"01:29.100","Text":"That comes out to 56."},{"Start":"01:29.100 ","End":"01:36.675","Text":"Now, what\u0027s the probability of X being 0?"},{"Start":"01:36.675 ","End":"01:39.240","Text":"Well, as we said,"},{"Start":"01:39.240 ","End":"01:44.085","Text":"we have 5 boys and 0 are chosen"},{"Start":"01:44.085 ","End":"01:53.015","Text":"times 3 girls and 3 of them are chosen divided by 56."},{"Start":"01:53.015 ","End":"01:59.580","Text":"That equals to 1/56."},{"Start":"01:59.840 ","End":"02:02.930","Text":"Now, let\u0027s do this."},{"Start":"02:02.930 ","End":"02:06.780","Text":"Probability of X equals 1,"},{"Start":"02:06.780 ","End":"02:08.040","Text":"1 boy is chosen."},{"Start":"02:08.040 ","End":"02:11.160","Text":"That means out of the 5 boys, 1 is chosen."},{"Start":"02:11.160 ","End":"02:13.305","Text":"That means that out of the 3 girls,"},{"Start":"02:13.305 ","End":"02:18.785","Text":"2 are chosen and we have to divide that by 56."},{"Start":"02:18.785 ","End":"02:23.975","Text":"That turns out to be 15/56."},{"Start":"02:23.975 ","End":"02:28.380","Text":"What\u0027s the probability of 2 boys chosen?"},{"Start":"02:28.380 ","End":"02:32.065","Text":"Well, that\u0027s 5/2. Now,"},{"Start":"02:32.065 ","End":"02:33.965","Text":"how many girls are chosen then?"},{"Start":"02:33.965 ","End":"02:39.730","Text":"3/1, 1 girl\u0027s chosen divided by 56."},{"Start":"02:39.730 ","End":"02:45.100","Text":"That equals to 30/56."},{"Start":"02:45.100 ","End":"02:49.670","Text":"The probability of 3 boys being chosen."},{"Start":"02:49.670 ","End":"02:58.065","Text":"Again that\u0027s 5/3 now times 3/0 divided by 56,"},{"Start":"02:58.065 ","End":"03:04.105","Text":"and that equals to 10/56."},{"Start":"03:04.105 ","End":"03:10.640","Text":"Now, we have all the probabilities that we need for each 1 of the values of X."},{"Start":"03:10.640 ","End":"03:13.310","Text":"Let\u0027s just put it back into the table."},{"Start":"03:13.310 ","End":"03:17.715","Text":"The probability of X where X equals 0,"},{"Start":"03:17.715 ","End":"03:24.975","Text":"that\u0027s 1/56, where X equals 1, that\u0027s 15/56."},{"Start":"03:24.975 ","End":"03:29.475","Text":"X equals 2 is 30/56."},{"Start":"03:29.475 ","End":"03:30.830","Text":"When X equals 3,"},{"Start":"03:30.830 ","End":"03:33.830","Text":"the probability is 10/56."},{"Start":"03:33.830 ","End":"03:36.890","Text":"Now the only thing that we need to do is to make sure"},{"Start":"03:36.890 ","End":"03:40.985","Text":"that all the probabilities add up to 1."},{"Start":"03:40.985 ","End":"03:43.730","Text":"That\u0027s 10 plus 30, that\u0027s 40,"},{"Start":"03:43.730 ","End":"03:48.635","Text":"that\u0027s 55 plus 1, that\u0027s 56/56."},{"Start":"03:48.635 ","End":"03:56.850","Text":"We\u0027re good. The probability function of X is this guy right here."}],"ID":12965},{"Watched":false,"Name":"Exercise 8","Duration":"8m 10s","ChapterTopicVideoID":12487,"CourseChapterTopicPlaylistID":64481,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.804","Text":"This question deals with TV ratings."},{"Start":"00:02.804 ","End":"00:07.690","Text":"Now, a survey examined whether people watched the news broadcasts of Channels 1,"},{"Start":"00:07.690 ","End":"00:11.624","Text":"2 and 3, and the following information was discovered."},{"Start":"00:11.624 ","End":"00:15.180","Text":"20 percent of the viewers watch Channel 2,"},{"Start":"00:15.180 ","End":"00:18.430","Text":"8 percent of the viewers watch Channel 1,"},{"Start":"00:18.430 ","End":"00:21.765","Text":"and 10 percent of the viewers watch Channel 3."},{"Start":"00:21.765 ","End":"00:28.275","Text":"Now in addition, 1 percent of the viewers watched all 3 channels together."},{"Start":"00:28.275 ","End":"00:32.570","Text":"10 percent of the viewers watched 2 of the 3 channels."},{"Start":"00:32.570 ","End":"00:39.845","Text":"Now, we define x as the number of news broadcasts watched by randomly selected viewer."},{"Start":"00:39.845 ","End":"00:45.065","Text":"Now, we\u0027re asked to construct a probability function of x."},{"Start":"00:45.065 ","End":"00:49.760","Text":"Now if x is defined as the number of news broadcasts that are watched,"},{"Start":"00:49.760 ","End":"00:52.580","Text":"well, what are the values that x can have?"},{"Start":"00:52.580 ","End":"00:55.230","Text":"Well, x can be 0,"},{"Start":"00:55.230 ","End":"01:00.590","Text":"that means that no one watches any of the channels."},{"Start":"01:00.590 ","End":"01:07.145","Text":"X can be 1, a person watches either Channel 1 or Channel 2 or Channel 3."},{"Start":"01:07.145 ","End":"01:11.720","Text":"X can be 2, a person can watch 2 of the 3 channels,"},{"Start":"01:11.720 ","End":"01:13.055","Text":"Channel 1 and 2,"},{"Start":"01:13.055 ","End":"01:16.625","Text":"Channels 1 and 3 or Channel 2 and 3,"},{"Start":"01:16.625 ","End":"01:22.450","Text":"and x can be 3 where a person watches all 3 of the channels."},{"Start":"01:22.450 ","End":"01:24.605","Text":"If that\u0027s the case,"},{"Start":"01:24.605 ","End":"01:30.755","Text":"let\u0027s now try to calculate or construct the probability function."},{"Start":"01:30.755 ","End":"01:33.170","Text":"Well, we have x right here."},{"Start":"01:33.170 ","End":"01:35.930","Text":"Now what are the probability of x?"},{"Start":"01:35.930 ","End":"01:39.458","Text":"Well, as we said, x can take on the values of 0,"},{"Start":"01:39.458 ","End":"01:42.830","Text":"1, 2 and 3."},{"Start":"01:42.830 ","End":"01:49.315","Text":"Now, we already have some of the information that\u0027s given to us right here."},{"Start":"01:49.315 ","End":"01:51.450","Text":"Let\u0027s look at this thing."},{"Start":"01:51.450 ","End":"01:55.160","Text":"1 percent of the viewers watch all 3 channels together."},{"Start":"01:55.160 ","End":"02:00.890","Text":"That means that the probability of x equals to 3. What\u0027s x?"},{"Start":"02:00.890 ","End":"02:02.945","Text":"That\u0027s the number of news broadcast."},{"Start":"02:02.945 ","End":"02:05.510","Text":"All 3 of them are being watched"},{"Start":"02:05.510 ","End":"02:08.450","Text":"and the probability of all 3 of them watched is 1 percent."},{"Start":"02:08.450 ","End":"02:11.250","Text":"That means that that equals 0.01."},{"Start":"02:11.470 ","End":"02:16.785","Text":"Likewise, 10 percent of the viewers watch 2 of the 3 channels."},{"Start":"02:16.785 ","End":"02:23.480","Text":"That means that the probability of x being equal to 2 equals 10 percent."},{"Start":"02:23.480 ","End":"02:27.170","Text":"Now, let\u0027s plug in these guys right back in here."},{"Start":"02:27.170 ","End":"02:29.960","Text":"Where x equals 3,"},{"Start":"02:29.960 ","End":"02:33.095","Text":"the probability is 0.01,"},{"Start":"02:33.095 ","End":"02:38.135","Text":"where x equals 2 the probability is 0.1, that\u0027s 10 percent."},{"Start":"02:38.135 ","End":"02:43.910","Text":"Now, all we have to do then is to calculate the probability of x,"},{"Start":"02:43.910 ","End":"02:46.520","Text":"where x equals 0 and x equals 1."},{"Start":"02:46.520 ","End":"02:52.745","Text":"Now let\u0027s try to calculate the probability of x equaling 1."},{"Start":"02:52.745 ","End":"02:54.395","Text":"Before we can do that,"},{"Start":"02:54.395 ","End":"02:57.110","Text":"let\u0027s just define some more variables."},{"Start":"02:57.110 ","End":"03:01.460","Text":"Well, let\u0027s define the A,"},{"Start":"03:01.460 ","End":"03:06.620","Text":"as the event that you viewed Channel 1,"},{"Start":"03:06.620 ","End":"03:15.190","Text":"and B, the event of viewing Channel 2,"},{"Start":"03:15.500 ","End":"03:23.181","Text":"and C, the event of viewing Channel 3,"},{"Start":"03:23.181 ","End":"03:27.155","Text":"and the probability of B,"},{"Start":"03:27.155 ","End":"03:28.220","Text":"well that\u0027s given to us,"},{"Start":"03:28.220 ","End":"03:32.410","Text":"that\u0027s right here, that\u0027s 20 percent."},{"Start":"03:32.410 ","End":"03:36.430","Text":"The probability of A,"},{"Start":"03:37.190 ","End":"03:40.110","Text":"that\u0027s 8 percent, that\u0027s right here,"},{"Start":"03:40.110 ","End":"03:45.200","Text":"and the probability of C is"},{"Start":"03:45.200 ","End":"03:52.620","Text":"10 percent, right here."},{"Start":"03:53.180 ","End":"04:00.875","Text":"I think we\u0027re now all ready to start calculating the probability that x equals 1."},{"Start":"04:00.875 ","End":"04:06.060","Text":"Now let\u0027s use a Venn diagram to help us out with our probabilities."},{"Start":"04:06.130 ","End":"04:10.865","Text":"Now, when x equals 3, what does that mean?"},{"Start":"04:10.865 ","End":"04:17.585","Text":"That means that people are watching all 3 broadcasts at the same time."},{"Start":"04:17.585 ","End":"04:19.685","Text":"Now, on our Venn diagram,"},{"Start":"04:19.685 ","End":"04:25.440","Text":"that\u0027s the intersection of all 3 events, that\u0027s right here."},{"Start":"04:25.900 ","End":"04:28.580","Text":"We know what the probability of that,"},{"Start":"04:28.580 ","End":"04:31.985","Text":"that\u0027s 1 percent, that\u0027s 0.01."},{"Start":"04:31.985 ","End":"04:36.650","Text":"Now, let\u0027s take a look at x equals 2."},{"Start":"04:36.650 ","End":"04:43.650","Text":"What does that mean? That means that people are watching 2 out of the 3 broadcasts."},{"Start":"04:43.650 ","End":"04:46.535","Text":"On the Venn diagram, we see that here,"},{"Start":"04:46.535 ","End":"04:51.395","Text":"as the intersection of 2 of the 3 events."},{"Start":"04:51.395 ","End":"04:59.410","Text":"That\u0027s this intersection right here and this intersection right here,"},{"Start":"04:59.410 ","End":"05:05.100","Text":"and this intersection right here."},{"Start":"05:05.100 ","End":"05:08.900","Text":"Now, what about x equals 1?"},{"Start":"05:08.900 ","End":"05:11.990","Text":"Well, what\u0027s the probability of x equal 1?"},{"Start":"05:11.990 ","End":"05:13.340","Text":"How can we see that?"},{"Start":"05:13.340 ","End":"05:16.100","Text":"How can we represent that on the Venn diagram?"},{"Start":"05:16.100 ","End":"05:19.770","Text":"Well, we can see that right here."},{"Start":"05:22.940 ","End":"05:27.120","Text":"This section right here, that\u0027s 1,"},{"Start":"05:27.120 ","End":"05:30.000","Text":"then it\u0027s this one right here,"},{"Start":"05:30.000 ","End":"05:36.963","Text":"and also this one right here."},{"Start":"05:36.963 ","End":"05:42.560","Text":"When we look or when"},{"Start":"05:42.560 ","End":"05:48.495","Text":"we want to calculate the probability of x being equal to 1,"},{"Start":"05:48.495 ","End":"05:51.395","Text":"well, that\u0027s basically saying,"},{"Start":"05:51.395 ","End":"05:54.980","Text":"I want to add up all the probabilities here."},{"Start":"05:54.980 ","End":"05:59.780","Text":"That\u0027s the probability of A plus the probability of B,"},{"Start":"05:59.780 ","End":"06:03.163","Text":"plus the probability of C. But hold on,"},{"Start":"06:03.163 ","End":"06:10.043","Text":"some of the sections we\u0027ve counted them twice or even thrice,"},{"Start":"06:10.043 ","End":"06:13.340","Text":"so the blue sections, we\u0027ve counted twice."},{"Start":"06:13.340 ","End":"06:14.929","Text":"Now what\u0027s the blue section?"},{"Start":"06:14.929 ","End":"06:17.030","Text":"That\u0027s x equals 2."},{"Start":"06:17.030 ","End":"06:24.680","Text":"That means that we have to take away 2 times the probability of x equaling 2."},{"Start":"06:24.680 ","End":"06:27.080","Text":"What about the red section?"},{"Start":"06:27.080 ","End":"06:30.275","Text":"Well, we\u0027ve counted that 3 times,"},{"Start":"06:30.275 ","End":"06:36.852","Text":"so we have to take away 3 times the probability of x equals 3,"},{"Start":"06:36.852 ","End":"06:44.000","Text":"and this expression equals the probability of x being equal to 1."},{"Start":"06:44.000 ","End":"06:48.206","Text":"Now, let\u0027s just plug in the numbers,"},{"Start":"06:48.206 ","End":"06:57.830","Text":"so we have probability of A is 0.08 plus the probability of B that\u0027s 0.2,"},{"Start":"06:57.830 ","End":"07:01.340","Text":"plus the probability of C that\u0027s 01."},{"Start":"07:01.340 ","End":"07:03.440","Text":"That\u0027s these guys right here,"},{"Start":"07:03.440 ","End":"07:07.415","Text":"minus 2 times the probability of x equals 2."},{"Start":"07:07.415 ","End":"07:16.535","Text":"That\u0027s here, 2 times 0.1 minus 3 times the probability of x equals 3,"},{"Start":"07:16.535 ","End":"07:20.060","Text":"that\u0027s this guy right here, 0.01,"},{"Start":"07:20.060 ","End":"07:28.810","Text":"and that equals to 0.15."},{"Start":"07:30.050 ","End":"07:37.715","Text":"Great. Let\u0027s plug that in right here, that\u0027s 0.15."},{"Start":"07:37.715 ","End":"07:45.295","Text":"Now, since we know that all the probabilities here have to sum up to 1,"},{"Start":"07:45.295 ","End":"07:51.769","Text":"well, then we automatically know what the probability of x equaling 0."},{"Start":"07:51.769 ","End":"08:00.200","Text":"That turns out to be 0.74."},{"Start":"08:00.200 ","End":"08:07.700","Text":"Here we go. We\u0027ve calculated or constructed the probability function of x."},{"Start":"08:07.700 ","End":"08:10.560","Text":"It\u0027s this guy right here."}],"ID":12966}],"Thumbnail":null,"ID":64481},{"Name":"Expectation, Variance, and Standard Deviation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"34s","ChapterTopicVideoID":12488,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.555","Text":"In this chapter, we\u0027ll be dealing with the expectation,"},{"Start":"00:03.555 ","End":"00:08.475","Text":"variance and standard deviation of a discrete random variable."},{"Start":"00:08.475 ","End":"00:12.350","Text":"Now, these estimates, the expectation,"},{"Start":"00:12.350 ","End":"00:16.800","Text":"variance and standard deviation, are very important."},{"Start":"00:16.800 ","End":"00:21.120","Text":"They\u0027re the main estimators that describe the central point of"},{"Start":"00:21.120 ","End":"00:27.510","Text":"the probability function and how scattered is the data around the central point."},{"Start":"00:27.510 ","End":"00:31.410","Text":"Instead of going into all the theory right now,"},{"Start":"00:31.410 ","End":"00:35.080","Text":"let\u0027s first start out with an example."}],"ID":12967},{"Watched":false,"Name":"Example 1","Duration":"10m 39s","ChapterTopicVideoID":12489,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.290","Text":"Now in our example,"},{"Start":"00:01.290 ","End":"00:02.760","Text":"we have a roulette wheel,"},{"Start":"00:02.760 ","End":"00:04.620","Text":"and it looks like this."},{"Start":"00:04.620 ","End":"00:10.155","Text":"Now, a person turns the roulette wheel in order to win."},{"Start":"00:10.155 ","End":"00:13.800","Text":"Now in our specific roulette wheel,"},{"Start":"00:13.800 ","End":"00:15.390","Text":"we have 3 numbers."},{"Start":"00:15.390 ","End":"00:18.105","Text":"We have 10, 20, and 30 bets."},{"Start":"00:18.105 ","End":"00:23.235","Text":"The definition of the winnings of the person turning the wheel."},{"Start":"00:23.235 ","End":"00:27.180","Text":"If we define X, the variable X,"},{"Start":"00:27.180 ","End":"00:30.820","Text":"as a person\u0027s winnings,"},{"Start":"00:31.250 ","End":"00:36.000","Text":"the values that X can have is either 10,"},{"Start":"00:36.000 ","End":"00:38.700","Text":"or 20, or 30."},{"Start":"00:38.700 ","End":"00:43.610","Text":"That\u0027s right here. That\u0027s the case."},{"Start":"00:43.610 ","End":"00:46.880","Text":"What\u0027s the probability of X equals 10?"},{"Start":"00:46.880 ","End":"00:51.035","Text":"What\u0027s the probability of someone winning 10 dollars?"},{"Start":"00:51.035 ","End":"00:56.100","Text":"Well, that\u0027s 0.25, that\u0027s 25 percent."},{"Start":"00:56.100 ","End":"01:01.910","Text":"How do I know that? Because the area that\u0027s defined by 10,"},{"Start":"01:01.910 ","End":"01:03.920","Text":"that\u0027s 1/4of the wheels,"},{"Start":"01:03.920 ","End":"01:05.570","Text":"so that\u0027s 25 percent."},{"Start":"01:05.570 ","End":"01:08.630","Text":"The probability of X equals 20."},{"Start":"01:08.630 ","End":"01:14.165","Text":"Well again, that\u0027s also 25 percent because it takes up 1/4 of the wheel."},{"Start":"01:14.165 ","End":"01:18.170","Text":"The probability of X equals 30, well,"},{"Start":"01:18.170 ","End":"01:23.480","Text":"that equals 50 percent because it takes up 1/2 of the wheel."},{"Start":"01:23.480 ","End":"01:28.700","Text":"Now that we have the values of X and their probabilities,"},{"Start":"01:28.700 ","End":"01:33.710","Text":"we can construct the probability function. Let\u0027s do that."},{"Start":"01:33.710 ","End":"01:35.570","Text":"That\u0027s X."},{"Start":"01:35.570 ","End":"01:38.735","Text":"That\u0027s the probability of X."},{"Start":"01:38.735 ","End":"01:42.575","Text":"Now, X can have a value of 10,"},{"Start":"01:42.575 ","End":"01:44.110","Text":"or a value of 20,"},{"Start":"01:44.110 ","End":"01:45.750","Text":"or a value of 30,"},{"Start":"01:45.750 ","End":"01:50.580","Text":"and the probabilities for each 1 is 0.25,"},{"Start":"01:50.580 ","End":"01:54.115","Text":"0.25, and 0.5."},{"Start":"01:54.115 ","End":"01:57.970","Text":"That\u0027s these probabilities right here."},{"Start":"01:57.970 ","End":"02:03.230","Text":"Right now, it was very easy to define the variable,"},{"Start":"02:03.230 ","End":"02:06.105","Text":"see what the values of the variables are,"},{"Start":"02:06.105 ","End":"02:12.225","Text":"calculate the probabilities, and build the probability function."},{"Start":"02:12.225 ","End":"02:16.825","Text":"Now that we\u0027ve calculated the probability function of X,"},{"Start":"02:16.825 ","End":"02:19.345","Text":"let\u0027s take 1 step forward."},{"Start":"02:19.345 ","End":"02:24.540","Text":"What we want to do right now is calculate the expectation of X."},{"Start":"02:24.540 ","End":"02:29.940","Text":"Now, the expectation of X, E of X,"},{"Start":"02:29.940 ","End":"02:38.270","Text":"or the expectation of X is defined as the sum of X times its probability."},{"Start":"02:39.170 ","End":"02:47.575","Text":"By the way, we can express the expectation of X as the Greek letter Mu."},{"Start":"02:47.575 ","End":"02:49.700","Text":"Now, in our case,"},{"Start":"02:49.700 ","End":"02:51.545","Text":"what\u0027s the expectation of X?"},{"Start":"02:51.545 ","End":"02:55.070","Text":"That\u0027s the sum of X times the probability of X,"},{"Start":"02:55.070 ","End":"02:59.265","Text":"and that would be 10 times 0.25,"},{"Start":"02:59.265 ","End":"03:02.300","Text":"that\u0027s X times its probability,"},{"Start":"03:02.300 ","End":"03:07.120","Text":"plus 20 times 0.25,"},{"Start":"03:07.120 ","End":"03:08.850","Text":"this guy right here,"},{"Start":"03:08.850 ","End":"03:12.375","Text":"plus 30 times 0.5,"},{"Start":"03:12.375 ","End":"03:14.225","Text":"that\u0027s this guy right here."},{"Start":"03:14.225 ","End":"03:22.725","Text":"That equals to 2.5 plus 5 plus 15,"},{"Start":"03:22.725 ","End":"03:26.850","Text":"and that equals to 22.5."},{"Start":"03:26.850 ","End":"03:32.090","Text":"Now, that is the calculation for the expectation of X."},{"Start":"03:32.090 ","End":"03:34.085","Text":"Now, what does that mean?"},{"Start":"03:34.085 ","End":"03:36.955","Text":"From a distribution perspective,"},{"Start":"03:36.955 ","End":"03:44.390","Text":"all that means is that this is the average of X or the distribution function."},{"Start":"03:44.390 ","End":"03:47.255","Text":"This is the average of the distribution function."},{"Start":"03:47.255 ","End":"03:51.895","Text":"From the question perspective of our winnings,"},{"Start":"03:51.895 ","End":"03:58.955","Text":"this is what we expect to win when we roll the roulette wheel over and over again."},{"Start":"03:58.955 ","End":"04:00.635","Text":"Over the long haul,"},{"Start":"04:00.635 ","End":"04:05.110","Text":"we\u0027ll be expected to win this amount right here."},{"Start":"04:05.110 ","End":"04:13.320","Text":"Now, obviously, we can earn on 1 turn of the wheel."},{"Start":"04:13.320 ","End":"04:17.360","Text":"We can earn 30 dollars or 20 dollars or 10 dollars."},{"Start":"04:17.360 ","End":"04:24.495","Text":"There\u0027s a certain amount of disbursement around this expectation, around this number."},{"Start":"04:24.495 ","End":"04:30.605","Text":"We\u0027re not always going to get this number right here."},{"Start":"04:30.605 ","End":"04:34.520","Text":"So that means that every time we turn the wheel,"},{"Start":"04:34.520 ","End":"04:39.805","Text":"there\u0027s a certain risk, the certain uncertainty about our winnings."},{"Start":"04:39.805 ","End":"04:41.300","Text":"Over the long haul,"},{"Start":"04:41.300 ","End":"04:43.490","Text":"we know that this is what we\u0027re going to earn."},{"Start":"04:43.490 ","End":"04:46.310","Text":"But for each individual turning the wheel,"},{"Start":"04:46.310 ","End":"04:49.475","Text":"we don\u0027t really know what we\u0027re going to earn."},{"Start":"04:49.475 ","End":"04:52.211","Text":"In that case,"},{"Start":"04:52.211 ","End":"05:01.205","Text":"we have to understand the average around the expectation."},{"Start":"05:01.205 ","End":"05:07.300","Text":"Now, this measure of disbursement is basically the variance of X."},{"Start":"05:07.300 ","End":"05:09.460","Text":"Now, the variance of X,"},{"Start":"05:09.460 ","End":"05:12.160","Text":"we can define it as Sigma squared."},{"Start":"05:12.160 ","End":"05:17.760","Text":"Sigma squared, that\u0027s a small Sigma,"},{"Start":"05:17.760 ","End":"05:19.650","Text":"a Greek letter, squared,"},{"Start":"05:19.650 ","End":"05:21.255","Text":"and that equals to,"},{"Start":"05:21.255 ","End":"05:22.910","Text":"that\u0027s the definition here."},{"Start":"05:22.910 ","End":"05:27.530","Text":"That\u0027s the sum of X minus Mu."},{"Start":"05:27.530 ","End":"05:37.225","Text":"Mu is the expectation of X. X minus the expectation squared times the probability of X."},{"Start":"05:37.225 ","End":"05:42.965","Text":"So basically what we\u0027re taking is we\u0027re taking the average distances of"},{"Start":"05:42.965 ","End":"05:49.410","Text":"all the X\u0027s from the expectation of X."},{"Start":"05:49.410 ","End":"05:54.185","Text":"So we\u0027re taking all the average distances and we\u0027re summing them."},{"Start":"05:54.185 ","End":"05:56.825","Text":"That equals, in our case,"},{"Start":"05:56.825 ","End":"06:06.105","Text":"they\u0027ll be 10 minus 22.5 squared times 0.25."},{"Start":"06:06.105 ","End":"06:09.660","Text":"That\u0027s 10 minus the expectation,"},{"Start":"06:09.660 ","End":"06:16.215","Text":"that\u0027s 22.5, times the probability of X plus,"},{"Start":"06:16.215 ","End":"06:22.005","Text":"there we go, 20 minus 22.5 times"},{"Start":"06:22.005 ","End":"06:32.295","Text":"0.25 plus 30 minus 22.5 squared,"},{"Start":"06:32.295 ","End":"06:38.010","Text":"I forgot the squared right here, times 0.5."},{"Start":"06:38.010 ","End":"06:46.570","Text":"Now, that turns out to be 68.75."},{"Start":"06:46.570 ","End":"06:50.135","Text":"Now, what does this number mean?"},{"Start":"06:50.135 ","End":"06:56.119","Text":"Well, this is the average distance of X"},{"Start":"06:56.119 ","End":"07:02.405","Text":"from the expectation of X, the distance squared."},{"Start":"07:02.405 ","End":"07:07.955","Text":"Now, this isn\u0027t as intuitive as it should be,"},{"Start":"07:07.955 ","End":"07:13.685","Text":"but we can think of it as the disbursement of all the data"},{"Start":"07:13.685 ","End":"07:20.315","Text":"of the probability function around the expectation."},{"Start":"07:20.315 ","End":"07:23.285","Text":"Now, this is a complicated expression."},{"Start":"07:23.285 ","End":"07:31.200","Text":"We can simplify the bit by saying that the variance of X"},{"Start":"07:31.200 ","End":"07:41.280","Text":"equals the expectation of X squared minus the expectation squared of X."},{"Start":"07:41.570 ","End":"07:49.925","Text":"Now, I won\u0027t try to prove that this expression equals to this expression."},{"Start":"07:49.925 ","End":"07:52.040","Text":"That\u0027s a little bit too complicated."},{"Start":"07:52.040 ","End":"07:53.810","Text":"You\u0027re going to have to take my word for it."},{"Start":"07:53.810 ","End":"07:55.430","Text":"But what I can tell you,"},{"Start":"07:55.430 ","End":"07:57.110","Text":"that in most cases,"},{"Start":"07:57.110 ","End":"08:01.295","Text":"this expression is much easier to work with."},{"Start":"08:01.295 ","End":"08:05.440","Text":"Let\u0027s just take a look at how this works."},{"Start":"08:05.440 ","End":"08:08.345","Text":"Now, in our case,"},{"Start":"08:08.345 ","End":"08:11.075","Text":"what\u0027s the expectation of X squared?"},{"Start":"08:11.075 ","End":"08:13.580","Text":"Well, that equals, now we take X,"},{"Start":"08:13.580 ","End":"08:15.945","Text":"10, we square it,"},{"Start":"08:15.945 ","End":"08:17.760","Text":"that\u0027s right here,"},{"Start":"08:17.760 ","End":"08:27.480","Text":"and we multiply by the probability 0.25 plus 20 squared,"},{"Start":"08:27.480 ","End":"08:29.030","Text":"that\u0027s this guy right here,"},{"Start":"08:29.030 ","End":"08:36.305","Text":"times 0.25 plus 30 squared times 0.5."},{"Start":"08:36.305 ","End":"08:44.165","Text":"Now, we have to take away the expectation squared of X,"},{"Start":"08:44.165 ","End":"08:47.360","Text":"and that\u0027s minus 22.5,"},{"Start":"08:47.360 ","End":"08:50.150","Text":"that\u0027s this guy right here squared."},{"Start":"08:50.150 ","End":"08:52.325","Text":"You\u0027re going to have to believe me,"},{"Start":"08:52.325 ","End":"08:57.280","Text":"that this turns out also to be 68.75."},{"Start":"08:57.280 ","End":"09:04.700","Text":"As I said, this is the measure of disbursement of the probability function."},{"Start":"09:04.700 ","End":"09:08.990","Text":"But it isn\u0027t that intuitive because it\u0027s in squared units,"},{"Start":"09:08.990 ","End":"09:11.285","Text":"it\u0027s the distance squared."},{"Start":"09:11.285 ","End":"09:15.635","Text":"So in order to make this a little bit more understandable,"},{"Start":"09:15.635 ","End":"09:19.265","Text":"we\u0027ve defined the standard deviation."},{"Start":"09:19.265 ","End":"09:20.690","Text":"Now the standard deviation,"},{"Start":"09:20.690 ","End":"09:22.790","Text":"you can see that right here."},{"Start":"09:22.790 ","End":"09:25.875","Text":"This is Sigma of X."},{"Start":"09:25.875 ","End":"09:29.330","Text":"Not Sigma squared like we have here, this is Sigma of X,"},{"Start":"09:29.330 ","End":"09:36.650","Text":"that\u0027s the square root of the variance and that\u0027s the square root of 68.75,"},{"Start":"09:36.650 ","End":"09:39.890","Text":"which turns out to be 8.29."},{"Start":"09:39.890 ","End":"09:42.095","Text":"Now, what\u0027s this number?"},{"Start":"09:42.095 ","End":"09:47.130","Text":"Again, if we are expected to"},{"Start":"09:47.130 ","End":"09:53.250","Text":"earn 22.5 dollars from this roulette wheel,"},{"Start":"09:53.250 ","End":"10:03.680","Text":"then our winning disbursement over time would be 8.28 dollars and 29 cents."},{"Start":"10:03.680 ","End":"10:06.980","Text":"That means we\u0027re going to be on average"},{"Start":"10:06.980 ","End":"10:15.725","Text":"8.29 dollars away from our average earnings, which is right here."},{"Start":"10:15.725 ","End":"10:21.935","Text":"Now, having understood what the expectation is,"},{"Start":"10:21.935 ","End":"10:24.015","Text":"what the variance is,"},{"Start":"10:24.015 ","End":"10:27.320","Text":"and what the standard deviation is,"},{"Start":"10:27.320 ","End":"10:32.510","Text":"your job is to start working on the problems by yourselves,"},{"Start":"10:32.510 ","End":"10:36.215","Text":"and only then after you\u0027ve had time to work on them,"},{"Start":"10:36.215 ","End":"10:39.720","Text":"look at the solution videos."}],"ID":12968},{"Watched":false,"Name":"Exercise 1","Duration":"4m 54s","ChapterTopicVideoID":12490,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"A person plays a game of chance."},{"Start":"00:02.415 ","End":"00:05.235","Text":"We define X as the amount he wins."},{"Start":"00:05.235 ","End":"00:08.595","Text":"The probability function of X is as follows."},{"Start":"00:08.595 ","End":"00:14.355","Text":"He wins minus 30 or he loses 30 with a probability of 40 percent."},{"Start":"00:14.355 ","End":"00:18.945","Text":"He doesn\u0027t win or he doesn\u0027t lose with probability of 0.1 percent."},{"Start":"00:18.945 ","End":"00:22.485","Text":"He wins 20 with a probability of 30 percent,"},{"Start":"00:22.485 ","End":"00:25.965","Text":"and he wins 40 with a probability of 20 percent."},{"Start":"00:25.965 ","End":"00:28.830","Text":"We\u0027re asked, what are the expectation,"},{"Start":"00:28.830 ","End":"00:32.445","Text":"variance, and standard deviation of X."},{"Start":"00:32.445 ","End":"00:35.535","Text":"Well, let\u0027s get started."},{"Start":"00:35.535 ","End":"00:42.335","Text":"The expectation of X is equal to"},{"Start":"00:42.335 ","End":"00:52.160","Text":"the sum of X times the probability for each x of each value."},{"Start":"00:52.160 ","End":"00:54.665","Text":"Now that equals to what?"},{"Start":"00:54.665 ","End":"01:05.265","Text":"Minus 30 times 0.4 plus 0 times 0.1 plus"},{"Start":"01:05.265 ","End":"01:12.105","Text":"20 times 0.3 plus 40 times"},{"Start":"01:12.105 ","End":"01:21.335","Text":"0.2 and that turns out to be 2 and that\u0027s the expectation."},{"Start":"01:21.335 ","End":"01:25.415","Text":"Now, let\u0027s look at the variance."},{"Start":"01:25.415 ","End":"01:34.730","Text":"The variance of X is equal to the sum over all the values of x of"},{"Start":"01:34.730 ","End":"01:44.660","Text":"the X minus the expectation squared times the probability of X_i."},{"Start":"01:44.660 ","End":"01:54.560","Text":"Now that also equals to the sum over all the values to the x of X"},{"Start":"01:54.560 ","End":"01:59.300","Text":"squared times the probability of"},{"Start":"01:59.300 ","End":"02:07.050","Text":"X minus the expectation squared of X."},{"Start":"02:08.690 ","End":"02:13.830","Text":"As I said, that equals to the variance, that\u0027s Sigma squared."},{"Start":"02:13.830 ","End":"02:17.045","Text":"Well, what does that equal to?"},{"Start":"02:17.045 ","End":"02:19.010","Text":"Let\u0027s go back to our table."},{"Start":"02:19.010 ","End":"02:28.970","Text":"That\u0027s minus 30 squared times 0.4 plus 0 squared times 0.1 plus"},{"Start":"02:28.970 ","End":"02:35.130","Text":"20 squared times 0.3 plus 40 squared times"},{"Start":"02:35.130 ","End":"02:41.960","Text":"0.2 and let\u0027s not forget to subtract 2 squared."},{"Start":"02:41.960 ","End":"02:46.652","Text":"That\u0027s our expectation squared and"},{"Start":"02:46.652 ","End":"02:53.820","Text":"that equals to 796."},{"Start":"02:53.820 ","End":"02:55.925","Text":"What\u0027s the standard deviation?"},{"Start":"02:55.925 ","End":"03:00.575","Text":"The standard deviation is the square root of the variance."},{"Start":"03:00.575 ","End":"03:05.610","Text":"That equals to square root of"},{"Start":"03:05.610 ","End":"03:15.130","Text":"796 and that equals to 28.21."},{"Start":"03:15.980 ","End":"03:23.375","Text":"Again, in this specific game with this specific probability function,"},{"Start":"03:23.375 ","End":"03:27.610","Text":"we can expect to win 2 dollars."},{"Start":"03:27.610 ","End":"03:30.690","Text":"But there\u0027s a specific risk or there is"},{"Start":"03:30.690 ","End":"03:34.040","Text":"a certain risk in playing this game because you can also"},{"Start":"03:34.040 ","End":"03:41.875","Text":"lose 30 dollars with a probability of 0.4 or not win or lose with a probability of 0.1."},{"Start":"03:41.875 ","End":"03:52.055","Text":"This risk right here is basically it\u0027s expressed as the standard deviation being 28.21."},{"Start":"03:52.055 ","End":"03:56.600","Text":"Now, most people, what they do is they add and"},{"Start":"03:56.600 ","End":"04:01.100","Text":"subtract the standard deviation from the expectation."},{"Start":"04:01.100 ","End":"04:04.355","Text":"Now, that\u0027s not really the correct thing to do"},{"Start":"04:04.355 ","End":"04:08.045","Text":"because the probability function isn\u0027t symmetrical."},{"Start":"04:08.045 ","End":"04:09.620","Text":"You can see that right here,"},{"Start":"04:09.620 ","End":"04:13.760","Text":"there is a 0.4 chance of losing"},{"Start":"04:13.760 ","End":"04:18.750","Text":"30 dollars but there\u0027s a 0.2 chance of winning 40 dollars."},{"Start":"04:18.750 ","End":"04:21.995","Text":"We can see that this isn\u0027t symmetrical and therefore,"},{"Start":"04:21.995 ","End":"04:28.490","Text":"it\u0027s not really correct to add and subtract the standard deviation from the expectation."},{"Start":"04:28.490 ","End":"04:30.235","Text":"What we can say though,"},{"Start":"04:30.235 ","End":"04:36.550","Text":"is that we are expected to win 2 dollars in this game"},{"Start":"04:36.550 ","End":"04:45.335","Text":"and that we can be as far away as 28.21 dollars from the expectation,"},{"Start":"04:45.335 ","End":"04:48.745","Text":"again, because of the risk we\u0027re taking right here."},{"Start":"04:48.745 ","End":"04:52.100","Text":"Great. We\u0027ve solved this problem."},{"Start":"04:52.100 ","End":"04:54.570","Text":"Let\u0027s go on to the next."}],"ID":12969},{"Watched":false,"Name":"Exercise 2","Duration":"7m 32s","ChapterTopicVideoID":12491,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.135","Text":"In this question, we\u0027ll be talking about bank accounts."},{"Start":"00:03.135 ","End":"00:07.169","Text":"Now there are 2 bank branches in a given community,"},{"Start":"00:07.169 ","End":"00:09.915","Text":"City Bank and Union Bank."},{"Start":"00:09.915 ","End":"00:12.510","Text":"50 percent of the population have an account at"},{"Start":"00:12.510 ","End":"00:16.140","Text":"City Bank and 40 percent have an account at Union Bank."},{"Start":"00:16.140 ","End":"00:20.205","Text":"20 percent of the populations have no bank account at all."},{"Start":"00:20.205 ","End":"00:25.200","Text":"Now, let X be the number of bank branches in which a person"},{"Start":"00:25.200 ","End":"00:30.930","Text":"has an account and we have to calculate the expectation of X."},{"Start":"00:30.930 ","End":"00:34.970","Text":"Now, in order to calculate the expectation of X,"},{"Start":"00:34.970 ","End":"00:38.335","Text":"we need to build a probability function for X."},{"Start":"00:38.335 ","End":"00:43.700","Text":"Before that, let\u0027s try to describe the data."},{"Start":"00:43.700 ","End":"00:49.835","Text":"Now we can use either Venn diagrams or probability matrices."},{"Start":"00:49.835 ","End":"00:54.380","Text":"I prefer to use a probability matrix in this problem,"},{"Start":"00:54.380 ","End":"01:00.350","Text":"so let\u0027s build 1. Here it is."},{"Start":"01:00.350 ","End":"01:04.025","Text":"Great. Now we have this,"},{"Start":"01:04.025 ","End":"01:09.170","Text":"but we still don\u0027t have our variables."},{"Start":"01:09.170 ","End":"01:15.800","Text":"Let\u0027s define A as a person that has"},{"Start":"01:15.800 ","End":"01:20.375","Text":"an account at"},{"Start":"01:20.375 ","End":"01:27.130","Text":"City Bank and B,"},{"Start":"01:27.130 ","End":"01:33.230","Text":"we\u0027ll define that as a person having an account at the Union Bank."},{"Start":"01:35.180 ","End":"01:40.530","Text":"Now, let\u0027s describe the data."},{"Start":"01:40.530 ","End":"01:42.740","Text":"Probability of A,"},{"Start":"01:42.740 ","End":"01:46.630","Text":"that means that the probability of a person having an account at"},{"Start":"01:46.630 ","End":"01:56.180","Text":"City Bank is 0.5 or 50 percent and that\u0027s given right here."},{"Start":"01:57.960 ","End":"02:07.415","Text":"The probability of a person having an account at Union Bank is 40 percent."},{"Start":"02:07.415 ","End":"02:12.265","Text":"The probability of a person not having an account at all,"},{"Start":"02:12.265 ","End":"02:19.200","Text":"not in the City Bank and not at the Union Bank is 20 percent."},{"Start":"02:19.200 ","End":"02:22.275","Text":"Again, we see that right here,"},{"Start":"02:22.275 ","End":"02:27.040","Text":"40 percent and 20 percent."},{"Start":"02:27.440 ","End":"02:32.545","Text":"Now, because we wanted to use the probability matrix,"},{"Start":"02:32.545 ","End":"02:35.905","Text":"let\u0027s fill in the probabilities that we already have."},{"Start":"02:35.905 ","End":"02:40.870","Text":"The probability of A we said is 50 percent."},{"Start":"02:40.870 ","End":"02:44.190","Text":"That\u0027s right here, 0.5."},{"Start":"02:44.190 ","End":"02:48.620","Text":"The probability of B is 0.4."},{"Start":"02:48.620 ","End":"02:49.700","Text":"Now that\u0027s right here."},{"Start":"02:49.700 ","End":"02:51.695","Text":"It\u0027s 0.4."},{"Start":"02:51.695 ","End":"02:55.400","Text":"The probability of not A and not B,"},{"Start":"02:55.400 ","End":"02:56.870","Text":"here\u0027s not A,"},{"Start":"02:56.870 ","End":"02:58.175","Text":"here\u0027s not B,"},{"Start":"02:58.175 ","End":"03:02.930","Text":"and this is the intersection right here, that\u0027s 0.2."},{"Start":"03:02.930 ","End":"03:07.505","Text":"Now, because we\u0027re so familiar with the probability matrix,"},{"Start":"03:07.505 ","End":"03:09.320","Text":"let\u0027s just fill in the rest pretty."},{"Start":"03:09.320 ","End":"03:10.865","Text":"It\u0027s very easy to fill it out."},{"Start":"03:10.865 ","End":"03:14.134","Text":"Here there\u0027ll be 0.6,"},{"Start":"03:14.134 ","End":"03:16.380","Text":"0.4 plus 0.6 equals 1."},{"Start":"03:16.380 ","End":"03:18.990","Text":"This equals to 0.5."},{"Start":"03:18.990 ","End":"03:20.925","Text":"Let\u0027s fill in the rest."},{"Start":"03:20.925 ","End":"03:28.060","Text":"That\u0027ll be 0.4, 0.1, 0.3."},{"Start":"03:28.340 ","End":"03:34.150","Text":"We pretty much have all the probabilities that we need."},{"Start":"03:34.430 ","End":"03:40.340","Text":"Right now because we\u0027ve defined everything that we can,"},{"Start":"03:40.340 ","End":"03:46.010","Text":"let\u0027s go and build our probability function for X."},{"Start":"03:46.010 ","End":"03:49.060","Text":"Now again, what\u0027s X?"},{"Start":"03:49.060 ","End":"03:54.070","Text":"X is the number of bank branches."},{"Start":"03:59.300 ","End":"04:02.505","Text":"What are the values of X?"},{"Start":"04:02.505 ","End":"04:07.755","Text":"We can have a person not having an account at all,"},{"Start":"04:07.755 ","End":"04:13.204","Text":"or we can have a person having an account at 1 of the 2 bank branches,"},{"Start":"04:13.204 ","End":"04:15.430","Text":"City Bank or Union Bank,"},{"Start":"04:15.430 ","End":"04:21.905","Text":"or we can have a person that has an account both at the City Bank and at the Union Bank."},{"Start":"04:21.905 ","End":"04:28.985","Text":"Great. Let\u0027s see what is our probability function."},{"Start":"04:28.985 ","End":"04:35.788","Text":"As we know, we\u0027ll write the probability function like this."},{"Start":"04:35.788 ","End":"04:38.940","Text":"Here we have the values of X,"},{"Start":"04:38.940 ","End":"04:42.720","Text":"there\u0027s 0, 1, and 2."},{"Start":"04:42.720 ","End":"04:48.775","Text":"Now, the probability that X equals 0, what does that mean?"},{"Start":"04:48.775 ","End":"04:55.510","Text":"That\u0027s the probability of a person not having any bank account in any branch."},{"Start":"04:55.510 ","End":"04:58.150","Text":"That\u0027s given, that\u0027s 0.2."},{"Start":"04:58.150 ","End":"05:01.065","Text":"That\u0027s right here, 0.2."},{"Start":"05:01.065 ","End":"05:05.870","Text":"Again, 20 percent of the population have no bank account."},{"Start":"05:06.430 ","End":"05:15.440","Text":"Now, what\u0027s the probability of X equals 2?"},{"Start":"05:15.440 ","End":"05:19.535","Text":"X equals 2 means that a person has a bank account"},{"Start":"05:19.535 ","End":"05:24.440","Text":"both at the Union Bank and at the City Bank."},{"Start":"05:24.440 ","End":"05:27.600","Text":"That\u0027s right here, that\u0027s 0.1."},{"Start":"05:30.170 ","End":"05:32.495","Text":"In order for everything to be equal to 1,"},{"Start":"05:32.495 ","End":"05:36.695","Text":"the probability equal to 1,"},{"Start":"05:36.695 ","End":"05:40.730","Text":"a person having only 1 bank account,"},{"Start":"05:40.730 ","End":"05:45.590","Text":"be at the Union Bank or at the City Bank, is 0.7."},{"Start":"05:45.590 ","End":"05:48.200","Text":"Now, again, let\u0027s look at it."},{"Start":"05:48.200 ","End":"05:56.690","Text":"This thing right here is the probability where X equals 0."},{"Start":"05:56.690 ","End":"06:03.140","Text":"He doesn\u0027t have a bank account at City Bank or at the Union Bank."},{"Start":"06:03.140 ","End":"06:10.010","Text":"Here that\u0027s the probability of X equals 2,"},{"Start":"06:10.010 ","End":"06:18.295","Text":"where a person has a bank account both at the City Bank and at the Union Bank."},{"Start":"06:18.295 ","End":"06:26.305","Text":"Right here, the sum of these probabilities is the probability of X equals 1,"},{"Start":"06:26.305 ","End":"06:34.865","Text":"where a person has a bank account at the Union Bank but not at the City Bank,"},{"Start":"06:34.865 ","End":"06:39.130","Text":"or a person having an account at the City Bank and not at the Union Bank."},{"Start":"06:39.130 ","End":"06:41.930","Text":"Here are the probabilities."},{"Start":"06:41.930 ","End":"06:46.385","Text":"What we have to do right now is just to calculate the expectation."},{"Start":"06:46.385 ","End":"06:50.840","Text":"Now the expectation is the multiplication of"},{"Start":"06:50.840 ","End":"06:59.420","Text":"each value of X by its probabilities and summing that up."},{"Start":"06:59.420 ","End":"07:02.790","Text":"Let\u0027s write this in the proper color."},{"Start":"07:05.030 ","End":"07:13.050","Text":"The expectation of X is equal to 0 times 0.2,"},{"Start":"07:13.050 ","End":"07:15.848","Text":"that\u0027s these guys right here,"},{"Start":"07:15.848 ","End":"07:24.525","Text":"plus 1 times 0.7 plus 2 times 0.1,"},{"Start":"07:24.525 ","End":"07:28.270","Text":"and that equals to 0.9."},{"Start":"07:28.730 ","End":"07:32.440","Text":"That\u0027s the expectation of X."}],"ID":12970},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"7m 6s","ChapterTopicVideoID":12492,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"This question will be calling families and"},{"Start":"00:02.640 ","End":"00:05.745","Text":"asking them about their satellite TV connection."},{"Start":"00:05.745 ","End":"00:08.580","Text":"Now, it\u0027s known that 20 percent of"},{"Start":"00:08.580 ","End":"00:11.745","Text":"families have a satellite TV connection in their homes."},{"Start":"00:11.745 ","End":"00:17.685","Text":"In a survey, a pollster seeks to interview a family with a satellite TV connection."},{"Start":"00:17.685 ","End":"00:20.730","Text":"He randomly telephones a family and continues"},{"Start":"00:20.730 ","End":"00:23.865","Text":"until he finds a family with a satellite TV connection."},{"Start":"00:23.865 ","End":"00:28.120","Text":"In any case, the pollster doesn\u0027t call more than 5 families."},{"Start":"00:28.120 ","End":"00:32.330","Text":"Now, we define X as the number of families called in the survey."},{"Start":"00:32.330 ","End":"00:36.110","Text":"We\u0027re asked first to construct the probability function of"},{"Start":"00:36.110 ","End":"00:41.650","Text":"X and second to calculate the expectation and standard deviation of X."},{"Start":"00:41.650 ","End":"00:43.635","Text":"Great."},{"Start":"00:43.635 ","End":"00:47.300","Text":"Now, before we can start answering these questions,"},{"Start":"00:47.300 ","End":"00:49.120","Text":"we have to describe the data."},{"Start":"00:49.120 ","End":"00:52.760","Text":"The best way to do this is with a probability tree."},{"Start":"00:52.760 ","End":"00:56.390","Text":"That\u0027s because the story that\u0027s given to us,"},{"Start":"00:56.390 ","End":"00:57.950","Text":"it\u0027s given to us in stages."},{"Start":"00:57.950 ","End":"01:00.196","Text":"A pollster calls the first family,"},{"Start":"01:00.196 ","End":"01:03.260","Text":"if he succeeds with a 20 percent chance, then he stops."},{"Start":"01:03.260 ","End":"01:09.290","Text":"If he doesn\u0027t, then he goes on to the next family and so on and so forth, 5 times."},{"Start":"01:09.290 ","End":"01:11.585","Text":"If that\u0027s the case,"},{"Start":"01:11.585 ","End":"01:16.115","Text":"let\u0027s start to construct our probability tree."},{"Start":"01:16.115 ","End":"01:18.455","Text":"Now, as we said,"},{"Start":"01:18.455 ","End":"01:20.510","Text":"here\u0027s the first level."},{"Start":"01:20.510 ","End":"01:22.520","Text":"Now, if he succeeds,"},{"Start":"01:22.520 ","End":"01:25.970","Text":"he succeeds with a 20 percent chance,"},{"Start":"01:25.970 ","End":"01:31.410","Text":"and if he doesn\u0027t with an 80 percent chance."},{"Start":"01:31.410 ","End":"01:33.690","Text":"Then he goes on to the next level."},{"Start":"01:33.690 ","End":"01:35.855","Text":"Again, if he succeeds,"},{"Start":"01:35.855 ","End":"01:38.180","Text":"he succeeds with a 20 percent chance,"},{"Start":"01:38.180 ","End":"01:44.120","Text":"and if he doesn\u0027t succeed in catching a family with a satellite connection,"},{"Start":"01:44.120 ","End":"01:50.825","Text":"then again, that\u0027s an 80 percent chance of not succeeding and so on and so forth."},{"Start":"01:50.825 ","End":"01:54.830","Text":"You get the picture. Having said that,"},{"Start":"01:54.830 ","End":"02:00.840","Text":"let\u0027s see how the whole tree looks and then we\u0027ll take it from there."},{"Start":"02:01.430 ","End":"02:05.555","Text":"This is how our probability tree looks."},{"Start":"02:05.555 ","End":"02:12.440","Text":"The next step is to see what the definition is of our variable."},{"Start":"02:12.440 ","End":"02:18.145","Text":"Now, X, our variable,"},{"Start":"02:18.145 ","End":"02:22.690","Text":"is defined as the number of families called in the survey."},{"Start":"02:22.690 ","End":"02:29.680","Text":"Now it\u0027s very important to understand what X is and what the values of X are so we"},{"Start":"02:29.680 ","End":"02:37.900","Text":"can construct the probability function and the expectation of standard deviations."},{"Start":"02:37.900 ","End":"02:40.855","Text":"Now, X is the number of families."},{"Start":"02:40.855 ","End":"02:43.840","Text":"Let\u0027s take a look at the tree."},{"Start":"02:43.840 ","End":"02:47.580","Text":"What is the first level?"},{"Start":"02:47.580 ","End":"02:52.510","Text":"Well, here is X equals 1."},{"Start":"02:53.180 ","End":"02:56.685","Text":"Here, X equals 2."},{"Start":"02:56.685 ","End":"02:59.115","Text":"What does that mean that X equals 1?"},{"Start":"02:59.115 ","End":"03:04.380","Text":"That the pollster called the first family and he succeeded."},{"Start":"03:05.570 ","End":"03:08.280","Text":"What does X equals 2 mean?"},{"Start":"03:08.280 ","End":"03:10.500","Text":"That he called the first family,"},{"Start":"03:10.500 ","End":"03:14.630","Text":"he didn\u0027t succeed in finding a family with a satellite connection,"},{"Start":"03:14.630 ","End":"03:17.390","Text":"he called the second family and he succeeded with"},{"Start":"03:17.390 ","End":"03:20.555","Text":"the second family and so on and so forth."},{"Start":"03:20.555 ","End":"03:23.915","Text":"X equals 3, X equals 4,"},{"Start":"03:23.915 ","End":"03:26.900","Text":"and X equals 5."},{"Start":"03:26.900 ","End":"03:34.130","Text":"Now, that means that X can take on the values of 1,"},{"Start":"03:34.130 ","End":"03:36.185","Text":"2, 3,"},{"Start":"03:36.185 ","End":"03:38.940","Text":"4, and 5."},{"Start":"03:38.940 ","End":"03:43.710","Text":"Now, what\u0027s the probability of X equals 1?"},{"Start":"03:43.710 ","End":"03:47.400","Text":"Again, X equals 1 means that he called the first family and he"},{"Start":"03:47.400 ","End":"03:53.275","Text":"succeeded on the first try in catching a family with a satellite connection."},{"Start":"03:53.275 ","End":"03:55.540","Text":"Well, that\u0027s 0.2, right?"},{"Start":"03:55.540 ","End":"03:58.345","Text":"He succeeded and he stops."},{"Start":"03:58.345 ","End":"04:02.770","Text":"What does the probability of X equals 2 means?"},{"Start":"04:02.770 ","End":"04:05.354","Text":"That means that he called the first family,"},{"Start":"04:05.354 ","End":"04:09.550","Text":"he didn\u0027t catch that family as having a satellite connection."},{"Start":"04:09.550 ","End":"04:10.990","Text":"But on the second try,"},{"Start":"04:10.990 ","End":"04:14.815","Text":"he caught a family that had a satellite connection."},{"Start":"04:14.815 ","End":"04:17.795","Text":"That means is that 0.8,"},{"Start":"04:17.795 ","End":"04:28.950","Text":"he missed out here and he succeeded here times 0.2 and that equals to 0.16."},{"Start":"04:30.050 ","End":"04:33.085","Text":"Now, when we continue on,"},{"Start":"04:33.085 ","End":"04:38.660","Text":"then these are the calculations of the probabilities for X equals 3,"},{"Start":"04:38.660 ","End":"04:41.940","Text":"where he didn\u0027t succeed on the first try,"},{"Start":"04:41.940 ","End":"04:43.785","Text":"didn\u0027t succeed on the second try,"},{"Start":"04:43.785 ","End":"04:45.660","Text":"and at the third try, he did succeed."},{"Start":"04:45.660 ","End":"04:50.480","Text":"That means that\u0027s 0.8 squared, excuse me,"},{"Start":"04:50.480 ","End":"04:55.640","Text":"that\u0027s 0.8 times 0.8 times 0.2, and so on and so forth."},{"Start":"04:55.640 ","End":"04:58.970","Text":"For X equals 4, he missed out on the first try,"},{"Start":"04:58.970 ","End":"05:00.290","Text":"second try, third try."},{"Start":"05:00.290 ","End":"05:04.880","Text":"That\u0027s 0.8 cubed times 0.2."},{"Start":"05:04.880 ","End":"05:10.310","Text":"Now, what happens when X equals 5?"},{"Start":"05:10.310 ","End":"05:12.980","Text":"Well, when x equals 5,"},{"Start":"05:12.980 ","End":"05:20.615","Text":"he basically missed out on all 4 families before this family,"},{"Start":"05:20.615 ","End":"05:24.310","Text":"he didn\u0027t catch a family with a satellite connection,"},{"Start":"05:24.310 ","End":"05:25.640","Text":"and on the fifth trial,"},{"Start":"05:25.640 ","End":"05:30.325","Text":"he either succeeded or again, he missed out."},{"Start":"05:30.325 ","End":"05:36.150","Text":"That means that that equals to 0.8 to the fifth,"},{"Start":"05:36.150 ","End":"05:38.280","Text":"that\u0027s this branch right here,"},{"Start":"05:38.280 ","End":"05:40.605","Text":"0.8 times 0.8 and so on,"},{"Start":"05:40.605 ","End":"05:46.635","Text":"5 times plus 0.8 to the fourth, right?"},{"Start":"05:46.635 ","End":"05:52.470","Text":"That he missed out on all the 4 previous families times 0.2,"},{"Start":"05:52.470 ","End":"05:56.410","Text":"he succeeded on the fifth family."},{"Start":"05:56.510 ","End":"06:00.430","Text":"That equals to 0.4096."},{"Start":"06:07.130 ","End":"06:09.875","Text":"Now that we\u0027ve calculated"},{"Start":"06:09.875 ","End":"06:15.947","Text":"all the probabilities from the data that we\u0027ve described here in the tree,"},{"Start":"06:15.947 ","End":"06:20.165","Text":"let\u0027s build the probability function for X."},{"Start":"06:20.165 ","End":"06:24.455","Text":"Excuse me. That means that we have X here,"},{"Start":"06:24.455 ","End":"06:27.485","Text":"and we have the probability of X."},{"Start":"06:27.485 ","End":"06:32.600","Text":"Now, X can equal 1,"},{"Start":"06:32.600 ","End":"06:37.040","Text":"2, 3, 4, or 5."},{"Start":"06:37.040 ","End":"06:41.140","Text":"Now if X equals 1 that\u0027s 0.2,"},{"Start":"06:41.140 ","End":"06:44.370","Text":"if X equals 2 that\u0027s 0.16,"},{"Start":"06:44.370 ","End":"06:49.900","Text":"excuse me, 0.128, that\u0027s 0.1024."},{"Start":"06:50.270 ","End":"06:56.700","Text":"When X equals 5, we have 0.4096."},{"Start":"06:56.700 ","End":"06:59.720","Text":"Here we basically answered"},{"Start":"06:59.720 ","End":"07:06.660","Text":"the first question of constructing a probability function for X."}],"ID":12971},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"2m 26s","ChapterTopicVideoID":12493,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"Now in section B, we want to calculate the expectation and standard deviation of X."},{"Start":"00:05.250 ","End":"00:09.690","Text":"Now, from section a, this is a probability function that we"},{"Start":"00:09.690 ","End":"00:15.600","Text":"calculated on this tree."},{"Start":"00:15.600 ","End":"00:19.710","Text":"Let\u0027s get started. The expectation of X is"},{"Start":"00:19.710 ","End":"00:25.215","Text":"basically summing of all of X times its probabilities."},{"Start":"00:25.215 ","End":"00:30.089","Text":"That equals to 1 times"},{"Start":"00:30.089 ","End":"00:36.990","Text":"0.2 plus 2 times 0.16 plus,"},{"Start":"00:36.990 ","End":"00:44.415","Text":"and so on and so forth is 3 times this plus 4 times this,"},{"Start":"00:44.415 ","End":"00:46.020","Text":"plus 5 times this."},{"Start":"00:46.020 ","End":"00:53.480","Text":"That ultimately comes out to 3.3616,"},{"Start":"00:53.480 ","End":"00:57.020","Text":"and that\u0027s the expectation of X."},{"Start":"00:57.020 ","End":"01:05.105","Text":"That means that the average number of families that the poster calls is 3.3616."},{"Start":"01:05.105 ","End":"01:09.695","Text":"Now let\u0027s calculate the variance of X,"},{"Start":"01:09.695 ","End":"01:15.875","Text":"but the variance of X is defined as the sum of X squared times"},{"Start":"01:15.875 ","End":"01:23.619","Text":"the probability for each X minus the expectation squared."},{"Start":"01:23.630 ","End":"01:33.860","Text":"That also equals to 1 squared times 0.2 plus 2 squared times 0.16."},{"Start":"01:33.860 ","End":"01:35.375","Text":"1 squared times this,"},{"Start":"01:35.375 ","End":"01:37.020","Text":"2 squared times this,"},{"Start":"01:37.020 ","End":"01:38.840","Text":"and again, 3 squared times this,"},{"Start":"01:38.840 ","End":"01:48.840","Text":"and so on and so forth minus 3.3616 squared."},{"Start":"01:48.840 ","End":"01:56.495","Text":"That equals to 2.57, that\u0027s the variance."},{"Start":"01:56.495 ","End":"01:58.390","Text":"But we didn\u0027t ask for the variance,"},{"Start":"01:58.390 ","End":"02:00.800","Text":"we asked for the standard deviation."},{"Start":"02:00.800 ","End":"02:07.575","Text":"Now, the standard deviation of X is simply the square root of the variance."},{"Start":"02:07.575 ","End":"02:11.825","Text":"That equals to square root of 2.57,"},{"Start":"02:11.825 ","End":"02:17.335","Text":"and that equals to 1.603."},{"Start":"02:17.335 ","End":"02:20.736","Text":"That\u0027s the standard deviation of X."},{"Start":"02:20.736 ","End":"02:26.440","Text":"This is the expectation of X."}],"ID":12972},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"5m 30s","ChapterTopicVideoID":12494,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"This question deals with opening doors."},{"Start":"00:02.280 ","End":"00:05.745","Text":"A person has a key ring with 5 keys,"},{"Start":"00:05.745 ","End":"00:08.910","Text":"only 1 of which opens the door to his home."},{"Start":"00:08.910 ","End":"00:11.250","Text":"He tries the keys randomly."},{"Start":"00:11.250 ","End":"00:12.930","Text":"After trying a given key,"},{"Start":"00:12.930 ","End":"00:16.335","Text":"he takes it off the ring in order to avoid using it again."},{"Start":"00:16.335 ","End":"00:21.014","Text":"Let X denote the number of attempts until the door opens."},{"Start":"00:21.014 ","End":"00:24.060","Text":"Now we\u0027re asked to construct a probability function of"},{"Start":"00:24.060 ","End":"00:28.935","Text":"X and to calculate the expectation and variance of X."},{"Start":"00:28.935 ","End":"00:33.920","Text":"The first thing that we need to do is to describe the data,"},{"Start":"00:33.920 ","End":"00:37.143","Text":"and what\u0027s the best technique to describe the data?"},{"Start":"00:37.143 ","End":"00:39.595","Text":"Here we\u0027ll use a probability tree."},{"Start":"00:39.595 ","End":"00:41.060","Text":"Again, why is that?"},{"Start":"00:41.060 ","End":"00:44.555","Text":"Because the story is given to us in stages,"},{"Start":"00:44.555 ","End":"00:48.770","Text":"the person chooses 1 of the 5 rings, if it works,"},{"Start":"00:48.770 ","End":"00:50.794","Text":"then great, if it doesn\u0027t,"},{"Start":"00:50.794 ","End":"00:53.960","Text":"then he takes it off the ring and then he has 4 keys left."},{"Start":"00:53.960 ","End":"00:55.565","Text":"He chooses 1 of the 4."},{"Start":"00:55.565 ","End":"00:58.610","Text":"He tries that out and if it works,"},{"Start":"00:58.610 ","End":"01:00.635","Text":"if it opens the doors and that\u0027s great."},{"Start":"01:00.635 ","End":"01:02.630","Text":"If it doesn\u0027t, then it goes down to the 3rd,"},{"Start":"01:02.630 ","End":"01:05.250","Text":"and 4th and 5th try."},{"Start":"01:05.990 ","End":"01:09.950","Text":"If we\u0027re going to use a probability tree,"},{"Start":"01:09.950 ","End":"01:12.050","Text":"then let\u0027s get to it."},{"Start":"01:12.050 ","End":"01:18.000","Text":"Again, the first stage he can try,"},{"Start":"01:18.000 ","End":"01:19.800","Text":"and if he succeeds on the first stage,"},{"Start":"01:19.800 ","End":"01:21.895","Text":"so what\u0027s the probability of him succeeding?"},{"Start":"01:21.895 ","End":"01:24.610","Text":"That\u0027s 1/5. Why is that?"},{"Start":"01:24.610 ","End":"01:30.840","Text":"Because only 1 opens the door and he has 5 keys in the key chain."},{"Start":"01:30.840 ","End":"01:33.180","Text":"So 1/5 opens the door."},{"Start":"01:33.180 ","End":"01:36.400","Text":"That means that 4/5 do not."},{"Start":"01:36.400 ","End":"01:38.155","Text":"That\u0027s on the first try."},{"Start":"01:38.155 ","End":"01:40.420","Text":"On the second try,"},{"Start":"01:40.420 ","End":"01:43.315","Text":"how many keys does he have in the second try?"},{"Start":"01:43.315 ","End":"01:45.265","Text":"Well, he has 4 keys,"},{"Start":"01:45.265 ","End":"01:48.460","Text":"and 1/4 will open the door."},{"Start":"01:48.460 ","End":"01:53.110","Text":"That means that 3/4 will not."},{"Start":"01:53.110 ","End":"01:54.480","Text":"On the third try,"},{"Start":"01:54.480 ","End":"01:56.390","Text":"how many keys does he have left?"},{"Start":"01:56.390 ","End":"01:58.078","Text":"He has 3 keys,"},{"Start":"01:58.078 ","End":"02:03.430","Text":"and 1/3 will open the door and 2/3 will not."},{"Start":"02:03.430 ","End":"02:06.545","Text":"On the fourth try, how many keys does he have?"},{"Start":"02:06.545 ","End":"02:12.320","Text":"He has 2, so 1/2 opens the door and 1/2 will not."},{"Start":"02:12.320 ","End":"02:14.855","Text":"On the last try,"},{"Start":"02:14.855 ","End":"02:17.445","Text":"he only has 1 key left."},{"Start":"02:17.445 ","End":"02:20.715","Text":"That will certainly open the door."},{"Start":"02:20.715 ","End":"02:27.215","Text":"Now that we\u0027ve constructed a probability tree right here and"},{"Start":"02:27.215 ","End":"02:33.650","Text":"we\u0027ve understood the logic behind the probabilities for each step of the way,"},{"Start":"02:33.650 ","End":"02:36.950","Text":"then first thing that we need to do is, again,"},{"Start":"02:36.950 ","End":"02:42.110","Text":"look at our variable and try to understand that"},{"Start":"02:42.110 ","End":"02:48.980","Text":"the variable is X and X is the number of attempts until the door opens."},{"Start":"02:48.980 ","End":"02:52.850","Text":"That\u0027s basically given to us right here."},{"Start":"02:52.850 ","End":"02:54.470","Text":"This is the first stage,"},{"Start":"02:54.470 ","End":"02:57.100","Text":"where x equals 1."},{"Start":"02:57.100 ","End":"02:59.570","Text":"If he succeeds, that\u0027s great."},{"Start":"02:59.570 ","End":"03:02.000","Text":"If he doesn\u0027t, then he goes on to the next phase."},{"Start":"03:02.000 ","End":"03:04.400","Text":"That means that\u0027s x equals 2."},{"Start":"03:04.400 ","End":"03:07.640","Text":"If he succeeds, not in the second,"},{"Start":"03:07.640 ","End":"03:10.130","Text":"if he succeeds on the third try,"},{"Start":"03:10.130 ","End":"03:11.600","Text":"then x equals 3,"},{"Start":"03:11.600 ","End":"03:12.823","Text":"the third attempt,"},{"Start":"03:12.823 ","End":"03:14.525","Text":"and the fourth attempt,"},{"Start":"03:14.525 ","End":"03:17.900","Text":"and finally the fifth attempt."},{"Start":"03:17.900 ","End":"03:20.195","Text":"So if that\u0027s the case,"},{"Start":"03:20.195 ","End":"03:25.620","Text":"what\u0027s the probability of x being equal to 1? What does that mean?"},{"Start":"03:25.620 ","End":"03:29.555","Text":"That he succeeds in opening the door on the first try?"},{"Start":"03:29.555 ","End":"03:33.810","Text":"Well, that equals to 1/5."},{"Start":"03:34.130 ","End":"03:37.825","Text":"That\u0027s this branch right here, and then it stops."},{"Start":"03:37.825 ","End":"03:43.440","Text":"What happens with a probability of x being equal to 2?"},{"Start":"03:43.440 ","End":"03:45.960","Text":"That means that he failed in"},{"Start":"03:45.960 ","End":"03:51.385","Text":"his first attempt and tried this on the second attempt and he succeeded."},{"Start":"03:51.385 ","End":"03:59.540","Text":"That means that it\u0027s 4/5 times 1/4."},{"Start":"03:59.540 ","End":"04:04.365","Text":"That\u0027s this branch and he succeeded. What does that equal?"},{"Start":"04:04.365 ","End":"04:07.090","Text":"Well, 4 and 4 cancel each other out,"},{"Start":"04:07.090 ","End":"04:09.610","Text":"and that equals to 1/ 5."},{"Start":"04:09.610 ","End":"04:15.235","Text":"Now, what happens with the third try?"},{"Start":"04:15.235 ","End":"04:19.310","Text":"Well, he didn\u0027t succeed on the first try."},{"Start":"04:19.310 ","End":"04:21.530","Text":"He didn\u0027t succeed on the second try,"},{"Start":"04:21.530 ","End":"04:23.980","Text":"but he did succeed on the third try."},{"Start":"04:23.980 ","End":"04:34.450","Text":"That\u0027s 4/5 times 3/4 times 1/3."},{"Start":"04:35.030 ","End":"04:37.260","Text":"What does that equal?"},{"Start":"04:37.260 ","End":"04:39.900","Text":"Well, again, this cancels out with this,"},{"Start":"04:39.900 ","End":"04:41.235","Text":"this cancels out with this,"},{"Start":"04:41.235 ","End":"04:43.140","Text":"and we have 1/5."},{"Start":"04:43.140 ","End":"04:45.740","Text":"As you can very well guess,"},{"Start":"04:45.740 ","End":"04:51.750","Text":"everything here, all the probabilities, are 1/5."},{"Start":"04:51.750 ","End":"04:58.570","Text":"We can construct our probability tree or a probability function very easily."},{"Start":"04:58.570 ","End":"05:07.025","Text":"We have X, we have the probability of x. X can be either 1,"},{"Start":"05:07.025 ","End":"05:08.705","Text":"2, 3, 4, 5."},{"Start":"05:08.705 ","End":"05:13.715","Text":"It\u0027s 1, 2, 3, 4, and 5."},{"Start":"05:13.715 ","End":"05:20.670","Text":"The probabilities are 1/5 for each of the values of X."},{"Start":"05:22.160 ","End":"05:30.010","Text":"There we go. Here we\u0027ve completed Section a."}],"ID":12973},{"Watched":false,"Name":"Exercise 4 - Part b","Duration":"1m 43s","ChapterTopicVideoID":12495,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.235","Text":"In Section B, we\u0027re asked to calculate the expectation and variance of X."},{"Start":"00:05.235 ","End":"00:06.540","Text":"Well, from Section A,"},{"Start":"00:06.540 ","End":"00:10.485","Text":"we\u0027ve calculated the probability function, that\u0027s over here."},{"Start":"00:10.485 ","End":"00:14.190","Text":"Let\u0027s get started with the expectation of X."},{"Start":"00:14.190 ","End":"00:22.635","Text":"Now, the expectation of X is defined as the sum of X times its probability."},{"Start":"00:22.635 ","End":"00:33.275","Text":"That equals to 1 times a fifth plus 2 times a fifth plus 3 times a fifth,"},{"Start":"00:33.275 ","End":"00:34.790","Text":"and so on and so forth,"},{"Start":"00:34.790 ","End":"00:38.120","Text":"and that equals to 3."},{"Start":"00:38.120 ","End":"00:40.715","Text":"That means on average,"},{"Start":"00:40.715 ","End":"00:45.420","Text":"he\u0027s expected to open the door on the third try."},{"Start":"00:45.770 ","End":"00:48.185","Text":"Now, if that\u0027s the case,"},{"Start":"00:48.185 ","End":"00:55.290","Text":"let\u0027s go on and calculate the variance."},{"Start":"00:55.790 ","End":"00:58.849","Text":"Now, what\u0027s the variance?"},{"Start":"00:58.849 ","End":"01:06.270","Text":"The variance is defined as the sum of X squared times"},{"Start":"01:06.270 ","End":"01:15.555","Text":"the probability of X minus the expectation squared of X."},{"Start":"01:15.555 ","End":"01:20.640","Text":"That equals to 1 squared times a fifth"},{"Start":"01:20.640 ","End":"01:27.960","Text":"plus 2 squared times a fifth plus and so on and so forth;"},{"Start":"01:27.960 ","End":"01:30.900","Text":"3 squared times a fifth, 4 squared times a fifth, and so on,"},{"Start":"01:30.900 ","End":"01:39.340","Text":"minus 3 squared and that comes out to 2."},{"Start":"01:39.350 ","End":"01:43.300","Text":"Here we\u0027ve solved Section B."}],"ID":12974},{"Watched":false,"Name":"Exercise 5","Duration":"4m 14s","ChapterTopicVideoID":12496,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this question, we have a probability function of"},{"Start":"00:03.060 ","End":"00:07.290","Text":"X which is a random variable and it\u0027s given to us like this."},{"Start":"00:07.290 ","End":"00:10.440","Text":"We have X with the values of 2,"},{"Start":"00:10.440 ","End":"00:11.490","Text":"4, 6,"},{"Start":"00:11.490 ","End":"00:14.640","Text":"and 8 and we have the probability of X,"},{"Start":"00:14.640 ","End":"00:17.745","Text":"where we see the 2 of the probabilities are missing."},{"Start":"00:17.745 ","End":"00:23.010","Text":"Now, we\u0027re given that the expectation of X is 4.2 and"},{"Start":"00:23.010 ","End":"00:28.335","Text":"we\u0027re asked to find the missing probabilities in the table right here and here."},{"Start":"00:28.335 ","End":"00:34.080","Text":"Secondly, we want to calculate the variance of X. Let\u0027s get at it."},{"Start":"00:34.080 ","End":"00:39.300","Text":"Let\u0027s first define as P,"},{"Start":"00:39.300 ","End":"00:42.820","Text":"the probability where X equals 2."},{"Start":"00:42.820 ","End":"00:45.485","Text":"Now, that\u0027s the case."},{"Start":"00:45.485 ","End":"00:49.830","Text":"What\u0027s the probability of X being equal to 6?"},{"Start":"00:50.320 ","End":"00:58.855","Text":"The probability of X being equal to 6 is 1 minus the sum of all the other probabilities."},{"Start":"00:58.855 ","End":"01:05.950","Text":"That\u0027s 1 minus P plus 0.3 plus 0.2."},{"Start":"01:06.680 ","End":"01:09.730","Text":"When we figure this out,"},{"Start":"01:09.730 ","End":"01:19.095","Text":"we see that the probability of X being equal to 6 is 0.5 minus P. First of all,"},{"Start":"01:19.095 ","End":"01:21.120","Text":"let\u0027s write that here,"},{"Start":"01:21.120 ","End":"01:25.905","Text":"0.5 minus P. Now we still don\u0027t know what P is."},{"Start":"01:25.905 ","End":"01:27.740","Text":"Let\u0027s calculate that."},{"Start":"01:27.740 ","End":"01:35.510","Text":"We\u0027re given that the expectation of X is 4.2. Let\u0027s use that."},{"Start":"01:35.510 ","End":"01:39.144","Text":"First of all, what\u0027s the definition of the expectation"},{"Start":"01:39.144 ","End":"01:44.940","Text":"that\u0027s the sum of x times the probability of x."},{"Start":"01:45.200 ","End":"01:47.975","Text":"Let\u0027s just plug the numbers in."},{"Start":"01:47.975 ","End":"01:50.430","Text":"That\u0027s 2 times P,"},{"Start":"01:51.490 ","End":"01:57.960","Text":"plus 4 times 0.3 plus 6 times"},{"Start":"01:57.960 ","End":"02:05.910","Text":"0.5 minus P plus 8 times 0.2,"},{"Start":"02:05.910 ","End":"02:11.220","Text":"and that equals to 2P plus 1.2"},{"Start":"02:11.220 ","End":"02:19.370","Text":"plus 3 minus 6P plus 1.6."},{"Start":"02:19.370 ","End":"02:22.970","Text":"Now let\u0027s just calculate this out."},{"Start":"02:22.970 ","End":"02:27.680","Text":"That\u0027s 2P minus 6P, that\u0027s minus 4P."},{"Start":"02:27.680 ","End":"02:30.455","Text":"1.2 plus 3 plus 4.2,"},{"Start":"02:30.455 ","End":"02:35.091","Text":"that\u0027s 5.8. Plus 5.8."},{"Start":"02:35.091 ","End":"02:37.930","Text":"Now we know that the expectation of x is 4.2,"},{"Start":"02:37.930 ","End":"02:41.095","Text":"so that equals to 4.2."},{"Start":"02:41.095 ","End":"02:43.455","Text":"Let\u0027s switch sides."},{"Start":"02:43.455 ","End":"02:52.960","Text":"4P equals to 1.6 and therefore P equals to 0.4."},{"Start":"02:53.300 ","End":"02:56.850","Text":"Excellent. If P equals 0.4,"},{"Start":"02:56.850 ","End":"02:58.365","Text":"we can plug that right here."},{"Start":"02:58.365 ","End":"03:05.780","Text":"That\u0027s 0.4 and here the probability for X equals 6,"},{"Start":"03:05.780 ","End":"03:13.360","Text":"that\u0027s 0.5 minus 0.4, that\u0027s 0.1."},{"Start":"03:13.360 ","End":"03:17.545","Text":"The next step is to calculate the variance of X."},{"Start":"03:17.545 ","End":"03:21.850","Text":"Now, what\u0027s the definition of the variance?"},{"Start":"03:21.850 ","End":"03:26.590","Text":"Well, that equals to the sum of x squared times"},{"Start":"03:26.590 ","End":"03:32.075","Text":"the probability of x minus the expectation squared of x."},{"Start":"03:32.075 ","End":"03:33.929","Text":"That\u0027s the definition."},{"Start":"03:33.929 ","End":"03:36.280","Text":"Let\u0027s plug in the numbers here that we have in"},{"Start":"03:36.280 ","End":"03:40.450","Text":"the probability function and we can calculate the variance."},{"Start":"03:40.450 ","End":"03:45.510","Text":"Well, that\u0027s 2 squared times 0.4,"},{"Start":"03:45.510 ","End":"03:47.010","Text":"that\u0027s this guy right here,"},{"Start":"03:47.010 ","End":"03:55.050","Text":"plus 4 squared times 0.3 plus 6 squared times 0.1 plus 8"},{"Start":"03:55.050 ","End":"04:04.065","Text":"squared times 0.2 minus 4.2 squared,"},{"Start":"04:04.065 ","End":"04:10.095","Text":"and that comes out to 5.16."},{"Start":"04:10.095 ","End":"04:14.350","Text":"That is the variance of x."}],"ID":12975},{"Watched":false,"Name":"Exercise 6","Duration":"3m 39s","ChapterTopicVideoID":12497,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.260","Text":"In this question, we have a discrete random variable that has the values of minus 5,"},{"Start":"00:06.260 ","End":"00:08.265","Text":"0, and plus 5."},{"Start":"00:08.265 ","End":"00:13.380","Text":"The variable\u0027s expectation is 0 and its variance is 10."},{"Start":"00:13.380 ","End":"00:17.660","Text":"Now, we have to find the probability function."},{"Start":"00:17.660 ","End":"00:21.335","Text":"Well, first of all, let\u0027s write down what we know."},{"Start":"00:21.335 ","End":"00:26.190","Text":"We know that the expectation of x equals 0,"},{"Start":"00:26.190 ","End":"00:30.820","Text":"and we know that the variance of x equals 10."},{"Start":"00:30.820 ","End":"00:35.790","Text":"Now, we want to find the probability function."},{"Start":"00:36.620 ","End":"00:38.865","Text":"Let\u0027s write this out."},{"Start":"00:38.865 ","End":"00:45.620","Text":"That\u0027s X, that\u0027s the probability in the x. X receives the values of minus 5,"},{"Start":"00:45.620 ","End":"00:48.635","Text":"0, and plus 5."},{"Start":"00:48.635 ","End":"00:54.440","Text":"Now, in order for the expectation to equal to 0,"},{"Start":"00:54.440 ","End":"01:01.085","Text":"that means that the distribution of the probability must be symmetrical."},{"Start":"01:01.085 ","End":"01:08.065","Text":"Why is that? Because the distance from this point to 0 and this point to 0 are equal."},{"Start":"01:08.065 ","End":"01:12.545","Text":"Let\u0027s write down here, that\u0027s p,"},{"Start":"01:12.545 ","End":"01:18.335","Text":"and define that as p, because they\u0027re symmetrical."},{"Start":"01:18.335 ","End":"01:21.590","Text":"What\u0027s the probability of 0?"},{"Start":"01:21.590 ","End":"01:25.140","Text":"Well, that\u0027s 1 minus 2p,"},{"Start":"01:25.140 ","End":"01:30.420","Text":"because the sum of other probabilities have to add up to 1."},{"Start":"01:31.690 ","End":"01:39.330","Text":"Now what we have to do is we have to calculate for p. Now,"},{"Start":"01:39.330 ","End":"01:43.815","Text":"we didn\u0027t use this data right here."},{"Start":"01:43.815 ","End":"01:47.750","Text":"Let\u0027s right now use this data to calculate what p is."},{"Start":"01:47.750 ","End":"01:51.755","Text":"Well, the variance of x is what?"},{"Start":"01:51.755 ","End":"01:54.920","Text":"That\u0027s equal to the sum of x"},{"Start":"01:54.920 ","End":"02:03.330","Text":"squared times p of x minus the expectation squared of x."},{"Start":"02:03.500 ","End":"02:06.255","Text":"Let\u0027s plug in the numbers."},{"Start":"02:06.255 ","End":"02:11.055","Text":"That\u0027s minus 5 squared times p,"},{"Start":"02:11.055 ","End":"02:13.305","Text":"that\u0027s this guy right here,"},{"Start":"02:13.305 ","End":"02:20.500","Text":"plus 0 squared times 1 minus 2p, that\u0027s this guy right here,"},{"Start":"02:20.500 ","End":"02:25.408","Text":"plus 5 squared times p,"},{"Start":"02:25.408 ","End":"02:29.853","Text":"that\u0027s this guy right here, minus 0 squared."},{"Start":"02:29.853 ","End":"02:31.790","Text":"That\u0027s the expectation squared."},{"Start":"02:31.790 ","End":"02:33.110","Text":"Our expectation equals 0,"},{"Start":"02:33.110 ","End":"02:34.746","Text":"so that\u0027s 0 squared."},{"Start":"02:34.746 ","End":"02:38.160","Text":"All this has to equal to what? To 10."},{"Start":"02:39.730 ","End":"02:42.830","Text":"We have minus 5 squared,"},{"Start":"02:42.830 ","End":"02:49.025","Text":"that\u0027s 25p plus all that\u0027s 0."},{"Start":"02:49.025 ","End":"02:50.840","Text":"That\u0027s 5 squared times p,"},{"Start":"02:50.840 ","End":"02:56.235","Text":"that\u0027s 25p minus 0 squared, that\u0027s 0."},{"Start":"02:56.235 ","End":"02:58.755","Text":"We have that equaling 10."},{"Start":"02:58.755 ","End":"03:06.195","Text":"That means that 50p equals to 10,"},{"Start":"03:06.195 ","End":"03:10.920","Text":"or P equals to 10 over 50,"},{"Start":"03:10.920 ","End":"03:15.450","Text":"and that equals to 0.2."},{"Start":"03:15.450 ","End":"03:19.805","Text":"If that\u0027s the case, then we\u0027ll go right back up here."},{"Start":"03:19.805 ","End":"03:22.100","Text":"P equals 0.2,"},{"Start":"03:22.100 ","End":"03:24.655","Text":"here, also 0.2,"},{"Start":"03:24.655 ","End":"03:30.270","Text":"and that means 1 minus 2 times 0.2 or 2 times 0.2, that\u0027s 0.4."},{"Start":"03:30.270 ","End":"03:34.530","Text":"So 1 minus 0.4 is 0.6."},{"Start":"03:34.530 ","End":"03:38.610","Text":"That is a probability function."}],"ID":12976},{"Watched":false,"Name":"Exercise 7","Duration":"8m 19s","ChapterTopicVideoID":12498,"CourseChapterTopicPlaylistID":64482,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"In this question, the probability distribution of X random variable is as follows."},{"Start":"00:06.030 ","End":"00:08.700","Text":"X has the values 1, 3,"},{"Start":"00:08.700 ","End":"00:12.825","Text":"and K with the probabilities of 1/4 when X equals 1,"},{"Start":"00:12.825 ","End":"00:14.745","Text":"1/2 when X equals 3,"},{"Start":"00:14.745 ","End":"00:20.980","Text":"and 1/4 when X equals K. We\u0027re asked what value of K,"},{"Start":"00:20.980 ","End":"00:26.150","Text":"this K right here will give the minimum value for the variance of x."},{"Start":"00:26.150 ","End":"00:29.120","Text":"Now, in order to calculate the variance,"},{"Start":"00:29.120 ","End":"00:32.330","Text":"let\u0027s first calculate the expectation."},{"Start":"00:32.330 ","End":"00:35.255","Text":"Now, what is the expectation of X?"},{"Start":"00:35.255 ","End":"00:40.055","Text":"Well, it\u0027s defined as the sum of x times the probability of x"},{"Start":"00:40.055 ","End":"00:46.410","Text":"and that equals to 1 times 1/4 plus 3 times 1/2,"},{"Start":"00:46.410 ","End":"00:52.380","Text":"plus k times 1/4,"},{"Start":"00:52.380 ","End":"00:55.954","Text":"and that equals to 0.25,"},{"Start":"00:55.954 ","End":"01:05.825","Text":"that\u0027s right here, and this is 1.5 right here and this is 0.25 times k. Now,"},{"Start":"01:05.825 ","End":"01:07.640","Text":"let\u0027s just add the things up."},{"Start":"01:07.640 ","End":"01:15.080","Text":"We have 1.75 plus 0.25k."},{"Start":"01:15.080 ","End":"01:18.880","Text":"Now, that\u0027s the expectation."},{"Start":"01:19.820 ","End":"01:23.620","Text":"Let\u0027s now calculate the variance."},{"Start":"01:23.620 ","End":"01:26.000","Text":"What\u0027s the variance of x?"},{"Start":"01:26.000 ","End":"01:32.420","Text":"Well, by definition, that\u0027s the sum of x squared times the probability"},{"Start":"01:32.420 ","End":"01:40.775","Text":"of x minus the expectation squared of x."},{"Start":"01:40.775 ","End":"01:43.610","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"01:43.610 ","End":"01:51.665","Text":"That\u0027s 1 squared times 1/4 plus 3 squared times 1/2,"},{"Start":"01:51.665 ","End":"02:00.700","Text":"plus k squared times 1/4 minus the expectation squared."},{"Start":"02:00.700 ","End":"02:10.280","Text":"Now, that\u0027s 1.75 plus right here is 0.25k squared."},{"Start":"02:11.510 ","End":"02:20.430","Text":"What does that equal to? That equals to 0.25 plus,"},{"Start":"02:20.430 ","End":"02:22.865","Text":"that\u0027s 3 squared a half of 9,"},{"Start":"02:22.865 ","End":"02:29.245","Text":"that\u0027s 4.5 plus 0.25k"},{"Start":"02:29.245 ","End":"02:33.620","Text":"squared minus again the expectation squared."},{"Start":"02:33.620 ","End":"02:43.300","Text":"We\u0027ll leave that this 1.75 plus 0.25k squared."},{"Start":"02:44.540 ","End":"02:49.085","Text":"Just to complete this calculation,"},{"Start":"02:49.085 ","End":"02:57.233","Text":"that equals to 4.75 plus 0.25k"},{"Start":"02:57.233 ","End":"03:06.910","Text":"squared minus 1.75 plus 0.25k squared."},{"Start":"03:07.820 ","End":"03:11.675","Text":"We found, we\u0027ve calculated,"},{"Start":"03:11.675 ","End":"03:15.080","Text":"what the variance of x was, it\u0027s right here."},{"Start":"03:15.080 ","End":"03:17.810","Text":"That\u0027s this long-expression right here."},{"Start":"03:17.810 ","End":"03:21.665","Text":"But we weren\u0027t asked to find the variance of x."},{"Start":"03:21.665 ","End":"03:27.980","Text":"We were asked what value of k will give the minimum value for the variance of x."},{"Start":"03:27.980 ","End":"03:31.729","Text":"Well, if we look at the expression for the variance,"},{"Start":"03:31.729 ","End":"03:36.920","Text":"we can look at it as a function of k."},{"Start":"03:36.920 ","End":"03:44.015","Text":"What we need to do is to find the minimum of that function."},{"Start":"03:44.015 ","End":"03:47.225","Text":"Well, let\u0027s write down this function."},{"Start":"03:47.225 ","End":"03:56.300","Text":"First of all, the function of k is equal to 4.75 plus"},{"Start":"03:56.300 ","End":"04:01.355","Text":"0.25k squared minus"},{"Start":"04:01.355 ","End":"04:08.530","Text":"1.75 plus 0.25k squared."},{"Start":"04:08.530 ","End":"04:10.240","Text":"Now, this is the same function."},{"Start":"04:10.240 ","End":"04:14.125","Text":"This is the same expression as we have here in the variance."},{"Start":"04:14.125 ","End":"04:19.120","Text":"But now we\u0027re looking at it in terms of k. Now,"},{"Start":"04:19.120 ","End":"04:23.455","Text":"in order to find a minimum or maximum of a function,"},{"Start":"04:23.455 ","End":"04:26.230","Text":"then we need to take the first derivative."},{"Start":"04:26.230 ","End":"04:32.895","Text":"Now, the first derivative of k. What we\u0027re doing is we\u0027re"},{"Start":"04:32.895 ","End":"04:40.430","Text":"taking the derivative with respect to k of this function."},{"Start":"04:43.100 ","End":"04:45.520","Text":"Once we have this derivative,"},{"Start":"04:45.520 ","End":"04:47.620","Text":"we have to equate that to 0."},{"Start":"04:47.620 ","End":"04:52.295","Text":"We have to solve for k and that will give us basically"},{"Start":"04:52.295 ","End":"05:00.810","Text":"the point at which the function f of k will be either at a minimum or a maximum."},{"Start":"05:00.810 ","End":"05:03.170","Text":"Let\u0027s start."},{"Start":"05:03.170 ","End":"05:09.140","Text":"The first derivative of the function is as follows."},{"Start":"05:09.140 ","End":"05:18.405","Text":"Well, this goes away and this is 0.5k minus 2 times"},{"Start":"05:18.405 ","End":"05:26.184","Text":"1.75 plus 0.25k times"},{"Start":"05:26.184 ","End":"05:30.290","Text":"the internal derivative right here, that\u0027s 0.25."},{"Start":"05:31.610 ","End":"05:35.320","Text":"Now, we have to equate that to 0,"},{"Start":"05:35.320 ","End":"05:38.245","Text":"but let\u0027s make this a little bit simpler first."},{"Start":"05:38.245 ","End":"05:48.325","Text":"That ultimately turns out to be 0.5k minus"},{"Start":"05:48.325 ","End":"05:54.100","Text":"0.875 minus"},{"Start":"05:54.100 ","End":"06:00.130","Text":"0.125k,"},{"Start":"06:00.130 ","End":"06:10.130","Text":"which is equal to 0.375k minus 0.875."},{"Start":"06:14.080 ","End":"06:16.475","Text":"Now, as we said,"},{"Start":"06:16.475 ","End":"06:25.665","Text":"we have to equate the first derivative to 0. f of k,"},{"Start":"06:25.665 ","End":"06:32.520","Text":"the first derivative, which equals to 0.375k,"},{"Start":"06:32.520 ","End":"06:38.675","Text":"that\u0027s right here, minus 0.875."},{"Start":"06:38.675 ","End":"06:41.440","Text":"We have to equate that to 0."},{"Start":"06:41.440 ","End":"06:48.270","Text":"That means that 0.375 times"},{"Start":"06:48.270 ","End":"06:55.200","Text":"k equals to 0.875."},{"Start":"06:55.200 ","End":"06:58.740","Text":"That means that k equals to"},{"Start":"06:58.740 ","End":"07:05.850","Text":"0.875 divided by 0.375,"},{"Start":"07:05.850 ","End":"07:10.780","Text":"and that equals to 2.33."},{"Start":"07:10.850 ","End":"07:16.730","Text":"Now, we figured out that the function f of k,"},{"Start":"07:16.730 ","End":"07:19.040","Text":"which is basically the variance,"},{"Start":"07:19.040 ","End":"07:25.045","Text":"gets its minimum or maximum when k equals 2.33."},{"Start":"07:25.045 ","End":"07:29.345","Text":"Now, we have to figure out whether it is a maximum or a minimum."},{"Start":"07:29.345 ","End":"07:33.480","Text":"What we need to do is take the second derivative."},{"Start":"07:33.830 ","End":"07:41.786","Text":"We take the second derivative with respect to k of the function,"},{"Start":"07:41.786 ","End":"07:45.715","Text":"that equals to this,"},{"Start":"07:45.715 ","End":"07:51.805","Text":"which equals 2 and here\u0027s the function right here."},{"Start":"07:51.805 ","End":"07:59.150","Text":"We have to take the derivative of this function and that equals to 0.375."},{"Start":"07:59.150 ","End":"08:01.085","Text":"It\u0027s greater than 0."},{"Start":"08:01.085 ","End":"08:04.810","Text":"That means that we have a minimum."},{"Start":"08:04.810 ","End":"08:14.150","Text":"We can say for sure that when k equals 2.33 in the probability function,"},{"Start":"08:14.150 ","End":"08:18.870","Text":"then the variance of x will be a minimum."}],"ID":12977}],"Thumbnail":null,"ID":64482},{"Name":"Linear Transformation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"6m 6s","ChapterTopicVideoID":12499,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.380 ","End":"00:03.420","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:03.420 ","End":"00:07.320","Text":"linear transformations for discrete random variables."},{"Start":"00:07.320 ","End":"00:09.855","Text":"What\u0027s a linear transformation?"},{"Start":"00:09.855 ","End":"00:17.115","Text":"Well, that\u0027s the addition and/or multiplication of a variable by a constant."},{"Start":"00:17.115 ","End":"00:19.920","Text":"When I talk about addition,"},{"Start":"00:19.920 ","End":"00:25.170","Text":"I also mean subtraction because subtraction is just the opposite operation of addition."},{"Start":"00:25.170 ","End":"00:31.200","Text":"Likewise for division, division is the opposite operation of multiplication."},{"Start":"00:31.200 ","End":"00:35.579","Text":"Again, when we take a random variable,"},{"Start":"00:35.579 ","End":"00:40.408","Text":"we either multiply it by a constant or we add a constant to it,"},{"Start":"00:40.408 ","End":"00:45.000","Text":"we\u0027re basically doing a linear transformation."},{"Start":"00:45.740 ","End":"00:48.020","Text":"We have a random variable,"},{"Start":"00:48.020 ","End":"00:53.450","Text":"let\u0027s call it X and we can either multiply it by constant,"},{"Start":"00:53.450 ","End":"00:55.250","Text":"let\u0027s call the constant a,"},{"Start":"00:55.250 ","End":"01:00.695","Text":"or we take X again and we add a constant to it,"},{"Start":"01:00.695 ","End":"01:02.135","Text":"let\u0027s called the constant b,"},{"Start":"01:02.135 ","End":"01:05.180","Text":"or we can do both at the same time,"},{"Start":"01:05.180 ","End":"01:08.420","Text":"we can have aX plus b."},{"Start":"01:08.420 ","End":"01:11.240","Text":"That means that we first multiply"},{"Start":"01:11.240 ","End":"01:14.450","Text":"X by a constant and then add a different constant to it."},{"Start":"01:14.450 ","End":"01:17.240","Text":"Basically what we\u0027re doing in all of these things,"},{"Start":"01:17.240 ","End":"01:20.930","Text":"we\u0027re creating a linear transformation of X."},{"Start":"01:20.930 ","End":"01:25.886","Text":"Having done that, we don\u0027t have X anymore,"},{"Start":"01:25.886 ","End":"01:28.865","Text":"we have a linear transformation of X."},{"Start":"01:28.865 ","End":"01:32.140","Text":"Let\u0027s call this transformation Y,"},{"Start":"01:32.140 ","End":"01:37.960","Text":"and Y would be aX plus b."},{"Start":"01:38.420 ","End":"01:41.915","Text":"Some of you may have seen this,"},{"Start":"01:41.915 ","End":"01:46.030","Text":"Y equals bX plus a."},{"Start":"01:46.030 ","End":"01:48.705","Text":"Basically these 2 things,"},{"Start":"01:48.705 ","End":"01:51.240","Text":"these 2 expressions are the same,"},{"Start":"01:51.240 ","End":"01:58.070","Text":"all we did is change the names of the multipliers and the addition constants."},{"Start":"01:58.070 ","End":"02:02.180","Text":"But basically this guy and this guy are totally the same,"},{"Start":"02:02.180 ","End":"02:04.310","Text":"and because they are the same,"},{"Start":"02:04.310 ","End":"02:06.080","Text":"let\u0027s just use this."},{"Start":"02:06.080 ","End":"02:12.060","Text":"I\u0027ll be using this expression to express the linear transformation of X."},{"Start":"02:12.800 ","End":"02:15.420","Text":"Having done that,"},{"Start":"02:15.420 ","End":"02:18.065","Text":"if Y equals aX plus b,"},{"Start":"02:18.065 ","End":"02:23.830","Text":"then what I want to do is I want to calculate the expectation of X and"},{"Start":"02:23.830 ","End":"02:33.470","Text":"the expectation of Y and the variance of Y just like we did in previous chapters with X,"},{"Start":"02:33.470 ","End":"02:37.895","Text":"where we calculated the expectation of the X and the variance of X."},{"Start":"02:37.895 ","End":"02:42.410","Text":"Now that we\u0027ve transformed X into something else,"},{"Start":"02:42.410 ","End":"02:45.290","Text":"we will call that something else Y,"},{"Start":"02:45.290 ","End":"02:48.980","Text":"we want to know what the expectation of the new variable is,"},{"Start":"02:48.980 ","End":"02:51.110","Text":"and the variance of the new variable,"},{"Start":"02:51.110 ","End":"02:53.230","Text":"we want to know what they are."},{"Start":"02:53.230 ","End":"02:55.600","Text":"Here are the rules."},{"Start":"02:55.600 ","End":"03:04.220","Text":"The rules are such that the expectation of Y equals a times the expectation of X plus b."},{"Start":"03:04.220 ","End":"03:09.185","Text":"This thing right here is the same as this guy right here,"},{"Start":"03:09.185 ","End":"03:12.980","Text":"and b, that\u0027s the same as this guy right here."},{"Start":"03:12.980 ","End":"03:18.530","Text":"The expectation of Y is a times the expectation of X plus b,"},{"Start":"03:18.530 ","End":"03:20.410","Text":"that\u0027s this guy right here."},{"Start":"03:20.410 ","End":"03:24.395","Text":"If we\u0027re looking at the variance of Y, well,"},{"Start":"03:24.395 ","End":"03:28.640","Text":"it\u0027s a squared times the variance of X,"},{"Start":"03:28.640 ","End":"03:31.615","Text":"it\u0027s the same a rate here."},{"Start":"03:31.615 ","End":"03:34.670","Text":"Once we\u0027ve calculated the variance,"},{"Start":"03:34.670 ","End":"03:36.650","Text":"then the standard deviation is easy."},{"Start":"03:36.650 ","End":"03:41.720","Text":"That\u0027s just the square root of the variance of Y which is"},{"Start":"03:41.720 ","End":"03:48.640","Text":"equal to the absolute value of a times the standard deviation of X."},{"Start":"03:48.640 ","End":"03:51.680","Text":"In order to make things simpler,"},{"Start":"03:51.680 ","End":"03:53.810","Text":"what I\u0027ve done is I\u0027ve prepared"},{"Start":"03:53.810 ","End":"03:59.360","Text":"a 4-step process that will help you solve the linear transformation problems."},{"Start":"03:59.360 ","End":"04:02.130","Text":"Here\u0027s the process."},{"Start":"04:02.270 ","End":"04:05.120","Text":"The first thing that you need to do is to"},{"Start":"04:05.120 ","End":"04:08.345","Text":"recognize that you\u0027re dealing with a linear transformation."},{"Start":"04:08.345 ","End":"04:10.550","Text":"That means that again,"},{"Start":"04:10.550 ","End":"04:13.865","Text":"we have a fixed change in all the observations."},{"Start":"04:13.865 ","End":"04:19.910","Text":"Whether the fixed change is multiplication of all the other observations by"},{"Start":"04:19.910 ","End":"04:23.420","Text":"a constant or the addition of a constant"},{"Start":"04:23.420 ","End":"04:27.650","Text":"to each 1 of the observations or maybe both things at the same time."},{"Start":"04:27.650 ","End":"04:37.650","Text":"So we know that we\u0027re taking X and we\u0027re transforming it in 1 way or another into Y."},{"Start":"04:37.870 ","End":"04:43.985","Text":"The second step is to write the transformation rules according to the data in question."},{"Start":"04:43.985 ","End":"04:52.570","Text":"We need to know how we\u0027re transforming the original variable into the different variable."},{"Start":"04:52.570 ","End":"04:56.820","Text":"The third step is to simplify the rules,"},{"Start":"04:56.820 ","End":"04:59.285","Text":"and identify the values of a and b."},{"Start":"04:59.285 ","End":"05:07.370","Text":"That means that once we know how we\u0027re going to transform X into Y,"},{"Start":"05:07.370 ","End":"05:17.060","Text":"then we have to bring this expression into the form of Y equals aX plus b,"},{"Start":"05:17.060 ","End":"05:21.050","Text":"where a would be the multiplier of X and b"},{"Start":"05:21.050 ","End":"05:25.915","Text":"will be the constant that we add or subtract to this expression."},{"Start":"05:25.915 ","End":"05:29.180","Text":"Once we\u0027ve identified a and b,"},{"Start":"05:29.180 ","End":"05:34.950","Text":"we can go ahead and calculate the expectation of Y,"},{"Start":"05:35.180 ","End":"05:38.260","Text":"and the variance of Y."},{"Start":"05:38.260 ","End":"05:45.095","Text":"Where the expectation of Y is a times the expectation of X plus b,"},{"Start":"05:45.095 ","End":"05:52.115","Text":"and the variance of Y is a squared times the variance of X,"},{"Start":"05:52.115 ","End":"05:56.135","Text":"where this guy a is this guy right here,"},{"Start":"05:56.135 ","End":"06:00.640","Text":"and this b is this b right here."},{"Start":"06:00.890 ","End":"06:06.120","Text":"Enough of theory, and let\u0027s take a look at an example."}],"ID":12978},{"Watched":false,"Name":"Example","Duration":"6m 53s","ChapterTopicVideoID":12500,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.460","Text":"In this example, we\u0027ll use the roulette example that we used in the last chapter."},{"Start":"00:05.460 ","End":"00:12.180","Text":"Now, we\u0027ll also assume that the cost of playing the game is $15. What does that mean?"},{"Start":"00:12.180 ","End":"00:14.670","Text":"That means that every time you turn the roulette wheel,"},{"Start":"00:14.670 ","End":"00:17.145","Text":"you have to pay the house $15."},{"Start":"00:17.145 ","End":"00:19.875","Text":"Now, what we\u0027re asked is,"},{"Start":"00:19.875 ","End":"00:24.735","Text":"what are the expectation and variance of the profit of the game?"},{"Start":"00:24.735 ","End":"00:26.969","Text":"Now, if you recall,"},{"Start":"00:26.969 ","End":"00:31.690","Text":"X was defined as your winnings,"},{"Start":"00:32.840 ","End":"00:39.140","Text":"and we\u0027ve calculated the expectation of X as being $22.5."},{"Start":"00:39.140 ","End":"00:45.440","Text":"The variance of X as being $68.75."},{"Start":"00:45.440 ","End":"00:49.895","Text":"Now, here, we\u0027re not talking about your winnings,"},{"Start":"00:49.895 ","End":"00:52.085","Text":"we\u0027re talking about profits."},{"Start":"00:52.085 ","End":"00:57.890","Text":"Now, profits is something totally different. What\u0027s the profit?"},{"Start":"00:57.890 ","End":"01:05.190","Text":"The profit is basically taking your winnings and taking away the expenses."},{"Start":"01:05.190 ","End":"01:07.790","Text":"What you have left is your profits."},{"Start":"01:07.790 ","End":"01:16.170","Text":"So that means that we have here a new variable which transforms X,"},{"Start":"01:16.170 ","End":"01:20.920","Text":"your winnings into profits."},{"Start":"01:22.670 ","End":"01:25.415","Text":"Once we\u0027ve understood that,"},{"Start":"01:25.415 ","End":"01:30.230","Text":"let\u0027s take a look at the 4-step process that we"},{"Start":"01:30.230 ","End":"01:35.680","Text":"used in order to help us solve these types of problems."},{"Start":"01:35.680 ","End":"01:39.980","Text":"Here it is. What\u0027s the first step?"},{"Start":"01:39.980 ","End":"01:43.775","Text":"The first step is to recognize that you\u0027re dealing with a linear transformation."},{"Start":"01:43.775 ","End":"01:45.850","Text":"Well, we\u0027ve already done that."},{"Start":"01:45.850 ","End":"01:50.570","Text":"We\u0027ve understood that we have to take her winnings,"},{"Start":"01:50.570 ","End":"01:57.890","Text":"we have to subtract our expenses from our winnings in order to get our profits."},{"Start":"01:57.890 ","End":"02:00.740","Text":"So for every observation,"},{"Start":"02:00.740 ","End":"02:04.310","Text":"for every time we turn the roulette wheel,"},{"Start":"02:04.310 ","End":"02:05.824","Text":"we have our winnings."},{"Start":"02:05.824 ","End":"02:09.915","Text":"From that, we have to subtract our expenses."},{"Start":"02:09.915 ","End":"02:13.294","Text":"The ability to play or to roll,"},{"Start":"02:13.294 ","End":"02:15.185","Text":"or to turn the wheel."},{"Start":"02:15.185 ","End":"02:18.440","Text":"That will create our profits."},{"Start":"02:18.440 ","End":"02:21.705","Text":"That\u0027s right here. Basically,"},{"Start":"02:21.705 ","End":"02:24.125","Text":"we\u0027ve done the first step."},{"Start":"02:24.125 ","End":"02:25.835","Text":"What\u0027s the second step?"},{"Start":"02:25.835 ","End":"02:29.750","Text":"Is reading the transformation rule according to the data."},{"Start":"02:29.750 ","End":"02:33.280","Text":"Well, again, because profits,"},{"Start":"02:33.280 ","End":"02:37.565","Text":"we\u0027ve defined profits as your winnings minus your expenses."},{"Start":"02:37.565 ","End":"02:46.320","Text":"So why are profits equals X our winnings minus 15?"},{"Start":"02:46.880 ","End":"02:50.370","Text":"Here we\u0027ve completed step number 2."},{"Start":"02:50.370 ","End":"02:52.475","Text":"Now, what\u0027s step number 3?"},{"Start":"02:52.475 ","End":"02:56.345","Text":"Simplify the rules and identify the values of a and b."},{"Start":"02:56.345 ","End":"02:59.225","Text":"Well here, there\u0027s not much to simplify,"},{"Start":"02:59.225 ","End":"03:01.655","Text":"but we can write this out differently."},{"Start":"03:01.655 ","End":"03:09.940","Text":"We can say that Y equals to 1 times X plus minus 15."},{"Start":"03:09.940 ","End":"03:14.630","Text":"Now, the minute you write it this way, it\u0027s very simple."},{"Start":"03:14.630 ","End":"03:17.900","Text":"It\u0027s very easy to identify a as being 1."},{"Start":"03:17.900 ","End":"03:24.720","Text":"That\u0027s the multiplier of X and b as being equal to minus 15."},{"Start":"03:25.840 ","End":"03:28.550","Text":"We\u0027ve done number 3."},{"Start":"03:28.550 ","End":"03:30.945","Text":"What steps number 4?"},{"Start":"03:30.945 ","End":"03:34.850","Text":"Substituting the above formulas according to the measures and questions."},{"Start":"03:34.850 ","End":"03:37.790","Text":"Now we have a and we have b,"},{"Start":"03:37.790 ","End":"03:39.440","Text":"and what do we ask?"},{"Start":"03:39.440 ","End":"03:43.925","Text":"We\u0027re asked what is the expectation and variance of the profit of the game."},{"Start":"03:43.925 ","End":"03:47.810","Text":"That means that we want to know the expectation of Y."},{"Start":"03:47.810 ","End":"03:49.970","Text":"Now, according to the formulas,"},{"Start":"03:49.970 ","End":"03:57.260","Text":"the expectation of Y is 8 times the expectation of X plus b."},{"Start":"03:57.260 ","End":"04:01.430","Text":"Now, let\u0027s just plug in our numbers here."},{"Start":"04:01.430 ","End":"04:07.560","Text":"A is equal to 1 times 22.5,"},{"Start":"04:07.560 ","End":"04:09.645","Text":"that\u0027s the expectation of X,"},{"Start":"04:09.645 ","End":"04:13.715","Text":"plus minus 15, divides minus 15."},{"Start":"04:13.715 ","End":"04:15.335","Text":"Now, what does that equal to?"},{"Start":"04:15.335 ","End":"04:18.535","Text":"That equals to 7.5."},{"Start":"04:18.535 ","End":"04:22.295","Text":"Now, let\u0027s see if it makes sense or not."},{"Start":"04:22.295 ","End":"04:26.690","Text":"Again, we\u0027re taking the expectation of her winnings,"},{"Start":"04:26.690 ","End":"04:28.220","Text":"which is 22.5,"},{"Start":"04:28.220 ","End":"04:31.370","Text":"for each gain, ultimately,"},{"Start":"04:31.370 ","End":"04:35.875","Text":"we\u0027re expected to win $22.5,"},{"Start":"04:35.875 ","End":"04:37.130","Text":"but for each game,"},{"Start":"04:37.130 ","End":"04:42.620","Text":"we also have to take away $15 just for the right to play."},{"Start":"04:42.620 ","End":"04:45.365","Text":"So for each game,"},{"Start":"04:45.365 ","End":"04:50.254","Text":"our expected profits are 22.5 minus 15."},{"Start":"04:50.254 ","End":"04:51.905","Text":"That\u0027s exactly what we have here."},{"Start":"04:51.905 ","End":"04:54.710","Text":"That\u0027s $7.5 right here."},{"Start":"04:54.710 ","End":"05:00.240","Text":"Let\u0027s take a look at the variance of Y. Variance of"},{"Start":"05:00.240 ","End":"05:06.485","Text":"Y is defined as a squared times the variance of X."},{"Start":"05:06.485 ","End":"05:09.350","Text":"Let\u0027s just write this out a little bit better."},{"Start":"05:09.350 ","End":"05:14.150","Text":"This is a squared times the variance of X."},{"Start":"05:14.150 ","End":"05:17.780","Text":"Well, a squared, this plug-in the numbers a squared equals 1,"},{"Start":"05:17.780 ","End":"05:24.695","Text":"so that\u0027s 1 squared times 68.75,"},{"Start":"05:24.695 ","End":"05:29.090","Text":"and that equals to 68.75."},{"Start":"05:29.090 ","End":"05:30.890","Text":"Now, what does that tell us?"},{"Start":"05:30.890 ","End":"05:35.185","Text":"That tells us that it really doesn\u0027t matter"},{"Start":"05:35.185 ","End":"05:41.955","Text":"about the constant that you add or subtract to the original variable."},{"Start":"05:41.955 ","End":"05:46.060","Text":"When you take the variance of a linear transformation,"},{"Start":"05:46.060 ","End":"05:51.595","Text":"you\u0027re always dealing with the square of the multiplier."},{"Start":"05:51.595 ","End":"05:55.975","Text":"Square of the, this guy right here,"},{"Start":"05:55.975 ","End":"05:58.725","Text":"or this guy right here, a."},{"Start":"05:58.725 ","End":"06:02.830","Text":"It makes sense because it really doesn\u0027t matter by how much you"},{"Start":"06:02.830 ","End":"06:07.620","Text":"subtract or you add a certain cost into to all the observation."},{"Start":"06:07.620 ","End":"06:16.420","Text":"What really matters to the variance is how the data is dispersed around the expectation."},{"Start":"06:16.420 ","End":"06:25.730","Text":"That\u0027s really dependent only on the multiplier of the original variable. Here we have it."},{"Start":"06:25.730 ","End":"06:28.595","Text":"We have solved the problem."},{"Start":"06:28.595 ","End":"06:32.495","Text":"We know what the expectation and variance of the profits."},{"Start":"06:32.495 ","End":"06:37.910","Text":"Now, again, the only thing that you need to do right now is just go ahead and solve"},{"Start":"06:37.910 ","End":"06:44.420","Text":"the problems in order to get better at solving that linear transformations."},{"Start":"06:44.420 ","End":"06:49.100","Text":"Only then I ask of you to view the solution videos."},{"Start":"06:49.100 ","End":"06:54.000","Text":"So you just make sure that you\u0027ve done things right. Good luck."}],"ID":12979},{"Watched":false,"Name":"Exercise 1","Duration":"5m 15s","ChapterTopicVideoID":12501,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this question we\u0027ll be calculating academic points."},{"Start":"00:04.080 ","End":"00:07.740","Text":"Now, Jack takes 5 courses in a semester."},{"Start":"00:07.740 ","End":"00:12.299","Text":"Assume that he receives 4 academic points for each course he completes."},{"Start":"00:12.299 ","End":"00:15.000","Text":"We\u0027re asked to calculate the expectation and"},{"Start":"00:15.000 ","End":"00:17.895","Text":"variance of the total number of points accumulated,"},{"Start":"00:17.895 ","End":"00:23.190","Text":"assuming that the expected number of courses he completes is 3.5,"},{"Start":"00:23.190 ","End":"00:25.780","Text":"with a variance of 2."},{"Start":"00:26.630 ","End":"00:32.555","Text":"First of all, we know several things and let\u0027s write them down."},{"Start":"00:32.555 ","End":"00:38.790","Text":"We know that x is the number of courses."},{"Start":"00:39.850 ","End":"00:51.245","Text":"We know that the expectation of x is 3.5 and we know that the variance of x is 2."},{"Start":"00:51.245 ","End":"00:53.360","Text":"Again, how do we know that?"},{"Start":"00:53.360 ","End":"00:55.325","Text":"Well, first of all,"},{"Start":"00:55.325 ","End":"00:58.760","Text":"the expectation is right here, 3.5,"},{"Start":"00:58.760 ","End":"01:01.595","Text":"and the variance is right here,"},{"Start":"01:01.595 ","End":"01:08.490","Text":"and we\u0027re looking at courses in a semester."},{"Start":"01:08.620 ","End":"01:12.410","Text":"Now, what do we have here?"},{"Start":"01:12.410 ","End":"01:19.390","Text":"Well, first of all, let\u0027s work with the 4-step process that we learned."},{"Start":"01:19.390 ","End":"01:21.935","Text":"Here it is right here."},{"Start":"01:21.935 ","End":"01:25.580","Text":"Now, the first thing that we need to do is we need to"},{"Start":"01:25.580 ","End":"01:29.960","Text":"recognize that we\u0027re dealing with a linear transformation."},{"Start":"01:29.960 ","End":"01:32.555","Text":"What\u0027s the linear transformation?"},{"Start":"01:32.555 ","End":"01:36.170","Text":"Again, we have to go from the number of courses,"},{"Start":"01:36.170 ","End":"01:38.780","Text":"and we\u0027re asked to"},{"Start":"01:38.780 ","End":"01:48.650","Text":"calculate the number of points that he accumulated."},{"Start":"01:48.650 ","End":"01:58.935","Text":"What does that mean? That we have to go from X to Y,"},{"Start":"01:58.935 ","End":"02:02.710","Text":"where X is the number of courses,"},{"Start":"02:04.700 ","End":"02:09.610","Text":"and Y is the number of academic points."},{"Start":"02:09.610 ","End":"02:12.730","Text":"Now, what\u0027s the linear transformation?"},{"Start":"02:12.730 ","End":"02:23.750","Text":"Well, we know that he receives 4 academic points for each course."},{"Start":"02:25.370 ","End":"02:28.630","Text":"We\u0027ve recognized first of all,"},{"Start":"02:28.630 ","End":"02:31.840","Text":"that we\u0027re dealing with a linear transformation."},{"Start":"02:31.840 ","End":"02:37.170","Text":"For each course he receives 4 academic points."},{"Start":"02:37.170 ","End":"02:38.930","Text":"Let\u0027s write this down right here."},{"Start":"02:38.930 ","End":"02:41.870","Text":"We\u0027ve finished this now we have to write this down."},{"Start":"02:41.870 ","End":"02:48.845","Text":"Y equals 4 times X, and why is that?"},{"Start":"02:48.845 ","End":"02:54.990","Text":"Because the number of points equals 4 times the number of courses he takes."},{"Start":"02:54.990 ","End":"02:56.930","Text":"If X is a number of courses he takes,"},{"Start":"02:56.930 ","End":"02:59.240","Text":"and for each course he gets 4 points,"},{"Start":"02:59.240 ","End":"03:04.650","Text":"then the number of points equals to 4 times X."},{"Start":"03:05.000 ","End":"03:10.490","Text":"Let\u0027s simplify this in order to identify a and b."},{"Start":"03:10.490 ","End":"03:18.940","Text":"Well, Y equals 4X plus 0, we can write this."},{"Start":"03:19.070 ","End":"03:21.500","Text":"Why did we write it like this?"},{"Start":"03:21.500 ","End":"03:25.250","Text":"Because we can easily see that a equals 4,"},{"Start":"03:25.250 ","End":"03:29.940","Text":"that\u0027s the multiplier, and b equals 0."},{"Start":"03:30.770 ","End":"03:35.215","Text":"Now we\u0027ve finished Step 3. Now, what\u0027s Step 4?"},{"Start":"03:35.215 ","End":"03:38.120","Text":"Is substitute a and b in the formulas."},{"Start":"03:38.120 ","End":"03:39.695","Text":"Now, what are we asked to do?"},{"Start":"03:39.695 ","End":"03:45.710","Text":"We\u0027re asked to calculate the expectation and variance of the total number of points."},{"Start":"03:45.710 ","End":"03:53.615","Text":"That means that we want to know what the expectation of Y is and the variance of Y."},{"Start":"03:53.615 ","End":"03:57.785","Text":"Well, the expectation of Y is,"},{"Start":"03:57.785 ","End":"03:59.585","Text":"if we remember the formula,"},{"Start":"03:59.585 ","End":"04:04.405","Text":"a times the expectation of X plus b."},{"Start":"04:04.405 ","End":"04:06.990","Text":"Now, let\u0027s plug in the numbers,"},{"Start":"04:06.990 ","End":"04:09.850","Text":"a equals 4 from right here,"},{"Start":"04:09.850 ","End":"04:12.680","Text":"times expectation of X,"},{"Start":"04:12.680 ","End":"04:16.590","Text":"that\u0027s 3.5 plus b,"},{"Start":"04:16.590 ","End":"04:22.420","Text":"plus 0, and that equals to 14."},{"Start":"04:23.350 ","End":"04:30.880","Text":"Jack is expected to earn 14 academic points in a semester."},{"Start":"04:30.880 ","End":"04:32.810","Text":"What\u0027s the variance?"},{"Start":"04:32.810 ","End":"04:39.070","Text":"Well, the variance equals a squared times the variance of X,"},{"Start":"04:39.070 ","End":"04:42.395","Text":"and that equals, let\u0027s plug in the numbers."},{"Start":"04:42.395 ","End":"04:44.360","Text":"What\u0027s a? Is 4,"},{"Start":"04:44.360 ","End":"04:49.770","Text":"that\u0027s 4 squared times the variance of X, times 2."},{"Start":"04:49.770 ","End":"04:54.150","Text":"That equals to 32."},{"Start":"04:54.290 ","End":"04:57.720","Text":"Which means that\u0027s 4 squared,"},{"Start":"04:57.720 ","End":"05:01.415","Text":"4 squared is 16 times 2 equals 32,"},{"Start":"05:01.415 ","End":"05:06.035","Text":"which means that on average,"},{"Start":"05:06.035 ","End":"05:09.545","Text":"the distribution or the dispersement of the points"},{"Start":"05:09.545 ","End":"05:15.270","Text":"around the expectation is 32 points squared."}],"ID":12980},{"Watched":false,"Name":"Exercise 2","Duration":"6m 7s","ChapterTopicVideoID":12502,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.285","Text":"This question will be talking about calculating the profits in a game of chance."},{"Start":"00:06.285 ","End":"00:09.890","Text":"Now, the expected price in a game of chance is 10,"},{"Start":"00:09.890 ","End":"00:11.860","Text":"with a variance of 3."},{"Start":"00:11.860 ","End":"00:15.375","Text":"It\u0027s decided to double the prize in the game."},{"Start":"00:15.375 ","End":"00:18.615","Text":"The cost of playing the game is 12, and we\u0027re asked,"},{"Start":"00:18.615 ","End":"00:23.560","Text":"what is the expectation and variance of the profits in the game?"},{"Start":"00:24.890 ","End":"00:29.250","Text":"Let\u0027s first look at a 4 step process and"},{"Start":"00:29.250 ","End":"00:34.120","Text":"see how we can use that to try to solve this question."},{"Start":"00:34.820 ","End":"00:37.605","Text":"Here are 4 steps."},{"Start":"00:37.605 ","End":"00:39.035","Text":"What\u0027s the first step?"},{"Start":"00:39.035 ","End":"00:44.760","Text":"The first step is to recognize that we\u0027re dealing with the linear transformation."},{"Start":"00:47.090 ","End":"00:49.985","Text":"What\u0027s our original variable?"},{"Start":"00:49.985 ","End":"00:54.980","Text":"Our original variable talks about the price or winnings in the game,"},{"Start":"00:54.980 ","End":"00:59.720","Text":"and we\u0027re given that the expectation of our winnings is 10,"},{"Start":"00:59.720 ","End":"01:03.110","Text":"and the variance of our winnings is 3."},{"Start":"01:03.110 ","End":"01:05.825","Text":"First of all, let\u0027s write this down."},{"Start":"01:05.825 ","End":"01:09.480","Text":"X would be our winnings."},{"Start":"01:11.080 ","End":"01:14.270","Text":"The expectation of X,"},{"Start":"01:14.270 ","End":"01:16.220","Text":"expectation our winning is 10,"},{"Start":"01:16.220 ","End":"01:21.390","Text":"and the variance of our winnings is 3."},{"Start":"01:21.470 ","End":"01:24.245","Text":"What are we looking for?"},{"Start":"01:24.245 ","End":"01:28.200","Text":"We\u0027re looking at profits."},{"Start":"01:28.200 ","End":"01:33.725","Text":"We want to know how to get from our winnings to profits."},{"Start":"01:33.725 ","End":"01:39.625","Text":"This is the second step of our process."},{"Start":"01:39.625 ","End":"01:43.880","Text":"We have to write the transformation rules according to the data and question."},{"Start":"01:43.880 ","End":"01:47.000","Text":"Excellent. Let\u0027s take an example."},{"Start":"01:47.000 ","End":"01:51.240","Text":"Assume that our winnings are 10."},{"Start":"01:51.260 ","End":"01:53.540","Text":"Now, what do we have to do?"},{"Start":"01:53.540 ","End":"01:56.060","Text":"We\u0027re given that we have to double the size,"},{"Start":"01:56.060 ","End":"01:57.665","Text":"the price in the game."},{"Start":"01:57.665 ","End":"02:00.620","Text":"That means we have to take our winnings,"},{"Start":"02:00.620 ","End":"02:02.675","Text":"multiply that by 2, we\u0027re doubling them,"},{"Start":"02:02.675 ","End":"02:06.900","Text":"but we also have to take away 12,"},{"Start":"02:06.900 ","End":"02:09.495","Text":"because that\u0027s the cost of playing the game."},{"Start":"02:09.495 ","End":"02:13.815","Text":"That\u0027s 20 minus 12, that\u0027s 8."},{"Start":"02:13.815 ","End":"02:16.710","Text":"If our winnings were 20,"},{"Start":"02:16.710 ","End":"02:20.624","Text":"it would be 2 times 20 minus 12."},{"Start":"02:20.624 ","End":"02:27.860","Text":"What would that be? That would be 40 minus 12, and that\u0027s 28."},{"Start":"02:27.860 ","End":"02:31.550","Text":"What would happen if our winnings were 30?"},{"Start":"02:31.550 ","End":"02:34.835","Text":"Again, that\u0027s 2 times 30 minus 12,"},{"Start":"02:34.835 ","End":"02:37.595","Text":"60 minus 12, that\u0027ll be 48,"},{"Start":"02:37.595 ","End":"02:39.890","Text":"and so on and so forth."},{"Start":"02:39.890 ","End":"02:45.245","Text":"We can see that the transformation rule is this."},{"Start":"02:45.245 ","End":"02:48.950","Text":"We have to double our winnings."},{"Start":"02:48.950 ","End":"02:51.365","Text":"That\u0027s 2 times X."},{"Start":"02:51.365 ","End":"02:55.350","Text":"We have to take away the cost of playing."},{"Start":"02:55.940 ","End":"02:59.815","Text":"That\u0027s the transformation rule right here."},{"Start":"02:59.815 ","End":"03:02.750","Text":"Let\u0027s look at the general transformation,"},{"Start":"03:02.750 ","End":"03:06.935","Text":"linear transformation from X to Y. What\u0027s Y?"},{"Start":"03:06.935 ","End":"03:10.620","Text":"Let\u0027s define Y as our profits."},{"Start":"03:12.220 ","End":"03:21.355","Text":"We can say that Y equals aX plus b."},{"Start":"03:21.355 ","End":"03:24.490","Text":"The minute we write this out,"},{"Start":"03:24.490 ","End":"03:28.900","Text":"the general way of writing the linear transformation on 1 hand,"},{"Start":"03:28.900 ","End":"03:32.740","Text":"and on the other hand, this is what we got when we mapped out"},{"Start":"03:32.740 ","End":"03:36.700","Text":"the transformation according to the problem that was given to us."},{"Start":"03:36.700 ","End":"03:42.700","Text":"It\u0027s very easy to see when we compare the 2 that a"},{"Start":"03:42.700 ","End":"03:50.540","Text":"equals to 2 and b equals to minus 12."},{"Start":"03:50.580 ","End":"03:59.290","Text":"Because there\u0027s no way to simplify this more."},{"Start":"03:59.750 ","End":"04:02.820","Text":"We\u0027ve d1 this,"},{"Start":"04:02.820 ","End":"04:05.340","Text":"we\u0027ve simplified the rules."},{"Start":"04:05.340 ","End":"04:07.965","Text":"We can\u0027t simplify this anymore."},{"Start":"04:07.965 ","End":"04:11.510","Text":"We were able to identify a and b."},{"Start":"04:11.510 ","End":"04:14.390","Text":"This is a, this is b,"},{"Start":"04:14.390 ","End":"04:18.310","Text":"a is equal to 2 and b is equal to minus 12."},{"Start":"04:18.310 ","End":"04:23.225","Text":"The last thing is to substitute a and b in the questions"},{"Start":"04:23.225 ","End":"04:28.040","Text":"or in the expressions for the expectation and variance of Y."},{"Start":"04:28.040 ","End":"04:31.220","Text":"This is basically what we\u0027ve been asked to do to"},{"Start":"04:31.220 ","End":"04:36.180","Text":"calculate the expectation and variance of the profit of Y."},{"Start":"04:36.550 ","End":"04:47.660","Text":"The expectation of Y equals a times the expectation of X plus b."},{"Start":"04:47.660 ","End":"04:49.940","Text":"Let\u0027s plug in the numbers,"},{"Start":"04:49.940 ","End":"04:53.810","Text":"a is equal to 2 times 10,"},{"Start":"04:53.810 ","End":"04:57.005","Text":"that\u0027s the expectation of X plus b."},{"Start":"04:57.005 ","End":"05:00.125","Text":"Now b is minus 12, that\u0027s minus 12."},{"Start":"05:00.125 ","End":"05:03.170","Text":"That equals to 20 minus 12,"},{"Start":"05:03.170 ","End":"05:05.710","Text":"that would equal to 8."},{"Start":"05:05.710 ","End":"05:09.660","Text":"That\u0027s the expectation of our profits,"},{"Start":"05:09.660 ","End":"05:18.085","Text":"our winnings, the expectation of our winnings minus the costs of the game."},{"Start":"05:18.085 ","End":"05:21.035","Text":"It\u0027s not the expectation of our winnings."},{"Start":"05:21.035 ","End":"05:23.630","Text":"It\u0027s basically doubling the price of the game,"},{"Start":"05:23.630 ","End":"05:28.325","Text":"doubling the winnings, and then taking away the cost."},{"Start":"05:28.325 ","End":"05:33.130","Text":"It\u0027s 2 times their winnings minus 12 and that equals to 8."},{"Start":"05:33.130 ","End":"05:36.585","Text":"What\u0027s the variance of Y?"},{"Start":"05:36.585 ","End":"05:42.520","Text":"The variance of Y is a squared times the variance of X."},{"Start":"05:42.520 ","End":"05:45.045","Text":"Let\u0027s plug in the numbers,"},{"Start":"05:45.045 ","End":"05:46.710","Text":"a equals 2,"},{"Start":"05:46.710 ","End":"05:49.700","Text":"so that\u0027s 2 squared times,"},{"Start":"05:49.700 ","End":"05:50.780","Text":"what\u0027s the variance of X?"},{"Start":"05:50.780 ","End":"05:52.540","Text":"That\u0027s here, that\u0027s 3."},{"Start":"05:52.540 ","End":"05:58.635","Text":"That equals to 2 squared is 4 times 3, that\u0027s 12."},{"Start":"05:58.635 ","End":"06:03.290","Text":"That would be the variance of our profits."},{"Start":"06:03.290 ","End":"06:07.860","Text":"This is the expectation of our profits."}],"ID":12981},{"Watched":false,"Name":"Exercise 3","Duration":"5m 7s","ChapterTopicVideoID":12503,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this question, we\u0027re given that the expectation of"},{"Start":"00:03.270 ","End":"00:08.185","Text":"a random variable is x and the standard deviation is 5."},{"Start":"00:08.185 ","End":"00:15.855","Text":"It\u0027s decided to add 2 to the variable and then to increase the whole thing by 10 percent."},{"Start":"00:15.855 ","End":"00:21.420","Text":"We\u0027re asked, what are the expectation and standard deviation after the change?"},{"Start":"00:21.420 ","End":"00:23.565","Text":"well, in order to do that,"},{"Start":"00:23.565 ","End":"00:30.310","Text":"Let\u0027s work with our 4 step process. Here it is."},{"Start":"00:31.520 ","End":"00:33.715","Text":"What\u0027s the first step?"},{"Start":"00:33.715 ","End":"00:38.405","Text":"The first step is to recognize that we\u0027re dealing with a linear transformation."},{"Start":"00:38.405 ","End":"00:40.400","Text":"Let\u0027s see what we have here."},{"Start":"00:40.400 ","End":"00:43.085","Text":"Our original variable x."},{"Start":"00:43.085 ","End":"00:48.575","Text":"We have the expectation of x and that\u0027s given to us, that\u0027s 10."},{"Start":"00:48.575 ","End":"00:54.065","Text":"That\u0027s right here and our standard deviation of x,"},{"Start":"00:54.065 ","End":"00:59.625","Text":"well that\u0027s 5, and what do we want to do?"},{"Start":"00:59.625 ","End":"01:06.260","Text":"Well, we want to add 2 to the variable and then to increase everything by 10 percent."},{"Start":"01:06.260 ","End":"01:08.885","Text":"Well, that\u0027s the second step. Let\u0027s write the cell."},{"Start":"01:08.885 ","End":"01:10.735","Text":"We have x,"},{"Start":"01:10.735 ","End":"01:14.205","Text":"we want to add 2 to it,"},{"Start":"01:14.205 ","End":"01:20.210","Text":"and then we take this whole thing and we increase it by 10 percent."},{"Start":"01:20.210 ","End":"01:27.540","Text":"Now that\u0027s just like multiplying it by 1.1. Now why is that?"},{"Start":"01:28.460 ","End":"01:30.770","Text":"Let\u0027s take an example."},{"Start":"01:30.770 ","End":"01:37.370","Text":"For example, if we have the number 100 and we want to increase it by 10 percent,"},{"Start":"01:37.370 ","End":"01:41.065","Text":"when we multiply it by 1.1,"},{"Start":"01:41.065 ","End":"01:43.665","Text":"we\u0027ll get 110,"},{"Start":"01:43.665 ","End":"01:46.845","Text":"where 10 would be 10 percent of 100."},{"Start":"01:46.845 ","End":"01:52.145","Text":"If we have 200 and we want to increase that by 10 percent,"},{"Start":"01:52.145 ","End":"01:54.230","Text":"when we multiply it by 1.1,"},{"Start":"01:54.230 ","End":"01:55.775","Text":"we\u0027ll get 220,"},{"Start":"01:55.775 ","End":"01:58.330","Text":"where 20 would be the 10 percent."},{"Start":"01:58.330 ","End":"02:02.900","Text":"In our case, we want to take this expression right here,"},{"Start":"02:02.900 ","End":"02:05.015","Text":"and we want to increase it by 10 percent."},{"Start":"02:05.015 ","End":"02:12.840","Text":"Again, we multiply it by 1.1 and that would be our linear transformation."},{"Start":"02:13.280 ","End":"02:17.660","Text":"We\u0027ve done step number 2."},{"Start":"02:17.660 ","End":"02:19.370","Text":"Let\u0027s go to step number 3."},{"Start":"02:19.370 ","End":"02:25.685","Text":"Let\u0027s simplify this and identify the values of a and b,"},{"Start":"02:25.685 ","End":"02:31.415","Text":"y equals 1.1 times x."},{"Start":"02:31.415 ","End":"02:39.255","Text":"We take this 1.1 multiply it by x plus 2 times 1.1,"},{"Start":"02:39.255 ","End":"02:45.580","Text":"and that equals to 1.1x plus 2.2."},{"Start":"02:47.480 ","End":"02:55.130","Text":"Now, let\u0027s compare this with our general expression of linear transformation."},{"Start":"02:55.130 ","End":"02:59.600","Text":"We know that y equals ax plus b."},{"Start":"02:59.600 ","End":"03:01.505","Text":"That\u0027s the general expression."},{"Start":"03:01.505 ","End":"03:05.014","Text":"Now when we compare this with this,"},{"Start":"03:05.014 ","End":"03:09.260","Text":"we can easily see that a equals 1.1,"},{"Start":"03:09.260 ","End":"03:15.480","Text":"it\u0027s right here, and b equals 2.2."},{"Start":"03:16.490 ","End":"03:20.365","Text":"Great. We\u0027ve done number 3."},{"Start":"03:20.365 ","End":"03:22.990","Text":"Let\u0027s go on to number 4."},{"Start":"03:22.990 ","End":"03:26.110","Text":"Let\u0027s take a and b and put that into"},{"Start":"03:26.110 ","End":"03:31.700","Text":"the expression for the expectation and standard deviation of y."},{"Start":"03:32.540 ","End":"03:41.770","Text":"The expectation of y equals a times the expectation of x plus b."},{"Start":"03:41.770 ","End":"03:46.665","Text":"Let\u0027s plug in the numbers. a equals 1.1."},{"Start":"03:46.665 ","End":"03:51.300","Text":"Expectation of x is 10 plus b,"},{"Start":"03:51.300 ","End":"03:59.105","Text":"that\u0027s 2.2 and that equals to 11 plus 2.2, that\u0027s 13.2."},{"Start":"03:59.105 ","End":"04:02.620","Text":"Now, what\u0027s the variance of y?"},{"Start":"04:02.620 ","End":"04:20.476","Text":"Well"},{"Start":"04:20.476 ","End":"04:24.940","Text":", not the variance, excuse me."},{"Start":"04:24.940 ","End":"04:30.115","Text":"Let\u0027s take the standard deviation of y."},{"Start":"04:30.115 ","End":"04:38.620","Text":"Now, that would equal to the absolute value of a times the standard deviation of x."},{"Start":"04:38.620 ","End":"04:41.134","Text":"Again, let\u0027s plug in the numbers."},{"Start":"04:41.134 ","End":"04:49.520","Text":"a is 1.1. Now, we\u0027ll take the absolute value of 1.1 times 5."},{"Start":"04:50.060 ","End":"04:57.690","Text":"That\u0027s the standard deviation of x and that equals to 5.5."},{"Start":"04:58.010 ","End":"05:06.630","Text":"Here we have the expectation of y and the standard deviation of y."}],"ID":12982},{"Watched":false,"Name":"Exercise 4","Duration":"3m 32s","ChapterTopicVideoID":12504,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this question, we\u0027re given that X is a random variable,"},{"Start":"00:04.230 ","End":"00:11.055","Text":"and we\u0027re also given that the expectation of X is 4 and the variance of X is 3."},{"Start":"00:11.055 ","End":"00:16.350","Text":"Now, we\u0027re also given a new variable Y,"},{"Start":"00:16.350 ","End":"00:19.340","Text":"that is a linear transformation of X,"},{"Start":"00:19.340 ","End":"00:22.415","Text":"where Y equals 7 minus X."},{"Start":"00:22.415 ","End":"00:27.195","Text":"We\u0027re asked to calculate the expectation of Y and the variance of Y."},{"Start":"00:27.195 ","End":"00:32.145","Text":"Well, again, let\u0027s look at our 4 step process"},{"Start":"00:32.145 ","End":"00:38.140","Text":"where we calculate the expectation of Y and the variance of Y."},{"Start":"00:38.810 ","End":"00:41.965","Text":"Here we have the 4 steps."},{"Start":"00:41.965 ","End":"00:47.435","Text":"Now, again, what we have is the first step,"},{"Start":"00:47.435 ","End":"00:50.750","Text":"which is to recognize that you\u0027re dealing with a linear transformation."},{"Start":"00:50.750 ","End":"00:53.930","Text":"Well, we don\u0027t need to recognize."},{"Start":"00:53.930 ","End":"00:55.835","Text":"It\u0027s given to us right here."},{"Start":"00:55.835 ","End":"01:01.025","Text":"That\u0027s Y being equal to 7 minus X."},{"Start":"01:01.025 ","End":"01:05.765","Text":"Now, do we have to write the transformation rule?"},{"Start":"01:05.765 ","End":"01:08.660","Text":"Well, no, again, that\u0027s given to us."},{"Start":"01:08.660 ","End":"01:10.940","Text":"Now, what do we need to do?"},{"Start":"01:10.940 ","End":"01:15.080","Text":"We need to simplify the rule and identify the value of a and b."},{"Start":"01:15.080 ","End":"01:22.335","Text":"Well, we can write this as minus X plus 7,"},{"Start":"01:22.335 ","End":"01:27.060","Text":"or like this, Y"},{"Start":"01:27.060 ","End":"01:33.515","Text":"equals minus 1 times X plus 7."},{"Start":"01:33.515 ","End":"01:39.635","Text":"Now if we look at the general equation for a linear transformation,"},{"Start":"01:39.635 ","End":"01:45.295","Text":"we know that Y equals aX plus b."},{"Start":"01:45.295 ","End":"01:51.095","Text":"Now again, why did I write this like this?"},{"Start":"01:51.095 ","End":"01:56.260","Text":"Well, the only reason why I wrote it like this was to bring it into"},{"Start":"01:56.260 ","End":"02:01.960","Text":"this form so I can easily identify what\u0027s a and what\u0027s b,"},{"Start":"02:01.960 ","End":"02:03.595","Text":"and in our case,"},{"Start":"02:03.595 ","End":"02:10.585","Text":"a equals minus 1 and b equals 7."},{"Start":"02:10.585 ","End":"02:14.250","Text":"That\u0027s pretty simple. Now,"},{"Start":"02:14.250 ","End":"02:16.420","Text":"all we have to do is plug in"},{"Start":"02:16.420 ","End":"02:20.920","Text":"these numbers when we calculate the expectation of Y and the variance of Y."},{"Start":"02:20.920 ","End":"02:24.985","Text":"Now, what\u0027s the expectation of Y?"},{"Start":"02:24.985 ","End":"02:30.610","Text":"Well, that\u0027s a times the expectation of X plus b."},{"Start":"02:30.610 ","End":"02:32.635","Text":"Let\u0027s plug in the numbers."},{"Start":"02:32.635 ","End":"02:38.755","Text":"A is minus 1 times the expectation of X."},{"Start":"02:38.755 ","End":"02:39.970","Text":"Well, that\u0027s given right here,"},{"Start":"02:39.970 ","End":"02:43.090","Text":"that\u0027s 4, plus b."},{"Start":"02:43.090 ","End":"02:45.005","Text":"Well, b is 7."},{"Start":"02:45.005 ","End":"02:50.220","Text":"That equals to minus 4 plus 7, that\u0027s plus 3."},{"Start":"02:50.220 ","End":"02:52.955","Text":"What about the variance of Y?"},{"Start":"02:52.955 ","End":"02:59.845","Text":"Well, the variance of Y is a squared times the variance of X."},{"Start":"02:59.845 ","End":"03:02.470","Text":"Now, what is a?"},{"Start":"03:02.470 ","End":"03:04.050","Text":"Again, a is minus 1."},{"Start":"03:04.050 ","End":"03:05.260","Text":"Let\u0027s plug in the numbers."},{"Start":"03:05.260 ","End":"03:10.210","Text":"That\u0027s minus 1 squared times the variance of X."},{"Start":"03:10.210 ","End":"03:12.265","Text":"Well, the variance of X is right here."},{"Start":"03:12.265 ","End":"03:14.270","Text":"That\u0027s 3 times 3."},{"Start":"03:14.270 ","End":"03:21.365","Text":"That means that minus 1 squared is 1 times 3 is 1 times 3 is 3. There we go."},{"Start":"03:21.365 ","End":"03:24.260","Text":"We\u0027ve solved this problem where"},{"Start":"03:24.260 ","End":"03:32.350","Text":"the expectation of Y is 3 and the variance of Y is also 3."}],"ID":12983},{"Watched":false,"Name":"Exercise 5 - Parts a-b","Duration":"4m 43s","ChapterTopicVideoID":12506,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"In this question we\u0027ll be talking about insurance."},{"Start":"00:02.865 ","End":"00:08.220","Text":"Now, a person decides to insure his car which is valued at $100,000."},{"Start":"00:08.220 ","End":"00:12.495","Text":"The following are the possible claims and the probabilities."},{"Start":"00:12.495 ","End":"00:17.610","Text":"The probability of a claim for the entire value of the car,"},{"Start":"00:17.610 ","End":"00:22.980","Text":"which means that this is a total write off, is 1/1,000."},{"Start":"00:22.980 ","End":"00:29.460","Text":"The probability of a claim for half the value of the car is 2 percent or 0.02."},{"Start":"00:29.460 ","End":"00:36.840","Text":"The probability of a claim for 1/4 of the value of the car is 5 percent or 0.05."},{"Start":"00:36.840 ","End":"00:39.190","Text":"Otherwise, there\u0027s no claim."},{"Start":"00:39.190 ","End":"00:44.145","Text":"Now, the insurance company allows 1 claim per year."},{"Start":"00:44.145 ","End":"00:50.045","Text":"We\u0027re given that X is the annual claim in thousands of dollars."},{"Start":"00:50.045 ","End":"00:54.080","Text":"Now here we have to pay attention because we\u0027re"},{"Start":"00:54.080 ","End":"00:58.985","Text":"given that the units of X is in thousands of dollars."},{"Start":"00:58.985 ","End":"01:02.180","Text":"But right here we\u0027re given that"},{"Start":"01:02.180 ","End":"01:08.330","Text":"the values of x that are given to us in the question are in actual dollars,"},{"Start":"01:08.330 ","End":"01:10.595","Text":"not in thousands of dollars."},{"Start":"01:10.595 ","End":"01:13.250","Text":"When we write a probability function,"},{"Start":"01:13.250 ","End":"01:17.070","Text":"we\u0027re going to have to divide by 1,000."},{"Start":"01:18.110 ","End":"01:21.740","Text":"Let\u0027s construct our probability function."},{"Start":"01:21.740 ","End":"01:24.180","Text":"We have X,"},{"Start":"01:24.520 ","End":"01:27.800","Text":"we have P of X."},{"Start":"01:27.800 ","End":"01:29.930","Text":"What are we given?"},{"Start":"01:29.930 ","End":"01:34.820","Text":"Well, the first thing is that the probability of the claim for the entire value"},{"Start":"01:34.820 ","End":"01:40.060","Text":"of the car is 1/1,000 that\u0027s 0.001."},{"Start":"01:40.060 ","End":"01:43.534","Text":"What\u0027s the value here? Well, that\u0027s $100,000."},{"Start":"01:43.534 ","End":"01:46.070","Text":"But since X is in thousands of dollars,"},{"Start":"01:46.070 ","End":"01:48.320","Text":"we\u0027ll write here 100."},{"Start":"01:48.320 ","End":"01:53.160","Text":"Now, what\u0027s the probability of the claim for half the value."},{"Start":"01:53.160 ","End":"01:58.050","Text":"We\u0027ll have the value of 100 is 50 and that\u0027s 0.02."},{"Start":"01:58.060 ","End":"02:02.270","Text":"That\u0027s right here. For 1/4 of the value,"},{"Start":"02:02.270 ","End":"02:05.075","Text":"well, 1/4 of 100 is 25."},{"Start":"02:05.075 ","End":"02:09.400","Text":"That\u0027s 0.05, that\u0027s 5 percent."},{"Start":"02:09.400 ","End":"02:12.825","Text":"Now, otherwise there\u0027s no claim."},{"Start":"02:12.825 ","End":"02:15.860","Text":"What\u0027s the probability of not having a claim?"},{"Start":"02:15.860 ","End":"02:18.800","Text":"Well, that\u0027s 1 minus the sum of these guys."},{"Start":"02:18.800 ","End":"02:22.350","Text":"That turns out to be 0.929."},{"Start":"02:22.720 ","End":"02:27.470","Text":"Here, we\u0027ve constructed the probability function of X."},{"Start":"02:27.470 ","End":"02:30.050","Text":"In this section, we\u0027re asked to calculate"},{"Start":"02:30.050 ","End":"02:32.675","Text":"the expectation variance of the amount of the claim."},{"Start":"02:32.675 ","End":"02:38.630","Text":"Now, if we recall from Section A we\u0027ve calculated probability function,"},{"Start":"02:38.630 ","End":"02:41.120","Text":"and here it is, right here."},{"Start":"02:41.120 ","End":"02:43.505","Text":"Now, what do we want to do?"},{"Start":"02:43.505 ","End":"02:46.639","Text":"We want to calculate the expectation variance."},{"Start":"02:46.639 ","End":"02:49.250","Text":"Now, what\u0027s the expectation of X?"},{"Start":"02:49.250 ","End":"02:55.235","Text":"That equals to the sum of X times the probability of X."},{"Start":"02:55.235 ","End":"03:05.795","Text":"That equals to 100 times 0.001 plus 50 times 0.02"},{"Start":"03:05.795 ","End":"03:11.810","Text":"plus 25 times 0.05"},{"Start":"03:11.810 ","End":"03:18.160","Text":"plus 0 times 0.929."},{"Start":"03:18.680 ","End":"03:20.945","Text":"After we do all the math,"},{"Start":"03:20.945 ","End":"03:24.650","Text":"that comes out to 2.35."},{"Start":"03:24.650 ","End":"03:26.285","Text":"Now, what\u0027s the units?"},{"Start":"03:26.285 ","End":"03:29.390","Text":"The units is in thousands of dollars."},{"Start":"03:29.390 ","End":"03:33.600","Text":"That\u0027s thousands of dollars came in thousands."},{"Start":"03:33.610 ","End":"03:37.400","Text":"Now, what\u0027s the variance of X?"},{"Start":"03:37.400 ","End":"03:42.560","Text":"Well, the definition of the variance is the sum of X squared times"},{"Start":"03:42.560 ","End":"03:48.455","Text":"its probability minus the expectation squared of X."},{"Start":"03:48.455 ","End":"03:58.775","Text":"Now, let\u0027s plug in the numbers that equals to 100 squared times 0.001 plus 50 squared"},{"Start":"03:58.775 ","End":"04:07.325","Text":"times 0.02 plus 25 squared times 0.05 plus"},{"Start":"04:07.325 ","End":"04:17.530","Text":"0 squared times 0.929 minus 2.35 squared."},{"Start":"04:17.540 ","End":"04:24.660","Text":"After doing the math, that comes out to 85.7275."},{"Start":"04:24.660 ","End":"04:26.700","Text":"Now, what\u0027s the units?"},{"Start":"04:26.700 ","End":"04:29.955","Text":"The unit is thousands of dollars squared."},{"Start":"04:29.955 ","End":"04:33.780","Text":"That\u0027s dollar squared thousands."},{"Start":"04:33.780 ","End":"04:43.320","Text":"Now, we\u0027ve basically calculated the expectation of X and the variance."}],"ID":12984},{"Watched":false,"Name":"Exercise 5 - Part c","Duration":"5m 20s","ChapterTopicVideoID":12505,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"In this section, we\u0027re given that the insurance premium is 4,000 dollars."},{"Start":"00:04.710 ","End":"00:07.530","Text":"We\u0027re asked, what are the expectation variance of"},{"Start":"00:07.530 ","End":"00:11.505","Text":"the insurance company\u0027s profits on this car\u0027s insurance?"},{"Start":"00:11.505 ","End":"00:15.585","Text":"Let\u0027s first recall what we did in section A and B."},{"Start":"00:15.585 ","End":"00:18.975","Text":"As we can see here, in section A,"},{"Start":"00:18.975 ","End":"00:23.820","Text":"we\u0027ve calculated the probability function of X."},{"Start":"00:23.820 ","End":"00:28.890","Text":"In section B, we\u0027ve calculated the expectation and variance of X."},{"Start":"00:28.890 ","End":"00:31.710","Text":"What was X?"},{"Start":"00:31.710 ","End":"00:33.975","Text":"X was basically the claim."},{"Start":"00:33.975 ","End":"00:39.845","Text":"Let\u0027s look at what we\u0027re asked to do."},{"Start":"00:39.845 ","End":"00:47.240","Text":"We\u0027re asked to calculate the expectation and variance of the insurance company\u0027s profits."},{"Start":"00:47.240 ","End":"00:49.655","Text":"The minute I see this,"},{"Start":"00:49.655 ","End":"00:52.520","Text":"I know that I\u0027m dealing here with some type of"},{"Start":"00:52.520 ","End":"00:56.330","Text":"transformation because we\u0027re given information about claims,"},{"Start":"00:56.330 ","End":"00:59.855","Text":"and we\u0027re talking about a company\u0027s profits."},{"Start":"00:59.855 ","End":"01:04.670","Text":"Let\u0027s take a look at our 4-step process,"},{"Start":"01:04.670 ","End":"01:08.285","Text":"and see how we can use that."},{"Start":"01:08.285 ","End":"01:13.310","Text":"Here\u0027s our 4-step process."},{"Start":"01:13.310 ","End":"01:15.290","Text":"The first thing we have to do is we have to"},{"Start":"01:15.290 ","End":"01:18.590","Text":"recognize that we\u0027re dealing with a linear transformation."},{"Start":"01:18.590 ","End":"01:20.420","Text":"Again, as we said,"},{"Start":"01:20.420 ","End":"01:23.960","Text":"we were talking about claims all along and all of a sudden,"},{"Start":"01:23.960 ","End":"01:26.240","Text":"we have to deal with profits."},{"Start":"01:26.240 ","End":"01:30.215","Text":"Whose profits are we dealing with?"},{"Start":"01:30.215 ","End":"01:32.705","Text":"The insurance company\u0027s profits."},{"Start":"01:32.705 ","End":"01:39.845","Text":"That means that we have to talk about the definition of a profit. What\u0027s a profit?"},{"Start":"01:39.845 ","End":"01:47.240","Text":"The definition of the profit is your income minus expenses."},{"Start":"01:47.240 ","End":"01:50.705","Text":"From an insurance company\u0027s perspective,"},{"Start":"01:50.705 ","End":"01:56.420","Text":"the income is the insurance premium and that\u0027s $4,000."},{"Start":"01:56.420 ","End":"01:59.990","Text":"Since X is in thousands of dollars,"},{"Start":"01:59.990 ","End":"02:04.684","Text":"that\u0027s the units, then we have to divide this by $1,000 as well."},{"Start":"02:04.684 ","End":"02:07.640","Text":"When we\u0027re writing the transformation,"},{"Start":"02:07.640 ","End":"02:13.475","Text":"that\u0027s step number 2, why that would equal to the income,"},{"Start":"02:13.475 ","End":"02:19.370","Text":"that\u0027s 4,000, we\u0027re dealing in units of thousands of dollars,"},{"Start":"02:19.370 ","End":"02:21.890","Text":"minus the claims,"},{"Start":"02:21.890 ","End":"02:24.650","Text":"the expenses of the insurance company,"},{"Start":"02:24.650 ","End":"02:29.080","Text":"how much the insurance company has to pay, that\u0027s X."},{"Start":"02:29.080 ","End":"02:33.880","Text":"Step number 2 is right here."},{"Start":"02:33.880 ","End":"02:37.910","Text":"That\u0027s Y equals 4,000 minus 4,000,"},{"Start":"02:37.910 ","End":"02:40.580","Text":"that\u0027s the income minus X,"},{"Start":"02:40.580 ","End":"02:42.755","Text":"the expenses, the claims."},{"Start":"02:42.755 ","End":"02:45.770","Text":"Let\u0027s go to step number 3."},{"Start":"02:45.770 ","End":"02:49.685","Text":"Simplify the rule and identify the values for a and b."},{"Start":"02:49.685 ","End":"02:54.620","Text":"The general expression for a linear expression"},{"Start":"02:54.620 ","End":"03:00.060","Text":"is Y equals aX plus b."},{"Start":"03:00.060 ","End":"03:03.840","Text":"Let\u0027s take this,"},{"Start":"03:03.840 ","End":"03:06.330","Text":"and try to write it, in this way."},{"Start":"03:06.330 ","End":"03:13.205","Text":"That\u0027s minus 1 times X plus 4."},{"Start":"03:13.205 ","End":"03:17.150","Text":"This and this are identical."},{"Start":"03:17.150 ","End":"03:22.280","Text":"This makes it very easy for us to identify what\u0027s a and b."},{"Start":"03:22.280 ","End":"03:27.740","Text":"In our case, a equals minus 1 and b equals 4."},{"Start":"03:27.740 ","End":"03:29.990","Text":"Minus 1, that\u0027s right here."},{"Start":"03:29.990 ","End":"03:31.625","Text":"That\u0027s the multiplier."},{"Start":"03:31.625 ","End":"03:35.585","Text":"B is plus 4, that\u0027s right here."},{"Start":"03:35.585 ","End":"03:39.905","Text":"Having identified what a and b are,"},{"Start":"03:39.905 ","End":"03:43.025","Text":"we can go on to step number 4."},{"Start":"03:43.025 ","End":"03:49.624","Text":"We\u0027ll substitute a and b in the expressions for expectation and variance."},{"Start":"03:49.624 ","End":"03:56.900","Text":"That means that when we\u0027re talking about the expectation of Y, what\u0027s that definition?"},{"Start":"03:56.900 ","End":"04:01.700","Text":"That\u0027s a times the expectation of X plus b."},{"Start":"04:01.700 ","End":"04:05.370","Text":"Let\u0027s plug in the numbers."},{"Start":"04:05.370 ","End":"04:07.150","Text":"A is minus 1,"},{"Start":"04:07.150 ","End":"04:10.310","Text":"times, what\u0027s the expectation of X? That\u0027s right here."},{"Start":"04:10.310 ","End":"04:15.260","Text":"That\u0027s 2.35, plus B,"},{"Start":"04:15.260 ","End":"04:18.530","Text":"plus 4, B\u0027s right here."},{"Start":"04:18.530 ","End":"04:23.960","Text":"That means that it\u0027s 4 minus 2.35."},{"Start":"04:23.960 ","End":"04:26.930","Text":"That\u0027s 1.65."},{"Start":"04:26.930 ","End":"04:30.770","Text":"Don\u0027t forget this is in thousands of dollars."},{"Start":"04:30.770 ","End":"04:40.379","Text":"The expectation for the insurance company\u0027s profit is $1,650."},{"Start":"04:40.379 ","End":"04:43.153","Text":"What\u0027s the variance of Y?"},{"Start":"04:43.153 ","End":"04:48.815","Text":"That\u0027s defined a squared times the variance of X."},{"Start":"04:48.815 ","End":"04:51.425","Text":"Again, let\u0027s plug in the numbers."},{"Start":"04:51.425 ","End":"04:53.695","Text":"What\u0027s a? A\u0027s minus 1."},{"Start":"04:53.695 ","End":"04:58.315","Text":"Minus 1 squared times the variance of X, that\u0027s right here,"},{"Start":"04:58.315 ","End":"05:05.550","Text":"85.7275, and minus 1 squared is just 1,"},{"Start":"05:05.550 ","End":"05:10.240","Text":"that equals to 85.7275."},{"Start":"05:12.190 ","End":"05:20.130","Text":"Here we\u0027ve calculated the expectation of Y and the variance of Y."}],"ID":12985},{"Watched":false,"Name":"Exercise 6 - Part c","Duration":"5m 49s","ChapterTopicVideoID":12507,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.659","Text":"The grade on the test is calculated as follows."},{"Start":"00:03.659 ","End":"00:10.275","Text":"Every correct answer is worth 10 points.1 point is deducted for every wrong answer."},{"Start":"00:10.275 ","End":"00:14.730","Text":"We\u0027re asked, what are the expectation and variance of the test mark?"},{"Start":"00:14.730 ","End":"00:16.725","Text":"Now, first of all,"},{"Start":"00:16.725 ","End":"00:20.440","Text":"let\u0027s recall what we did in the previous sections."},{"Start":"00:21.260 ","End":"00:24.225","Text":"Here it is. What do we have?"},{"Start":"00:24.225 ","End":"00:27.885","Text":"We have the probability function of x,"},{"Start":"00:27.885 ","End":"00:34.200","Text":"and we\u0027ve calculated the expectation of x and the variance of x. Now, what was x?"},{"Start":"00:34.200 ","End":"00:38.195","Text":"X was defined as the number of correct answers."},{"Start":"00:38.195 ","End":"00:40.720","Text":"Now, what are we asked?"},{"Start":"00:40.720 ","End":"00:43.725","Text":"We\u0027re asked about the test mark."},{"Start":"00:43.725 ","End":"00:46.955","Text":"That\u0027s a completely different variable,"},{"Start":"00:46.955 ","End":"00:52.350","Text":"but it is a linear transformation of x."},{"Start":"00:52.350 ","End":"00:56.840","Text":"Let\u0027s just use our 4-point process to see how we can"},{"Start":"00:56.840 ","End":"01:02.100","Text":"calculate the expectation and variance of the test mark."},{"Start":"01:02.120 ","End":"01:04.715","Text":"This is our process."},{"Start":"01:04.715 ","End":"01:07.700","Text":"Now, the first thing that we have to do is we have to"},{"Start":"01:07.700 ","End":"01:11.420","Text":"recognize that we\u0027re dealing with linear transformation."},{"Start":"01:11.420 ","End":"01:15.859","Text":"Basically what we have or what we were given is x,"},{"Start":"01:15.859 ","End":"01:21.064","Text":"which was the number of correct answers and we\u0027re given some sort"},{"Start":"01:21.064 ","End":"01:27.485","Text":"of rule that translate the correct answers to points."},{"Start":"01:27.485 ","End":"01:30.785","Text":"Every correct answer is worth 10 points."},{"Start":"01:30.785 ","End":"01:35.275","Text":"On the other hand, 1 point is deducted for every wrong answer."},{"Start":"01:35.275 ","End":"01:39.830","Text":"We\u0027re asked, what are the expectation and variance of the test mark?"},{"Start":"01:39.830 ","End":"01:46.145","Text":"We want to translate the number of correct answers to y,"},{"Start":"01:46.145 ","End":"01:51.550","Text":"which is basically the test marks."},{"Start":"01:52.940 ","End":"01:55.415","Text":"Now how do we do that?"},{"Start":"01:55.415 ","End":"02:00.350","Text":"Well, let\u0027s write the transformation according to the data in the question."},{"Start":"02:00.350 ","End":"02:02.375","Text":"Let\u0027s take an example."},{"Start":"02:02.375 ","End":"02:10.259","Text":"Assume that the student had 8 correct answers. X equals 8."},{"Start":"02:10.259 ","End":"02:13.440","Text":"Now, every correct answer is worth 10."},{"Start":"02:13.440 ","End":"02:16.485","Text":"We have to multiply that by 10."},{"Start":"02:16.485 ","End":"02:21.545","Text":"But 1 point is deducted for every wrong answer."},{"Start":"02:21.545 ","End":"02:23.825","Text":"How many wrong answers do we have?"},{"Start":"02:23.825 ","End":"02:25.805","Text":"If we have 8 correct ones,"},{"Start":"02:25.805 ","End":"02:28.415","Text":"then we have 2 wrong ones."},{"Start":"02:28.415 ","End":"02:32.570","Text":"Because we only have 10 questions, times 1."},{"Start":"02:32.570 ","End":"02:35.555","Text":"That\u0027s 1 point is deducted."},{"Start":"02:35.555 ","End":"02:39.275","Text":"Now let\u0027s say we have 7 correct answers."},{"Start":"02:39.275 ","End":"02:42.570","Text":"Again, we have to multiply that by 10."},{"Start":"02:42.570 ","End":"02:45.230","Text":"How many wrong answers do we have?"},{"Start":"02:45.230 ","End":"02:48.020","Text":"Here, since we have 7 correct answers,"},{"Start":"02:48.020 ","End":"02:53.060","Text":"we have 3 wrong answers times 1 point that\u0027s deducted."},{"Start":"02:53.060 ","End":"02:58.640","Text":"Now, let\u0027s take a look at the general expression."},{"Start":"02:58.640 ","End":"03:02.955","Text":"If x is the number of correct answers,"},{"Start":"03:02.955 ","End":"03:05.495","Text":"then we have to multiply that by 10."},{"Start":"03:05.495 ","End":"03:07.535","Text":"That\u0027s right here, 10 points."},{"Start":"03:07.535 ","End":"03:14.835","Text":"But we have to take away 10 minus x times 1."},{"Start":"03:14.835 ","End":"03:17.270","Text":"That\u0027s 10 minus 8 equals 2,"},{"Start":"03:17.270 ","End":"03:19.145","Text":"10 minus 7 equals 3."},{"Start":"03:19.145 ","End":"03:26.050","Text":"So this will be our linear transformation."},{"Start":"03:26.330 ","End":"03:30.990","Text":"Basically we\u0027ve done step number 2."},{"Start":"03:30.990 ","End":"03:33.735","Text":"Let\u0027s look at step number 3."},{"Start":"03:33.735 ","End":"03:37.635","Text":"Step number 3 is just to simplify this."},{"Start":"03:37.635 ","End":"03:46.440","Text":"We can bring it to the form of y equals ax plus b."},{"Start":"03:47.590 ","End":"03:52.450","Text":"Let\u0027s just do a little bit of algebra here."},{"Start":"03:52.450 ","End":"04:00.595","Text":"Now, y equals 10x minus 10 plus 1."},{"Start":"04:00.595 ","End":"04:05.915","Text":"That equals to 11x plus 1x."},{"Start":"04:05.915 ","End":"04:08.930","Text":"That\u0027s 11x minus 10."},{"Start":"04:09.710 ","End":"04:15.595","Text":"The minute we brought this expression into this form,"},{"Start":"04:15.595 ","End":"04:23.015","Text":"we know that a equals 11 and b equals minus 10."},{"Start":"04:23.015 ","End":"04:33.135","Text":"Now we can easily calculate the expectation and variance of the test mark. Here we go."},{"Start":"04:33.135 ","End":"04:42.400","Text":"The expectation of y equals a times the expectation of x plus b."},{"Start":"04:42.400 ","End":"04:44.745","Text":"Let\u0027s plug in the numbers,"},{"Start":"04:44.745 ","End":"04:50.040","Text":"a equals 11 times the expectation of x,"},{"Start":"04:50.040 ","End":"04:54.080","Text":"that\u0027s right here, that\u0027s 7.35."},{"Start":"04:54.080 ","End":"04:56.465","Text":"That was given to us in the question,"},{"Start":"04:56.465 ","End":"05:01.680","Text":"minus b plus, minus 10."},{"Start":"05:01.870 ","End":"05:07.260","Text":"That turns out to be 70.85."},{"Start":"05:10.190 ","End":"05:13.000","Text":"What\u0027s the variance of y?"},{"Start":"05:13.000 ","End":"05:18.695","Text":"Well, that\u0027s defined as a squared times the variance of x."},{"Start":"05:18.695 ","End":"05:20.610","Text":"Now, what\u0027s a?"},{"Start":"05:20.610 ","End":"05:21.705","Text":"a is 11."},{"Start":"05:21.705 ","End":"05:25.520","Text":"That\u0027s 11 squared times the variance right here."},{"Start":"05:25.520 ","End":"05:31.490","Text":"That\u0027s 1.8275, and that turns"},{"Start":"05:31.490 ","End":"05:38.340","Text":"out to be 221.1275."},{"Start":"05:38.630 ","End":"05:43.970","Text":"Now we\u0027ve calculated the expectation of y and the variance of y,"},{"Start":"05:43.970 ","End":"05:48.510","Text":"where y is the test mark."}],"ID":12986},{"Watched":false,"Name":"Exercise 6 - Parts a-b","Duration":"4m 33s","ChapterTopicVideoID":12508,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.165","Text":"In this question we\u0027ll be talking about test results."},{"Start":"00:03.165 ","End":"00:08.025","Text":"Let X be the number of correct answers on a test with 10 questions."},{"Start":"00:08.025 ","End":"00:11.895","Text":"The probability function of X is given in the following table."},{"Start":"00:11.895 ","End":"00:14.505","Text":"We\u0027ll see here that when X equals 5,"},{"Start":"00:14.505 ","End":"00:16.320","Text":"the probability is 0.1,"},{"Start":"00:16.320 ","End":"00:17.790","Text":"when X equals 6,"},{"Start":"00:17.790 ","End":"00:20.595","Text":"the probability 0.2 and so on and so forth."},{"Start":"00:20.595 ","End":"00:24.705","Text":"But we have 2 missing probabilities."},{"Start":"00:24.705 ","End":"00:30.015","Text":"We assume that the expected number of correct answer on the test is 7.35."},{"Start":"00:30.015 ","End":"00:34.630","Text":"The first thing we\u0027re asked to do is to complete the probability function."},{"Start":"00:35.660 ","End":"00:39.930","Text":"Let\u0027s call the probability when x equals 9,"},{"Start":"00:39.930 ","End":"00:48.430","Text":"let\u0027s call that p. Since we know that the sum of all the probabilities have to equal 1,"},{"Start":"00:48.430 ","End":"00:51.650","Text":"that means that the probability when X equals 10,"},{"Start":"00:51.650 ","End":"00:57.885","Text":"is 1 minus P minus the sum of all the known probabilities."},{"Start":"00:57.885 ","End":"00:59.925","Text":"That comes out to 0.8."},{"Start":"00:59.925 ","End":"01:09.800","Text":"That means that the probability of X equaling 10 is 0.2 minus P. What else are we given?"},{"Start":"01:09.800 ","End":"01:15.630","Text":"We\u0027re given that the expectation of X is 7.35."},{"Start":"01:16.130 ","End":"01:20.610","Text":"The expectation of X, how is that defined?"},{"Start":"01:20.610 ","End":"01:26.155","Text":"That\u0027s defined as the sum of x times the probability of x."},{"Start":"01:26.155 ","End":"01:37.245","Text":"That equals to 5 times 0.1 plus 6 times 0.2 plus 7 times 0.2"},{"Start":"01:37.245 ","End":"01:47.190","Text":"plus 8 times 0.3 plus 9 times p plus 10 times"},{"Start":"01:47.190 ","End":"01:56.865","Text":"0.2 minus p. That equals to 0.5"},{"Start":"01:56.865 ","End":"02:03.495","Text":"plus 1.2 plus 1.4 plus 2.4"},{"Start":"02:03.495 ","End":"02:12.480","Text":"plus 9p plus 2 minus 10p."},{"Start":"02:12.480 ","End":"02:15.070","Text":"Let\u0027s just figure that out."},{"Start":"02:15.070 ","End":"02:21.160","Text":"That basically equals to 7.5 minus"},{"Start":"02:21.160 ","End":"02:27.690","Text":"p. We did say that the expectation is equal to 7.35,"},{"Start":"02:27.690 ","End":"02:30.750","Text":"so that equals to 7.35."},{"Start":"02:30.750 ","End":"02:39.210","Text":"That means that p equals to 0.15."},{"Start":"02:39.210 ","End":"02:46.440","Text":"Now, we can complete the probability function where p equals 0.15."},{"Start":"02:48.470 ","End":"02:55.120","Text":"This means that this equals to 0.05."},{"Start":"02:55.340 ","End":"02:58.520","Text":"In this question, we\u0027re asked to calculate"},{"Start":"02:58.520 ","End":"03:02.650","Text":"the variance of the number of correct answers on the test."},{"Start":"03:02.650 ","End":"03:08.155","Text":"Let\u0027s just recall what we did in section A."},{"Start":"03:08.155 ","End":"03:16.610","Text":"Here\u0027s the full probability function that includes the missing probabilities."},{"Start":"03:16.610 ","End":"03:20.725","Text":"We\u0027re asked to calculate the variance."},{"Start":"03:20.725 ","End":"03:23.850","Text":"The variance of x,"},{"Start":"03:23.850 ","End":"03:30.170","Text":"that\u0027s defined as the sum of x squared times"},{"Start":"03:30.170 ","End":"03:37.350","Text":"the probability of x minus the expectation squared of x."},{"Start":"03:37.390 ","End":"03:45.755","Text":"We are given that the expectation of x equals 7.35."},{"Start":"03:45.755 ","End":"03:47.810","Text":"That was in the question."},{"Start":"03:47.810 ","End":"03:53.095","Text":"If that\u0027s the case, let\u0027s just go ahead and plug in the numbers."},{"Start":"03:53.095 ","End":"03:57.000","Text":"That equals to 5 squared times 0.1,"},{"Start":"03:57.000 ","End":"03:58.695","Text":"it\u0027s this guy right her,"},{"Start":"03:58.695 ","End":"04:09.785","Text":"plus 6 squared times 0.2 plus 7 squared times 0.2 plus 8 squared times 0.3"},{"Start":"04:09.785 ","End":"04:15.710","Text":"plus 9 squared times 0.15 plus 10 squared times"},{"Start":"04:15.710 ","End":"04:22.575","Text":"0.05 minus 7.35 squared."},{"Start":"04:22.575 ","End":"04:25.475","Text":"After doing all the math,"},{"Start":"04:25.475 ","End":"04:32.940","Text":"this comes out to 1.8275."}],"ID":12987},{"Watched":false,"Name":"Exercise 7 - Parts a-b","Duration":"3m 16s","ChapterTopicVideoID":12509,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"In this question, we\u0027re given that"},{"Start":"00:02.280 ","End":"00:06.030","Text":"the probability function of a random variable is this;"},{"Start":"00:06.030 ","End":"00:10.890","Text":"the probability of X when it equals k, equals k/A."},{"Start":"00:10.890 ","End":"00:18.220","Text":"K runs from 1-4 and we\u0027re asked to find the value of A."},{"Start":"00:19.190 ","End":"00:27.315","Text":"Let\u0027s write out the probability function of X and see if we can calculate what A is."},{"Start":"00:27.315 ","End":"00:28.860","Text":"Well, here we go."},{"Start":"00:28.860 ","End":"00:31.710","Text":"That\u0027s X, that\u0027s P of"},{"Start":"00:31.710 ","End":"00:39.980","Text":"x. X can take on the value of k when k goes from 1-4,"},{"Start":"00:39.980 ","End":"00:41.675","Text":"so X equals 1,"},{"Start":"00:41.675 ","End":"00:44.480","Text":"2, 3, and 4."},{"Start":"00:44.480 ","End":"00:47.510","Text":"What\u0027s the probability of X?"},{"Start":"00:47.510 ","End":"00:49.430","Text":"Well, when X equals 1,"},{"Start":"00:49.430 ","End":"00:54.038","Text":"the probability of X is 1/A"},{"Start":"00:54.038 ","End":"01:01.455","Text":"because we substitute 1 instead of k. When X equals 2,"},{"Start":"01:01.455 ","End":"01:04.470","Text":"well, that means that it\u0027s 2/A,"},{"Start":"01:04.470 ","End":"01:08.775","Text":"and 3/A, and 4/A."},{"Start":"01:08.775 ","End":"01:16.565","Text":"We know that the sum of all the probabilities have to add up to 1."},{"Start":"01:16.565 ","End":"01:22.220","Text":"That means that 1/A, plus 2/A,"},{"Start":"01:22.220 ","End":"01:28.695","Text":"plus 3/A, plus 4/A have to equal to 1."},{"Start":"01:28.695 ","End":"01:36.260","Text":"That means that A equals to 1 plus 2 plus 3 plus 4,"},{"Start":"01:36.260 ","End":"01:38.870","Text":"and that equals to 10."},{"Start":"01:38.870 ","End":"01:43.490","Text":"A equals 10. In this section,"},{"Start":"01:43.490 ","End":"01:47.990","Text":"we\u0027re asked to calculate the expectation and variance of the variable in question."},{"Start":"01:47.990 ","End":"01:52.295","Text":"Let\u0027s just recall what we did in Section a."},{"Start":"01:52.295 ","End":"01:58.465","Text":"We\u0027ve calculated the probability function of the variable X."},{"Start":"01:58.465 ","End":"02:03.580","Text":"We\u0027re asked to find the expectation and variance of X."},{"Start":"02:03.580 ","End":"02:10.905","Text":"Well, if we recall the expectation of X equals the sum of X times its probability."},{"Start":"02:10.905 ","End":"02:14.220","Text":"That equals to 1 times 0.1,"},{"Start":"02:14.220 ","End":"02:17.445","Text":"plus 2 times 0.2,"},{"Start":"02:17.445 ","End":"02:20.190","Text":"plus 3 times 0.3,"},{"Start":"02:20.190 ","End":"02:23.460","Text":"plus 4 times 0.4,"},{"Start":"02:23.460 ","End":"02:28.000","Text":"and that equals to 3."},{"Start":"02:29.420 ","End":"02:33.330","Text":"What\u0027s the variance of X?"},{"Start":"02:33.330 ","End":"02:38.020","Text":"Again, the definition of the variance is X squared times"},{"Start":"02:38.020 ","End":"02:43.935","Text":"the probability of X minus the expectation squared of X."},{"Start":"02:43.935 ","End":"02:47.945","Text":"Don\u0027t forget that. Let\u0027s just plug in the numbers."},{"Start":"02:47.945 ","End":"02:51.975","Text":"That\u0027s 1 squared times 0.1,"},{"Start":"02:51.975 ","End":"02:54.945","Text":"plus 2 squared times 0.2,"},{"Start":"02:54.945 ","End":"02:58.275","Text":"plus 3 squared times 0.3,"},{"Start":"02:58.275 ","End":"03:08.100","Text":"plus 4 squared times 0.4 minus 3 squared, that\u0027s expectation squared."},{"Start":"03:08.100 ","End":"03:10.710","Text":"That equals to 1."},{"Start":"03:10.710 ","End":"03:13.590","Text":"That\u0027s the variance of X right here,"},{"Start":"03:13.590 ","End":"03:16.810","Text":"and this is the expectation of X."}],"ID":12988},{"Watched":false,"Name":"Exercise 7 - Parts c-d","Duration":"6m 44s","ChapterTopicVideoID":12510,"CourseChapterTopicPlaylistID":64483,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.635","Text":"In this section, we\u0027re asked to calculate the expectation of x cubed."},{"Start":"00:04.635 ","End":"00:10.095","Text":"Well, first of all, let\u0027s take a look at what we\u0027ve done in the previous sections."},{"Start":"00:10.095 ","End":"00:19.440","Text":"Here we go. What we did was to calculate the probability function of x."},{"Start":"00:19.440 ","End":"00:23.180","Text":"Now, we\u0027re not asked to calculate the expectation of x,"},{"Start":"00:23.180 ","End":"00:25.380","Text":"but of x cubed."},{"Start":"00:25.640 ","End":"00:30.545","Text":"Let\u0027s write down the probability function of x cubed."},{"Start":"00:30.545 ","End":"00:33.560","Text":"Now, if x equals 1,"},{"Start":"00:33.560 ","End":"00:36.335","Text":"then x cubed is also equals 1."},{"Start":"00:36.335 ","End":"00:38.450","Text":"When x equals 2,"},{"Start":"00:38.450 ","End":"00:41.500","Text":"the x cubed equals 8,"},{"Start":"00:41.500 ","End":"00:45.090","Text":"27, and when x equals 4,"},{"Start":"00:45.090 ","End":"00:48.770","Text":"then x cubed equals 64."},{"Start":"00:48.770 ","End":"00:55.305","Text":"Now, probability of x cubed"},{"Start":"00:55.305 ","End":"01:01.715","Text":"remains the same because when x equals 1 with a probability of 01,"},{"Start":"01:01.715 ","End":"01:05.915","Text":"when you raise x to the third power,"},{"Start":"01:05.915 ","End":"01:08.015","Text":"the probability stay the same thing."},{"Start":"01:08.015 ","End":"01:11.200","Text":"Don\u0027t raise the probabilities to the power of 3,"},{"Start":"01:11.200 ","End":"01:15.680","Text":"that\u0027s wrong, so that remains the same, that\u0027s 0.1."},{"Start":"01:15.680 ","End":"01:21.450","Text":"Here is 0.2, here is 0.3, and here is 0.4."},{"Start":"01:21.450 ","End":"01:24.830","Text":"Now, remember, by raising x to the power of 3,"},{"Start":"01:24.830 ","End":"01:27.605","Text":"we\u0027re not doing the linear transformation."},{"Start":"01:27.605 ","End":"01:32.090","Text":"A linear transformation happens when you multiply x by a"},{"Start":"01:32.090 ","End":"01:36.890","Text":"constant or you add or subtract a value,"},{"Start":"01:36.890 ","End":"01:38.570","Text":"a constant from x."},{"Start":"01:38.570 ","End":"01:44.490","Text":"Now, when you raise it to the third power or to the fourth power or to any power."},{"Start":"01:44.780 ","End":"01:52.580","Text":"Now, the minute that we\u0027ve constructed our probability function,"},{"Start":"01:52.580 ","End":"01:57.845","Text":"let\u0027s calculate the expectation of the x cubed."},{"Start":"01:57.845 ","End":"02:02.860","Text":"Well, again, that\u0027s 1 times 0.1 right here now,"},{"Start":"02:02.860 ","End":"02:13.925","Text":"plus 8 times 0.2 plus 27 times 0.3 plus 64 times 0.4,"},{"Start":"02:13.925 ","End":"02:18.280","Text":"and that equals to 35.4."},{"Start":"02:19.690 ","End":"02:22.640","Text":"In this section, we\u0027re asked to calculate"},{"Start":"02:22.640 ","End":"02:25.700","Text":"the expectation and variance of the following variable,"},{"Start":"02:25.700 ","End":"02:27.980","Text":"x over 2 minus 4."},{"Start":"02:27.980 ","End":"02:29.855","Text":"Now, before we go on,"},{"Start":"02:29.855 ","End":"02:33.425","Text":"let\u0027s just recall what we did in the previous sections."},{"Start":"02:33.425 ","End":"02:38.785","Text":"Thus we can see, we have here the probability function of x,"},{"Start":"02:38.785 ","End":"02:43.955","Text":"and we\u0027ve also calculated the expectation of x and the variance of x."},{"Start":"02:43.955 ","End":"02:52.820","Text":"Now, we can easily see that this guy right here is a linear transformation of x."},{"Start":"02:52.820 ","End":"02:54.033","Text":"Why is that?"},{"Start":"02:54.033 ","End":"02:56.480","Text":"Well, we\u0027re taking a constant,"},{"Start":"02:56.480 ","End":"02:59.555","Text":"2, and we\u0027re dividing x by that constant."},{"Start":"02:59.555 ","End":"03:02.750","Text":"Apart from that, once we divided x by a constant,"},{"Start":"03:02.750 ","End":"03:07.590","Text":"we were also taking away a constant minus 4."},{"Start":"03:07.590 ","End":"03:12.225","Text":"Not like in Section C above,"},{"Start":"03:12.225 ","End":"03:18.290","Text":"we\u0027re not taking x and we\u0027re cubing it or raising it to any form of power."},{"Start":"03:18.290 ","End":"03:23.405","Text":"Here, we\u0027re taking constants and we\u0027re manipulating x with these constants,"},{"Start":"03:23.405 ","End":"03:27.550","Text":"that\u0027s the basis of a linear transformation."},{"Start":"03:27.550 ","End":"03:32.840","Text":"Let\u0027s call this linear transformation y and write this out like this,"},{"Start":"03:32.840 ","End":"03:35.645","Text":"this is x over 2 minus 4."},{"Start":"03:35.645 ","End":"03:42.150","Text":"Now, before we can continue on to calculate the expectation and variance,"},{"Start":"03:42.150 ","End":"03:45.665","Text":"let\u0027s just look at our 4-step process."},{"Start":"03:45.665 ","End":"03:49.485","Text":"Again, let\u0027s look at step number 1."},{"Start":"03:49.485 ","End":"03:52.790","Text":"We have to recognize that we\u0027re dealing with the linear transformation."},{"Start":"03:52.790 ","End":"03:56.255","Text":"Well, we are. This is it right here."},{"Start":"03:56.255 ","End":"04:01.325","Text":"Now, we can also write this as"},{"Start":"04:01.325 ","End":"04:08.035","Text":"1/2 times x plus minus 4."},{"Start":"04:08.035 ","End":"04:11.210","Text":"They\u0027re totally identical. We\u0027re taking x,"},{"Start":"04:11.210 ","End":"04:14.150","Text":"we\u0027re multiplying it by a half or dividing by 2,"},{"Start":"04:14.150 ","End":"04:15.440","Text":"that\u0027s this guy right here,"},{"Start":"04:15.440 ","End":"04:19.835","Text":"and we\u0027re taking away 4 or we\u0027re adding minus 4."},{"Start":"04:19.835 ","End":"04:25.310","Text":"Now, we can take this guy right here and we can"},{"Start":"04:25.310 ","End":"04:31.340","Text":"use a probability function to calculate the expectation variance."},{"Start":"04:31.340 ","End":"04:36.665","Text":"But why do that? We already have the expectation and variance of x,"},{"Start":"04:36.665 ","End":"04:44.115","Text":"and we know that a linear transformation is ax plus b,"},{"Start":"04:44.115 ","End":"04:45.755","Text":"that the general expression,"},{"Start":"04:45.755 ","End":"04:51.890","Text":"we know how to deal with calculating expectations variance of linear transformations."},{"Start":"04:51.890 ","End":"04:59.650","Text":"We don\u0027t need to create a new probability function, so let\u0027s go on."},{"Start":"05:00.230 ","End":"05:03.830","Text":"Now step number 3,"},{"Start":"05:03.830 ","End":"05:07.790","Text":"we have to simplify the rules and identify the values of a and b."},{"Start":"05:07.790 ","End":"05:10.895","Text":"Well, we\u0027ve already done that right here."},{"Start":"05:10.895 ","End":"05:17.075","Text":"Once we brought the transformation to this form,"},{"Start":"05:17.075 ","End":"05:21.275","Text":"which in essence is the general form of the linear transformation,"},{"Start":"05:21.275 ","End":"05:27.720","Text":"we can easily identify a as being 1/2, that\u0027s here,"},{"Start":"05:27.720 ","End":"05:35.650","Text":"that\u0027s the multiplier, and b being equal to minus 4, that\u0027s right here."},{"Start":"05:36.020 ","End":"05:38.590","Text":"Now, that we have a and b,"},{"Start":"05:38.590 ","End":"05:42.810","Text":"we can go ahead and calculate the expectation variance of y."},{"Start":"05:43.130 ","End":"05:51.820","Text":"The expectation of y equals a times the expectation of x plus b."},{"Start":"05:51.820 ","End":"05:55.290","Text":"Now, let\u0027s plug in the numbers, a is 1/2,"},{"Start":"05:55.290 ","End":"05:59.490","Text":"expectation of x is 3, and what\u0027s b?"},{"Start":"05:59.490 ","End":"06:04.200","Text":"That\u0027s b minus 4, that\u0027s right here."},{"Start":"06:04.200 ","End":"06:08.115","Text":"That equals to minus 2.5."},{"Start":"06:08.115 ","End":"06:10.810","Text":"What\u0027s the variance of y?"},{"Start":"06:10.810 ","End":"06:17.615","Text":"Well, that\u0027s a squared times the variance of x. What\u0027s a?"},{"Start":"06:17.615 ","End":"06:18.980","Text":"Again, that\u0027s 1/2,"},{"Start":"06:18.980 ","End":"06:23.690","Text":"so that\u0027s 1/2 squared times the variance of x,"},{"Start":"06:23.690 ","End":"06:25.550","Text":"which is right here, times 1,"},{"Start":"06:25.550 ","End":"06:28.950","Text":"and that equals to a 1/4."},{"Start":"06:29.450 ","End":"06:36.485","Text":"Now, we\u0027ve calculated the expectation which is minus 2.5 and variance,"},{"Start":"06:36.485 ","End":"06:40.835","Text":"1/4 of the new value y,"},{"Start":"06:40.835 ","End":"06:44.520","Text":"which is a linear transformation of x."}],"ID":12989}],"Thumbnail":null,"ID":64483},{"Name":"Expectation and Variance of a Sum of Random Variables","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"4m 14s","ChapterTopicVideoID":12511,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In this chapter, I want to talk about the expectation"},{"Start":"00:03.120 ","End":"00:06.390","Text":"and variance of the sum of random variables."},{"Start":"00:06.390 ","End":"00:10.200","Text":"Now, assume that I have n random variables,"},{"Start":"00:10.200 ","End":"00:12.360","Text":"X_1, X_2, X_3,"},{"Start":"00:12.360 ","End":"00:15.045","Text":"and so forth until X_n."},{"Start":"00:15.045 ","End":"00:19.470","Text":"Now what I want to do is I want to talk about the sum of these guys."},{"Start":"00:19.470 ","End":"00:26.595","Text":"So if I have X_1 plus X_2 plus and so on and so forth until X_n,"},{"Start":"00:26.595 ","End":"00:29.575","Text":"let\u0027s just call that T for convenience sake."},{"Start":"00:29.575 ","End":"00:31.130","Text":"What I want to do,"},{"Start":"00:31.130 ","End":"00:40.520","Text":"I want to calculate the expectation of T and the variance of T. Let\u0027s take a look at"},{"Start":"00:40.520 ","End":"00:45.380","Text":"the expectation of T. The expectation of T is"},{"Start":"00:45.380 ","End":"00:51.440","Text":"the expectation of X_1 plus X_2 until X_n."},{"Start":"00:51.440 ","End":"00:59.105","Text":"Now, that would equal the expectation of X_1 plus the expectation of X_2,"},{"Start":"00:59.105 ","End":"01:04.685","Text":"and so on and so forth until we add up the expectation of X_n."},{"Start":"01:04.685 ","End":"01:08.730","Text":"So we can say that"},{"Start":"01:08.730 ","End":"01:18.544","Text":"the expectation of the sum of x,"},{"Start":"01:18.544 ","End":"01:25.585","Text":"well that would equal to the sum of the expectation of X."},{"Start":"01:25.585 ","End":"01:30.290","Text":"Now, let\u0027s just take a quick example."},{"Start":"01:30.290 ","End":"01:39.165","Text":"Assuming that X_i are my winnings for game number i,"},{"Start":"01:39.165 ","End":"01:43.755","Text":"and I played a series of games from 1 till n,"},{"Start":"01:43.755 ","End":"01:49.075","Text":"and I want to know what\u0027s my total expected winnings."},{"Start":"01:49.075 ","End":"01:51.740","Text":"Well, that\u0027s exactly this thing right here."},{"Start":"01:51.740 ","End":"01:54.130","Text":"What\u0027s my total expected winnings?"},{"Start":"01:54.130 ","End":"02:00.860","Text":"Well, that would be the expected winnings of X_1 plus X_2 plus,"},{"Start":"02:00.860 ","End":"02:02.900","Text":"and so on and so forth until X_n,"},{"Start":"02:02.900 ","End":"02:06.860","Text":"expected winnings of all the games together."},{"Start":"02:06.860 ","End":"02:13.700","Text":"Well, that would equal to the expected winnings of X_1, the first game,"},{"Start":"02:13.700 ","End":"02:17.465","Text":"plus the expected winnings of X_2, the second game,"},{"Start":"02:17.465 ","End":"02:24.690","Text":"and so on and so forth until I add up the expected winning of the nth game."},{"Start":"02:24.950 ","End":"02:29.935","Text":"So let\u0027s now talk about the variance."},{"Start":"02:29.935 ","End":"02:33.500","Text":"Now, if I have the same series,"},{"Start":"02:33.500 ","End":"02:35.660","Text":"X_1, X_2, and X_n,"},{"Start":"02:35.660 ","End":"02:38.855","Text":"and if they\u0027re independent of each other,"},{"Start":"02:38.855 ","End":"02:45.125","Text":"that means that X_ij is independent."},{"Start":"02:45.125 ","End":"02:50.840","Text":"Now what does that mean? That means that X_1 is independent of X_2, X_3,"},{"Start":"02:50.840 ","End":"02:52.765","Text":"X_4 until X_n,"},{"Start":"02:52.765 ","End":"02:55.835","Text":"X_2 would be independent of X_1,"},{"Start":"02:55.835 ","End":"02:58.235","Text":"X_3, X_4 until X_n."},{"Start":"02:58.235 ","End":"03:02.180","Text":"Until X_n, which would be independent of X_1,"},{"Start":"03:02.180 ","End":"03:04.325","Text":"X_2, X3, and so on and so forth."},{"Start":"03:04.325 ","End":"03:09.350","Text":"So every variable isn\u0027t as independent of the other."},{"Start":"03:09.350 ","End":"03:11.975","Text":"Then what do I have here?"},{"Start":"03:11.975 ","End":"03:19.820","Text":"I still want to know what the variance of the sum of these variables."},{"Start":"03:19.820 ","End":"03:27.050","Text":"Now, the variance of the total is equal to the variance of X_1 plus X_2 and so on"},{"Start":"03:27.050 ","End":"03:30.080","Text":"and so forth until X_n and that would equal to"},{"Start":"03:30.080 ","End":"03:34.295","Text":"the variance of X_1 plus the variance of X_2,"},{"Start":"03:34.295 ","End":"03:35.525","Text":"and so on and so forth,"},{"Start":"03:35.525 ","End":"03:38.390","Text":"plus the variance of X_n,"},{"Start":"03:38.390 ","End":"03:43.835","Text":"or we can write this out as the variance of the sum of"},{"Start":"03:43.835 ","End":"03:49.955","Text":"X equals the sum of the variance of X."},{"Start":"03:49.955 ","End":"03:54.635","Text":"Now again, here, when we dealt with the expectation,"},{"Start":"03:54.635 ","End":"03:59.580","Text":"this is correct always."},{"Start":"03:59.580 ","End":"04:08.380","Text":"Here, this is correct only when X_ij are independent."},{"Start":"04:08.470 ","End":"04:11.945","Text":"Now, enough of the theory,"},{"Start":"04:11.945 ","End":"04:14.370","Text":"let\u0027s go to an example."}],"ID":12990},{"Watched":false,"Name":"Example","Duration":"4m 39s","ChapterTopicVideoID":12512,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"Now in this game, a person plays 2 independent games of chance."},{"Start":"00:04.905 ","End":"00:09.585","Text":"The expected price from the first game is 7 with a standard deviation of 3,"},{"Start":"00:09.585 ","End":"00:15.480","Text":"and the expected price from the second game is minus 2 with a standard deviation of 4."},{"Start":"00:15.480 ","End":"00:21.525","Text":"We\u0027re asked, what is the expectation and variance of the total prizes from the 2 games?"},{"Start":"00:21.525 ","End":"00:24.570","Text":"Well, let\u0027s first write down our data."},{"Start":"00:24.570 ","End":"00:33.910","Text":"Let\u0027s define X_1 as our winnings from the first game,"},{"Start":"00:37.070 ","End":"00:43.980","Text":"and X_2 would be our winnings from the second game."},{"Start":"00:43.980 ","End":"00:49.730","Text":"So we know that the expectation of our winnings for"},{"Start":"00:49.730 ","End":"00:57.480","Text":"the first game is 7 and the standard deviation is 3."},{"Start":"00:57.590 ","End":"00:59.670","Text":"For the second game,"},{"Start":"00:59.670 ","End":"01:04.429","Text":"the expectation of our winnings from the second game is minus"},{"Start":"01:04.429 ","End":"01:11.380","Text":"2 and the standard deviation of X_2 or winnings is 4."},{"Start":"01:11.380 ","End":"01:14.510","Text":"Again, how do I know this?"},{"Start":"01:14.510 ","End":"01:20.250","Text":"Well, the expectation of the first game right here, that\u0027s 7,"},{"Start":"01:20.250 ","End":"01:22.200","Text":"the standard deviation is 3,"},{"Start":"01:22.200 ","End":"01:26.885","Text":"and the expectation of the second game is minus 2,"},{"Start":"01:26.885 ","End":"01:29.225","Text":"the standard deviation 4."},{"Start":"01:29.225 ","End":"01:33.965","Text":"Now, another thing that we have to take into consideration is given to us"},{"Start":"01:33.965 ","End":"01:39.335","Text":"that the 2 games are independent of each other, which is important."},{"Start":"01:39.335 ","End":"01:42.800","Text":"Now, we\u0027re asked to find"},{"Start":"01:42.800 ","End":"01:49.100","Text":"the variance and expectation of the total prices from the 2 games."},{"Start":"01:49.100 ","End":"01:59.479","Text":"That means that I can define T as X_1 plus X_2 and we\u0027re asked"},{"Start":"01:59.479 ","End":"02:09.680","Text":"to calculate the expectation of T and the variance of T. So we"},{"Start":"02:09.680 ","End":"02:13.790","Text":"know that the expectation of T equals"},{"Start":"02:13.790 ","End":"02:20.670","Text":"the expectation of X_1 plus X_2."},{"Start":"02:20.670 ","End":"02:25.490","Text":"Now, what do we know about the expectation of the sum?"},{"Start":"02:25.490 ","End":"02:28.310","Text":"It equals to the sum of the expectation."},{"Start":"02:28.310 ","End":"02:36.170","Text":"So that means that the expectation of X_1 plus the expectation of X_2."},{"Start":"02:36.170 ","End":"02:41.900","Text":"Now let\u0027s plug in the numbers that will equal to the expectation of X_1 well,"},{"Start":"02:41.900 ","End":"02:44.515","Text":"that\u0027s here, that\u0027s 7."},{"Start":"02:44.515 ","End":"02:47.935","Text":"The expectation of X_2,"},{"Start":"02:47.935 ","End":"02:49.850","Text":"that\u0027s here, that\u0027s minus 2."},{"Start":"02:49.850 ","End":"02:51.860","Text":"So 7 minus 2,"},{"Start":"02:51.860 ","End":"02:54.535","Text":"and that equals to 5."},{"Start":"02:54.535 ","End":"03:00.485","Text":"That\u0027s the expectation of our winnings when we play both games."},{"Start":"03:00.485 ","End":"03:03.530","Text":"Now, let\u0027s take a look at the variance."},{"Start":"03:03.530 ","End":"03:05.795","Text":"What\u0027s the variance of T?"},{"Start":"03:05.795 ","End":"03:13.670","Text":"Well, we know that the variance of T equals the variance of X_1 plus X_2."},{"Start":"03:13.670 ","End":"03:16.325","Text":"But what do we also know?"},{"Start":"03:16.325 ","End":"03:21.530","Text":"We know here we\u0027re given that the 2 games are independent of each other."},{"Start":"03:21.530 ","End":"03:27.320","Text":"Because they\u0027re independent, then the variance of the sum,"},{"Start":"03:27.320 ","End":"03:29.450","Text":"equals the sum of the variance,"},{"Start":"03:29.450 ","End":"03:35.525","Text":"that\u0027s variance of X_1 plus the variance of X_2."},{"Start":"03:35.525 ","End":"03:38.870","Text":"Now, let\u0027s plug in the numbers."},{"Start":"03:38.870 ","End":"03:42.765","Text":"The variance of X_1 is what?"},{"Start":"03:42.765 ","End":"03:44.390","Text":"That\u0027s 3 squared."},{"Start":"03:44.390 ","End":"03:47.240","Text":"We\u0027re given the standard deviation, not the variance."},{"Start":"03:47.240 ","End":"03:54.050","Text":"We have to square this plus the variance of X_2, that\u0027s 4 squared."},{"Start":"03:54.050 ","End":"03:56.240","Text":"So that\u0027s 9 plus 16,"},{"Start":"03:56.240 ","End":"03:59.945","Text":"that equals to 25, and that\u0027s the variance."},{"Start":"03:59.945 ","End":"04:04.100","Text":"Now, if we were asked about the standard deviation of T,"},{"Start":"04:04.100 ","End":"04:07.205","Text":"well, that would be the square root of the variance."},{"Start":"04:07.205 ","End":"04:10.670","Text":"The standard deviation of T would equal"},{"Start":"04:10.670 ","End":"04:15.795","Text":"the square root of the variance of T. That would equal to 5,"},{"Start":"04:15.795 ","End":"04:18.275","Text":"the square root of 25 is 5."},{"Start":"04:18.275 ","End":"04:20.510","Text":"So here we have it."},{"Start":"04:20.510 ","End":"04:24.740","Text":"We have the expectation of T,"},{"Start":"04:24.740 ","End":"04:29.240","Text":"and we have the variance of T. Now,"},{"Start":"04:29.240 ","End":"04:33.605","Text":"all we need to do is to go to our problems, solve the problems,"},{"Start":"04:33.605 ","End":"04:39.390","Text":"and only then look at our solution videos. So good luck."}],"ID":12991},{"Watched":false,"Name":"Exercise 1","Duration":"3m 54s","ChapterTopicVideoID":12513,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this question, we\u0027ll be talking about the profits in the stock exchange."},{"Start":"00:03.990 ","End":"00:06.720","Text":"Now in the Dow Jones Stock Exchange,"},{"Start":"00:06.720 ","End":"00:12.135","Text":"the profit of stock A has an expectation of 5 and a variance of 10."},{"Start":"00:12.135 ","End":"00:17.655","Text":"The profit of stock B has an expectation of 4 and a variance of 5."},{"Start":"00:17.655 ","End":"00:21.165","Text":"It\u0027s known that all investments in the 2 stocks"},{"Start":"00:21.165 ","End":"00:24.465","Text":"are independent of each other and we\u0027re asked,"},{"Start":"00:24.465 ","End":"00:26.460","Text":"what are the expectation and variance of"},{"Start":"00:26.460 ","End":"00:31.320","Text":"the total profits of the investment in the 2 stocks combined."},{"Start":"00:31.320 ","End":"00:34.230","Text":"Let\u0027s first write down our data."},{"Start":"00:34.230 ","End":"00:43.995","Text":"Let\u0027s define x_1 as the profits from stock A,"},{"Start":"00:43.995 ","End":"00:49.755","Text":"and x_2 would be the profits from stock B."},{"Start":"00:49.755 ","End":"00:55.895","Text":"Now we know what the expectation of the first of stock A is,"},{"Start":"00:55.895 ","End":"01:04.040","Text":"that\u0027s 5 and the variance is 10."},{"Start":"01:04.040 ","End":"01:14.175","Text":"For stock B, the expectation of stock B is 4 and the variance is 5."},{"Start":"01:14.175 ","End":"01:18.055","Text":"Now again, let\u0027s go to our data and make sure."},{"Start":"01:18.055 ","End":"01:23.610","Text":"Profits of stock A has an expectation of 5 and a variance of 10,"},{"Start":"01:23.610 ","End":"01:25.295","Text":"that\u0027s 5 and 10."},{"Start":"01:25.295 ","End":"01:30.050","Text":"The profit for stock B has an expectation of 4 and a variance of 5,"},{"Start":"01:30.050 ","End":"01:32.035","Text":"that\u0027s 4 and 5."},{"Start":"01:32.035 ","End":"01:34.445","Text":"Now what do we also know?"},{"Start":"01:34.445 ","End":"01:38.810","Text":"We\u0027re given that the 2 stocks are independent of each other."},{"Start":"01:38.810 ","End":"01:41.090","Text":"Again, this is important."},{"Start":"01:41.090 ","End":"01:45.360","Text":"What are we asked?"},{"Start":"01:45.360 ","End":"01:49.370","Text":"We\u0027re asked about the total profits of the 2 investments."},{"Start":"01:49.370 ","End":"01:55.610","Text":"That means that we can define T as the total profits."},{"Start":"01:55.610 ","End":"02:00.650","Text":"That means that T equals x_1 plus x_2."},{"Start":"02:00.650 ","End":"02:05.405","Text":"We\u0027re asked to calculate the expectation of"},{"Start":"02:05.405 ","End":"02:11.450","Text":"T and the variance of T. Now let\u0027s get started."},{"Start":"02:11.450 ","End":"02:13.850","Text":"What\u0027s the expectation of T?"},{"Start":"02:13.850 ","End":"02:20.060","Text":"That equals to the expectation of x_1 plus x_2"},{"Start":"02:20.060 ","End":"02:23.540","Text":"and we know that the expectation of"},{"Start":"02:23.540 ","End":"02:27.200","Text":"the sum is always equals to the sum of the expectation,"},{"Start":"02:27.200 ","End":"02:34.435","Text":"that\u0027s expectation of x_1 plus the expectation of x_2."},{"Start":"02:34.435 ","End":"02:36.710","Text":"Let\u0027s plug in the numbers,"},{"Start":"02:36.710 ","End":"02:38.210","Text":"the expectation of x_1,"},{"Start":"02:38.210 ","End":"02:43.775","Text":"that\u0027s 5, and the expectation of x_2, that\u0027s 4."},{"Start":"02:43.775 ","End":"02:51.480","Text":"That means that the expectation of the total investment is 9,"},{"Start":"02:51.480 ","End":"02:55.350","Text":"that\u0027s the total profit of the investment."},{"Start":"02:55.350 ","End":"02:59.805","Text":"That\u0027s the expected profits from the 2 investments."},{"Start":"02:59.805 ","End":"03:03.405","Text":"What about the variance of T?"},{"Start":"03:03.405 ","End":"03:09.990","Text":"That equals to the variance of x_1 plus x_2."},{"Start":"03:09.990 ","End":"03:13.220","Text":"Now what do we know about the variance of the sum?"},{"Start":"03:13.220 ","End":"03:16.985","Text":"It does equal to the variance to the sum of the variance,"},{"Start":"03:16.985 ","End":"03:21.690","Text":"that\u0027s the variance of x_1 plus the variance of x_2."},{"Start":"03:21.690 ","End":"03:23.580","Text":"But under which conditions,"},{"Start":"03:23.580 ","End":"03:27.170","Text":"when x_1 and x_2 are independent of each other,"},{"Start":"03:27.170 ","End":"03:28.400","Text":"we have that right here,"},{"Start":"03:28.400 ","End":"03:29.830","Text":"it was given to us."},{"Start":"03:29.830 ","End":"03:31.685","Text":"Let\u0027s plug in the numbers."},{"Start":"03:31.685 ","End":"03:34.730","Text":"The variance of x_1 is 10."},{"Start":"03:34.730 ","End":"03:39.445","Text":"That\u0027s right here, and the variance of x_2 is 5,"},{"Start":"03:39.445 ","End":"03:47.390","Text":"so that means that the variance of the total profits of the 2 investments equals 15."},{"Start":"03:47.390 ","End":"03:53.760","Text":"There you have it, the expectation of T and the variance of T."}],"ID":12992},{"Watched":false,"Name":"Exercise 2","Duration":"3m 22s","ChapterTopicVideoID":12514,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.045","Text":"In this question, we\u0027re given X and Y that are independent variables."},{"Start":"00:06.045 ","End":"00:09.600","Text":"Now we\u0027re also given that the standard deviation of X is"},{"Start":"00:09.600 ","End":"00:13.515","Text":"3 and the standard deviation of Y is 4."},{"Start":"00:13.515 ","End":"00:18.645","Text":"We\u0027re asked, what is the standard deviation of X plus Y?"},{"Start":"00:18.645 ","End":"00:22.335","Text":"Let\u0027s get started. First of all,"},{"Start":"00:22.335 ","End":"00:24.075","Text":"let\u0027s write down the data."},{"Start":"00:24.075 ","End":"00:27.810","Text":"The standard deviation of X equals"},{"Start":"00:27.810 ","End":"00:35.055","Text":"3 and the standard deviation of Y equals 4,"},{"Start":"00:35.055 ","End":"00:36.960","Text":"that\u0027s given to us."},{"Start":"00:36.960 ","End":"00:42.570","Text":"Now, we\u0027re also given that X and Y are independent."},{"Start":"00:42.570 ","End":"00:48.025","Text":"X and Y are independent."},{"Start":"00:48.025 ","End":"00:54.385","Text":"Now we\u0027re asked about the standard deviation of X plus Y."},{"Start":"00:54.385 ","End":"01:01.375","Text":"We want to know what is the standard deviation of X plus Y."},{"Start":"01:01.375 ","End":"01:08.740","Text":"Well, what do we know about the sum of 2 random variables?"},{"Start":"01:08.740 ","End":"01:14.784","Text":"Specifically about the variance of the sum of 2 variables."},{"Start":"01:14.784 ","End":"01:23.420","Text":"Well, let\u0027s first write down or define T as being equal to X plus Y."},{"Start":"01:23.420 ","End":"01:30.350","Text":"Now, we know that the variance of T equals what?"},{"Start":"01:30.350 ","End":"01:34.495","Text":"That\u0027s the variance of X plus Y."},{"Start":"01:34.495 ","End":"01:41.885","Text":"That\u0027s the variance of the sum and we know that that equals to the sum of the variance."},{"Start":"01:41.885 ","End":"01:47.315","Text":"The variance of X plus the variance of Y, under which conditions,"},{"Start":"01:47.315 ","End":"01:50.659","Text":"where X and Y are independent variables,"},{"Start":"01:50.659 ","End":"01:52.880","Text":"that means that that equals that only when they\u0027re"},{"Start":"01:52.880 ","End":"01:56.215","Text":"independent and that\u0027s given to us right here."},{"Start":"01:56.215 ","End":"02:00.530","Text":"That\u0027s great. But what are we given?"},{"Start":"02:00.530 ","End":"02:03.230","Text":"We\u0027re not given the variance of X and the variance of Y,"},{"Start":"02:03.230 ","End":"02:12.140","Text":"we\u0027re given the standard deviation so what is the variance of X?"},{"Start":"02:12.140 ","End":"02:14.750","Text":"Well, let\u0027s plug in the numbers."},{"Start":"02:14.750 ","End":"02:16.745","Text":"That\u0027s 3 squared,"},{"Start":"02:16.745 ","End":"02:18.635","Text":"which equals to 9."},{"Start":"02:18.635 ","End":"02:21.625","Text":"What\u0027s the variance of Y?"},{"Start":"02:21.625 ","End":"02:26.350","Text":"Well, that\u0027s 4 squared, that equals to 16."},{"Start":"02:27.080 ","End":"02:31.355","Text":"Let\u0027s plug in these numbers right in here."},{"Start":"02:31.355 ","End":"02:36.920","Text":"That equals to 9 plus 16,"},{"Start":"02:36.920 ","End":"02:38.825","Text":"which equals to 25."},{"Start":"02:38.825 ","End":"02:45.110","Text":"Now again, 25 is the variance of T or the variance of the sum of X and Y."},{"Start":"02:45.110 ","End":"02:50.299","Text":"But we\u0027re not asked to find the variance but the standard deviation."},{"Start":"02:50.299 ","End":"02:56.600","Text":"We note that the standard deviation of T equals the square root of"},{"Start":"02:56.600 ","End":"03:03.605","Text":"the variance of T. But we calculated the variance of T, that\u0027s 25."},{"Start":"03:03.605 ","End":"03:05.435","Text":"Let\u0027s plug that in."},{"Start":"03:05.435 ","End":"03:13.680","Text":"The standard deviation of T equals the square root of 25, and that equals to 5."},{"Start":"03:14.150 ","End":"03:16.190","Text":"To answer the question,"},{"Start":"03:16.190 ","End":"03:19.130","Text":"what\u0027s the standard deviation of X plus Y?"},{"Start":"03:19.130 ","End":"03:22.200","Text":"There it is. It\u0027s 5."}],"ID":12993},{"Watched":false,"Name":"Exercise 3","Duration":"3m 14s","ChapterTopicVideoID":12515,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.345","Text":"A person plays 2 independent games of chance."},{"Start":"00:03.345 ","End":"00:06.000","Text":"X is the amount won in the first game,"},{"Start":"00:06.000 ","End":"00:08.834","Text":"and Y is the amount won in the second game."},{"Start":"00:08.834 ","End":"00:14.880","Text":"We\u0027re given that the expectation of x is 10 and the standard deviation of x is 3;"},{"Start":"00:14.880 ","End":"00:17.250","Text":"and the expectation of y is 12,"},{"Start":"00:17.250 ","End":"00:19.770","Text":"and the standard deviation of y is 4."},{"Start":"00:19.770 ","End":"00:22.260","Text":"We\u0027re asked, what are the expectation and"},{"Start":"00:22.260 ","End":"00:26.650","Text":"standard deviation of the amounts won in both games?"},{"Start":"00:26.900 ","End":"00:31.500","Text":"Let\u0027s define t as the total amount,"},{"Start":"00:31.500 ","End":"00:37.795","Text":"or the sum, of our winnings from the first game and the second game."},{"Start":"00:37.795 ","End":"00:42.130","Text":"We\u0027re asked, what\u0027s the expectation of t,"},{"Start":"00:42.130 ","End":"00:48.995","Text":"and what\u0027s the standard deviation of t? Let\u0027s get started."},{"Start":"00:48.995 ","End":"00:52.790","Text":"We know that the expectation of t,"},{"Start":"00:52.790 ","End":"00:57.520","Text":"well, that equals to the expectation of x plus y."},{"Start":"00:57.520 ","End":"01:01.160","Text":"What do we know about the expectation of the sum?"},{"Start":"01:01.160 ","End":"01:05.239","Text":"It always equals to the sum of the expectations;"},{"Start":"01:05.239 ","End":"01:10.595","Text":"that\u0027s the expectation of x plus the expectation of y."},{"Start":"01:10.595 ","End":"01:12.605","Text":"Now, let\u0027s plug in the numbers."},{"Start":"01:12.605 ","End":"01:15.805","Text":"Expectation of x is 10, that\u0027s right here."},{"Start":"01:15.805 ","End":"01:19.115","Text":"The expectation of y is 12."},{"Start":"01:19.115 ","End":"01:22.040","Text":"That means that the expectation of t or"},{"Start":"01:22.040 ","End":"01:29.465","Text":"the expectation of the total amounts won in both game is 22."},{"Start":"01:29.465 ","End":"01:35.510","Text":"Now let\u0027s take a look at the variance of t. Now, don\u0027t forget,"},{"Start":"01:35.510 ","End":"01:37.985","Text":"we\u0027re asked about the standard deviation,"},{"Start":"01:37.985 ","End":"01:42.440","Text":"but we can never add standard deviations;"},{"Start":"01:42.440 ","End":"01:44.765","Text":"we have to go through variances."},{"Start":"01:44.765 ","End":"01:47.660","Text":"Now, what do we know about variance of t,"},{"Start":"01:47.660 ","End":"01:51.755","Text":"or the variance of a sum of random variables?"},{"Start":"01:51.755 ","End":"02:01.315","Text":"Well, the variance of a sum of random variables equals to the sum of the variance,"},{"Start":"02:01.315 ","End":"02:04.925","Text":"v of x plus v of y,"},{"Start":"02:04.925 ","End":"02:06.725","Text":"under which conditions,"},{"Start":"02:06.725 ","End":"02:13.060","Text":"the 2 games are independent,"},{"Start":"02:13.060 ","End":"02:15.245","Text":"and we\u0027re given that right here."},{"Start":"02:15.245 ","End":"02:17.615","Text":"Let\u0027s plug in the numbers."},{"Start":"02:17.615 ","End":"02:19.220","Text":"What\u0027s the appearance of x?"},{"Start":"02:19.220 ","End":"02:22.940","Text":"Well, we\u0027re given the standard deviation of x being 3,"},{"Start":"02:22.940 ","End":"02:26.360","Text":"so the variance of x is 3 squared,"},{"Start":"02:26.360 ","End":"02:30.359","Text":"the standard deviation squared is the variance;"},{"Start":"02:30.359 ","End":"02:34.760","Text":"plus the variance of y,"},{"Start":"02:34.760 ","End":"02:38.270","Text":"that\u0027s the standard deviation of y squared,"},{"Start":"02:38.270 ","End":"02:39.650","Text":"that\u0027s 4 squared;"},{"Start":"02:39.650 ","End":"02:42.185","Text":"that equals 9 plus 16,"},{"Start":"02:42.185 ","End":"02:43.940","Text":"and that equals to 25."},{"Start":"02:43.940 ","End":"02:45.020","Text":"Now that\u0027s the variance,"},{"Start":"02:45.020 ","End":"02:48.095","Text":"and we\u0027re asked about the standard deviation."},{"Start":"02:48.095 ","End":"02:55.295","Text":"Now we know that the standard deviation is equal to the square root of the variance."},{"Start":"02:55.295 ","End":"02:59.285","Text":"In our case, that\u0027s the square root of 25,"},{"Start":"02:59.285 ","End":"03:01.370","Text":"that equals to 5."},{"Start":"03:01.370 ","End":"03:03.740","Text":"To answer the question,"},{"Start":"03:03.740 ","End":"03:09.305","Text":"the expectation of the sum is 22,"},{"Start":"03:09.305 ","End":"03:13.890","Text":"and the standard deviation of the sum is 5."}],"ID":12994},{"Watched":false,"Name":"Exercise 4","Duration":"7m 15s","ChapterTopicVideoID":12516,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In roulette, the chances of winning $30 are 50 percent,"},{"Start":"00:04.320 ","End":"00:09.085","Text":"and the chances of winning $10 and $20 are 25 percent each."},{"Start":"00:09.085 ","End":"00:12.680","Text":"We\u0027re asked what are the expectation and variance of"},{"Start":"00:12.680 ","End":"00:17.520","Text":"the total amount won by a person who plays roulette 4 times?"},{"Start":"00:17.520 ","End":"00:20.160","Text":"Well, before we can answer that,"},{"Start":"00:20.160 ","End":"00:24.300","Text":"let\u0027s take a look at what are the expectation and variance of"},{"Start":"00:24.300 ","End":"00:30.210","Text":"the total amount won in 1 game by 1 turn of the roulette wheel."},{"Start":"00:30.210 ","End":"00:32.460","Text":"Well, in order to do that,"},{"Start":"00:32.460 ","End":"00:36.150","Text":"let\u0027s write out the probability function of x."},{"Start":"00:36.150 ","End":"00:41.865","Text":"Now, x would be the winnings in 1 game of roulette;"},{"Start":"00:41.865 ","End":"00:43.320","Text":"1 turn of the wheel."},{"Start":"00:43.320 ","End":"00:48.060","Text":"Now, a person can win either $10,"},{"Start":"00:48.060 ","End":"00:56.315","Text":"$20 dollars or $30 with a probability of 25 percent here,"},{"Start":"00:56.315 ","End":"01:00.845","Text":"25 percent here, and 50 percent here."},{"Start":"01:00.845 ","End":"01:06.845","Text":"Now, let\u0027s take a look at the expectation of x."},{"Start":"01:06.845 ","End":"01:14.645","Text":"Well, if we recall the expectation of x is defined as the sum of x times its probability."},{"Start":"01:14.645 ","End":"01:20.855","Text":"That equals to 10 times 0.25 plus"},{"Start":"01:20.855 ","End":"01:29.230","Text":"20 times 0.25 plus 30 times 0.5."},{"Start":"01:29.230 ","End":"01:31.635","Text":"After doing the math,"},{"Start":"01:31.635 ","End":"01:36.145","Text":"this comes out to 22.5."},{"Start":"01:36.145 ","End":"01:40.465","Text":"Great. Now let\u0027s take a look at the variance of x."},{"Start":"01:40.465 ","End":"01:46.310","Text":"Again, the variance of x is defined as the sum of x squared times"},{"Start":"01:46.310 ","End":"01:52.840","Text":"the probability of x minus the expectation squared of x."},{"Start":"01:52.840 ","End":"02:01.670","Text":"Let\u0027s write that out. That\u0027s 10 squared times 0.25 plus 20 squared times"},{"Start":"02:01.670 ","End":"02:09.530","Text":"0.25 plus 30 squared times 0.5 minus,"},{"Start":"02:09.530 ","End":"02:15.870","Text":"let\u0027s not forget this, minus 22.5 squared."},{"Start":"02:16.340 ","End":"02:25.970","Text":"After doing the math, that comes out to 68.75 squared dollars,"},{"Start":"02:25.970 ","End":"02:28.740","Text":"that\u0027s the unit, dollar squared."},{"Start":"02:29.230 ","End":"02:38.085","Text":"Now we figured out the expectation and variance of 1 game of roulette,"},{"Start":"02:38.085 ","End":"02:40.540","Text":"but we\u0027re not asked about that."},{"Start":"02:40.540 ","End":"02:48.290","Text":"We\u0027re asked about the total amount won by a person who plays the roulette 4 times."},{"Start":"02:48.290 ","End":"02:53.725","Text":"That means that we have a new random variable T,"},{"Start":"02:53.725 ","End":"03:04.080","Text":"which is equal to x_1 plus x_2 plus x_3 plus x_4."},{"Start":"03:04.080 ","End":"03:08.160","Text":"X being the amount won in each game."},{"Start":"03:08.160 ","End":"03:14.755","Text":"T is the total amount won in 4 games where x_1 is the amount won in the first game,"},{"Start":"03:14.755 ","End":"03:17.610","Text":"x_2 is the amount won in the second game,"},{"Start":"03:17.610 ","End":"03:20.890","Text":"the third game, and in the fourth game respectively."},{"Start":"03:20.890 ","End":"03:31.300","Text":"Again, we\u0027re asked what\u0027s the expectation of T and what\u0027s the variance of T?"},{"Start":"03:32.690 ","End":"03:35.400","Text":"Let\u0027s get to work."},{"Start":"03:35.400 ","End":"03:40.065","Text":"The expectation of T equals what?"},{"Start":"03:40.065 ","End":"03:49.035","Text":"That\u0027s the expectation of x_1 plus x_2 plus x_3 plus x_4."},{"Start":"03:49.035 ","End":"03:58.640","Text":"Now, we\u0027ve learned that the expectation of the sum equals the sum of the expectation."},{"Start":"03:58.640 ","End":"04:05.645","Text":"That means that that equals to the expectation of x_1 plus the expectation of"},{"Start":"04:05.645 ","End":"04:13.765","Text":"x_2 plus the expectation of x_3 plus the expectation of x_4."},{"Start":"04:13.765 ","End":"04:17.780","Text":"Now, I\u0027m not saying that the results of"},{"Start":"04:17.780 ","End":"04:22.130","Text":"the first game would be equal to the results in the second game,"},{"Start":"04:22.130 ","End":"04:26.000","Text":"but the expectation of each game is the same."},{"Start":"04:26.000 ","End":"04:27.290","Text":"We\u0027ve calculated that,"},{"Start":"04:27.290 ","End":"04:29.120","Text":"that\u0027s 22 and 1/2."},{"Start":"04:29.120 ","End":"04:34.960","Text":"That means that we have 22 and 1/2,"},{"Start":"04:34.960 ","End":"04:37.775","Text":"that\u0027s the expectation for the first game."},{"Start":"04:37.775 ","End":"04:40.550","Text":"Now, the expectation for the second game is the same,"},{"Start":"04:40.550 ","End":"04:45.000","Text":"that\u0027s 22.5, and the third game, that\u0027s the same,"},{"Start":"04:45.000 ","End":"04:48.575","Text":"22.5, and the fourth game again,"},{"Start":"04:48.575 ","End":"04:51.430","Text":"that\u0027s the same 22.5,"},{"Start":"04:51.430 ","End":"04:54.640","Text":"that equals to 90."},{"Start":"04:54.680 ","End":"05:03.780","Text":"What we\u0027re saying is after 4 games we\u0027re expected to win $90."},{"Start":"05:04.330 ","End":"05:09.830","Text":"Now, let\u0027s take a look at the variance of T. The variance of"},{"Start":"05:09.830 ","End":"05:15.300","Text":"T equals the variance of the sum."},{"Start":"05:15.300 ","End":"05:23.130","Text":"That\u0027s variance of x_1 plus x_2 plus x_3 plus x_4."},{"Start":"05:23.130 ","End":"05:26.645","Text":"Now, what does that equal to?"},{"Start":"05:26.645 ","End":"05:31.175","Text":"Well, we know that the variance of the sum equals the sum of the variances,"},{"Start":"05:31.175 ","End":"05:36.540","Text":"that\u0027s variance of x_1 plus the variance of x_2 plus"},{"Start":"05:36.540 ","End":"05:42.330","Text":"the variance of x_3 plus the variance of x_4."},{"Start":"05:42.330 ","End":"05:49.200","Text":"Now, although it hasn\u0027t been explicitly stated that x_1, x_2, x_3,"},{"Start":"05:49.200 ","End":"05:52.650","Text":"and x_4 are independent of each other, well,"},{"Start":"05:52.650 ","End":"05:58.995","Text":"we can assume that each turn of the roulette is independent of the other."},{"Start":"05:58.995 ","End":"06:03.640","Text":"Having said that, we can assume independence,"},{"Start":"06:03.640 ","End":"06:10.130","Text":"and therefore the variance of the sum does equal to the sum of the variance."},{"Start":"06:10.130 ","End":"06:13.835","Text":"Now, again, just like with the expectation,"},{"Start":"06:13.835 ","End":"06:16.240","Text":"what\u0027s the variance of 1 game?"},{"Start":"06:16.240 ","End":"06:18.360","Text":"Well, that\u0027s 68.75."},{"Start":"06:18.360 ","End":"06:19.980","Text":"We calculated that."},{"Start":"06:19.980 ","End":"06:27.630","Text":"But that equals to the variance of every turn of the roulette wheel."},{"Start":"06:27.630 ","End":"06:35.240","Text":"That equals to 68.75 for the variance of the first game,"},{"Start":"06:35.240 ","End":"06:39.800","Text":"plus 68.75 for the second game,"},{"Start":"06:39.800 ","End":"06:42.920","Text":"68.75 for the third game,"},{"Start":"06:42.920 ","End":"06:46.095","Text":"68.75 for the fourth game,"},{"Start":"06:46.095 ","End":"06:52.260","Text":"and that comes out to 275."},{"Start":"06:52.260 ","End":"06:56.460","Text":"Now we\u0027ve calculated"},{"Start":"06:56.460 ","End":"07:03.030","Text":"our expected winnings"},{"Start":"07:03.030 ","End":"07:06.035","Text":"of playing 4 games,"},{"Start":"07:06.035 ","End":"07:14.280","Text":"that\u0027s $90, with a variance of the 4 games being $275 squared."}],"ID":12995},{"Watched":false,"Name":"Exercise 5 - Parts a-b","Duration":"4m 29s","ChapterTopicVideoID":12518,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this question, we\u0027re given that"},{"Start":"00:01.980 ","End":"00:05.879","Text":"the probability function of a random variable is as follows."},{"Start":"00:05.879 ","End":"00:08.820","Text":"The probability of X when it equals K,"},{"Start":"00:08.820 ","End":"00:13.650","Text":"equals A over K minus 1 where K equals 2,"},{"Start":"00:13.650 ","End":"00:15.615","Text":"3, 4, and 5."},{"Start":"00:15.615 ","End":"00:21.760","Text":"Otherwise, it\u0027s 0. Now the first thing that we need to do is find the value of A."},{"Start":"00:21.760 ","End":"00:28.695","Text":"The easiest way to do that is to take this function and put it into table form."},{"Start":"00:28.695 ","End":"00:34.605","Text":"Let\u0027s write here X and the probability of x."},{"Start":"00:34.605 ","End":"00:39.710","Text":"Now X can equal 2,"},{"Start":"00:39.710 ","End":"00:42.810","Text":"where K equals 2,"},{"Start":"00:42.810 ","End":"00:47.165","Text":"3, 4, and 5."},{"Start":"00:47.165 ","End":"00:51.410","Text":"Now, what\u0027s the probability of x when x equals 2?"},{"Start":"00:51.410 ","End":"00:58.880","Text":"That\u0027s A over 2 minus 1 and the probability of x where x equals 3,"},{"Start":"00:58.880 ","End":"01:03.530","Text":"that\u0027s A over 3 minus 1 and likewise,"},{"Start":"01:03.530 ","End":"01:10.585","Text":"that\u0027ll be A over 4 minus 1 and A over 5 minus 1."},{"Start":"01:10.585 ","End":"01:16.460","Text":"Now, we know that the sum of the probabilities have to add up to 1."},{"Start":"01:16.460 ","End":"01:19.745","Text":"Let\u0027s just calculate that."},{"Start":"01:19.745 ","End":"01:25.290","Text":"That\u0027s A over 1 plus A over"},{"Start":"01:25.290 ","End":"01:31.615","Text":"2 plus A over 3 plus A over 4."},{"Start":"01:31.615 ","End":"01:33.665","Text":"Now that equals to 1."},{"Start":"01:33.665 ","End":"01:40.490","Text":"Let\u0027s just multiply both sides by 12 so we get this."},{"Start":"01:40.490 ","End":"01:44.895","Text":"We get 12A plus"},{"Start":"01:44.895 ","End":"01:52.950","Text":"6A plus 4A plus 3A and that equals to 12."},{"Start":"01:52.950 ","End":"01:57.900","Text":"That means if this is 25A equals to 12,"},{"Start":"01:57.900 ","End":"02:10.600","Text":"that means that A equals to 12 over 25 and that equals to 0.48."},{"Start":"02:10.960 ","End":"02:16.295","Text":"In this section, we\u0027re asked to calculate the expectation of variance of X."},{"Start":"02:16.295 ","End":"02:17.915","Text":"Well, first of all,"},{"Start":"02:17.915 ","End":"02:22.310","Text":"let\u0027s write out again the probability function of x."},{"Start":"02:22.310 ","End":"02:29.525","Text":"We have X, we have the probability of x and x equal to 2,"},{"Start":"02:29.525 ","End":"02:32.696","Text":"3, 4, and 5."},{"Start":"02:32.696 ","End":"02:40.340","Text":"Now, the probability of x when x equals 2 was 0.48."},{"Start":"02:40.340 ","End":"02:44.840","Text":"When x equals 3, that was 0.24."},{"Start":"02:44.840 ","End":"02:46.760","Text":"When x equals 4,"},{"Start":"02:46.760 ","End":"02:53.705","Text":"that will be 0.16 and when x equals 5 that will be 0.12."},{"Start":"02:53.705 ","End":"02:56.180","Text":"Now what we\u0027re asked to do,"},{"Start":"02:56.180 ","End":"02:59.900","Text":"we\u0027re asked to calculate the expectation of X."},{"Start":"02:59.900 ","End":"03:04.940","Text":"Well, that\u0027s the sum of x times"},{"Start":"03:04.940 ","End":"03:11.190","Text":"its probability and that equals to 2 times 0.48,"},{"Start":"03:11.190 ","End":"03:14.625","Text":"plus 3 times 0.24,"},{"Start":"03:14.625 ","End":"03:17.910","Text":"plus 4 times 0.16,"},{"Start":"03:17.910 ","End":"03:23.070","Text":"plus 5 times 0.12,"},{"Start":"03:23.070 ","End":"03:31.340","Text":"and that equals to 2.92, it\u0027s the expectation."},{"Start":"03:31.340 ","End":"03:34.030","Text":"Now, what\u0027s the variance of X?"},{"Start":"03:34.030 ","End":"03:40.476","Text":"Well, that\u0027s the sum of x squared times its probability."},{"Start":"03:40.476 ","End":"03:45.265","Text":"Again minus, the expectation squared of x."},{"Start":"03:45.265 ","End":"03:48.495","Text":"Never forget that, and lots of people do."},{"Start":"03:48.495 ","End":"03:51.120","Text":"Let\u0027s plug in the numbers."},{"Start":"03:51.120 ","End":"03:55.770","Text":"That\u0027s 2 squared times 0.48,"},{"Start":"03:55.770 ","End":"03:59.685","Text":"plus 3 squared times 0.24,"},{"Start":"03:59.685 ","End":"04:04.280","Text":"plus 4 squared times 0.16,"},{"Start":"04:04.280 ","End":"04:13.360","Text":"plus 5 squared times 0.12, minus 2.92 squared."},{"Start":"04:13.360 ","End":"04:16.385","Text":"After doing the math,"},{"Start":"04:16.385 ","End":"04:22.175","Text":"that comes out to 1.1136."},{"Start":"04:22.175 ","End":"04:28.860","Text":"Now we\u0027ve calculated the expectation of X and the variance of X."}],"ID":12997},{"Watched":false,"Name":"Exercise 5 - Part c","Duration":"4m 46s","ChapterTopicVideoID":12517,"CourseChapterTopicPlaylistID":64484,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.500","Text":"In this section, we have"},{"Start":"00:01.500 ","End":"00:07.260","Text":"n independent random variables that are taken from the above probability distribution."},{"Start":"00:07.260 ","End":"00:10.920","Text":"We\u0027re asked to write the equation for the expectation and variance of"},{"Start":"00:10.920 ","End":"00:14.655","Text":"the sum of the variables as a function of n. Well,"},{"Start":"00:14.655 ","End":"00:18.714","Text":"first of all, let\u0027s recall what we did in the previous sections."},{"Start":"00:18.714 ","End":"00:21.740","Text":"We know that the variance of X,"},{"Start":"00:21.740 ","End":"00:27.100","Text":"we\u0027ve calculated that to be 2.92."},{"Start":"00:27.100 ","End":"00:29.490","Text":"The variance of X,"},{"Start":"00:29.490 ","End":"00:35.820","Text":"we\u0027ve calculated that to be 1.1136."},{"Start":"00:35.820 ","End":"00:40.430","Text":"Now, what do we have? What\u0027s given to us?"},{"Start":"00:40.430 ","End":"00:43.870","Text":"We\u0027re given n independent random variables"},{"Start":"00:43.870 ","End":"00:48.245","Text":"that are taken from the above probability distribution. What does that mean?"},{"Start":"00:48.245 ","End":"00:52.730","Text":"That means that they all have the same probability distribution."},{"Start":"00:52.730 ","End":"00:58.420","Text":"That means that they all have the same expectation and the same variance."},{"Start":"00:58.420 ","End":"00:59.970","Text":"That\u0027s very important."},{"Start":"00:59.970 ","End":"01:02.900","Text":"Now, what we have,"},{"Start":"01:02.900 ","End":"01:05.510","Text":"we have n independent random variables."},{"Start":"01:05.510 ","End":"01:07.250","Text":"That means we have X_1,"},{"Start":"01:07.250 ","End":"01:11.000","Text":"X_2, and so on and so forth, and X_n."},{"Start":"01:11.000 ","End":"01:16.150","Text":"Each X has the same expectation and the same variance."},{"Start":"01:16.150 ","End":"01:20.595","Text":"We\u0027re asked about the sum of the variables."},{"Start":"01:20.595 ","End":"01:24.980","Text":"Let\u0027s define T as the total or the sum of the variables."},{"Start":"01:24.980 ","End":"01:32.560","Text":"That\u0027ll be X_1 plus X_2 plus and so on and so forth until X_n."},{"Start":"01:32.560 ","End":"01:43.385","Text":"We\u0027re asked about the expectation of T and the variance of T. Let\u0027s get started."},{"Start":"01:43.385 ","End":"01:46.550","Text":"What\u0027s the expectation of T here?"},{"Start":"01:46.550 ","End":"01:51.195","Text":"Well, that\u0027s the expectation of X_1 plus,"},{"Start":"01:51.195 ","End":"01:55.090","Text":"and so on and so forth plus X_n."},{"Start":"01:55.100 ","End":"01:59.165","Text":"Now, what do we know about the expectation of the sum?"},{"Start":"01:59.165 ","End":"02:02.255","Text":"That equals to the sum of the expectations."},{"Start":"02:02.255 ","End":"02:05.840","Text":"That equals to the expectation of X_1 plus"},{"Start":"02:05.840 ","End":"02:10.460","Text":"the expectation of X_2 plus and so on and so forth,"},{"Start":"02:10.460 ","End":"02:14.820","Text":"plus the expectation of X_n."},{"Start":"02:15.190 ","End":"02:20.690","Text":"But we know that for each of the X\u0027s,"},{"Start":"02:20.690 ","End":"02:23.495","Text":"the expectation is the same."},{"Start":"02:23.495 ","End":"02:25.900","Text":"It\u0027s this guy right here."},{"Start":"02:25.900 ","End":"02:28.170","Text":"Let\u0027s take the first 1."},{"Start":"02:28.170 ","End":"02:32.700","Text":"The expectation of X_1, that\u0027s 2.92."},{"Start":"02:32.700 ","End":"02:35.570","Text":"The expectation of X_2, again,"},{"Start":"02:35.570 ","End":"02:40.920","Text":"because it comes from the same distribution, it\u0027s 2.92."},{"Start":"02:41.450 ","End":"02:46.265","Text":"Likewise, for every other expectation of X,"},{"Start":"02:46.265 ","End":"02:54.235","Text":"that\u0027s 2.92 and that equals to n times 2.92."},{"Start":"02:54.235 ","End":"03:02.860","Text":"That is the expectation for the sum of n variables."},{"Start":"03:04.490 ","End":"03:07.955","Text":"Let\u0027s go on to the variance."},{"Start":"03:07.955 ","End":"03:11.630","Text":"What\u0027s the variance of T?"},{"Start":"03:11.630 ","End":"03:21.155","Text":"Well, that equals to the variance of X_1 plus so on and so forth plus X_n."},{"Start":"03:21.155 ","End":"03:25.610","Text":"Now, what do we know about the variance of the sum?"},{"Start":"03:25.610 ","End":"03:28.730","Text":"That equals to the sum of the variance."},{"Start":"03:28.730 ","End":"03:31.985","Text":"That\u0027s variance of X_1 plus the variance of"},{"Start":"03:31.985 ","End":"03:38.075","Text":"X_2 plus and so on and so forth until we get to the variance of X_n."},{"Start":"03:38.075 ","End":"03:44.885","Text":"Now, under what conditions where each X is independent of each other?"},{"Start":"03:44.885 ","End":"03:46.580","Text":"But we have that right here."},{"Start":"03:46.580 ","End":"03:50.310","Text":"That\u0027s given to us, so that\u0027s independent."},{"Start":"03:50.500 ","End":"03:53.450","Text":"Now, what else do we know?"},{"Start":"03:53.450 ","End":"03:55.370","Text":"We know that since X_1,"},{"Start":"03:55.370 ","End":"03:59.075","Text":"X_2 until X_n came from the same distribution,"},{"Start":"03:59.075 ","End":"04:02.765","Text":"that means that the variance is the same as well."},{"Start":"04:02.765 ","End":"04:08.760","Text":"We have here 1.1136,"},{"Start":"04:08.760 ","End":"04:13.110","Text":"that\u0027s for X_1 plus 1.1136,"},{"Start":"04:13.110 ","End":"04:21.210","Text":"that\u0027s for X_2 and basically for every X until X_n plus 1.1136."},{"Start":"04:21.210 ","End":"04:24.350","Text":"How many times do we have to add the variance?"},{"Start":"04:24.350 ","End":"04:26.230","Text":"Well, n times."},{"Start":"04:26.230 ","End":"04:33.135","Text":"That means that it\u0027s n times 1.1136."},{"Start":"04:33.135 ","End":"04:38.570","Text":"That is the variance of T as a function of"},{"Start":"04:38.570 ","End":"04:46.260","Text":"n. Here is the expectation of T as a function of n."}],"ID":12996}],"Thumbnail":null,"ID":64484},{"Name":"Special Discrete Probability Distributions - Binomial Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"8m 25s","ChapterTopicVideoID":12519,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:03.180 ","End":"00:08.085","Text":"special probability distributions, specifically binomial distribution."},{"Start":"00:08.085 ","End":"00:11.820","Text":"Now, this will be the first of a number of chapters that will"},{"Start":"00:11.820 ","End":"00:16.200","Text":"deal with various special probability distributions."},{"Start":"00:16.200 ","End":"00:21.465","Text":"Now, in order for us to work with these distributions,"},{"Start":"00:21.465 ","End":"00:23.505","Text":"we have to recognize them."},{"Start":"00:23.505 ","End":"00:27.360","Text":"Now, it\u0027ll be up to you to identify them by yourselves through"},{"Start":"00:27.360 ","End":"00:31.905","Text":"the context of the question and so the data that\u0027s provided to you."},{"Start":"00:31.905 ","End":"00:35.355","Text":"Now, once you\u0027ve identified these distributions,"},{"Start":"00:35.355 ","End":"00:40.830","Text":"you can use either the predefined expression or equation for"},{"Start":"00:40.830 ","End":"00:47.434","Text":"the distribution and you can also calculate the expectation of variance."},{"Start":"00:47.434 ","End":"00:51.410","Text":"Again, they\u0027re predefined equations so you don\u0027t have to worry about them,"},{"Start":"00:51.410 ","End":"00:54.710","Text":"all you need to do is just to identify"},{"Start":"00:54.710 ","End":"01:02.195","Text":"the special distribution and know what the expectation and variance equations are."},{"Start":"01:02.195 ","End":"01:04.505","Text":"Now, having said that,"},{"Start":"01:04.505 ","End":"01:10.560","Text":"I want to introduce to you a very special concept that\u0027s called a Bernoulli trial."},{"Start":"01:10.560 ","End":"01:14.360","Text":"Now, a Bernoulli trial is a trial in which"},{"Start":"01:14.360 ","End":"01:17.959","Text":"there are 2 possible results and only 2 possible results,"},{"Start":"01:17.959 ","End":"01:19.880","Text":"and we\u0027ll call them success or failures."},{"Start":"01:19.880 ","End":"01:23.930","Text":"Now, example of that could be either a product coming"},{"Start":"01:23.930 ","End":"01:28.730","Text":"out of the production line which is either good or faulty,"},{"Start":"01:28.730 ","End":"01:32.015","Text":"a person that is employed or unemployed,"},{"Start":"01:32.015 ","End":"01:37.340","Text":"a coin toss that produces heads or tails and so on and so forth."},{"Start":"01:37.340 ","End":"01:41.870","Text":"Now, the binomial distribution calculates"},{"Start":"01:41.870 ","End":"01:47.495","Text":"the probability of success when n independent Bernoulli trials are repeated."},{"Start":"01:47.495 ","End":"01:50.565","Text":"Now, that means,"},{"Start":"01:50.565 ","End":"01:52.230","Text":"well, let\u0027s take an example."},{"Start":"01:52.230 ","End":"01:59.250","Text":"Let\u0027s say we have a coin toss and we want to toss the coin 5 times,"},{"Start":"01:59.250 ","End":"02:02.915","Text":"so that means that n, in our case, equals 5."},{"Start":"02:02.915 ","End":"02:06.050","Text":"Now, we know that each toss is independent of each"},{"Start":"02:06.050 ","End":"02:11.525","Text":"other and we also know that the probability of,"},{"Start":"02:11.525 ","End":"02:18.735","Text":"let\u0027s say getting a head not a tail but a head is equal to 50 percent,"},{"Start":"02:18.735 ","End":"02:21.705","Text":"so p equals 0.5."},{"Start":"02:21.705 ","End":"02:25.520","Text":"In 1 Bernoulli trial,"},{"Start":"02:25.520 ","End":"02:28.445","Text":"the probability is 0.5."},{"Start":"02:28.445 ","End":"02:33.500","Text":"When we want to know what are the chances,"},{"Start":"02:33.500 ","End":"02:42.810","Text":"what is the number of heads in 5 tosses,"},{"Start":"02:42.810 ","End":"02:46.355","Text":"well, that\u0027s basically the definition of our success."},{"Start":"02:46.355 ","End":"02:50.060","Text":"X would be the number of heads in 5 tosses."},{"Start":"02:50.060 ","End":"02:52.835","Text":"When we have that,"},{"Start":"02:52.835 ","End":"03:00.480","Text":"we can define X as being distributed with a binomial distribution."},{"Start":"03:02.330 ","End":"03:05.945","Text":"As we said, the binomial distribution"},{"Start":"03:05.945 ","End":"03:11.210","Text":"calculates the probability of success when n independent Bernoulli trials are repeated."},{"Start":"03:11.210 ","End":"03:15.650","Text":"Let\u0027s define X just like in our coin toss example."},{"Start":"03:15.650 ","End":"03:17.810","Text":"Let\u0027s define X as the total number of"},{"Start":"03:17.810 ","End":"03:22.360","Text":"successes when repeating n independent Bernoulli trials,"},{"Start":"03:22.360 ","End":"03:28.325","Text":"and let p be the chance of success and q is the chance of failure in an individual trial."},{"Start":"03:28.325 ","End":"03:38.515","Text":"Now, let\u0027s define p as the probability of success in 1 trial."},{"Start":"03:38.515 ","End":"03:48.160","Text":"Since there are only 2 possible answers or 2 possible results in a Bernoulli trial,"},{"Start":"03:48.160 ","End":"03:57.480","Text":"then the probability of failure is equal to 1 minus p,"},{"Start":"03:57.480 ","End":"04:00.360","Text":"and we\u0027ll call that q."},{"Start":"04:00.360 ","End":"04:03.870","Text":"Now, so again,"},{"Start":"04:03.870 ","End":"04:10.280","Text":"let p be the chances of success and q the chances of failure,"},{"Start":"04:10.280 ","End":"04:13.410","Text":"so now we can define x as"},{"Start":"04:13.410 ","End":"04:20.760","Text":"the total number of successes that will be distributed,"},{"Start":"04:20.760 ","End":"04:22.945","Text":"that\u0027s the sign for distributed binomial,"},{"Start":"04:22.945 ","End":"04:26.200","Text":"B for binomial with 2 parameters n and p,"},{"Start":"04:26.200 ","End":"04:29.484","Text":"n being the number of independent trials,"},{"Start":"04:29.484 ","End":"04:36.460","Text":"and p being the probability of success in a Bernoulli trial."},{"Start":"04:36.460 ","End":"04:43.900","Text":"Once we\u0027ve defined X as being binomially distributed with n and p,"},{"Start":"04:43.900 ","End":"04:48.025","Text":"we can write out the probability function."},{"Start":"04:48.025 ","End":"04:52.555","Text":"Now, the probability function of X is this."},{"Start":"04:52.555 ","End":"04:57.580","Text":"The probability of X equaling k is equal to n over"},{"Start":"04:57.580 ","End":"05:04.870","Text":"k times p to the power of k times 1 minus p to the power of n minus k,"},{"Start":"05:04.870 ","End":"05:08.200","Text":"where k equals 0, 1, 2, 3,"},{"Start":"05:08.200 ","End":"05:12.530","Text":"and so on, so forth until n, that\u0027s n trials."},{"Start":"05:12.530 ","End":"05:18.880","Text":"Now, let\u0027s take a look at the components of the distribution function."},{"Start":"05:18.890 ","End":"05:22.450","Text":"Let\u0027s take a look at n over k. Now what does that mean?"},{"Start":"05:22.450 ","End":"05:23.740","Text":"What\u0027s that defined?"},{"Start":"05:23.740 ","End":"05:32.545","Text":"Well, n over k is defined as n factorial over k factorial times n minus k factorial."},{"Start":"05:32.545 ","End":"05:41.610","Text":"Now, n factorial just means that it\u0027s the multiplication of all the numbers from 1"},{"Start":"05:41.610 ","End":"05:46.580","Text":"till n. That means it\u0027s 1 times 2 times 3 times n minus 2 times n"},{"Start":"05:46.580 ","End":"05:52.280","Text":"minus 1 times n. A special definition is where n equals 0,"},{"Start":"05:52.280 ","End":"05:54.970","Text":"0 factorial equals 1."},{"Start":"05:54.970 ","End":"06:00.500","Text":"Now, you\u0027re not expected to calculate this thing by yourselves."},{"Start":"06:00.500 ","End":"06:03.530","Text":"You have a calculator and you can do"},{"Start":"06:03.530 ","End":"06:07.100","Text":"that once you know what the function in the calculator is,"},{"Start":"06:07.100 ","End":"06:13.320","Text":"you just plug in the numbers and it\u0027ll spew out the numbers, the results."},{"Start":"06:13.320 ","End":"06:22.110","Text":"Now, having identified the expression for the binomial distribution,"},{"Start":"06:22.110 ","End":"06:26.285","Text":"we can calculate or it\u0027s given to us that"},{"Start":"06:26.285 ","End":"06:32.750","Text":"the expression for the expectation of X is equals to n times p. Now,"},{"Start":"06:32.750 ","End":"06:35.015","Text":"n being the number of times repeat"},{"Start":"06:35.015 ","End":"06:40.115","Text":"a Bernoulli trial and p is the probability of success for each trial."},{"Start":"06:40.115 ","End":"06:49.065","Text":"Now, n times p would be the expectation of getting a success over all the trials."},{"Start":"06:49.065 ","End":"06:51.060","Text":"What\u0027s the variance? Well,"},{"Start":"06:51.060 ","End":"06:55.080","Text":"the variance of X is equal to n times p times q."},{"Start":"06:55.080 ","End":"07:01.580","Text":"Number of trials times the probability of success times the probability of failure."},{"Start":"07:01.580 ","End":"07:08.780","Text":"Let\u0027s sum up the idea of binomial distribution."},{"Start":"07:08.780 ","End":"07:10.985","Text":"Now, as we said previously,"},{"Start":"07:10.985 ","End":"07:14.975","Text":"that you have to identify these types of"},{"Start":"07:14.975 ","End":"07:19.640","Text":"distributions through the context of the question or through the data."},{"Start":"07:19.640 ","End":"07:24.745","Text":"That means we\u0027re looking for a set of criteria that will allow us to"},{"Start":"07:24.745 ","End":"07:32.840","Text":"identify this distribution as being the binomial distribution,"},{"Start":"07:32.840 ","End":"07:35.270","Text":"and this is it right here."},{"Start":"07:35.270 ","End":"07:42.350","Text":"In order for a binomial distribution to be relevant to our question,"},{"Start":"07:42.350 ","End":"07:45.095","Text":"the following conditions must be met."},{"Start":"07:45.095 ","End":"07:52.180","Text":"First of all, the same Bernoulli trial has to be repeated independently of each other."},{"Start":"07:52.180 ","End":"07:56.565","Text":"Secondly, the trials is repeated n times,"},{"Start":"07:56.565 ","End":"08:01.675","Text":"and thirdly, x is defined as the total number of successes obtained."},{"Start":"08:01.675 ","End":"08:05.120","Text":"Now, if these 3 conditions are met,"},{"Start":"08:05.120 ","End":"08:12.650","Text":"then we have a binomial distribution and we can use this function"},{"Start":"08:12.650 ","End":"08:20.900","Text":"right here and we can use these expressions to calculate the expectation and variance."},{"Start":"08:20.900 ","End":"08:25.230","Text":"Enough of theory, let\u0027s take a look at an example."}],"ID":12998},{"Watched":false,"Name":"Example 1 Part a","Duration":"7m 35s","ChapterTopicVideoID":12520,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"In our example, 80 percent of the people in a given country have a driving license."},{"Start":"00:05.535 ","End":"00:09.923","Text":"Now, 10 people are randomly chosen from the country,"},{"Start":"00:09.923 ","End":"00:15.720","Text":"and we\u0027re asked, what\u0027s the probability that exactly 9 of them have a driving license?"},{"Start":"00:15.720 ","End":"00:21.229","Text":"Secondly, what\u0027s the probability that at least 9 of them have a driving license?"},{"Start":"00:21.229 ","End":"00:24.000","Text":"Thirdly, what is the expectation and"},{"Start":"00:24.000 ","End":"00:28.260","Text":"standard deviation of the number of people sampled having a driving license?"},{"Start":"00:28.260 ","End":"00:31.175","Text":"Let\u0027s get started."},{"Start":"00:31.175 ","End":"00:33.890","Text":"The first thing that we want to do is,"},{"Start":"00:33.890 ","End":"00:38.680","Text":"to ask whether we have a binomial distribution or not."},{"Start":"00:38.680 ","End":"00:40.745","Text":"When we do that,"},{"Start":"00:40.745 ","End":"00:45.500","Text":"we have to see if the conditions for the binomial distributions are met."},{"Start":"00:45.500 ","End":"00:50.090","Text":"Let\u0027s take a look at the conditions. Here they are."},{"Start":"00:50.090 ","End":"00:58.085","Text":"The first condition talks about the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:58.085 ","End":"00:59.855","Text":"Well, first of all,"},{"Start":"00:59.855 ","End":"01:03.050","Text":"let\u0027s talk about independence."},{"Start":"01:03.050 ","End":"01:10.145","Text":"Now, when we sample a small number of people from a very large population,"},{"Start":"01:10.145 ","End":"01:15.890","Text":"we can assume that the people are independent of each other."},{"Start":"01:15.890 ","End":"01:21.185","Text":"In all cases where we have a small sample size,"},{"Start":"01:21.185 ","End":"01:23.180","Text":"from a large population,"},{"Start":"01:23.180 ","End":"01:26.495","Text":"we can assume independence. That\u0027s the rule."},{"Start":"01:26.495 ","End":"01:29.300","Text":"Now, the second part."},{"Start":"01:29.300 ","End":"01:32.075","Text":"Let\u0027s talk about Bernoulli trials."},{"Start":"01:32.075 ","End":"01:35.165","Text":"Do we have Bernoulli trials here?"},{"Start":"01:35.165 ","End":"01:36.980","Text":"Well, I believe that we do."},{"Start":"01:36.980 ","End":"01:41.665","Text":"Why is that? Well, first of all, let\u0027s define success."},{"Start":"01:41.665 ","End":"01:49.450","Text":"Let\u0027s define success as a person having a license,"},{"Start":"01:50.210 ","End":"01:56.610","Text":"and failure as a person not having a license, no license."},{"Start":"01:56.610 ","End":"02:03.550","Text":"Now, what\u0027s the probability of a person having a license?"},{"Start":"02:03.550 ","End":"02:05.275","Text":"That\u0027s 80 percent."},{"Start":"02:05.275 ","End":"02:06.565","Text":"We have that right here."},{"Start":"02:06.565 ","End":"02:08.875","Text":"They have a driving license,"},{"Start":"02:08.875 ","End":"02:17.140","Text":"so that means that the probability of success is 0.8."},{"Start":"02:17.140 ","End":"02:19.440","Text":"Now, what\u0027s the probability of failure?"},{"Start":"02:19.440 ","End":"02:21.420","Text":"Well, that\u0027s 1 minus p,"},{"Start":"02:21.420 ","End":"02:23.160","Text":"and we call that q,"},{"Start":"02:23.160 ","End":"02:26.375","Text":"that equals to 0.2."},{"Start":"02:26.375 ","End":"02:29.845","Text":"Now, what else do we have?"},{"Start":"02:29.845 ","End":"02:33.540","Text":"Well, how do we find success?"},{"Start":"02:33.540 ","End":"02:36.345","Text":"Success here, as we said,"},{"Start":"02:36.345 ","End":"02:46.805","Text":"is the number of people having a license."},{"Start":"02:46.805 ","End":"02:53.459","Text":"When we sample 10 people."},{"Start":"02:53.459 ","End":"02:57.700","Text":"This is number 3."},{"Start":"02:57.700 ","End":"02:59.495","Text":"Now, what about number 2?"},{"Start":"02:59.495 ","End":"03:01.220","Text":"The trial is repeated 10 times."},{"Start":"03:01.220 ","End":"03:03.560","Text":"Well, obviously we take 10 people,"},{"Start":"03:03.560 ","End":"03:05.480","Text":"we sample one after the other."},{"Start":"03:05.480 ","End":"03:10.189","Text":"Each sample basically is a Bernoulli trial here."},{"Start":"03:10.189 ","End":"03:15.620","Text":"Having the probability of success or having a license is"},{"Start":"03:15.620 ","End":"03:21.785","Text":"0.8 and the probability of failure of person not having a license is 0.2,"},{"Start":"03:21.785 ","End":"03:26.540","Text":"so the conditions for a binomial distribution is met."},{"Start":"03:26.540 ","End":"03:28.430","Text":"We can say,"},{"Start":"03:28.430 ","End":"03:33.710","Text":"now that X is distributed binomial,"},{"Start":"03:33.710 ","End":"03:42.000","Text":"where n equals 10 and p equals 0.8."},{"Start":"03:42.700 ","End":"03:49.940","Text":"Now that we know that X is distributed binomially with n equals 10 and p equals 0.8."},{"Start":"03:49.940 ","End":"03:53.990","Text":"We can write the expression for the binomial distribution."},{"Start":"03:53.990 ","End":"04:01.760","Text":"We say that the probability of X being equal to k equals n over"},{"Start":"04:01.760 ","End":"04:06.530","Text":"k times p to the power of"},{"Start":"04:06.530 ","End":"04:15.076","Text":"k times q to the power of n minus k. That\u0027s the probability distribution."},{"Start":"04:15.076 ","End":"04:18.267","Text":"Let\u0027s plug in the numbers for Section 8."},{"Start":"04:18.267 ","End":"04:22.480","Text":"We\u0027re talking about k being equal to 9"},{"Start":"04:22.480 ","End":"04:28.083","Text":"and we\u0027re asked what\u0027s the probability where x equals 9?"},{"Start":"04:28.083 ","End":"04:32.125","Text":"Again, let\u0027s plug in the numbers."},{"Start":"04:32.125 ","End":"04:34.210","Text":"n equals 10,"},{"Start":"04:34.210 ","End":"04:38.170","Text":"that\u0027s right here, 10 people. What\u0027s k?"},{"Start":"04:38.170 ","End":"04:39.760","Text":"k equals 9."},{"Start":"04:39.760 ","End":"04:44.295","Text":"It\u0027s n over 9,"},{"Start":"04:44.295 ","End":"04:53.115","Text":"or basically it\u0027s 10 over 9 times 0.8, p equals 0.8."},{"Start":"04:53.115 ","End":"05:00.375","Text":"0.8 to the power of 9 times 0.2,"},{"Start":"05:00.375 ","End":"05:02.395","Text":"that\u0027s q. That\u0027s comes here."},{"Start":"05:02.395 ","End":"05:04.480","Text":"n minus k. n minus k,"},{"Start":"05:04.480 ","End":"05:07.315","Text":"that\u0027s 10 minus 9. That\u0027s 1."},{"Start":"05:07.315 ","End":"05:11.830","Text":"Now, when we do the calculation,"},{"Start":"05:11.830 ","End":"05:16.570","Text":"that comes out to 0.268."},{"Start":"05:16.570 ","End":"05:19.569","Text":"Now, we can do this mechanically,"},{"Start":"05:19.569 ","End":"05:22.315","Text":"but let\u0027s see what\u0027s going on here."},{"Start":"05:22.315 ","End":"05:29.210","Text":"Let\u0027s try to understand this outside of the framework of the expression."},{"Start":"05:29.400 ","End":"05:34.045","Text":"Well, here I\u0027ve written a series of letters,"},{"Start":"05:34.045 ","End":"05:38.065","Text":"S being success and F being a failure."},{"Start":"05:38.065 ","End":"05:39.645","Text":"Now, what was asked?"},{"Start":"05:39.645 ","End":"05:44.010","Text":"What\u0027s the probability that exactly 9 of them have a driver\u0027s license?"},{"Start":"05:44.010 ","End":"05:46.725","Text":"Well, if 9 of them have a driver\u0027s license,"},{"Start":"05:46.725 ","End":"05:48.850","Text":"that means that 1 does not."},{"Start":"05:48.850 ","End":"05:50.715","Text":"We have 9 successes,"},{"Start":"05:50.715 ","End":"05:55.485","Text":"9 people having a driving license and the last 1 doesn\u0027t."},{"Start":"05:55.485 ","End":"06:00.470","Text":"What\u0027s the probability of a success as 0.8?"},{"Start":"06:00.470 ","End":"06:03.860","Text":"That\u0027s basically 0.8 times 0.8 times 0.8,"},{"Start":"06:03.860 ","End":"06:07.009","Text":"9 times, times a failure."},{"Start":"06:07.009 ","End":"06:10.640","Text":"Well, that\u0027s 0.2 to the power of 1."},{"Start":"06:10.640 ","End":"06:13.025","Text":"You only have 1 failure."},{"Start":"06:13.025 ","End":"06:17.975","Text":"But again, we\u0027re not asked what\u0027s the probability that"},{"Start":"06:17.975 ","End":"06:23.804","Text":"the last person doesn\u0027t have a driver\u0027s license."},{"Start":"06:23.804 ","End":"06:25.680","Text":"We\u0027re asked that one of them doesn\u0027t."},{"Start":"06:25.680 ","End":"06:28.550","Text":"Now really doesn\u0027t matter whether it\u0027s the last person"},{"Start":"06:28.550 ","End":"06:33.635","Text":"or any person within the series of the 10 people."},{"Start":"06:33.635 ","End":"06:36.290","Text":"Here we see an example of that."},{"Start":"06:36.290 ","End":"06:38.180","Text":"He doesn\u0027t have to be the last person here."},{"Start":"06:38.180 ","End":"06:39.650","Text":"You can be this person,"},{"Start":"06:39.650 ","End":"06:41.000","Text":"the third person here,"},{"Start":"06:41.000 ","End":"06:43.070","Text":"and so on and so forth."},{"Start":"06:43.070 ","End":"06:46.670","Text":"Now, how many options do we have here?"},{"Start":"06:46.670 ","End":"06:48.755","Text":"Well, we have 10 options."},{"Start":"06:48.755 ","End":"06:56.645","Text":"Now, this 10 is exactly this component right here."},{"Start":"06:56.645 ","End":"07:01.040","Text":"When we calculate what 10 over 9 is,"},{"Start":"07:01.040 ","End":"07:03.275","Text":"that comes out to 10."},{"Start":"07:03.275 ","End":"07:12.860","Text":"This component of the distribution talks about how many combinations"},{"Start":"07:12.860 ","End":"07:22.505","Text":"do we have of 9 people having a license and 1 person not having a license."},{"Start":"07:22.505 ","End":"07:27.440","Text":"I hope that made things a little bit clearer."},{"Start":"07:27.440 ","End":"07:30.170","Text":"If that\u0027s the case,"},{"Start":"07:30.170 ","End":"07:34.350","Text":"let\u0027s just go on to the next question."}],"ID":12999},{"Watched":false,"Name":"Example 1 Parts b-c","Duration":"4m 13s","ChapterTopicVideoID":12521,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.170","Text":"Now, here we have the results of what we did in Section A."},{"Start":"00:07.170 ","End":"00:13.230","Text":"We\u0027ve understood that x is distributed binomially with n equals 10 and p equals 0.8."},{"Start":"00:13.230 ","End":"00:18.930","Text":"We\u0027ve calculated that the probability of k being equal to 9,"},{"Start":"00:18.930 ","End":"00:21.555","Text":"well, that equals to 0.268."},{"Start":"00:21.555 ","End":"00:29.550","Text":"Again, that\u0027s the probability of having 9 people exactly who have a driver\u0027s license."},{"Start":"00:29.550 ","End":"00:31.725","Text":"Now, in Section B, we\u0027re asked,"},{"Start":"00:31.725 ","End":"00:36.990","Text":"what\u0027s the probability that at least 9 of them have a driver license."},{"Start":"00:36.990 ","End":"00:46.830","Text":"Now, that means we\u0027re looking at the probability of x being greater or equal to 9,"},{"Start":"00:46.830 ","End":"00:50.599","Text":"at least 9 of them have a driver\u0027s license."},{"Start":"00:50.599 ","End":"00:56.746","Text":"Well, that means that we want to know what\u0027s the probability of x"},{"Start":"00:56.746 ","End":"01:04.725","Text":"equals 9 plus the probability of x being equal to 10."},{"Start":"01:04.725 ","End":"01:11.615","Text":"Now, we already know what the probability of x equals 9 is,"},{"Start":"01:11.615 ","End":"01:14.560","Text":"we calculated that right here,"},{"Start":"01:14.560 ","End":"01:20.540","Text":"and that equals to 0.268."},{"Start":"01:20.540 ","End":"01:26.605","Text":"Now, all we have to do is calculate the probability of x being equal to 10."},{"Start":"01:26.605 ","End":"01:29.120","Text":"Now, let\u0027s plug in the numbers."},{"Start":"01:29.120 ","End":"01:32.205","Text":"The probability of x being equal to 10,"},{"Start":"01:32.205 ","End":"01:39.525","Text":"that\u0027s 10/10 times 0.8"},{"Start":"01:39.525 ","End":"01:46.730","Text":"to the power of 10 times 0.2 to the power of 0."},{"Start":"01:46.730 ","End":"01:53.825","Text":"Now, if we take out our calculator and do the calculations, with 10/10,"},{"Start":"01:53.825 ","End":"01:56.000","Text":"that equals to 1,"},{"Start":"01:56.000 ","End":"02:01.460","Text":"and 0.2 to the power of 0 that also equals to 1."},{"Start":"02:01.460 ","End":"02:07.760","Text":"So basically the probability of x being greater or equal to 9, well,"},{"Start":"02:07.760 ","End":"02:15.750","Text":"that equals to 0.268 plus 0.8 to the power of 10."},{"Start":"02:16.000 ","End":"02:21.060","Text":"Now, that equals to 0.376."},{"Start":"02:25.520 ","End":"02:29.360","Text":"In Section C, we\u0027re asked what are the expectations and"},{"Start":"02:29.360 ","End":"02:33.920","Text":"standard deviation of the number of people sampled having a driving license."},{"Start":"02:33.920 ","End":"02:38.480","Text":"Well, if we recall the expectation of x where x is distributed"},{"Start":"02:38.480 ","End":"02:42.875","Text":"binomially is equals to n times p. Now,"},{"Start":"02:42.875 ","End":"02:44.545","Text":"let\u0027s plug in the numbers,"},{"Start":"02:44.545 ","End":"02:48.270","Text":"and that\u0027s 10 times 0.8,"},{"Start":"02:48.270 ","End":"02:50.385","Text":"that equals to 8."},{"Start":"02:50.385 ","End":"02:54.485","Text":"It makes sense, and why is that?"},{"Start":"02:54.485 ","End":"02:59.829","Text":"Well, we\u0027re told that 80 percent of the people have a driving license."},{"Start":"02:59.829 ","End":"03:03.570","Text":"On the other hand, we\u0027ve sampled 10 people,"},{"Start":"03:03.570 ","End":"03:07.250","Text":"so out of the 10 people were expected to have"},{"Start":"03:07.250 ","End":"03:11.950","Text":"8 of the 10 people to have a driving license."},{"Start":"03:11.950 ","End":"03:15.110","Text":"Now let\u0027s look at the standard deviation."},{"Start":"03:15.110 ","End":"03:18.770","Text":"Well, again, we don\u0027t have the standard deviation off the bat,"},{"Start":"03:18.770 ","End":"03:21.695","Text":"but we do have the equation for the variance,"},{"Start":"03:21.695 ","End":"03:27.565","Text":"and the variance of x is n times p times q."},{"Start":"03:27.565 ","End":"03:30.800","Text":"Now that equals, let\u0027s plug in the numbers,"},{"Start":"03:30.800 ","End":"03:34.160","Text":"10 times 0.8 times q,"},{"Start":"03:34.160 ","End":"03:39.440","Text":"which is 0.2, and that turns out to be 1.6."},{"Start":"03:39.440 ","End":"03:42.520","Text":"Now again, we\u0027re asked for the standard deviation."},{"Start":"03:42.520 ","End":"03:50.000","Text":"The standard deviation of x is known to be equal to the square root of the variance 1.6,"},{"Start":"03:50.000 ","End":"03:54.900","Text":"and that turns out to be 1.26."},{"Start":"03:55.040 ","End":"03:57.930","Text":"We\u0027ve answered the question,"},{"Start":"03:57.930 ","End":"04:04.465","Text":"with the expectation of x is 8 people and the standard deviation is"},{"Start":"04:04.465 ","End":"04:12.660","Text":"1.26 and 1.26 is the average disbursement of the number of people around the expectation."}],"ID":13000},{"Watched":false,"Name":"Exercise 1 Parts a-c","Duration":"6m 58s","ChapterTopicVideoID":12522,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this question, we\u0027ll be talking about unemployment."},{"Start":"00:03.660 ","End":"00:08.460","Text":"10 percent of the population in a country are unemployed."},{"Start":"00:08.460 ","End":"00:12.675","Text":"5 people are randomly selected from that population."},{"Start":"00:12.675 ","End":"00:16.500","Text":"Now, X is defined as the number of unemployed people in our sample,"},{"Start":"00:16.500 ","End":"00:20.310","Text":"and we\u0027re asked, what\u0027s the probability distribution of X?"},{"Start":"00:20.310 ","End":"00:22.950","Text":"Well, my answer is that this is"},{"Start":"00:22.950 ","End":"00:28.685","Text":"a binomial distribution and not only because we\u0027re dealing with it here in this chapter,"},{"Start":"00:28.685 ","End":"00:35.645","Text":"but we have the definitions and we have other conditions."},{"Start":"00:35.645 ","End":"00:38.660","Text":"Now, let\u0027s just check it out."},{"Start":"00:38.660 ","End":"00:43.475","Text":"Let\u0027s look at the conditions for a binomial distribution that\u0027s here."},{"Start":"00:43.475 ","End":"00:46.850","Text":"Here we\u0027re talking about first condition being"},{"Start":"00:46.850 ","End":"00:50.120","Text":"the same Bernoulli trials of period independently."},{"Start":"00:50.120 ","End":"00:57.020","Text":"Again, when we have a small or very small sample size from a very large population"},{"Start":"00:57.020 ","End":"01:01.640","Text":"we\u0027re sampling people from"},{"Start":"01:01.640 ","End":"01:07.170","Text":"countries\u0027 population then we can assume that they\u0027re independent of each other."},{"Start":"01:07.170 ","End":"01:10.495","Text":"Now, do we have a Bernoulli trial?"},{"Start":"01:10.495 ","End":"01:12.505","Text":"Well, yes, we do."},{"Start":"01:12.505 ","End":"01:21.790","Text":"Let\u0027s define success as a person who\u0027s unemployed,"},{"Start":"01:23.810 ","End":"01:31.100","Text":"and failure as a person who\u0027s employed."},{"Start":"01:31.100 ","End":"01:36.280","Text":"Now, what\u0027s the probability of a person being unemployed?"},{"Start":"01:36.280 ","End":"01:38.170","Text":"Well, that\u0027s 10 percent."},{"Start":"01:38.170 ","End":"01:41.680","Text":"We\u0027re taking the relative frequency of"},{"Start":"01:41.680 ","End":"01:49.710","Text":"the unemployed people in the population so that means p equals 0.1."},{"Start":"01:49.710 ","End":"01:53.930","Text":"Now, that means that the probability of"},{"Start":"01:53.930 ","End":"01:58.770","Text":"a person being employed is 1 minus p and we\u0027ll call that q,"},{"Start":"01:58.770 ","End":"02:02.510","Text":"and that equals to 0.9."},{"Start":"02:02.510 ","End":"02:05.750","Text":"Now, the trial is repeated n times so obviously,"},{"Start":"02:05.750 ","End":"02:08.735","Text":"we\u0027re taking 5 people 1 at a time,"},{"Start":"02:08.735 ","End":"02:10.835","Text":"and we\u0027re sampling each person."},{"Start":"02:10.835 ","End":"02:18.740","Text":"Now each person has a probability of 0.01 to be unemployed and 0.9 to be employed."},{"Start":"02:18.740 ","End":"02:23.975","Text":"N in our case equals 5."},{"Start":"02:23.975 ","End":"02:29.297","Text":"Now, x is defined as the total number of successes obtained,"},{"Start":"02:29.297 ","End":"02:35.655","Text":"so x equals the number of success or"},{"Start":"02:35.655 ","End":"02:44.345","Text":"the number of unemployed, people."},{"Start":"02:44.345 ","End":"02:47.930","Text":"We\u0027ve met other conditions here now we know that"},{"Start":"02:47.930 ","End":"02:52.890","Text":"X is distributed binomially where n equals"},{"Start":"02:52.890 ","End":"03:02.615","Text":"5 and p equals 0.1 and here we\u0027ve answered question or section a."},{"Start":"03:02.615 ","End":"03:04.835","Text":"In this section, we\u0027re asked,"},{"Start":"03:04.835 ","End":"03:09.470","Text":"what\u0027s the probability that exactly 1 unemployed person was selected?"},{"Start":"03:09.470 ","End":"03:13.519","Text":"Now, we know that X is distributed"},{"Start":"03:13.519 ","End":"03:21.090","Text":"binomially with n equals to 5 and p equals to 0.1."},{"Start":"03:21.090 ","End":"03:26.570","Text":"Now, the probability of X being equal to k,"},{"Start":"03:26.570 ","End":"03:31.610","Text":"that equals to n over k,"},{"Start":"03:31.610 ","End":"03:33.485","Text":"p to the power of k,"},{"Start":"03:33.485 ","End":"03:40.310","Text":"q to the power of n minus k. In our situation let\u0027s just plug in the numbers."},{"Start":"03:40.310 ","End":"03:45.875","Text":"The probability where X equals to 1,"},{"Start":"03:45.875 ","End":"03:49.160","Text":"we\u0027re asked 1 unemployed person right here."},{"Start":"03:49.160 ","End":"03:53.690","Text":"That equals to n equals 5 over 1,"},{"Start":"03:53.690 ","End":"04:03.180","Text":"p is 0.1 to the power of k to the power of 1 times q."},{"Start":"04:03.180 ","End":"04:05.660","Text":"Now q is 0.9."},{"Start":"04:05.660 ","End":"04:11.840","Text":"That\u0027s 1 minus 0.1 that\u0027s 0.9 to the power of n minus k. Well,"},{"Start":"04:11.840 ","End":"04:14.990","Text":"n minus k is 5 minus 1, that\u0027s 4."},{"Start":"04:14.990 ","End":"04:18.590","Text":"Now, we don\u0027t have to use a lot of"},{"Start":"04:18.590 ","End":"04:22.190","Text":"brainpower all we need to do right now is to plug in all these numbers"},{"Start":"04:22.190 ","End":"04:28.490","Text":"into our calculator and figure out what this expression is and"},{"Start":"04:28.490 ","End":"04:36.180","Text":"that comes out to 0.32805."},{"Start":"04:38.050 ","End":"04:44.705","Text":"In this section, we\u0027re asked what\u0027s the probability that all of the selected people work?"},{"Start":"04:44.705 ","End":"04:46.460","Text":"Now, let\u0027s recall."},{"Start":"04:46.460 ","End":"04:54.335","Text":"X is distributed binomially with n equals 5 and p equals 0.1,"},{"Start":"04:54.335 ","End":"04:57.820","Text":"where the probability that X equals K"},{"Start":"04:57.820 ","End":"05:02.750","Text":"that\u0027s a general formula for the binomial distribution."},{"Start":"05:02.750 ","End":"05:07.420","Text":"That\u0027s n over k, p to the power of k,"},{"Start":"05:07.420 ","End":"05:11.320","Text":"q to the power of n minus k. Now,"},{"Start":"05:11.320 ","End":"05:17.035","Text":"in our case, we\u0027re looking for the probability that all the people work."},{"Start":"05:17.035 ","End":"05:22.270","Text":"Now, X is defined as the number of unemployed people."},{"Start":"05:22.270 ","End":"05:23.800","Text":"If that\u0027s the case,"},{"Start":"05:23.800 ","End":"05:27.895","Text":"then we\u0027re looking for k being equal to 0,"},{"Start":"05:27.895 ","End":"05:30.745","Text":"that none of the people in the sample are"},{"Start":"05:30.745 ","End":"05:35.335","Text":"unemployed or that all of the selected people work."},{"Start":"05:35.335 ","End":"05:41.875","Text":"We\u0027re basically looking for the probability of X being equal to 0."},{"Start":"05:41.875 ","End":"05:44.665","Text":"Now, let\u0027s plug in the numbers."},{"Start":"05:44.665 ","End":"05:50.000","Text":"That\u0027s 5 over 0 times 0.1 to"},{"Start":"05:50.000 ","End":"05:56.420","Text":"the power of 0 times 0.9 to the power of n minus k,"},{"Start":"05:56.420 ","End":"06:05.360","Text":"that\u0027s 5 minus 0, that\u0027s 5 and that turns out to be 0.59049."},{"Start":"06:05.360 ","End":"06:09.860","Text":"Now, let\u0027s just take a look at the components,"},{"Start":"06:09.860 ","End":"06:13.480","Text":"5 over 0, that turns out to be 1."},{"Start":"06:13.480 ","End":"06:17.765","Text":"Any number to the power of 0 that turns out to be 1"},{"Start":"06:17.765 ","End":"06:22.715","Text":"so we have 1 times 1 times 0.9 to the power of 5."},{"Start":"06:22.715 ","End":"06:28.340","Text":"Now intuitively, it makes sense because what\u0027s the probability of a person working?"},{"Start":"06:28.340 ","End":"06:33.125","Text":"Well, that\u0027s 0.9 and how many people do we have in the sample?"},{"Start":"06:33.125 ","End":"06:35.330","Text":"We have 5 people in the sample,"},{"Start":"06:35.330 ","End":"06:37.760","Text":"and they\u0027re independent of each other."},{"Start":"06:37.760 ","End":"06:41.465","Text":"If all the people are working,"},{"Start":"06:41.465 ","End":"06:49.175","Text":"then the probability of them doing that is 0.9 times 0.9 and so on and so forth, 5 times."},{"Start":"06:49.175 ","End":"06:57.000","Text":"That\u0027s 0.9 to the power of 5 and that equals 0.59049."}],"ID":13001},{"Watched":false,"Name":"Exercise 1 Parts d-f","Duration":"5m 13s","ChapterTopicVideoID":12523,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:02.400","Text":"In this section we\u0027re asked,"},{"Start":"00:02.400 ","End":"00:06.195","Text":"what\u0027s the probability that 3 of the people in the sample work?"},{"Start":"00:06.195 ","End":"00:09.360","Text":"Well, if 3 of the people in the sample work,"},{"Start":"00:09.360 ","End":"00:11.955","Text":"that means that 2 do not work."},{"Start":"00:11.955 ","End":"00:17.445","Text":"That means that in our case, k equals 2."},{"Start":"00:17.445 ","End":"00:28.305","Text":"Again, we have x is distributed binomially with n equals 5 and p equals 0.1."},{"Start":"00:28.305 ","End":"00:36.030","Text":"Where the probability where x equals k equals n over k,"},{"Start":"00:36.030 ","End":"00:38.040","Text":"p to the power of k,"},{"Start":"00:38.040 ","End":"00:44.615","Text":"q to the power of n minus k. We said that k equals 2."},{"Start":"00:44.615 ","End":"00:51.485","Text":"We want to find out what\u0027s the probability of 3 people working."},{"Start":"00:51.485 ","End":"00:57.860","Text":"Which means that we\u0027re looking at 2 people that are not working"},{"Start":"00:57.860 ","End":"01:04.285","Text":"because x again is defined as the number of unemployed people."},{"Start":"01:04.285 ","End":"01:11.630","Text":"We\u0027re looking at the probability of x equaling 2 is plug in the numbers."},{"Start":"01:11.630 ","End":"01:13.870","Text":"That\u0027s 5 over 2,"},{"Start":"01:13.870 ","End":"01:17.280","Text":"0.1 to the power of 2,"},{"Start":"01:17.280 ","End":"01:20.250","Text":"0.9 to the power of 3,"},{"Start":"01:20.250 ","End":"01:21.975","Text":"that\u0027s 5 minus 2."},{"Start":"01:21.975 ","End":"01:29.320","Text":"That comes out to 0.07 to 9."},{"Start":"01:29.920 ","End":"01:32.195","Text":"In this section, we\u0027re asked,"},{"Start":"01:32.195 ","End":"01:36.950","Text":"what\u0027s the probability that at least 1 person in the sample is unemployed?"},{"Start":"01:36.950 ","End":"01:42.890","Text":"Well, when we\u0027re dealing with the question of at least 1 person,"},{"Start":"01:42.890 ","End":"01:48.300","Text":"we\u0027re looking at x being greater or equal to 1."},{"Start":"01:49.130 ","End":"01:54.724","Text":"Now let\u0027s take a look at the values of X. X can take on the values of 0,"},{"Start":"01:54.724 ","End":"01:57.540","Text":"1, 2, 3,"},{"Start":"01:57.540 ","End":"01:59.580","Text":"4, and 5."},{"Start":"01:59.580 ","End":"02:02.660","Text":"Where x is the number of unemployed people."},{"Start":"02:02.660 ","End":"02:05.245","Text":"Now we\u0027re asked, what\u0027s the probability"},{"Start":"02:05.245 ","End":"02:08.630","Text":"that at least 1 person in the samples and employed."},{"Start":"02:08.630 ","End":"02:11.900","Text":"We\u0027re looking at these guys right here,"},{"Start":"02:11.900 ","End":"02:16.070","Text":"where x equals 1 or 2 or 3 or 4 or 5."},{"Start":"02:16.070 ","End":"02:19.595","Text":"Now, instead of calculating the probabilities for"},{"Start":"02:19.595 ","End":"02:23.135","Text":"each 1 of these values and then adding them up."},{"Start":"02:23.135 ","End":"02:26.855","Text":"It\u0027s much easier to take the complementary set."},{"Start":"02:26.855 ","End":"02:34.780","Text":"That would equal to 1 minus the probability of x being equal to 0."},{"Start":"02:34.970 ","End":"02:39.205","Text":"Now, if we remember,"},{"Start":"02:39.205 ","End":"02:45.225","Text":"we\u0027ve already calculated the probability of x being equal to 0."},{"Start":"02:45.225 ","End":"02:49.620","Text":"That equals 0.59049."},{"Start":"02:53.200 ","End":"02:59.075","Text":"1 minus 0.59049."},{"Start":"02:59.075 ","End":"03:02.430","Text":"Well, that equals to 0.40954."},{"Start":"03:06.950 ","End":"03:14.879","Text":"That\u0027s the probability of at least 1 person in the sample being unemployed."},{"Start":"03:15.370 ","End":"03:19.070","Text":"In this section, we\u0027re asked what are the expectation and"},{"Start":"03:19.070 ","End":"03:22.370","Text":"variance of the number of unemployed people in the sample?"},{"Start":"03:22.370 ","End":"03:27.500","Text":"Well, again, let\u0027s recall x is distributed binomially,"},{"Start":"03:27.500 ","End":"03:33.425","Text":"where n equals 5 and p equals 0.1."},{"Start":"03:33.425 ","End":"03:38.570","Text":"Now, remember that the expectation of x is"},{"Start":"03:38.570 ","End":"03:46.210","Text":"defined as n times p. Let\u0027s plug in the numbers n equals 5,"},{"Start":"03:46.210 ","End":"03:53.565","Text":"P equals 0.1, and that comes out to 0.5."},{"Start":"03:53.565 ","End":"03:56.235","Text":"In a sample of 5,"},{"Start":"03:56.235 ","End":"04:01.610","Text":"with probability of 0.1 being unemployed,"},{"Start":"04:01.610 ","End":"04:06.275","Text":"then we are expected that half a person will be unemployed in that sample."},{"Start":"04:06.275 ","End":"04:08.990","Text":"Let\u0027s take a look at the variance."},{"Start":"04:08.990 ","End":"04:13.940","Text":"The variance is defined as n times p times q."},{"Start":"04:13.940 ","End":"04:18.710","Text":"Again, that\u0027s the definition in a binomial distribution."},{"Start":"04:18.710 ","End":"04:20.390","Text":"Let\u0027s plug in the numbers."},{"Start":"04:20.390 ","End":"04:24.860","Text":"That\u0027s 5 times 0.1 times 0.9,"},{"Start":"04:24.860 ","End":"04:29.900","Text":"and that turns out to be 0.45."},{"Start":"04:29.900 ","End":"04:35.570","Text":"Now, there\u0027s an alternative way of calculating the expectation and variance."},{"Start":"04:35.570 ","End":"04:38.690","Text":"Let\u0027s remember how we did this previously."},{"Start":"04:38.690 ","End":"04:42.665","Text":"We had x, we had p of x,"},{"Start":"04:42.665 ","End":"04:48.650","Text":"and we calculated the probabilities for each value of x,"},{"Start":"04:48.650 ","End":"04:51.020","Text":"where x could have been 1, 0,"},{"Start":"04:51.020 ","End":"04:52.400","Text":"1, 2, 3,"},{"Start":"04:52.400 ","End":"04:54.365","Text":"4, and 5 in our case."},{"Start":"04:54.365 ","End":"04:57.985","Text":"Now, that\u0027s too much work."},{"Start":"04:57.985 ","End":"05:00.065","Text":"Then, why should we do this?"},{"Start":"05:00.065 ","End":"05:02.540","Text":"We already have the shortcuts right here."},{"Start":"05:02.540 ","End":"05:04.610","Text":"We have the proper equations for"},{"Start":"05:04.610 ","End":"05:09.305","Text":"the expectation and variance of the binomial distributions."},{"Start":"05:09.305 ","End":"05:13.440","Text":"There it is. Good luck."}],"ID":13002},{"Watched":false,"Name":"Exercise 2","Duration":"8m 27s","ChapterTopicVideoID":12524,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this question, we\u0027ll be talking about people who have smartphones."},{"Start":"00:03.960 ","End":"00:08.370","Text":"Now, according to the data provided by the Ministry of Communications,"},{"Start":"00:08.370 ","End":"00:11.655","Text":"70 percent of the population have a smartphone."},{"Start":"00:11.655 ","End":"00:14.715","Text":"Now, 10 people are randomly selected."},{"Start":"00:14.715 ","End":"00:18.990","Text":"We define X as the number of people sampled who have smartphones."},{"Start":"00:18.990 ","End":"00:23.070","Text":"We\u0027re asked, what\u0027s the probability distribution of X?"},{"Start":"00:23.070 ","End":"00:26.715","Text":"Well, X has a binomial distribution."},{"Start":"00:26.715 ","End":"00:28.050","Text":"Now how do I know that?"},{"Start":"00:28.050 ","End":"00:31.320","Text":"Because it meets the criteria for binomial distribution."},{"Start":"00:31.320 ","End":"00:35.460","Text":"But let\u0027s take a look at the criteria. Here it is."},{"Start":"00:35.460 ","End":"00:40.310","Text":"Now, the first criteria right here states"},{"Start":"00:40.310 ","End":"00:45.470","Text":"that we have the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:45.470 ","End":"00:49.820","Text":"Now, are we talking about a Bernoulli trial? Well, of course."},{"Start":"00:49.820 ","End":"00:51.155","Text":"Why is that?"},{"Start":"00:51.155 ","End":"00:54.380","Text":"Because we\u0027re talking about success and failures."},{"Start":"00:54.380 ","End":"01:03.375","Text":"The success is defined as having a smartphone. Here it is."},{"Start":"01:03.375 ","End":"01:06.620","Text":"What\u0027s the probability of having a smartphone?"},{"Start":"01:06.620 ","End":"01:10.130","Text":"That\u0027s 0.7. It\u0027s given right here."},{"Start":"01:10.130 ","End":"01:12.900","Text":"What\u0027s failure?"},{"Start":"01:13.600 ","End":"01:18.630","Text":"The failure is not having a smartphone."},{"Start":"01:19.900 ","End":"01:28.730","Text":"Now, that means that the probability of not having a smartphone is 0.3."},{"Start":"01:28.730 ","End":"01:32.825","Text":"Now, let\u0027s talk about independence."},{"Start":"01:32.825 ","End":"01:40.295","Text":"Again. If we\u0027re sampling a very small number from a very large population,"},{"Start":"01:40.295 ","End":"01:46.355","Text":"then the assumption is that all the objects in the sample are independent."},{"Start":"01:46.355 ","End":"01:51.395","Text":"In this case, we\u0027re sampling 10 people from a population."},{"Start":"01:51.395 ","End":"01:56.285","Text":"That means that the people are independent of each other."},{"Start":"01:56.285 ","End":"01:59.090","Text":"The first criteria is met."},{"Start":"01:59.090 ","End":"02:02.285","Text":"Now the trial is repeated n times."},{"Start":"02:02.285 ","End":"02:06.295","Text":"Here n equals how much? 10 people."},{"Start":"02:06.295 ","End":"02:12.310","Text":"Why is that? Because we\u0027re sampling 1 person after the other with the same probability."},{"Start":"02:12.310 ","End":"02:14.345","Text":"If we sample 1 person,"},{"Start":"02:14.345 ","End":"02:16.970","Text":"there\u0027s a probability of 70 percent and you\u0027ll"},{"Start":"02:16.970 ","End":"02:20.930","Text":"have a smartphone and 30 percent, that you won\u0027t."},{"Start":"02:20.930 ","End":"02:27.755","Text":"What\u0027s the third criteria that x is defined as the total number of successes obtained."},{"Start":"02:27.755 ","End":"02:29.840","Text":"What do we asked for?"},{"Start":"02:29.840 ","End":"02:34.310","Text":"We define X as the number of people sampled to have smartphones."},{"Start":"02:34.310 ","End":"02:45.280","Text":"Excellent. X is defined as number of people who have a smartphone."},{"Start":"02:49.130 ","End":"02:54.210","Text":"We have here all the conditions."},{"Start":"02:54.210 ","End":"03:06.350","Text":"That point to a binomial distribution where n equals 10 and p equals 0.7."},{"Start":"03:06.350 ","End":"03:10.325","Text":"There we go. In this section, we\u0027re asked,"},{"Start":"03:10.325 ","End":"03:14.885","Text":"what\u0027s the probability that 8 people in the sample have a smartphone?"},{"Start":"03:14.885 ","End":"03:17.900","Text":"Well, let\u0027s write this out again."},{"Start":"03:17.900 ","End":"03:27.080","Text":"X is distributed with a binomial distribution where n equals 10 and p equals 0.7."},{"Start":"03:27.080 ","End":"03:34.480","Text":"We\u0027re asked, what\u0027s the probability where x equals 8."},{"Start":"03:34.480 ","End":"03:40.010","Text":"Now, the probability x equaling k,"},{"Start":"03:40.010 ","End":"03:43.879","Text":"that\u0027s a general equation for the binomial distribution."},{"Start":"03:43.879 ","End":"03:46.655","Text":"That equals to n over k,"},{"Start":"03:46.655 ","End":"03:49.670","Text":"p to the power of k,"},{"Start":"03:49.670 ","End":"03:55.625","Text":"q to the power of n minus k. If we\u0027re asked,"},{"Start":"03:55.625 ","End":"03:59.300","Text":"what\u0027s the probability of x equaling 8?"},{"Start":"03:59.300 ","End":"04:01.880","Text":"Well, let\u0027s plug in the numbers."},{"Start":"04:01.880 ","End":"04:07.070","Text":"That equals to 10 over 8,"},{"Start":"04:07.070 ","End":"04:11.830","Text":"10 is n and 8 is k. Now,"},{"Start":"04:11.830 ","End":"04:16.220","Text":"p is 0.7 to the power of k to the power of 8"},{"Start":"04:16.220 ","End":"04:22.365","Text":"times 0.3 to the power of 10 minus 8, that\u0027s 2."},{"Start":"04:22.365 ","End":"04:27.450","Text":"That equals to 0.2335."},{"Start":"04:30.460 ","End":"04:34.325","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"04:34.325 ","End":"04:37.790","Text":"that at least 9 people in the sample have a smartphone?"},{"Start":"04:37.790 ","End":"04:40.550","Text":"Well, let\u0027s write out our data again."},{"Start":"04:40.550 ","End":"04:43.865","Text":"X is distributed binomially,"},{"Start":"04:43.865 ","End":"04:48.830","Text":"where n equals 10 and p equals 0.7."},{"Start":"04:48.830 ","End":"04:57.470","Text":"We know that the general equation of the binomial distribution is n over k,"},{"Start":"04:57.470 ","End":"05:05.645","Text":"p to the power of k times q to the power of n minus k. Now, we\u0027re asked,"},{"Start":"05:05.645 ","End":"05:10.430","Text":"what\u0027s the probability that at least 9 people in the sample,"},{"Start":"05:10.430 ","End":"05:14.420","Text":"that means that x is greater or equal to 9."},{"Start":"05:14.420 ","End":"05:19.595","Text":"Now, because we\u0027ve sampled only 10 people,"},{"Start":"05:19.595 ","End":"05:24.440","Text":"this probability equals to the probability of x being equal"},{"Start":"05:24.440 ","End":"05:30.540","Text":"to 9 plus the probability of x being equal to 10."},{"Start":"05:30.590 ","End":"05:33.290","Text":"Let\u0027s plug in the numbers."},{"Start":"05:33.290 ","End":"05:36.320","Text":"That equals to n over k,"},{"Start":"05:36.320 ","End":"05:38.850","Text":"that\u0027s 10 over 9."},{"Start":"05:38.850 ","End":"05:48.840","Text":"That\u0027s for this guy right here times 0.7 to the power of 9 times 0.3 to the power of 1."},{"Start":"05:48.840 ","End":"05:53.005","Text":"K is 9 and this 1 is 10 minus 9,"},{"Start":"05:53.005 ","End":"05:55.653","Text":"n minus k plus,"},{"Start":"05:55.653 ","End":"05:57.755","Text":"now let\u0027s figure this one out."},{"Start":"05:57.755 ","End":"06:00.930","Text":"That\u0027s 10 over 10,"},{"Start":"06:00.930 ","End":"06:10.555","Text":"0.7 to the power of 10 times 0.3 to the power of 0."},{"Start":"06:10.555 ","End":"06:13.360","Text":"Now, when you calculate the cell,"},{"Start":"06:13.360 ","End":"06:19.760","Text":"this comes out to 0.1493."},{"Start":"06:19.980 ","End":"06:24.040","Text":"In this section, we\u0027re asked what are the expectations and"},{"Start":"06:24.040 ","End":"06:28.149","Text":"standard deviation of the number of people sampled to have a smartphone."},{"Start":"06:28.149 ","End":"06:38.920","Text":"Well again, x is distributed binomially with n equaling 10 and p equaling 0.7."},{"Start":"06:38.920 ","End":"06:43.285","Text":"Now, we know what the expectation of x is,"},{"Start":"06:43.285 ","End":"06:48.260","Text":"that equals to n times p. Let\u0027s just plug in"},{"Start":"06:48.260 ","End":"06:54.470","Text":"the numbers n equals 10 and times p, which is 0.7."},{"Start":"06:54.470 ","End":"06:57.755","Text":"That means that we\u0027re talking about 7."},{"Start":"06:57.755 ","End":"07:05.340","Text":"Now. The expectation for people who have smartphones in our sample, are 7 people."},{"Start":"07:05.500 ","End":"07:09.515","Text":"Now, let\u0027s take a look at the variance."},{"Start":"07:09.515 ","End":"07:15.835","Text":"The variance of x is defined as n times p times q."},{"Start":"07:15.835 ","End":"07:17.760","Text":"Let\u0027s plug in the numbers,"},{"Start":"07:17.760 ","End":"07:20.100","Text":"n is 10 times p,"},{"Start":"07:20.100 ","End":"07:23.130","Text":"that\u0027s 0.7 times q,"},{"Start":"07:23.130 ","End":"07:29.045","Text":"that\u0027s 0.3 and that equals to 2.1."},{"Start":"07:29.045 ","End":"07:30.530","Text":"But what are the units?"},{"Start":"07:30.530 ","End":"07:35.690","Text":"The units are people squared."},{"Start":"07:35.690 ","End":"07:38.104","Text":"Now this isn\u0027t very intuitive."},{"Start":"07:38.104 ","End":"07:40.400","Text":"Let\u0027s take the standard deviation."},{"Start":"07:40.400 ","End":"07:41.840","Text":"That\u0027s what we\u0027re asked for."},{"Start":"07:41.840 ","End":"07:48.845","Text":"Now, the standard deviation of x is equal to the square root of the variance,"},{"Start":"07:48.845 ","End":"07:56.090","Text":"square root of 2.1 and that equals to 1.449."},{"Start":"07:56.090 ","End":"07:58.415","Text":"The units here are people."},{"Start":"07:58.415 ","End":"07:59.870","Text":"Much more intuitive."},{"Start":"07:59.870 ","End":"08:01.460","Text":"What does this mean?"},{"Start":"08:01.460 ","End":"08:07.885","Text":"It means that the expectation of people who have a smartphone in the sample are 7 people."},{"Start":"08:07.885 ","End":"08:09.965","Text":"What we have here,"},{"Start":"08:09.965 ","End":"08:13.460","Text":"the standard deviation is the average disbursement of people"},{"Start":"08:13.460 ","End":"08:18.050","Text":"around the expectation that\u0027s 1.449 people."},{"Start":"08:18.050 ","End":"08:20.120","Text":"Here we have it."},{"Start":"08:20.120 ","End":"08:28.170","Text":"We\u0027ve answered both the questions for the expectation and the variance of x."}],"ID":13003},{"Watched":false,"Name":"Exercise 3 Parts a-b","Duration":"5m 17s","ChapterTopicVideoID":12525,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:04.260","Text":"In this question, we\u0027ll be talking about slot machines."},{"Start":"00:04.260 ","End":"00:06.800","Text":"In a casino, there\u0027s 1 row,"},{"Start":"00:06.800 ","End":"00:08.970","Text":"6 identical slot machines."},{"Start":"00:08.970 ","End":"00:12.135","Text":"Playing a game on 1 of the machine costs $5."},{"Start":"00:12.135 ","End":"00:16.980","Text":"The probability of winning $20 on each machine is 10 percent,"},{"Start":"00:16.980 ","End":"00:19.875","Text":"and the probability of losing is 90 percent."},{"Start":"00:19.875 ","End":"00:22.890","Text":"Now, a gambler enters the casino and places"},{"Start":"00:22.890 ","End":"00:26.085","Text":"$5 in each of the 6 machines and we\u0027re asked,"},{"Start":"00:26.085 ","End":"00:31.305","Text":"what\u0027s the probability that the gambler will lose on all the machines?"},{"Start":"00:31.305 ","End":"00:35.830","Text":"Well, here we see that we\u0027re dealing with a binomial distribution."},{"Start":"00:35.830 ","End":"00:39.275","Text":"Now, in order to just prove that,"},{"Start":"00:39.275 ","End":"00:45.495","Text":"we have to make sure that it meets all the criteria that we talked about previously."},{"Start":"00:45.495 ","End":"00:47.610","Text":"Let\u0027s look at the criteria,"},{"Start":"00:47.610 ","End":"00:49.775","Text":"the conditions, and here they are."},{"Start":"00:49.775 ","End":"00:56.105","Text":"The first 1 states that we have Bernoulli trials and they\u0027re independent of each other."},{"Start":"00:56.105 ","End":"00:59.150","Text":"Well, when we\u0027re talking about a Bernoulli trial,"},{"Start":"00:59.150 ","End":"01:04.440","Text":"we\u0027re talking about success and failure."},{"Start":"01:04.630 ","End":"01:09.620","Text":"Now, success and failure of y in this game,"},{"Start":"01:09.620 ","End":"01:11.990","Text":"in our situation, we\u0027re talking about winning or"},{"Start":"01:11.990 ","End":"01:15.500","Text":"losing when playing the slot machine. That\u0027s a win."},{"Start":"01:15.500 ","End":"01:19.625","Text":"Success is a win and failure is a lose."},{"Start":"01:19.625 ","End":"01:21.980","Text":"Now, what\u0027s the probability of winning?"},{"Start":"01:21.980 ","End":"01:26.990","Text":"Well, probability is 0.1 to win,"},{"Start":"01:26.990 ","End":"01:31.399","Text":"and Q, which is the probability of using,"},{"Start":"01:31.399 ","End":"01:35.165","Text":"that\u0027s 1 minus p, that\u0027s 0.9."},{"Start":"01:35.165 ","End":"01:37.730","Text":"Now, are they independent?"},{"Start":"01:37.730 ","End":"01:41.210","Text":"Well, the slot machines do work independent of each"},{"Start":"01:41.210 ","End":"01:45.065","Text":"other and we can infer that they are independent."},{"Start":"01:45.065 ","End":"01:47.846","Text":"Now, the trial is repeated n times."},{"Start":"01:47.846 ","End":"01:53.220","Text":"Here, we\u0027re playing the 6 slot machines."},{"Start":"01:53.220 ","End":"01:57.690","Text":"That means that n equals 6 right here."},{"Start":"01:57.690 ","End":"02:02.480","Text":"The last criteria, X is defined as the total number of successes."},{"Start":"02:02.480 ","End":"02:09.405","Text":"I want to know how much I won when I play the 6th slot machines."},{"Start":"02:09.405 ","End":"02:15.540","Text":"That means that X is defined as the number of wins."},{"Start":"02:17.510 ","End":"02:21.980","Text":"Because I\u0027ve met all the criteria,"},{"Start":"02:21.980 ","End":"02:26.450","Text":"I know that x is distributed"},{"Start":"02:26.450 ","End":"02:33.595","Text":"binomially with n equals 6 and p equals what?"},{"Start":"02:33.595 ","End":"02:36.680","Text":"p equals 0.1."},{"Start":"02:38.660 ","End":"02:41.185","Text":"What are we asked in section a?"},{"Start":"02:41.185 ","End":"02:47.185","Text":"We\u0027re asked, what\u0027s the probability that the gambler will lose on all the machines?"},{"Start":"02:47.185 ","End":"02:52.355","Text":"We\u0027re defining x as the number of wins and we"},{"Start":"02:52.355 ","End":"02:57.310","Text":"want to know what\u0027s the probability that the gambler will lose an all machines."},{"Start":"02:57.310 ","End":"03:06.070","Text":"That means that we\u0027re looking at the probability where x equals 0. Why is that?"},{"Start":"03:06.070 ","End":"03:08.185","Text":"Because we don\u0027t want any wins."},{"Start":"03:08.185 ","End":"03:10.480","Text":"We want all the machines to lose."},{"Start":"03:10.480 ","End":"03:17.920","Text":"Now, do we remember what the equation is for the binomial probability?"},{"Start":"03:17.920 ","End":"03:21.665","Text":"Well, that equals to n over k,"},{"Start":"03:21.665 ","End":"03:24.320","Text":"p to the power of k,"},{"Start":"03:24.320 ","End":"03:27.500","Text":"q to the power of n minus k. Well,"},{"Start":"03:27.500 ","End":"03:31.050","Text":"in our case, k equals 0."},{"Start":"03:31.050 ","End":"03:34.070","Text":"The probability of X equaling 0,"},{"Start":"03:34.070 ","End":"03:35.660","Text":"that\u0027s n over k,"},{"Start":"03:35.660 ","End":"03:40.190","Text":"n is 6 and k is 0."},{"Start":"03:40.190 ","End":"03:45.200","Text":"The probability of win is 0.1_k,"},{"Start":"03:45.200 ","End":"03:51.260","Text":"that\u0027s 0 times 0.9 to the power of n minus k,"},{"Start":"03:51.260 ","End":"03:53.450","Text":"that\u0027s 6 minus 0,"},{"Start":"03:53.450 ","End":"03:59.400","Text":"that\u0027s 6, and that equals to 0.5314."},{"Start":"04:05.300 ","End":"04:07.690","Text":"In this section, we\u0027re asked,"},{"Start":"04:07.690 ","End":"04:13.025","Text":"what\u0027s the probability that the gambler will win on exactly 2 machines?"},{"Start":"04:13.025 ","End":"04:15.230","Text":"Well, let\u0027s recall,"},{"Start":"04:15.230 ","End":"04:18.920","Text":"X is distributed binomially with n equals"},{"Start":"04:18.920 ","End":"04:24.040","Text":"6 and the probability of winning on a machine is 0.1."},{"Start":"04:24.040 ","End":"04:27.590","Text":"We know that the probability of X equaling k,"},{"Start":"04:27.590 ","End":"04:31.218","Text":"that\u0027s a general equation for the binomial distribution,"},{"Start":"04:31.218 ","End":"04:34.970","Text":"that\u0027s n on k, p_k,"},{"Start":"04:34.970 ","End":"04:39.215","Text":"q_n minus k. Now, what are we asked?"},{"Start":"04:39.215 ","End":"04:46.296","Text":"We\u0027re asked, what\u0027s the probability that the gambler will win on exactly 2 machines?"},{"Start":"04:46.296 ","End":"04:47.945","Text":"That means that here,"},{"Start":"04:47.945 ","End":"04:50.693","Text":"k equals to 2."},{"Start":"04:50.693 ","End":"04:52.965","Text":"X has to equal to 2."},{"Start":"04:52.965 ","End":"04:54.350","Text":"Now, that equals,"},{"Start":"04:54.350 ","End":"04:55.760","Text":"let\u0027s plug in the numbers."},{"Start":"04:55.760 ","End":"05:01.290","Text":"n equals 6 onto 0.1."},{"Start":"05:01.290 ","End":"05:09.375","Text":"That\u0027s p_2 times 0.9_6 minus 2, that\u0027s 4."},{"Start":"05:09.375 ","End":"05:16.400","Text":"That equals to 0.0984."}],"ID":13004},{"Watched":false,"Name":"Exercise 3 Parts c-d","Duration":"12m 24s","ChapterTopicVideoID":12526,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:02.460","Text":"In this section we\u0027re asked,"},{"Start":"00:02.460 ","End":"00:06.675","Text":"what\u0027s the probability that the gambler will win more money than he gambles?"},{"Start":"00:06.675 ","End":"00:09.330","Text":"Well, before we can answer this question,"},{"Start":"00:09.330 ","End":"00:12.480","Text":"let\u0027s just recall some of the data that we were given."},{"Start":"00:12.480 ","End":"00:18.870","Text":"The number of machines equals 6."},{"Start":"00:18.870 ","End":"00:21.750","Text":"We have 6 machines that the gambler plays simultaneously."},{"Start":"00:21.750 ","End":"00:30.711","Text":"The cost of playing 1 machine,"},{"Start":"00:30.711 ","End":"00:35.040","Text":"that equals to $5."},{"Start":"00:35.040 ","End":"00:39.090","Text":"If you recall, he just puts in $5 in each of the machines,"},{"Start":"00:39.090 ","End":"00:41.460","Text":"and he plays the machines simultaneously."},{"Start":"00:41.460 ","End":"00:45.000","Text":"Now, what about his winnings?"},{"Start":"00:45.000 ","End":"00:55.935","Text":"The wins for 1 machine is $20,"},{"Start":"00:55.935 ","End":"01:00.980","Text":"and the probability of winning this $20 is 0.1 for 1 machine."},{"Start":"01:00.980 ","End":"01:04.085","Text":"Now,"},{"Start":"01:04.085 ","End":"01:11.510","Text":"if the gambler plays 6 machines simultaneously,"},{"Start":"01:11.510 ","End":"01:14.390","Text":"and he pays $5 per machine,"},{"Start":"01:14.390 ","End":"01:18.829","Text":"then the cost of playing"},{"Start":"01:18.829 ","End":"01:27.245","Text":"6 machines equals $30."},{"Start":"01:27.245 ","End":"01:30.020","Text":"Now, we\u0027re asked again,"},{"Start":"01:30.020 ","End":"01:34.550","Text":"what\u0027s the probability that the gambler will win more money than he gambles?"},{"Start":"01:34.550 ","End":"01:38.924","Text":"Now, if x is the number of wins,"},{"Start":"01:38.924 ","End":"01:47.200","Text":"how many wins does the gambler need to have in order to get his investment back?"},{"Start":"01:47.200 ","End":"01:51.115","Text":"Well, if he wins $20 on 1 machine,"},{"Start":"01:51.115 ","End":"01:54.100","Text":"then he needs to win at least than 2 machines"},{"Start":"01:54.100 ","End":"01:57.895","Text":"in order to get $40 where 40 is greater than 30."},{"Start":"01:57.895 ","End":"02:03.250","Text":"Basically, we\u0027re looking for the probability of x,"},{"Start":"02:03.250 ","End":"02:08.215","Text":"the number of wins being greater or equal to 2. Again, why?"},{"Start":"02:08.215 ","End":"02:10.960","Text":"Because if he wins 2 games or more,"},{"Start":"02:10.960 ","End":"02:16.080","Text":"then he\u0027ll win $40, 2 times $20."},{"Start":"02:16.080 ","End":"02:17.740","Text":"That\u0027s $40 or more,"},{"Start":"02:17.740 ","End":"02:21.265","Text":"and that\u0027s greater than his investment of $30."},{"Start":"02:21.265 ","End":"02:23.665","Text":"This is what we\u0027re looking for."},{"Start":"02:23.665 ","End":"02:25.880","Text":"Now, as we said,"},{"Start":"02:25.880 ","End":"02:29.495","Text":"x can have the values of 0,"},{"Start":"02:29.495 ","End":"02:32.900","Text":"1, 2, 3, 4,"},{"Start":"02:32.900 ","End":"02:35.790","Text":"5, and 6."},{"Start":"02:36.680 ","End":"02:40.650","Text":"We\u0027re looking at this set right here,"},{"Start":"02:40.650 ","End":"02:45.490","Text":"where x is either 2 or 3 or 4 or 5 or 6."},{"Start":"02:45.490 ","End":"02:48.830","Text":"Now, in order to calculate that,"},{"Start":"02:48.830 ","End":"02:50.945","Text":"that takes a lot of effort and energy,"},{"Start":"02:50.945 ","End":"02:53.480","Text":"I prefer taking the complementary set,"},{"Start":"02:53.480 ","End":"03:01.020","Text":"which is 1 minus the probability of x being less than or equal to 1."},{"Start":"03:03.250 ","End":"03:05.840","Text":"What does that equal to?"},{"Start":"03:05.840 ","End":"03:12.185","Text":"That\u0027s 1 minus the probability of x equaling 0,"},{"Start":"03:12.185 ","End":"03:16.920","Text":"plus the probability of x equaling 1."},{"Start":"03:17.710 ","End":"03:27.860","Text":"Now, we know that the probability of x equaling k equals n over k,"},{"Start":"03:27.860 ","End":"03:29.990","Text":"p to the power of k,"},{"Start":"03:29.990 ","End":"03:35.730","Text":"q to the power of n minus k. That\u0027s the binomial distribution."},{"Start":"03:36.290 ","End":"03:39.635","Text":"We need to remember that, and why is that?"},{"Start":"03:39.635 ","End":"03:42.410","Text":"Because when we plug in the numbers here,"},{"Start":"03:42.410 ","End":"03:44.525","Text":"we\u0027ll see how we can calculate things out."},{"Start":"03:44.525 ","End":"03:49.120","Text":"Now, what\u0027s the probability of x equaling 0?"},{"Start":"03:49.120 ","End":"03:53.015","Text":"Well, if you recall, we\u0027ve calculated that in Section A,"},{"Start":"03:53.015 ","End":"03:58.490","Text":"that equals to 0.5314."},{"Start":"03:58.490 ","End":"04:01.085","Text":"Let\u0027s plug in this number."},{"Start":"04:01.085 ","End":"04:02.300","Text":"That equals right here,"},{"Start":"04:02.300 ","End":"04:07.820","Text":"1 minus the probability of x equaling 0,"},{"Start":"04:07.820 ","End":"04:14.780","Text":"that\u0027s 0.5314, plus the probability of x equaling 1."},{"Start":"04:14.780 ","End":"04:18.685","Text":"Well, let\u0027s plug in the numbers right here."},{"Start":"04:18.685 ","End":"04:20.955","Text":"N is 6,"},{"Start":"04:20.955 ","End":"04:25.460","Text":"k is 1, that\u0027s 6 over 1 then, 0.1,"},{"Start":"04:25.460 ","End":"04:30.030","Text":"that\u0027s the probability of a win to the power of 1 times 0.9,"},{"Start":"04:30.030 ","End":"04:32.205","Text":"that\u0027s our q,"},{"Start":"04:32.205 ","End":"04:34.470","Text":"to the power of n minus k,"},{"Start":"04:34.470 ","End":"04:37.600","Text":"that 6 minus 1, that\u0027s 5."},{"Start":"04:38.240 ","End":"04:40.970","Text":"Having done the calculation,"},{"Start":"04:40.970 ","End":"04:44.160","Text":"this comes out to 0.1143."},{"Start":"04:46.300 ","End":"04:49.910","Text":"This is the probability of"},{"Start":"04:49.910 ","End":"04:55.740","Text":"the gambler winning more money than he invests in playing the 6 games."},{"Start":"04:56.500 ","End":"04:58.805","Text":"In this section we\u0027re asked,"},{"Start":"04:58.805 ","End":"05:02.810","Text":"what are the expectation and standard deviation of the gambler\u0027s net profits,"},{"Start":"05:02.810 ","End":"05:05.490","Text":"the prices minus the investment?"},{"Start":"05:05.870 ","End":"05:09.600","Text":"First, let\u0027s go over some data."},{"Start":"05:09.600 ","End":"05:11.640","Text":"Well, we have 6 machines,"},{"Start":"05:11.640 ","End":"05:14.630","Text":"that\u0027s n. We have x,"},{"Start":"05:14.630 ","End":"05:17.509","Text":"which is the number of wins."},{"Start":"05:17.509 ","End":"05:23.390","Text":"The probability of winning is 0.1,"},{"Start":"05:23.390 ","End":"05:34.090","Text":"and we know that x is binomially distributed with n equals 6 and p equals 0.1."},{"Start":"05:34.090 ","End":"05:42.364","Text":"Now, we know that the cost of playing"},{"Start":"05:42.364 ","End":"05:52.140","Text":"1 game equals $5 per game or per machine."},{"Start":"05:52.270 ","End":"05:58.740","Text":"The cost of playing 6 games,"},{"Start":"06:02.030 ","End":"06:05.085","Text":"that equals to $30."},{"Start":"06:05.085 ","End":"06:07.640","Text":"Why is that? So $5 per game,"},{"Start":"06:07.640 ","End":"06:12.325","Text":"per machine, times 6 machines that\u0027s 6 times 5, that\u0027s $30."},{"Start":"06:12.325 ","End":"06:14.865","Text":"Now, how much do we win?"},{"Start":"06:14.865 ","End":"06:21.190","Text":"What we win is $20 per game."},{"Start":"06:22.820 ","End":"06:25.820","Text":"Now, what else do we know?"},{"Start":"06:25.820 ","End":"06:32.415","Text":"We know that the expectation of x is n times p,"},{"Start":"06:32.415 ","End":"06:33.840","Text":"that equals 2,"},{"Start":"06:33.840 ","End":"06:36.915","Text":"because x is binomially distributed."},{"Start":"06:36.915 ","End":"06:43.895","Text":"That\u0027s 6 times 0.1 and that equals to 0.6 and the variance of"},{"Start":"06:43.895 ","End":"06:52.550","Text":"x equals n times p times q and that equals to 6 times 0.1 times 0.9,"},{"Start":"06:52.550 ","End":"06:56.220","Text":"and that equals to 0.54."},{"Start":"06:56.260 ","End":"07:02.780","Text":"This is basically all the information that we have to date."},{"Start":"07:02.780 ","End":"07:04.520","Text":"Now, what are we asked?"},{"Start":"07:04.520 ","End":"07:08.270","Text":"We\u0027re asked about the gambler\u0027s net profit,"},{"Start":"07:08.270 ","End":"07:10.190","Text":"not about his wins."},{"Start":"07:10.190 ","End":"07:16.272","Text":"That means that I have some transformation,"},{"Start":"07:16.272 ","End":"07:22.255","Text":"that I have to take the number of wins and I have to transform them into profits,"},{"Start":"07:22.255 ","End":"07:26.580","Text":"and that reminds me of a linear transformation."},{"Start":"07:26.580 ","End":"07:30.320","Text":"Let\u0027s just take a look at the steps required for"},{"Start":"07:30.320 ","End":"07:35.080","Text":"a linear transformation. Here are the steps."},{"Start":"07:35.080 ","End":"07:37.730","Text":"Well, as we\u0027ve said, the first step is to"},{"Start":"07:37.730 ","End":"07:41.315","Text":"recognize that we\u0027re dealing with a linear transformation, and I think we are."},{"Start":"07:41.315 ","End":"07:47.675","Text":"We have to take our winnings and we have to transform them into net profits."},{"Start":"07:47.675 ","End":"07:50.480","Text":"Now, what are the transformation?"},{"Start":"07:50.480 ","End":"07:53.290","Text":"What is the transformation rule?"},{"Start":"07:53.290 ","End":"07:56.465","Text":"If y is the transformation,"},{"Start":"07:56.465 ","End":"08:04.365","Text":"we have to take the number of wins and we have to transform that into money."},{"Start":"08:04.365 ","End":"08:09.060","Text":"How much did I win according to the number of wins?"},{"Start":"08:09.060 ","End":"08:12.960","Text":"Well, I know that I win $20 per game,"},{"Start":"08:12.960 ","End":"08:16.200","Text":"so it has to be 20 times x,"},{"Start":"08:16.200 ","End":"08:17.880","Text":"where x is the number of wins."},{"Start":"08:17.880 ","End":"08:21.559","Text":"But because I\u0027m dealing with profits,"},{"Start":"08:21.559 ","End":"08:25.285","Text":"I have to take away from my winnings my expenses."},{"Start":"08:25.285 ","End":"08:26.930","Text":"Now, what are my expenses?"},{"Start":"08:26.930 ","End":"08:31.620","Text":"Well, that\u0027s $30. Why is that?"},{"Start":"08:31.620 ","End":"08:37.725","Text":"Because I\u0027m playing 6 machines at $5 a game."},{"Start":"08:37.725 ","End":"08:42.570","Text":"This looks like a good linear transformation,"},{"Start":"08:42.570 ","End":"08:49.970","Text":"but we have to compare that to our general form of a linear transformation."},{"Start":"08:49.970 ","End":"08:57.720","Text":"That looks like, y equals ax plus b."},{"Start":"08:58.190 ","End":"09:01.530","Text":"Is this in the form of this?"},{"Start":"09:01.530 ","End":"09:03.120","Text":"Well, yes, obviously."},{"Start":"09:03.120 ","End":"09:07.070","Text":"We can easily identify that a equals 20,"},{"Start":"09:07.070 ","End":"09:09.470","Text":"a is the multiplier of x,"},{"Start":"09:09.470 ","End":"09:14.135","Text":"so a equals 20, and b equals minus 30."},{"Start":"09:14.135 ","End":"09:21.936","Text":"Excellent. We\u0027ve recognized that we have a linear transformation,"},{"Start":"09:21.936 ","End":"09:25.880","Text":"we wrote the rules, we simplified everything,"},{"Start":"09:25.880 ","End":"09:29.783","Text":"although there wasn\u0027t that much to simplify,"},{"Start":"09:29.783 ","End":"09:33.420","Text":"but we did identify a and b."},{"Start":"09:33.420 ","End":"09:36.560","Text":"Now, let\u0027s get back to our question."},{"Start":"09:36.560 ","End":"09:42.500","Text":"We\u0027re asked, what is the expectation of y?"},{"Start":"09:42.500 ","End":"09:47.920","Text":"Well, if we recall, the expectation of y, looks like this."},{"Start":"09:47.920 ","End":"09:53.240","Text":"That\u0027s a times the expectation of x plus b."},{"Start":"09:53.240 ","End":"09:57.275","Text":"That\u0027s the expectation of the linear transformation."},{"Start":"09:57.275 ","End":"10:00.680","Text":"Now, that equals, let\u0027s plug in the numbers,"},{"Start":"10:00.680 ","End":"10:04.605","Text":"a equals 20, what\u0027s the expectation of x?"},{"Start":"10:04.605 ","End":"10:08.070","Text":"Well, that\u0027s 0.6, that\u0027s right here."},{"Start":"10:08.070 ","End":"10:11.415","Text":"What\u0027s b? B is minus 30."},{"Start":"10:11.415 ","End":"10:15.080","Text":"It\u0027s 20 times 0.6 minus 30,"},{"Start":"10:15.080 ","End":"10:19.100","Text":"and that equals to minus 18."},{"Start":"10:19.100 ","End":"10:22.410","Text":"Now, what\u0027s this minus 18?"},{"Start":"10:22.410 ","End":"10:24.445","Text":"That\u0027s minus $18,"},{"Start":"10:24.445 ","End":"10:33.880","Text":"which basically means that if I have 6 machines and I have to pay $5 a game,"},{"Start":"10:33.880 ","End":"10:35.970","Text":"or $5 per machine,"},{"Start":"10:35.970 ","End":"10:39.165","Text":"and I\u0027m winning $20 per game,"},{"Start":"10:39.165 ","End":"10:41.855","Text":"but at a probability of 0.1,"},{"Start":"10:41.855 ","End":"10:46.970","Text":"then I\u0027m expected to lose minus $18,"},{"Start":"10:46.970 ","End":"10:51.140","Text":"so I\u0027m expecting to lose $18."},{"Start":"10:51.140 ","End":"10:54.910","Text":"Now, let\u0027s take a look at the variance of y."},{"Start":"10:54.910 ","End":"10:57.320","Text":"Well, the variance of y, or the variance of"},{"Start":"10:57.320 ","End":"11:03.030","Text":"a linear transformation is a squared times the variance of x."},{"Start":"11:03.140 ","End":"11:07.370","Text":"Now, that equals, what\u0027s a?"},{"Start":"11:07.370 ","End":"11:13.640","Text":"A is 20, that\u0027s 20 squared times the variance of x, times 0.54."},{"Start":"11:13.640 ","End":"11:20.300","Text":"That\u0027s right here. That equals to 215."},{"Start":"11:20.300 ","End":"11:22.085","Text":"Now, what are the units here?"},{"Start":"11:22.085 ","End":"11:24.350","Text":"The units here are in dollars squared."},{"Start":"11:24.350 ","End":"11:27.805","Text":"That\u0027s not intuitive at all."},{"Start":"11:27.805 ","End":"11:33.065","Text":"Also, we\u0027re asked for the standard deviation, not the variance."},{"Start":"11:33.065 ","End":"11:42.900","Text":"The standard deviation of y is the square root of the variance,"},{"Start":"11:42.900 ","End":"11:50.190","Text":"215, and that equals to 14.697, and that\u0027s okay."},{"Start":"11:50.190 ","End":"11:53.760","Text":"Right here, we\u0027re talking about this being the units,"},{"Start":"11:53.760 ","End":"11:57.125","Text":"dollars instead of dollar square right here."},{"Start":"11:57.125 ","End":"11:59.475","Text":"This is dollars."},{"Start":"11:59.475 ","End":"12:02.075","Text":"Again, we\u0027ve answered the question."},{"Start":"12:02.075 ","End":"12:05.090","Text":"We\u0027ve calculated the expectation of y,"},{"Start":"12:05.090 ","End":"12:09.800","Text":"the linear transformation or the profits of the gambler."},{"Start":"12:09.800 ","End":"12:14.375","Text":"That\u0027s minus $18, he\u0027s expected to lose $18."},{"Start":"12:14.375 ","End":"12:19.879","Text":"We\u0027ve calculated the standard deviation of the linear transformation or the profits,"},{"Start":"12:19.879 ","End":"12:24.540","Text":"and that equals to $14.697."}],"ID":13005},{"Watched":false,"Name":"Exercise 4","Duration":"7m 12s","ChapterTopicVideoID":12527,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this question, we\u0027ll be talking about levels of education."},{"Start":"00:03.750 ","End":"00:07.260","Text":"Now, in a given country, the distribution of the education of"},{"Start":"00:07.260 ","End":"00:10.620","Text":"people over the age of 30 is as follows;"},{"Start":"00:10.620 ","End":"00:14.400","Text":"10 percent of the population have a low level of education,"},{"Start":"00:14.400 ","End":"00:17.175","Text":"60 percent have a high-school education,"},{"Start":"00:17.175 ","End":"00:21.345","Text":"20 percent are first-degree graduates from university,"},{"Start":"00:21.345 ","End":"00:25.285","Text":"and 10 percent are second degree or higher graduates."},{"Start":"00:25.285 ","End":"00:30.895","Text":"Now, 20 people over the age of 30 are randomly selected and we\u0027re asked,"},{"Start":"00:30.895 ","End":"00:37.637","Text":"what\u0027s the probability that 5 of them are university graduates.?"},{"Start":"00:37.637 ","End":"00:40.880","Text":"In order to answer this question,"},{"Start":"00:40.880 ","End":"00:44.305","Text":"well, let\u0027s first do a little bit of definitions."},{"Start":"00:44.305 ","End":"00:48.050","Text":"Let\u0027s see about our Bernoulli trials."},{"Start":"00:48.050 ","End":"00:49.880","Text":"When we talk about Bernoulli trials,"},{"Start":"00:49.880 ","End":"00:52.730","Text":"we are talking about success and failure,"},{"Start":"00:52.730 ","End":"00:55.501","Text":"so in our case, what\u0027s success?"},{"Start":"00:55.501 ","End":"00:57.735","Text":"Well, success is being"},{"Start":"00:57.735 ","End":"01:07.810","Text":"a graduate and failure is not being a graduate."},{"Start":"01:08.540 ","End":"01:14.170","Text":"All right? Graduate. Now, what do we know?"},{"Start":"01:14.170 ","End":"01:18.737","Text":"The probability of being a graduate,"},{"Start":"01:18.737 ","End":"01:22.050","Text":"that\u0027s 0.2 plus 0.1,"},{"Start":"01:22.050 ","End":"01:23.700","Text":"that equals to 0.3."},{"Start":"01:23.700 ","End":"01:28.320","Text":"Why is that? Well, 0.2 are people with first degrees and"},{"Start":"01:28.320 ","End":"01:34.470","Text":"0.1 are people with secondary degrees, that\u0027s the probabilities."},{"Start":"01:34.470 ","End":"01:37.360","Text":"Now, the probability of failure,"},{"Start":"01:37.360 ","End":"01:40.885","Text":"that\u0027s q equals 1 minus p,"},{"Start":"01:40.885 ","End":"01:42.700","Text":"that equals 1 minus 0,"},{"Start":"01:42.700 ","End":"01:47.510","Text":"3, that equals to 0.7."},{"Start":"01:47.570 ","End":"01:55.910","Text":"We have here our success and failure where we\u0027ve defined success,"},{"Start":"01:55.910 ","End":"01:57.200","Text":"we\u0027ve defined failure,"},{"Start":"01:57.200 ","End":"02:01.165","Text":"we\u0027ve defined the probability of success and failure."},{"Start":"02:01.165 ","End":"02:03.585","Text":"What do we want to know?"},{"Start":"02:03.585 ","End":"02:07.250","Text":"Well, the next thing that we want to know is how many people we have."},{"Start":"02:07.250 ","End":"02:12.410","Text":"Well, we have 20 in our sample where each 1 of the people that are"},{"Start":"02:12.410 ","End":"02:18.930","Text":"sampled is considered a Bernoulli trial with these probabilities."},{"Start":"02:18.970 ","End":"02:25.510","Text":"Probability of 0.3 of having a degree of some sorts and 0.7 of none."},{"Start":"02:25.510 ","End":"02:29.195","Text":"Now, each 1 of the people are independent of each other."},{"Start":"02:29.195 ","End":"02:30.800","Text":"Again, why is that?"},{"Start":"02:30.800 ","End":"02:36.997","Text":"Because we have a small number of people from a whole population of the country,"},{"Start":"02:36.997 ","End":"02:40.940","Text":"so again, that\u0027s the assumption is that they\u0027re independent."},{"Start":"02:40.940 ","End":"02:44.224","Text":"The last thing to define is x,"},{"Start":"02:44.224 ","End":"02:49.650","Text":"where x could be the number of graduates,"},{"Start":"02:50.600 ","End":"02:54.323","Text":"the number of graduates in our sample,"},{"Start":"02:54.323 ","End":"02:56.435","Text":"and what are we looking for?"},{"Start":"02:56.435 ","End":"03:04.545","Text":"We\u0027re looking for the probability of X being equal to, here it is."},{"Start":"03:04.545 ","End":"03:07.130","Text":"That\u0027s 5 of them are university graduates."},{"Start":"03:07.130 ","End":"03:09.605","Text":"That means that x equals 5."},{"Start":"03:09.605 ","End":"03:18.035","Text":"Now, we know that X because it met all the criteria here,"},{"Start":"03:18.035 ","End":"03:23.255","Text":"we know that X is distributed binomially."},{"Start":"03:23.255 ","End":"03:26.130","Text":"So let\u0027s just write this out."},{"Start":"03:26.290 ","End":"03:32.750","Text":"X is distributed binomially with N equals"},{"Start":"03:32.750 ","End":"03:37.160","Text":"20 and p that equals to 0.3"},{"Start":"03:37.160 ","End":"03:43.535","Text":"and we know that the probability where x equals K,"},{"Start":"03:43.535 ","End":"03:46.550","Text":"we know that equals n over k,"},{"Start":"03:46.550 ","End":"03:48.920","Text":"p to the power of k,"},{"Start":"03:48.920 ","End":"03:52.460","Text":"q to the power of n minus k. In our case,"},{"Start":"03:52.460 ","End":"03:54.320","Text":"let\u0027s just plug in the numbers."},{"Start":"03:54.320 ","End":"03:59.135","Text":"The probability of X being equal to 5,"},{"Start":"03:59.135 ","End":"04:02.780","Text":"that equals to n is 20,"},{"Start":"04:02.780 ","End":"04:04.535","Text":"K is 5, right?"},{"Start":"04:04.535 ","End":"04:07.470","Text":"20 here, 20 here,"},{"Start":"04:07.470 ","End":"04:08.940","Text":"and 5 of them, right?"},{"Start":"04:08.940 ","End":"04:11.395","Text":"So we\u0027re looking at 5 successes."},{"Start":"04:11.395 ","End":"04:13.670","Text":"So that\u0027s N over K,"},{"Start":"04:13.670 ","End":"04:16.490","Text":"20 over 5 times p. What is p?"},{"Start":"04:16.490 ","End":"04:23.390","Text":"0.3 to the power of k to the power of 5 times 0.7, that\u0027s our q."},{"Start":"04:23.390 ","End":"04:25.040","Text":"To the power of n minus k,"},{"Start":"04:25.040 ","End":"04:27.551","Text":"that\u0027s 20 minus 5, that\u0027s 15."},{"Start":"04:27.551 ","End":"04:33.960","Text":"That comes out to 0.1789."},{"Start":"04:34.900 ","End":"04:38.270","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"04:38.270 ","End":"04:42.485","Text":"the expectation of the number of poorly educated people in the sample?"},{"Start":"04:42.485 ","End":"04:45.080","Text":"Well, as we said in Section A,"},{"Start":"04:45.080 ","End":"04:50.630","Text":"each person in the sample is like a Bernoulli trial."},{"Start":"04:50.630 ","End":"04:58.350","Text":"So we have to define our success and our failure."},{"Start":"04:59.210 ","End":"05:01.925","Text":"Now, what\u0027s our success here?"},{"Start":"05:01.925 ","End":"05:08.660","Text":"That success is a person who is poorly educated, let\u0027s define that."},{"Start":"05:08.660 ","End":"05:13.130","Text":"Educated; as opposed to a person"},{"Start":"05:13.130 ","End":"05:18.005","Text":"having a higher education or university graduate in Section 8,"},{"Start":"05:18.005 ","End":"05:24.350","Text":"right now we define our success as a person being poorly educated and failure"},{"Start":"05:24.350 ","End":"05:32.430","Text":"means that he has a high school education or more."},{"Start":"05:32.600 ","End":"05:39.020","Text":"Now, what else do we know? We know that n equals 20 because that\u0027s the sample size,"},{"Start":"05:39.020 ","End":"05:41.285","Text":"that\u0027s the number of people that we\u0027ve sampled."},{"Start":"05:41.285 ","End":"05:46.160","Text":"Now, what\u0027s the probability of success of being poorly educated,"},{"Start":"05:46.160 ","End":"05:48.860","Text":"that equals 10 percent, right?"},{"Start":"05:48.860 ","End":"05:51.640","Text":"This is what we got from the data."},{"Start":"05:51.640 ","End":"05:54.000","Text":"Now, what else do we know?"},{"Start":"05:54.000 ","End":"05:58.955","Text":"Let\u0027s define our w as a random variable that"},{"Start":"05:58.955 ","End":"06:05.190","Text":"counts the number of people that have a poor education."},{"Start":"06:10.730 ","End":"06:13.875","Text":"Education. There we go."},{"Start":"06:13.875 ","End":"06:16.935","Text":"So what do we want to know?"},{"Start":"06:16.935 ","End":"06:22.325","Text":"First of all, we know that w is distributed binomially"},{"Start":"06:22.325 ","End":"06:28.600","Text":"with n equaling 20 and p equaling 0.1."},{"Start":"06:28.600 ","End":"06:31.010","Text":"That\u0027s what\u0027s given to us right here."},{"Start":"06:31.010 ","End":"06:32.420","Text":"P equals 1, right?"},{"Start":"06:32.420 ","End":"06:39.005","Text":"That\u0027s the probability of success where success is a person that is poorly educated."},{"Start":"06:39.005 ","End":"06:43.669","Text":"So that means that the expectation of w,"},{"Start":"06:43.669 ","End":"06:47.870","Text":"because w is distributed binomially,"},{"Start":"06:47.870 ","End":"06:51.230","Text":"that\u0027s n times p. That equals,"},{"Start":"06:51.230 ","End":"06:52.355","Text":"let\u0027s plug in the numbers,"},{"Start":"06:52.355 ","End":"06:55.835","Text":"n is 20 and p is 0.1."},{"Start":"06:55.835 ","End":"06:58.650","Text":"That equals to 2 people."},{"Start":"07:00.020 ","End":"07:06.380","Text":"All this means is that in the sample of 20 people,"},{"Start":"07:06.380 ","End":"07:12.480","Text":"We can expect that 2 people will be poorly educated."}],"ID":13006},{"Watched":false,"Name":"Exercise 5","Duration":"8m 25s","ChapterTopicVideoID":12528,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this question, we\u0027ll be talking about students and cars."},{"Start":"00:03.630 ","End":"00:08.190","Text":"20 percent of the students attending NYU actually live in New York."},{"Start":"00:08.190 ","End":"00:10.125","Text":"Of the students living in New York,"},{"Start":"00:10.125 ","End":"00:12.180","Text":"30 percent arrive by car."},{"Start":"00:12.180 ","End":"00:14.955","Text":"Of the students who do not live in New York,"},{"Start":"00:14.955 ","End":"00:16.830","Text":"50 percent come by car."},{"Start":"00:16.830 ","End":"00:22.710","Text":"We\u0027re asked, a guard checks the bags of each student entering the campus."},{"Start":"00:22.710 ","End":"00:27.840","Text":"What\u0027s the probability that among 5 students checked by the guard,"},{"Start":"00:27.840 ","End":"00:30.915","Text":"only 1 came by car?"},{"Start":"00:30.915 ","End":"00:34.500","Text":"The first thing that we have to do is"},{"Start":"00:34.500 ","End":"00:40.230","Text":"to calculate the probability of a student coming by car."},{"Start":"00:40.230 ","End":"00:42.175","Text":"If we look at the question,"},{"Start":"00:42.175 ","End":"00:45.935","Text":"we see that we can solve this by probability tree."},{"Start":"00:45.935 ","End":"00:50.300","Text":"Let\u0027s just build a probability tree."},{"Start":"00:50.300 ","End":"00:54.630","Text":"Here it is. As we can see,"},{"Start":"00:54.630 ","End":"00:57.440","Text":"we have here the 2 levels."},{"Start":"00:57.440 ","End":"01:00.680","Text":"The first level is the level of the students,"},{"Start":"01:00.680 ","End":"01:03.155","Text":"whether they live in New York or not."},{"Start":"01:03.155 ","End":"01:10.010","Text":"Now we said that 20 percent of the students attending NYU actually live in New York."},{"Start":"01:10.010 ","End":"01:12.485","Text":"Here it is. That\u0027s the 20 percent."},{"Start":"01:12.485 ","End":"01:16.535","Text":"That means that 80 percent do not live in New York."},{"Start":"01:16.535 ","End":"01:21.795","Text":"What else? Of the students living in New York,30 percent arrive by car."},{"Start":"01:21.795 ","End":"01:23.960","Text":"That\u0027s the second level right here,"},{"Start":"01:23.960 ","End":"01:27.895","Text":"who has a car or who arrives by car and who doesn\u0027t."},{"Start":"01:27.895 ","End":"01:30.080","Text":"Out of the people living in New York,"},{"Start":"01:30.080 ","End":"01:32.150","Text":"30 percent arrive by car."},{"Start":"01:32.150 ","End":"01:35.030","Text":"That means that 70 percent do not."},{"Start":"01:35.030 ","End":"01:38.270","Text":"Of the people who do not live in New York,"},{"Start":"01:38.270 ","End":"01:41.180","Text":"50 percent of them come by car."},{"Start":"01:41.180 ","End":"01:44.085","Text":"That means that 50 percent don\u0027t."},{"Start":"01:44.085 ","End":"01:50.916","Text":"If we remember how to calculate the probabilities in the probability tree,"},{"Start":"01:50.916 ","End":"02:00.710","Text":"within a tree we multiply the probabilities and amongst the branches we add."},{"Start":"02:00.710 ","End":"02:03.290","Text":"That means that what do we have to do?"},{"Start":"02:03.290 ","End":"02:08.395","Text":"We have to see what are the branches of students coming by car?"},{"Start":"02:08.395 ","End":"02:10.610","Text":"Of the people who are not,"},{"Start":"02:10.610 ","End":"02:13.385","Text":"of the students who are not living in New York City,"},{"Start":"02:13.385 ","End":"02:16.955","Text":"this is the branch that we\u0027re interested in,"},{"Start":"02:16.955 ","End":"02:21.491","Text":"and of the people that are living in New York City,"},{"Start":"02:21.491 ","End":"02:26.610","Text":"this is a branch that we\u0027re interested in,"},{"Start":"02:26.610 ","End":"02:37.440","Text":"so the probability of a student coming by car is"},{"Start":"02:37.440 ","End":"02:45.270","Text":"0.8 times 0.5 plus 0.2 times"},{"Start":"02:45.270 ","End":"02:55.150","Text":"0.3 and that turns out to be 0.46."},{"Start":"02:56.660 ","End":"02:59.985","Text":"Let\u0027s get back to our question."},{"Start":"02:59.985 ","End":"03:05.870","Text":"We\u0027re asked, what\u0027s the probability that among 5 students checked by the guard,"},{"Start":"03:05.870 ","End":"03:08.410","Text":"only 1 came by car?"},{"Start":"03:08.410 ","End":"03:13.880","Text":"We\u0027re interested in 5 students,"},{"Start":"03:13.880 ","End":"03:17.740","Text":"so our sample is 5."},{"Start":"03:17.740 ","End":"03:23.380","Text":"We\u0027re interested to find out the probability,"},{"Start":"03:23.380 ","End":"03:28.320","Text":"where our random variable X equals 1."},{"Start":"03:28.320 ","End":"03:29.900","Text":"What\u0027s a random variable?"},{"Start":"03:29.900 ","End":"03:31.715","Text":"A random variable here,"},{"Start":"03:31.715 ","End":"03:34.370","Text":"we\u0027ll define that as the number of"},{"Start":"03:34.370 ","End":"03:42.390","Text":"students who came by car and that\u0027s what we want to know."},{"Start":"03:42.390 ","End":"03:45.545","Text":"Are the students independent of each other?"},{"Start":"03:45.545 ","End":"03:53.480","Text":"Obviously. Do we have a Bernoulli trial here?"},{"Start":"03:53.480 ","End":"03:55.737","Text":"Yes, the success,"},{"Start":"03:55.737 ","End":"04:02.220","Text":"we define success as a person coming by car and we have a probability for that."},{"Start":"04:02.830 ","End":"04:10.760","Text":"We know that X is distributed binomially"},{"Start":"04:10.760 ","End":"04:18.750","Text":"with n equals 5 and p being equal to 0.46."},{"Start":"04:20.630 ","End":"04:23.225","Text":"If that\u0027s the case,"},{"Start":"04:23.225 ","End":"04:30.095","Text":"let\u0027s just calculate what\u0027s the probability of X being equal to 1?"},{"Start":"04:30.095 ","End":"04:36.455","Text":"If we have a binomial distribution with the probability of X being equal to k,"},{"Start":"04:36.455 ","End":"04:38.870","Text":"that\u0027s easy for us right now."},{"Start":"04:38.870 ","End":"04:44.330","Text":"That\u0027s n over k times p to the power of k times q to"},{"Start":"04:44.330 ","End":"04:50.945","Text":"the power of n minus k. Let\u0027s just plug in the numbers."},{"Start":"04:50.945 ","End":"04:58.005","Text":"That\u0027s n, that\u0027s 5 and k is 1. What\u0027s our p?"},{"Start":"04:58.005 ","End":"05:06.645","Text":"Our p is 0.46 to the power of 1 times, what\u0027s our q?"},{"Start":"05:06.645 ","End":"05:09.270","Text":"Q is 1 minus p,"},{"Start":"05:09.270 ","End":"05:11.880","Text":"which is 1 minus 0.46,"},{"Start":"05:11.880 ","End":"05:15.750","Text":"that\u0027s 0.54 to the power of 4."},{"Start":"05:15.750 ","End":"05:18.810","Text":"4 is n minus k,"},{"Start":"05:18.810 ","End":"05:19.890","Text":"that\u0027s 5 minus 1,"},{"Start":"05:19.890 ","End":"05:28.210","Text":"that\u0027s 4 and that comes out to 0.1956."},{"Start":"05:28.210 ","End":"05:35.495","Text":"That\u0027s the probability of 1 student coming by car out of 5."},{"Start":"05:35.495 ","End":"05:38.930","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"05:38.930 ","End":"05:43.630","Text":"the majority of the 5 students came to NYU by car?"},{"Start":"05:43.630 ","End":"05:46.445","Text":"Basically what we\u0027re asking is,"},{"Start":"05:46.445 ","End":"05:49.250","Text":"what is the probability that X,"},{"Start":"05:49.250 ","End":"05:53.120","Text":"the number of students come by car, is the majority."},{"Start":"05:53.120 ","End":"05:57.355","Text":"That means it has to be greater or equal to 3."},{"Start":"05:57.355 ","End":"06:00.350","Text":"2/3 of the 5 students isn\u0027t a majority,"},{"Start":"06:00.350 ","End":"06:02.860","Text":"but 3 out of the 5 is."},{"Start":"06:03.560 ","End":"06:05.840","Text":"If that\u0027s the case,"},{"Start":"06:05.840 ","End":"06:09.590","Text":"this equals to the probability of X being equal to"},{"Start":"06:09.590 ","End":"06:15.150","Text":"3 plus the probability of X being equal to 4,"},{"Start":"06:15.150 ","End":"06:20.040","Text":"plus the probability of X being equal to 5."},{"Start":"06:20.040 ","End":"06:26.690","Text":"We know that X is binomially distributed with n"},{"Start":"06:26.690 ","End":"06:33.185","Text":"being equal to 5 and p being equal to 0.46."},{"Start":"06:33.185 ","End":"06:37.130","Text":"We know that the definition of"},{"Start":"06:37.130 ","End":"06:44.255","Text":"the binomial distribution is the probability of X being equal to k is n over k,"},{"Start":"06:44.255 ","End":"06:46.655","Text":"p to the power of k,"},{"Start":"06:46.655 ","End":"06:51.350","Text":"q to the power of n minus k. If that\u0027s the case,"},{"Start":"06:51.350 ","End":"06:54.710","Text":"let\u0027s just plug in the numbers and calculate things out."},{"Start":"06:54.710 ","End":"07:00.350","Text":"That equals, that where the probability of x equals 3,"},{"Start":"07:00.350 ","End":"07:03.690","Text":"well that\u0027s 5 over 3."},{"Start":"07:03.690 ","End":"07:05.390","Text":"K equals 3 here,"},{"Start":"07:05.390 ","End":"07:10.580","Text":"0.46 to the power of 3 times"},{"Start":"07:10.580 ","End":"07:17.100","Text":"0.54 to the power of 2 plus,"},{"Start":"07:17.100 ","End":"07:21.070","Text":"now here we\u0027re calculating where k equals 4."},{"Start":"07:21.070 ","End":"07:24.125","Text":"That\u0027s 5 over 4,"},{"Start":"07:24.125 ","End":"07:34.190","Text":"0.46 to the power of 4 times 0.54 to the power of 1,"},{"Start":"07:34.190 ","End":"07:38.425","Text":"n minus k, 5 minus 4 plus,"},{"Start":"07:38.425 ","End":"07:43.140","Text":"what\u0027s the probability of X when X equals 5?"},{"Start":"07:43.140 ","End":"07:45.493","Text":"That\u0027s 5 over 5,"},{"Start":"07:45.493 ","End":"07:55.120","Text":"0.460 to the power of 5 times 0.54 to the power of 0."},{"Start":"07:55.790 ","End":"08:03.770","Text":"This section calculates the probability when X equals 3."},{"Start":"08:03.770 ","End":"08:08.840","Text":"This section right here calculates the probability when X equals 4."},{"Start":"08:08.840 ","End":"08:13.680","Text":"This section calculates the probability when X equals 5."},{"Start":"08:13.850 ","End":"08:16.385","Text":"When we calculate everything,"},{"Start":"08:16.385 ","End":"08:23.220","Text":"this comes out to 0.4253."}],"ID":13007},{"Watched":false,"Name":"Exercise 6 Parts a-b","Duration":"5m 8s","ChapterTopicVideoID":12530,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.125","Text":"In this question, we\u0027ll be talking about getting the right answer on an exam."},{"Start":"00:04.125 ","End":"00:09.090","Text":"Now, John takes a multiple choice exam having 20 questions."},{"Start":"00:09.090 ","End":"00:13.170","Text":"The chances of his knowing the answer to a question are 80 percent."},{"Start":"00:13.170 ","End":"00:15.900","Text":"If he doesn\u0027t know the answer, he guesses."},{"Start":"00:15.900 ","End":"00:18.555","Text":"Each question has 4 possible answers,"},{"Start":"00:18.555 ","End":"00:20.520","Text":"only 1 of which is correct,"},{"Start":"00:20.520 ","End":"00:25.770","Text":"and we\u0027re asked, what is the chances of answering a given question correctly?"},{"Start":"00:25.770 ","End":"00:31.095","Text":"Now, this question has nothing to do with the binomial distribution."},{"Start":"00:31.095 ","End":"00:36.930","Text":"We\u0027re asked to find the probability of answering a given question correctly,"},{"Start":"00:36.930 ","End":"00:42.815","Text":"when we\u0027re given the procedure of how he answers the questions."},{"Start":"00:42.815 ","End":"00:46.730","Text":"Now, this basically looks like a probability tree."},{"Start":"00:46.730 ","End":"00:52.580","Text":"Where the first-level of whether John knows or doesn\u0027t know the correct answer,"},{"Start":"00:52.580 ","End":"00:57.415","Text":"and the second level is whether he guesses correctly or not."},{"Start":"00:57.415 ","End":"01:02.040","Text":"Let\u0027s build a probability tree and here it is."},{"Start":"01:02.040 ","End":"01:06.305","Text":"As we said, the first level"},{"Start":"01:06.305 ","End":"01:11.060","Text":"is whether John knows or doesn\u0027t know and he knows with the probability of 80 percent."},{"Start":"01:11.060 ","End":"01:13.066","Text":"That\u0027s right here,"},{"Start":"01:13.066 ","End":"01:17.620","Text":"so that means that he doesn\u0027t know the probability of 20 percent."},{"Start":"01:17.620 ","End":"01:21.590","Text":"Now, if we go to the second level where John guesses,"},{"Start":"01:21.590 ","End":"01:25.970","Text":"now there are 4 possible answers to each question,"},{"Start":"01:25.970 ","End":"01:28.355","Text":"only 1 of which is correct."},{"Start":"01:28.355 ","End":"01:33.775","Text":"If he guesses, he can guess with a probability of 25 percent,"},{"Start":"01:33.775 ","End":"01:37.065","Text":"1 over 4 of guessing right."},{"Start":"01:37.065 ","End":"01:42.335","Text":"That means that he can guess wrong with a probability of 75 percent."},{"Start":"01:42.335 ","End":"01:44.720","Text":"Now, what do we ask?"},{"Start":"01:44.720 ","End":"01:48.800","Text":"We\u0027re asked, what are the chances of answering the given questions correctly,"},{"Start":"01:48.800 ","End":"01:55.110","Text":"the probability of getting the question right?"},{"Start":"01:56.040 ","End":"02:01.480","Text":"Now, when we take a look at our probability tree,"},{"Start":"02:01.480 ","End":"02:09.421","Text":"we can get the question right on this branch or right here,"},{"Start":"02:09.421 ","End":"02:15.520","Text":"so that means that we can right now, calculate the probability."},{"Start":"02:15.520 ","End":"02:20.195","Text":"Well that\u0027s 0.8, that\u0027s this branch right here,"},{"Start":"02:20.195 ","End":"02:25.120","Text":"plus this branch right here,"},{"Start":"02:25.120 ","End":"02:32.050","Text":"which is 0.2 times 0.25."},{"Start":"02:32.050 ","End":"02:35.240","Text":"That equals to 0.85."},{"Start":"02:36.620 ","End":"02:38.945","Text":"In this section we\u0027re asked,"},{"Start":"02:38.945 ","End":"02:43.129","Text":"what are the chances of answering exactly 16 questions correctly?"},{"Start":"02:43.129 ","End":"02:52.270","Text":"Well, let\u0027s define X as the number of correct answers."},{"Start":"02:53.300 ","End":"02:56.285","Text":"Now, how many questions do we have?"},{"Start":"02:56.285 ","End":"02:58.159","Text":"We have 20 questions,"},{"Start":"02:58.159 ","End":"03:00.080","Text":"so n equals 20."},{"Start":"03:00.080 ","End":"03:04.970","Text":"What\u0027s the probability of answering a question correctly?"},{"Start":"03:04.970 ","End":"03:08.965","Text":"Well, that equals to 0.85."},{"Start":"03:08.965 ","End":"03:11.625","Text":"We\u0027ve answered that in Section A."},{"Start":"03:11.625 ","End":"03:16.100","Text":"Now, because a question can be answered either correctly or incorrectly,"},{"Start":"03:16.100 ","End":"03:21.759","Text":"and all questions have the same probability of being answered correctly,"},{"Start":"03:21.759 ","End":"03:25.930","Text":"then we do have a Bernoulli trial,"},{"Start":"03:25.940 ","End":"03:29.060","Text":"the questions are independent."},{"Start":"03:29.060 ","End":"03:31.249","Text":"Obviously answering the questions,"},{"Start":"03:31.249 ","End":"03:35.870","Text":"1 question is independent of answering a different question."},{"Start":"03:35.870 ","End":"03:38.765","Text":"We have 20 of these trials,"},{"Start":"03:38.765 ","End":"03:44.990","Text":"and we want to know what is the total number of correct answers."},{"Start":"03:44.990 ","End":"03:49.535","Text":"We can honestly say that X has"},{"Start":"03:49.535 ","End":"03:58.535","Text":"a binomial distribution where n equals 20 and p equals 0.85."},{"Start":"03:58.535 ","End":"04:06.830","Text":"Now, we know that the general equation for a binomial probability is this,"},{"Start":"04:06.830 ","End":"04:10.100","Text":"where the probability of X being equal to k,"},{"Start":"04:10.100 ","End":"04:12.025","Text":"that\u0027s n over k,"},{"Start":"04:12.025 ","End":"04:17.190","Text":"p^k q^n minus k,"},{"Start":"04:17.190 ","End":"04:23.255","Text":"where q is 1 minus p. Let\u0027s answer the question."},{"Start":"04:23.255 ","End":"04:28.310","Text":"What\u0027s the probability of X being equal to 16?"},{"Start":"04:28.310 ","End":"04:33.470","Text":"We want to know what are the chances of answering exactly 16 questions."},{"Start":"04:33.470 ","End":"04:36.860","Text":"Well, let\u0027s plug in the numbers. N equals 20."},{"Start":"04:36.860 ","End":"04:38.360","Text":"That\u0027s 20 over what?"},{"Start":"04:38.360 ","End":"04:42.605","Text":"16. What\u0027s p?"},{"Start":"04:42.605 ","End":"04:49.890","Text":"P is 0.85^16 times q."},{"Start":"04:49.890 ","End":"04:51.840","Text":"Well, q is 1 minus p,"},{"Start":"04:51.840 ","End":"04:54.270","Text":"so that\u0027s 1 minus 0.85."},{"Start":"04:54.270 ","End":"04:58.875","Text":"That\u0027s 0.15^n minus k,"},{"Start":"04:58.875 ","End":"05:01.665","Text":"n minus k is 4."},{"Start":"05:01.665 ","End":"05:06.040","Text":"That turns out to be 0.1821."}],"ID":13008},{"Watched":false,"Name":"Exercise 6 Part c","Duration":"6m 10s","ChapterTopicVideoID":12529,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this section, we\u0027re given that John receives"},{"Start":"00:02.940 ","End":"00:06.105","Text":"5 points for each question he answers correctly,"},{"Start":"00:06.105 ","End":"00:10.995","Text":"and 1 point is deducted from his mark for each question he answers incorrectly."},{"Start":"00:10.995 ","End":"00:16.270","Text":"We\u0027re asked, what are the expectation and variance of the student\u0027s mark?"},{"Start":"00:16.400 ","End":"00:19.065","Text":"Well, first of all,"},{"Start":"00:19.065 ","End":"00:23.490","Text":"I believe that we\u0027re dealing with a linear transformation here. Why is that?"},{"Start":"00:23.490 ","End":"00:25.515","Text":"Because we\u0027re taking x,"},{"Start":"00:25.515 ","End":"00:27.615","Text":"the number of correct answers,"},{"Start":"00:27.615 ","End":"00:30.885","Text":"and we\u0027re transforming them into y,"},{"Start":"00:30.885 ","End":"00:34.545","Text":"the final mark that John receives."},{"Start":"00:34.545 ","End":"00:36.255","Text":"Now if that\u0027s the case,"},{"Start":"00:36.255 ","End":"00:44.190","Text":"then let\u0027s take a look at the 4 stages of a linear transformation. Here they are."},{"Start":"00:44.190 ","End":"00:46.365","Text":"Well, first of all,"},{"Start":"00:46.365 ","End":"00:49.545","Text":"we\u0027ve answered the first step."},{"Start":"00:49.545 ","End":"00:53.195","Text":"We recognize that we\u0027re dealing with a linear transformation."},{"Start":"00:53.195 ","End":"00:56.285","Text":"Now, let\u0027s go to the second step."},{"Start":"00:56.285 ","End":"01:01.835","Text":"We have to write the transformation according to what\u0027s given to us."},{"Start":"01:01.835 ","End":"01:04.720","Text":"Well, if that\u0027s the case, why?"},{"Start":"01:04.720 ","End":"01:08.130","Text":"The final mark is equal to y?"},{"Start":"01:08.130 ","End":"01:09.995","Text":"Now, what\u0027s given to us?"},{"Start":"01:09.995 ","End":"01:15.740","Text":"We\u0027re given that John receives 5 point for each question he answers correctly,"},{"Start":"01:15.740 ","End":"01:23.735","Text":"and 1 point is deducted for each question he answers incorrectly."},{"Start":"01:23.735 ","End":"01:27.020","Text":"What does that mean? That means that if"},{"Start":"01:27.020 ","End":"01:30.170","Text":"x is the number of questions he answers correctly,"},{"Start":"01:30.170 ","End":"01:33.590","Text":"then I have to multiply that by 5 to"},{"Start":"01:33.590 ","End":"01:42.555","Text":"transform the number of questions to points."},{"Start":"01:42.555 ","End":"01:44.900","Text":"But I have to take away what?"},{"Start":"01:44.900 ","End":"01:46.460","Text":"I have to take away 1,"},{"Start":"01:46.460 ","End":"01:47.960","Text":"that\u0027s 1 point,"},{"Start":"01:47.960 ","End":"01:52.580","Text":"times the number of questions he answers incorrectly."},{"Start":"01:52.580 ","End":"01:54.965","Text":"Now, we have 20 questions."},{"Start":"01:54.965 ","End":"01:58.130","Text":"We know that n equals 20."},{"Start":"01:58.130 ","End":"02:01.715","Text":"If he answers x questions correctly,"},{"Start":"02:01.715 ","End":"02:08.215","Text":"then he must answer 20 minus x questions incorrectly."},{"Start":"02:08.215 ","End":"02:16.010","Text":"This would be the linear transformation according to this question."},{"Start":"02:16.010 ","End":"02:19.130","Text":"Now, what\u0027s the general form of a linear transformation?"},{"Start":"02:19.130 ","End":"02:23.670","Text":"That\u0027s y equals ax plus b."},{"Start":"02:23.670 ","End":"02:31.575","Text":"All we have to do is bring this transformation into this form. Let\u0027s do that."},{"Start":"02:31.575 ","End":"02:36.400","Text":"Well, that equals to 5x minus 20,"},{"Start":"02:36.400 ","End":"02:38.195","Text":"minus 1 times 20,"},{"Start":"02:38.195 ","End":"02:41.990","Text":"minus 1 times minus x, that\u0027s plus x."},{"Start":"02:41.990 ","End":"02:48.440","Text":"That equals to 6x minus 20."},{"Start":"02:48.440 ","End":"02:54.080","Text":"Now here we\u0027ve brought the linear transformation into the general form."},{"Start":"02:54.080 ","End":"02:57.505","Text":"Now we can identify a and b,"},{"Start":"02:57.505 ","End":"02:58.965","Text":"where a equals what?"},{"Start":"02:58.965 ","End":"03:00.810","Text":"That\u0027s the multiplier here."},{"Start":"03:00.810 ","End":"03:06.790","Text":"That\u0027s 6. B equals minus 20."},{"Start":"03:07.190 ","End":"03:11.235","Text":"Excellent. We\u0027ve done that."},{"Start":"03:11.235 ","End":"03:15.320","Text":"Now we can start and answer the question."},{"Start":"03:15.320 ","End":"03:18.935","Text":"The question being, what are the expectation and variance of the student\u0027s mark?"},{"Start":"03:18.935 ","End":"03:23.290","Text":"We\u0027re asked what are the expectation and variance of y."},{"Start":"03:23.290 ","End":"03:26.025","Text":"But before we do that,"},{"Start":"03:26.025 ","End":"03:33.095","Text":"let\u0027s first see about x and the expectation and variance of x."},{"Start":"03:33.095 ","End":"03:35.420","Text":"Now, if we recall,"},{"Start":"03:35.420 ","End":"03:41.480","Text":"x was distributed binomially"},{"Start":"03:41.480 ","End":"03:49.100","Text":"with n equaling 20 and p being equal to 0.85."},{"Start":"03:49.100 ","End":"03:50.630","Text":"Now if that\u0027s the case,"},{"Start":"03:50.630 ","End":"03:54.830","Text":"then the expectation of x is n times p,"},{"Start":"03:54.830 ","End":"03:59.885","Text":"which is 20 times 0.85."},{"Start":"03:59.885 ","End":"04:02.230","Text":"That equals to 17."},{"Start":"04:02.230 ","End":"04:08.360","Text":"The variance of x equals n times p times q."},{"Start":"04:08.360 ","End":"04:15.480","Text":"That equals to 20 times 0.85 times 0.15."},{"Start":"04:16.870 ","End":"04:22.950","Text":"That equals to 2.55."},{"Start":"04:23.110 ","End":"04:27.560","Text":"Excellent. Now, having done that,"},{"Start":"04:27.560 ","End":"04:35.970","Text":"we can go ahead and answer the question about the expectation and variance of y."},{"Start":"04:36.610 ","End":"04:40.430","Text":"Now, what is the expectation of y?"},{"Start":"04:40.430 ","End":"04:43.480","Text":"Y as we said is a linear transformation of x."},{"Start":"04:43.480 ","End":"04:49.070","Text":"If we recall how we get the expectation of y,"},{"Start":"04:49.070 ","End":"04:56.570","Text":"that equals to a times the expectation of x plus b."},{"Start":"04:56.570 ","End":"05:01.300","Text":"In our case, a is equal to 6."},{"Start":"05:01.300 ","End":"05:03.440","Text":"What\u0027s the expectation of x?"},{"Start":"05:03.440 ","End":"05:05.300","Text":"Well, we just figured it out right here."},{"Start":"05:05.300 ","End":"05:07.340","Text":"That\u0027s 17."},{"Start":"05:07.820 ","End":"05:11.180","Text":"We have to add b. Now b is minus 20,"},{"Start":"05:11.180 ","End":"05:18.515","Text":"so we subtract 20 from that and we get 82."},{"Start":"05:18.515 ","End":"05:20.390","Text":"Now, that\u0027s his final grade."},{"Start":"05:20.390 ","End":"05:22.865","Text":"That\u0027s his expected final grade."},{"Start":"05:22.865 ","End":"05:27.055","Text":"Now let\u0027s take a look at the variance of y. Variance"},{"Start":"05:27.055 ","End":"05:32.285","Text":"of y is defined as a squared times the variance of x."},{"Start":"05:32.285 ","End":"05:35.120","Text":"Again, let\u0027s plug in the numbers."},{"Start":"05:35.120 ","End":"05:39.600","Text":"A squared is 6 squared, times what?"},{"Start":"05:39.600 ","End":"05:41.105","Text":"The variance of x."},{"Start":"05:41.105 ","End":"05:47.480","Text":"Now the variance of x is 2.55 and that"},{"Start":"05:47.480 ","End":"05:56.730","Text":"equals to 91.8 points squared."},{"Start":"05:56.750 ","End":"06:02.540","Text":"Here we\u0027ve answered the question where we calculated"},{"Start":"06:02.540 ","End":"06:09.510","Text":"the expected value of y and the variance of y."}],"ID":13009},{"Watched":false,"Name":"Exercise 7","Duration":"7m 15s","ChapterTopicVideoID":12531,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this question, we\u0027ll be talking about faulty products being packaged."},{"Start":"00:05.070 ","End":"00:10.425","Text":"Now, we\u0027re given that 5 percent of products coming out of a production line are faulty."},{"Start":"00:10.425 ","End":"00:14.160","Text":"The products are packaged in a cardboard box."},{"Start":"00:14.160 ","End":"00:16.680","Text":"Each box has 10 different products."},{"Start":"00:16.680 ","End":"00:19.790","Text":"Now, the boxes are packaged in a container."},{"Start":"00:19.790 ","End":"00:23.490","Text":"Each container has 20 boxes."},{"Start":"00:23.490 ","End":"00:25.830","Text":"We\u0027re asked, what is the probability of"},{"Start":"00:25.830 ","End":"00:30.180","Text":"at least 1 faulty product in a randomly selected box?"},{"Start":"00:30.180 ","End":"00:37.980","Text":"Now, here we\u0027re talking about a faulty product within a box."},{"Start":"00:37.980 ","End":"00:48.760","Text":"That means that we define our success as being a faulty product,"},{"Start":"00:53.330 ","End":"01:00.545","Text":"and the probability of a product being faulty is 0.05,"},{"Start":"01:00.545 ","End":"01:03.360","Text":"that\u0027s the 5 percent right here."},{"Start":"01:03.590 ","End":"01:07.220","Text":"Now, how many products are in a box?"},{"Start":"01:07.220 ","End":"01:12.440","Text":"Well, there are 10 different products."},{"Start":"01:12.440 ","End":"01:13.850","Text":"N equals 10."},{"Start":"01:13.850 ","End":"01:17.635","Text":"Now, the products are independent of each other, obviously,"},{"Start":"01:17.635 ","End":"01:23.195","Text":"and each box is basically a Bernoulli trial."},{"Start":"01:23.195 ","End":"01:31.168","Text":"Whether it\u0027s a faulty product or not a faulty product with the same probability,"},{"Start":"01:31.168 ","End":"01:39.150","Text":"and we\u0027re asked, what\u0027s the number of faulty products?"},{"Start":"01:43.130 ","End":"01:48.745","Text":"Excellent, so that means that we have X"},{"Start":"01:48.745 ","End":"01:55.295","Text":"and it has a binomial distribution where n equals 10,"},{"Start":"01:55.295 ","End":"02:00.640","Text":"and p equals 0.05."},{"Start":"02:00.640 ","End":"02:04.080","Text":"Now, let\u0027s get back to our question."},{"Start":"02:04.080 ","End":"02:08.630","Text":"We\u0027re asked, what\u0027s the probability of at least 1 faulty product?"},{"Start":"02:08.630 ","End":"02:11.750","Text":"That means that we want to know"},{"Start":"02:11.750 ","End":"02:20.340","Text":"whether probability of X being greater or equal to 1,"},{"Start":"02:20.340 ","End":"02:24.110","Text":"because X is the number of faulty products,"},{"Start":"02:24.110 ","End":"02:28.270","Text":"and we want a box that has at least 1 faulty product."},{"Start":"02:28.270 ","End":"02:33.500","Text":"Well, we know that X can have the values of 0, 1, 2,"},{"Start":"02:33.500 ","End":"02:35.934","Text":"and so on and so forth until 10,"},{"Start":"02:35.934 ","End":"02:39.055","Text":"so we\u0027re looking at this set."},{"Start":"02:39.055 ","End":"02:42.900","Text":"But again, it\u0027s very complicated to figure this out,"},{"Start":"02:42.900 ","End":"02:44.875","Text":"so let\u0027s take the complimentary set."},{"Start":"02:44.875 ","End":"02:52.010","Text":"That equals to 1 minus the probability of X being equal to 0."},{"Start":"02:54.080 ","End":"03:00.025","Text":"Now, what\u0027s the probability of X being equal to 0?"},{"Start":"03:00.025 ","End":"03:03.490","Text":"Well, X is binomially distributed,"},{"Start":"03:03.490 ","End":"03:08.295","Text":"so that\u0027s 10 over 0, n equals 10."},{"Start":"03:08.295 ","End":"03:19.060","Text":"P is 0.05 to the power of 0 times 0.95 to the power of 10,"},{"Start":"03:19.060 ","End":"03:25.910","Text":"and that equals to 0.401."},{"Start":"03:26.680 ","End":"03:29.465","Text":"In this section, we\u0027re asked,"},{"Start":"03:29.465 ","End":"03:32.540","Text":"what are the expectation and standard deviation of the number of"},{"Start":"03:32.540 ","End":"03:37.509","Text":"boxes in the container with at least 1 damaged product?"},{"Start":"03:37.509 ","End":"03:39.800","Text":"Now, remember in section A,"},{"Start":"03:39.800 ","End":"03:43.385","Text":"we were talking about products within a box,"},{"Start":"03:43.385 ","End":"03:48.515","Text":"and here we\u0027re talking about boxes within a container."},{"Start":"03:48.515 ","End":"03:56.870","Text":"We have a binomial distribution within a binomial distribution. Let\u0027s get to work."},{"Start":"03:56.870 ","End":"04:00.540","Text":"Let\u0027s define success."},{"Start":"04:01.100 ","End":"04:03.750","Text":"Success as what?"},{"Start":"04:03.750 ","End":"04:14.529","Text":"As a box with at least 1 faulty product."},{"Start":"04:18.470 ","End":"04:23.710","Text":"Now, how many of these boxes do we have in a container?"},{"Start":"04:23.710 ","End":"04:26.240","Text":"Well, we have 20."},{"Start":"04:26.570 ","End":"04:34.595","Text":"Again, now we have a Bernoulli trial where we\u0027re looking at boxes instead of products."},{"Start":"04:34.595 ","End":"04:42.190","Text":"Now, that means that we can right now define a new random variable w,"},{"Start":"04:42.190 ","End":"04:43.698","Text":"let\u0027s call it,"},{"Start":"04:43.698 ","End":"04:53.370","Text":"and it\u0027ll be the number of boxes with at least 1 faulty product."},{"Start":"04:53.370 ","End":"04:56.035","Text":"It\u0027s the same thing, just like here."},{"Start":"04:56.035 ","End":"04:58.175","Text":"Now, because of that,"},{"Start":"04:58.175 ","End":"05:04.370","Text":"we know that w is distributed binomially where n equals 20."},{"Start":"05:04.370 ","End":"05:09.170","Text":"Now, we\u0027re looking at boxes within a container but with the same probability."},{"Start":"05:09.170 ","End":"05:13.140","Text":"The probability is 0.401"},{"Start":"05:14.660 ","End":"05:21.410","Text":"and we know that the probability of w being equal to k,"},{"Start":"05:21.410 ","End":"05:24.260","Text":"that equals to n over k,"},{"Start":"05:24.260 ","End":"05:33.410","Text":"p^k, q^n-k that\u0027s the equation for a binomial distribution."},{"Start":"05:33.410 ","End":"05:37.400","Text":"Well, if that\u0027s the case, let\u0027s just put in the numbers right here."},{"Start":"05:37.400 ","End":"05:41.465","Text":"What we\u0027re looking for is"},{"Start":"05:41.465 ","End":"05:47.150","Text":"what are the expectations standard deviation as the number of boxes in a container?"},{"Start":"05:47.150 ","End":"05:50.945","Text":"Well, we know that n equals 20,"},{"Start":"05:50.945 ","End":"05:55.415","Text":"and we know that p equals 0.401."},{"Start":"05:55.415 ","End":"05:58.174","Text":"The expectation of w,"},{"Start":"05:58.174 ","End":"06:03.470","Text":"because w is distributed binomially that equals"},{"Start":"06:03.470 ","End":"06:09.095","Text":"to n times p. N equals 20 as we said,"},{"Start":"06:09.095 ","End":"06:13.823","Text":"p is 0.401,"},{"Start":"06:13.823 ","End":"06:19.600","Text":"and that equals to 8.025 boxes."},{"Start":"06:20.750 ","End":"06:24.030","Text":"What\u0027s the variance of w?"},{"Start":"06:24.030 ","End":"06:28.575","Text":"Well, the variance of w is n times p times q."},{"Start":"06:28.575 ","End":"06:31.860","Text":"That equals to 20 times"},{"Start":"06:31.860 ","End":"06:40.215","Text":"0.401 times 1 minus 0.401,"},{"Start":"06:40.215 ","End":"06:50.155","Text":"and that equals to 4.867 boxes squared."},{"Start":"06:50.155 ","End":"06:54.290","Text":"Now, what\u0027s the standard deviation of w?"},{"Start":"06:54.290 ","End":"06:58.835","Text":"Well, that\u0027s the square root of 4.867,"},{"Start":"06:58.835 ","End":"07:05.970","Text":"and that equals to 2.193 boxes."},{"Start":"07:05.970 ","End":"07:08.030","Text":"Here, we have it,"},{"Start":"07:08.030 ","End":"07:14.920","Text":"the standard deviation of w and the expectation of w."}],"ID":13010},{"Watched":false,"Name":"Exercise 8","Duration":"2m 30s","ChapterTopicVideoID":12532,"CourseChapterTopicPlaylistID":64485,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.785","Text":"In this question, we\u0027re given that a balanced coin is tossed 5 times,"},{"Start":"00:04.785 ","End":"00:09.000","Text":"we define X as the number of times that heads appear and we\u0027re"},{"Start":"00:09.000 ","End":"00:13.630","Text":"asked to calculate the expectation of X squared,"},{"Start":"00:13.630 ","End":"00:15.285","Text":"so let\u0027s write this out."},{"Start":"00:15.285 ","End":"00:20.950","Text":"X is the number of times that heads appear, number of heads."},{"Start":"00:22.490 ","End":"00:25.830","Text":"We\u0027re tossing a coin 5 times,"},{"Start":"00:25.830 ","End":"00:27.605","Text":"that means n equals 5,"},{"Start":"00:27.605 ","End":"00:30.965","Text":"and because it\u0027s a balanced coin,"},{"Start":"00:30.965 ","End":"00:36.095","Text":"the probability of getting a heads is 0.5."},{"Start":"00:36.095 ","End":"00:41.735","Text":"Now, this is a classic Bernoulli trial where"},{"Start":"00:41.735 ","End":"00:47.000","Text":"each task has a probability of 0.5 of getting heads,"},{"Start":"00:47.000 ","End":"00:51.590","Text":"and we\u0027re asked how many heads do we have after 5 tosses?"},{"Start":"00:51.590 ","End":"00:53.105","Text":"Well, if that\u0027s the case,"},{"Start":"00:53.105 ","End":"01:03.330","Text":"then X has a binomial distribution where n equals 5 and p equals 0.5."},{"Start":"01:03.980 ","End":"01:07.785","Text":"Let\u0027s see what the expectation of X is."},{"Start":"01:07.785 ","End":"01:12.300","Text":"Now, that equals to n times p. Now if we plug in"},{"Start":"01:12.300 ","End":"01:18.205","Text":"the numbers that\u0027s 5 times 0.5 and that equals to 2.5."},{"Start":"01:18.205 ","End":"01:21.620","Text":"What\u0027s the variance of X?"},{"Start":"01:21.620 ","End":"01:26.725","Text":"Well, that\u0027s defined as n times p times q."},{"Start":"01:26.725 ","End":"01:33.060","Text":"That is 5 times 0.5 times 0.5,"},{"Start":"01:33.060 ","End":"01:36.830","Text":"and that equals to 1.25."},{"Start":"01:36.830 ","End":"01:40.220","Text":"Now, what else do we know about the variance?"},{"Start":"01:40.220 ","End":"01:44.705","Text":"We know that the variance of X equals"},{"Start":"01:44.705 ","End":"01:53.215","Text":"the expectation of X squared minus the expectation squared of X."},{"Start":"01:53.215 ","End":"01:56.494","Text":"Now if we plug in these numbers,"},{"Start":"01:56.494 ","End":"01:59.660","Text":"well, the variance of X that\u0027s 1.25."},{"Start":"01:59.660 ","End":"02:02.530","Text":"We calculated that that\u0027s 1.25,"},{"Start":"02:02.530 ","End":"02:06.470","Text":"and that equals to the expectation of X squared."},{"Start":"02:06.470 ","End":"02:07.910","Text":"That\u0027s what we\u0027re trying to find out."},{"Start":"02:07.910 ","End":"02:13.220","Text":"Calculate expectation of X squared minus the expectation squared of X."},{"Start":"02:13.220 ","End":"02:19.470","Text":"Well, the expectation of X is 2.5 so that\u0027s 2.5 squared,"},{"Start":"02:19.580 ","End":"02:29.670","Text":"and that means that the expectation of X squared equals 7.5."}],"ID":13011}],"Thumbnail":null,"ID":64485},{"Name":"Special Discrete Probability Distributions - Geometric Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"3m 57s","ChapterTopicVideoID":12533,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this chapter, we\u0027ll be talking about special discrete probability distributions,"},{"Start":"00:05.580 ","End":"00:08.505","Text":"specifically the geometric distribution."},{"Start":"00:08.505 ","End":"00:12.270","Text":"Now, what makes this distribution special?"},{"Start":"00:12.270 ","End":"00:13.665","Text":"Well, first of all,"},{"Start":"00:13.665 ","End":"00:18.990","Text":"we\u0027re given the probability function as an equation and secondly,"},{"Start":"00:18.990 ","End":"00:22.650","Text":"we\u0027re also given the equations for"},{"Start":"00:22.650 ","End":"00:27.165","Text":"the expectation and the variance of these distributions."},{"Start":"00:27.165 ","End":"00:29.730","Text":"So we don\u0027t have to calculate them by hand."},{"Start":"00:29.730 ","End":"00:33.310","Text":"Let\u0027s dive in to their geometric distribution."},{"Start":"00:33.310 ","End":"00:38.975","Text":"Well, here we have the same Bernoulli trial that\u0027s repeated n times,"},{"Start":"00:38.975 ","End":"00:41.360","Text":"where each trial is independent of each other."},{"Start":"00:41.360 ","End":"00:44.210","Text":"Now if you recall from the binomial distribution,"},{"Start":"00:44.210 ","End":"00:49.400","Text":"a Bernoulli trial is a trial where we have either a success or"},{"Start":"00:49.400 ","End":"00:53.105","Text":"a failure and we have a probability of success"},{"Start":"00:53.105 ","End":"00:58.080","Text":"and a probability of failure which is 1 minus the probability of success."},{"Start":"00:58.810 ","End":"01:05.990","Text":"We have x, that\u0027s defined as the number of trials carried out until the first success."},{"Start":"01:05.990 ","End":"01:09.860","Text":"Now, this is the key point in a geometric distribution."},{"Start":"01:09.860 ","End":"01:13.790","Text":"We keep on going until we get to our first success,"},{"Start":"01:13.790 ","End":"01:15.530","Text":"whatever that was defined."},{"Start":"01:15.530 ","End":"01:21.830","Text":"Now, let p be the chances of success in an individual trial and q,"},{"Start":"01:21.830 ","End":"01:22.940","Text":"the chances of failure."},{"Start":"01:22.940 ","End":"01:26.150","Text":"Again, we have p,"},{"Start":"01:26.150 ","End":"01:29.840","Text":"the probability of success and we have q,"},{"Start":"01:29.840 ","End":"01:31.460","Text":"the probability of failure,"},{"Start":"01:31.460 ","End":"01:35.120","Text":"but that\u0027s 1 minus p. Let\u0027s not forget that."},{"Start":"01:35.120 ","End":"01:40.344","Text":"If X has a geometric distribution,"},{"Start":"01:40.344 ","End":"01:41.600","Text":"we write it like this."},{"Start":"01:41.600 ","End":"01:44.075","Text":"X is distributed,"},{"Start":"01:44.075 ","End":"01:45.830","Text":"this is Sanford distributed,"},{"Start":"01:45.830 ","End":"01:49.500","Text":"with a distribution,"},{"Start":"01:49.500 ","End":"01:51.605","Text":"g for geometric with parameter p,"},{"Start":"01:51.605 ","End":"01:54.605","Text":"p being the probability of success."},{"Start":"01:54.605 ","End":"02:01.775","Text":"The function or the equation for the geometric distribution is this."},{"Start":"02:01.775 ","End":"02:10.685","Text":"The probability where X equals k. Now that equals to p times q to the k minus 1,"},{"Start":"02:10.685 ","End":"02:12.860","Text":"where k equals 1, 2,"},{"Start":"02:12.860 ","End":"02:15.680","Text":"3 and so forth and so forth until infinity,"},{"Start":"02:15.680 ","End":"02:18.320","Text":"case basically the number that we reached,"},{"Start":"02:18.320 ","End":"02:21.785","Text":"where we reached our first success."},{"Start":"02:21.785 ","End":"02:25.830","Text":"Now, this equation does make sense where"},{"Start":"02:25.830 ","End":"02:30.875","Text":"we\u0027re doing the trials until the first success and why is that?"},{"Start":"02:30.875 ","End":"02:37.925","Text":"Because each trial is independent and q is the probability of failures."},{"Start":"02:37.925 ","End":"02:47.585","Text":"Well, excuse me, if we have to do k trials where the kth trial is the success,"},{"Start":"02:47.585 ","End":"02:50.184","Text":"that means that k minus 1,"},{"Start":"02:50.184 ","End":"02:53.195","Text":"trials before that were failures."},{"Start":"02:53.195 ","End":"03:00.820","Text":"We had to multiply the probability of failure k minus 1 time,"},{"Start":"03:00.820 ","End":"03:06.350","Text":"so that\u0027s q to the power of k minus 1 times the probability of success,"},{"Start":"03:06.350 ","End":"03:15.960","Text":"which was p. That\u0027s why this is the probability function for the geometric distribution."},{"Start":"03:17.800 ","End":"03:26.195","Text":"Now the equations for the expectation of variance of the geometric distribution is this."},{"Start":"03:26.195 ","End":"03:29.960","Text":"The expectation of X equals 1 over p,"},{"Start":"03:29.960 ","End":"03:33.605","Text":"1 over the probability of success,"},{"Start":"03:33.605 ","End":"03:38.840","Text":"and the equation for the variance is q,"},{"Start":"03:38.840 ","End":"03:42.380","Text":"the probability of failure over p squared,"},{"Start":"03:42.380 ","End":"03:44.900","Text":"the probability of success squared."},{"Start":"03:44.900 ","End":"03:51.500","Text":"Now, there are special characteristics for this distribution,"},{"Start":"03:51.500 ","End":"03:56.910","Text":"but we\u0027ll get to them once we do the examples."}],"ID":13012},{"Watched":false,"Name":"Example 1 Parts a-c","Duration":"6m 51s","ChapterTopicVideoID":12534,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.295","Text":"In this example, we have the basket that contains 10 balls,"},{"Start":"00:05.295 ","End":"00:07.230","Text":"3 which are green."},{"Start":"00:07.230 ","End":"00:13.440","Text":"A person randomly removes 1 ball after the other until he takes out a green ball."},{"Start":"00:13.440 ","End":"00:18.390","Text":"Each ball is placed back in the basket before the next ball is chosen."},{"Start":"00:18.390 ","End":"00:24.400","Text":"We\u0027re asked, what\u0027s the probability distribution of the number of balls removed?"},{"Start":"00:24.680 ","End":"00:29.760","Text":"Here, we obviously have a geometric distribution. Why is that?"},{"Start":"00:29.760 ","End":"00:36.405","Text":"We have basically Bernoulli trials that are repeated over and over again."},{"Start":"00:36.405 ","End":"00:41.040","Text":"When we have a Bernoulli trial, what\u0027s the success?"},{"Start":"00:41.040 ","End":"00:43.650","Text":"We need to have success and failure."},{"Start":"00:43.650 ","End":"00:50.740","Text":"Well the success is basically getting a green ball."},{"Start":"00:50.740 ","End":"00:53.705","Text":"What\u0027s the probability of success?"},{"Start":"00:53.705 ","End":"00:56.450","Text":"What\u0027s the probability of getting a green ball?"},{"Start":"00:56.450 ","End":"01:01.300","Text":"We have 3 green balls out of a total of 10."},{"Start":"01:01.300 ","End":"01:06.790","Text":"That\u0027s 3/10 that\u0027s equals to 0.3 under 30 percent."},{"Start":"01:07.400 ","End":"01:11.375","Text":"What about the question of independence?"},{"Start":"01:11.375 ","End":"01:14.015","Text":"When we remove 1 ball,"},{"Start":"01:14.015 ","End":"01:19.685","Text":"is it independent from the next time we remove the ball? Well, yes it is."},{"Start":"01:19.685 ","End":"01:24.860","Text":"Because what happens is once we remove ball we look at it we see what\u0027s its color."},{"Start":"01:24.860 ","End":"01:34.226","Text":"Then we place it back into the basket before we pick the next ball. That\u0027s right here."},{"Start":"01:34.226 ","End":"01:40.105","Text":"That means that each time that we remove the ball,"},{"Start":"01:40.105 ","End":"01:43.595","Text":"the probability of getting a green ball remains the same,"},{"Start":"01:43.595 ","End":"01:47.345","Text":"0.3 and the probability of failure,"},{"Start":"01:47.345 ","End":"01:54.150","Text":"that\u0027s q equals 1 minus p, that equals 0.7."},{"Start":"01:55.310 ","End":"01:59.450","Text":"Now, what do we know about our random variable?"},{"Start":"01:59.450 ","End":"02:06.455","Text":"Well, a random variable is the number of balls that we"},{"Start":"02:06.455 ","End":"02:14.760","Text":"removed until we get a green 1."},{"Start":"02:19.670 ","End":"02:25.415","Text":"Here as we said, we have x and we know now that it\u0027s"},{"Start":"02:25.415 ","End":"02:32.480","Text":"distributed geometrically with the probability of 0.3."},{"Start":"02:32.480 ","End":"02:35.800","Text":"That\u0027s a probability of success."},{"Start":"02:35.800 ","End":"02:41.765","Text":"In Section b, we\u0027re asked what\u0027s the probability that 5 balls were removed?"},{"Start":"02:41.765 ","End":"02:46.660","Text":"Well, again, what\u0027s the probability where x equals k,"},{"Start":"02:46.660 ","End":"02:51.491","Text":"where x is distributed with the geometric distribution?"},{"Start":"02:51.491 ","End":"02:59.795","Text":"Well, that equals to p times q to the power of k minus 1."},{"Start":"02:59.795 ","End":"03:03.410","Text":"If k equals 5,"},{"Start":"03:03.410 ","End":"03:08.225","Text":"then we\u0027re looking for the probability where x equals 5."},{"Start":"03:08.225 ","End":"03:13.550","Text":"Let\u0027s just plug in the numbers, p is 0.3,"},{"Start":"03:13.550 ","End":"03:21.465","Text":"so that\u0027s 0.3 times q 0.7 to the power of k minus 1."},{"Start":"03:21.465 ","End":"03:22.905","Text":"K is 5,"},{"Start":"03:22.905 ","End":"03:26.160","Text":"that\u0027ll be to the power of 5 minus 1,"},{"Start":"03:26.160 ","End":"03:28.660","Text":"that will be to the power of 4."},{"Start":"03:28.700 ","End":"03:37.805","Text":"We can easily calculate this without knowing the geometric distribution. Why is that?"},{"Start":"03:37.805 ","End":"03:40.865","Text":"Because we\u0027re looking at"},{"Start":"03:40.865 ","End":"03:47.420","Text":"the probability of taking out 5 balls where only the last ball was green."},{"Start":"03:47.420 ","End":"03:52.150","Text":"That means that the first 4 balls had to be another color."},{"Start":"03:52.150 ","End":"03:55.310","Text":"Because the other colors have failures,"},{"Start":"03:55.310 ","End":"03:58.820","Text":"then we have to multiply the probability of failure 4"},{"Start":"03:58.820 ","End":"04:03.830","Text":"times before multiplying the probability of success."},{"Start":"04:03.830 ","End":"04:06.170","Text":"That\u0027s 0.7 times 0.7,"},{"Start":"04:06.170 ","End":"04:08.390","Text":"4 times 0.7 to the fourth,"},{"Start":"04:08.390 ","End":"04:13.740","Text":"times 0.3 which is the probability of success."},{"Start":"04:13.990 ","End":"04:20.945","Text":"In section c, we\u0027re asked what\u0027s the probability that more than 5 balls were removed?"},{"Start":"04:20.945 ","End":"04:27.455","Text":"Basically we\u0027re asked, what\u0027s the probability that x is greater than 5?"},{"Start":"04:27.455 ","End":"04:29.300","Text":"In order to answer that,"},{"Start":"04:29.300 ","End":"04:33.875","Text":"let\u0027s take a look at 1 of the special characteristics of the geometric distribution."},{"Start":"04:33.875 ","End":"04:38.120","Text":"Here it is right here. Let\u0027s take a look at it."},{"Start":"04:38.120 ","End":"04:40.870","Text":"It\u0027s this thing right here."},{"Start":"04:41.450 ","End":"04:47.060","Text":"That\u0027s the probability of x being greater than k that equals to q,"},{"Start":"04:47.060 ","End":"04:54.579","Text":"the probability of failure to the power of k. Let\u0027s copy this back into our example."},{"Start":"04:54.579 ","End":"05:01.165","Text":"That\u0027s the probability of x being greater than k,"},{"Start":"05:01.165 ","End":"05:08.259","Text":"equals to q to the power of k. Let\u0027s just answer this question,"},{"Start":"05:08.259 ","End":"05:12.680","Text":"the probability of x being greater than 5, that\u0027s q,"},{"Start":"05:12.680 ","End":"05:18.480","Text":"that\u0027s 0.7 to the power of 5."},{"Start":"05:20.200 ","End":"05:24.680","Text":"Let\u0027s take a look at the rationale behind this equation right here."},{"Start":"05:24.680 ","End":"05:29.835","Text":"We\u0027re asked, what\u0027s the probability of removing more than 5 balls?"},{"Start":"05:29.835 ","End":"05:31.625","Text":"I\u0027m going to have to ask myself,"},{"Start":"05:31.625 ","End":"05:34.265","Text":"when do I remove more than 5 balls."},{"Start":"05:34.265 ","End":"05:38.480","Text":"What\u0027s basically when the first 5 balls weren\u0027t green?"},{"Start":"05:38.480 ","End":"05:41.720","Text":"If I told you that I removed 5 balls that aren\u0027t green?"},{"Start":"05:41.720 ","End":"05:45.530","Text":"What does that tell you about the number of balls that I did remove?"},{"Start":"05:45.530 ","End":"05:47.900","Text":"It\u0027s definitely more than 5,"},{"Start":"05:47.900 ","End":"05:50.315","Text":"I don\u0027t know if it\u0027s 6 or 7 or 8,"},{"Start":"05:50.315 ","End":"05:58.250","Text":"but that\u0027s because I keep on removing balls until I succeed or until I get the green 1."},{"Start":"05:58.250 ","End":"06:04.920","Text":"But since I don\u0027t know how many balls I will remove until I succeed,"},{"Start":"06:04.920 ","End":"06:10.945","Text":"all I do know is that I removed 5 balls and I haven\u0027t succeeded."},{"Start":"06:10.945 ","End":"06:18.575","Text":"The probability of removing more than 5 balls would be 0.7 to the power of 5."},{"Start":"06:18.575 ","End":"06:23.090","Text":"That\u0027s the rationale behind this."},{"Start":"06:23.090 ","End":"06:24.455","Text":"Now if I was asked,"},{"Start":"06:24.455 ","End":"06:27.620","Text":"what\u0027s the probability of x equaling 5?"},{"Start":"06:27.620 ","End":"06:32.455","Text":"Then I would go back to this equation right here."},{"Start":"06:32.455 ","End":"06:37.250","Text":"But no, I was asked about this equation where x is"},{"Start":"06:37.250 ","End":"06:41.495","Text":"greater than 5 so I had to refer to this characteristic,"},{"Start":"06:41.495 ","End":"06:45.671","Text":"the special characteristic of the geometric distribution."},{"Start":"06:45.671 ","End":"06:50.130","Text":"This would be the answer."}],"ID":13013},{"Watched":false,"Name":"Example 1 Parts d-e","Duration":"8m 33s","ChapterTopicVideoID":12535,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.860","Text":"In Section d, we\u0027re asked,"},{"Start":"00:01.860 ","End":"00:04.214","Text":"if more than 3 balls were removed,"},{"Start":"00:04.214 ","End":"00:07.890","Text":"what are the chances that exactly 5 balls were removed?"},{"Start":"00:07.890 ","End":"00:13.695","Text":"Well, this looks like a conditional probability question. Let\u0027s get to it."},{"Start":"00:13.695 ","End":"00:17.310","Text":"We\u0027re asked, what\u0027s the probability,"},{"Start":"00:17.310 ","End":"00:21.060","Text":"what are the chances that exactly 5 balls were removed?"},{"Start":"00:21.060 ","End":"00:23.220","Text":"That means x equals 5."},{"Start":"00:23.220 ","End":"00:25.050","Text":"Given what?"},{"Start":"00:25.050 ","End":"00:30.945","Text":"If more than 3 balls were removed given that x is greater than 3."},{"Start":"00:30.945 ","End":"00:33.825","Text":"We know how to solve these guys."},{"Start":"00:33.825 ","End":"00:42.725","Text":"That\u0027s equals to the probability of the intersect of x equals 5 intersect,"},{"Start":"00:42.725 ","End":"00:53.580","Text":"x is greater than 3 divided by the probability that x is greater than 3."},{"Start":"00:55.690 ","End":"00:58.310","Text":"Now, what does that equal to?"},{"Start":"00:58.310 ","End":"01:01.590","Text":"Well, the probability of x being greater than 3,"},{"Start":"01:01.590 ","End":"01:08.810","Text":"we\u0027ve solved that in Section c. We use this characteristic of the geometric distribution."},{"Start":"01:08.810 ","End":"01:11.465","Text":"That equals to q^k,"},{"Start":"01:11.465 ","End":"01:16.865","Text":"the probability of x is greater than k is equal to q^k."},{"Start":"01:16.865 ","End":"01:19.380","Text":"Here, k is 3."},{"Start":"01:20.300 ","End":"01:23.150","Text":"In the denominator, we have q,"},{"Start":"01:23.150 ","End":"01:27.710","Text":"which is 0.7^k, k is 3."},{"Start":"01:27.710 ","End":"01:33.000","Text":"Now, what\u0027s this thing right here."},{"Start":"01:33.000 ","End":"01:38.010","Text":"The intersect between x equals 5 and x is greater than 3."},{"Start":"01:38.010 ","End":"01:39.825","Text":"Well, let\u0027s take a look."},{"Start":"01:39.825 ","End":"01:41.630","Text":"If x is greater than 3,"},{"Start":"01:41.630 ","End":"01:43.445","Text":"then x would equal 4,"},{"Start":"01:43.445 ","End":"01:46.490","Text":"5, 6, and so on and so forth."},{"Start":"01:46.490 ","End":"01:50.635","Text":"Now, when x equals 5, that\u0027s right here."},{"Start":"01:50.635 ","End":"01:54.440","Text":"The intersect of the numbers 4,"},{"Start":"01:54.440 ","End":"01:56.075","Text":"5, 6 to infinity,"},{"Start":"01:56.075 ","End":"01:58.310","Text":"and x being equal to 5."},{"Start":"01:58.310 ","End":"02:00.500","Text":"Well, that\u0027s x being equal to 5."},{"Start":"02:00.500 ","End":"02:05.065","Text":"That\u0027s the probability of x being equal to 5."},{"Start":"02:05.065 ","End":"02:07.300","Text":"This is the intersect."},{"Start":"02:07.300 ","End":"02:11.270","Text":"Now, what\u0027s the probability of x being equal to 5"},{"Start":"02:11.270 ","End":"02:16.295","Text":"when x is distributed with the geometric distribution."},{"Start":"02:16.295 ","End":"02:20.400","Text":"Well, that\u0027s this equation right here."},{"Start":"02:20.400 ","End":"02:26.930","Text":"The probability of x being equal to k equals p times q^k minus 1,"},{"Start":"02:26.930 ","End":"02:30.565","Text":"where k here equals to 5,"},{"Start":"02:30.565 ","End":"02:35.100","Text":"p is 0.3 and q is 0.7."},{"Start":"02:35.100 ","End":"02:36.900","Text":"We have p,"},{"Start":"02:36.900 ","End":"02:41.870","Text":"that\u0027s 0.3 times 0.7^k minus 1,"},{"Start":"02:41.870 ","End":"02:47.845","Text":"that\u0027s 4, divided by 0.7^3."},{"Start":"02:47.845 ","End":"02:51.180","Text":"Well, that equals to 0.3 times,"},{"Start":"02:51.180 ","End":"02:52.940","Text":"this cancels each other out,"},{"Start":"02:52.940 ","End":"02:59.670","Text":"so that\u0027s 0.7 and that equals to 0.21."},{"Start":"03:00.030 ","End":"03:05.350","Text":"Now, we can solve this question a little bit differently because we"},{"Start":"03:05.350 ","End":"03:10.210","Text":"can use 1 of the special characteristics of the geometric distribution."},{"Start":"03:10.210 ","End":"03:12.370","Text":"Let\u0027s take a look at it."},{"Start":"03:12.370 ","End":"03:14.950","Text":"Here it is, right here."},{"Start":"03:14.950 ","End":"03:18.895","Text":"The probability where x equals n plus k,"},{"Start":"03:18.895 ","End":"03:21.355","Text":"given that x is greater than k,"},{"Start":"03:21.355 ","End":"03:24.660","Text":"well that equals the probability of x being equal to"},{"Start":"03:24.660 ","End":"03:31.130","Text":"n. Let\u0027s just copy this back down and see how we can work with this."},{"Start":"03:31.190 ","End":"03:42.203","Text":"We have the probability of x being equal to n plus k given that x is greater than k."},{"Start":"03:42.203 ","End":"03:47.555","Text":"That equals to the probability of x being equal to"},{"Start":"03:47.555 ","End":"03:55.950","Text":"n. If k equals 3 from here,"},{"Start":"03:55.950 ","End":"04:01.940","Text":"n would equal to 2 because 2 plus 3 equals 5."},{"Start":"04:01.940 ","End":"04:04.610","Text":"That would equal to the probability of x being equal to"},{"Start":"04:04.610 ","End":"04:08.570","Text":"n. The probability of x being equal to n,"},{"Start":"04:08.570 ","End":"04:14.900","Text":"that\u0027s 0.3 times 0.7^n equals 2."},{"Start":"04:14.900 ","End":"04:17.240","Text":"It\u0027s 2 minus 1, that\u0027ll be 1."},{"Start":"04:17.240 ","End":"04:22.205","Text":"That equals to 0.21. There you go."},{"Start":"04:22.205 ","End":"04:25.700","Text":"I have the same answer as I did when I did it the"},{"Start":"04:25.700 ","End":"04:29.375","Text":"long way instead of using the characteristic."},{"Start":"04:29.375 ","End":"04:33.515","Text":"Now, let\u0027s just see what\u0027s the rationale behind this."},{"Start":"04:33.515 ","End":"04:38.875","Text":"Now, assume that I have 5 balls,"},{"Start":"04:38.875 ","End":"04:41.295","Text":"1, 2,"},{"Start":"04:41.295 ","End":"04:45.250","Text":"3, 4 and 5."},{"Start":"04:45.250 ","End":"04:48.980","Text":"I say, what\u0027s the probability of getting it"},{"Start":"04:48.980 ","End":"04:53.240","Text":"right on the 5th ball when x is greater than 3?"},{"Start":"04:53.240 ","End":"04:55.445","Text":"What does x is greater than 3 mean?"},{"Start":"04:55.445 ","End":"04:58.865","Text":"It means that in the first 3 balls,"},{"Start":"04:58.865 ","End":"05:00.725","Text":"I didn\u0027t get a green 1,"},{"Start":"05:00.725 ","End":"05:07.855","Text":"so in essence, I can take these guys right here and I can disregard them."},{"Start":"05:07.855 ","End":"05:11.765","Text":"Now, having disregarded them, what I\u0027m asked,"},{"Start":"05:11.765 ","End":"05:16.675","Text":"I\u0027m asked what\u0027s the probability of getting this guy right?"},{"Start":"05:16.675 ","End":"05:26.295","Text":"Well, that\u0027s exactly the probability of getting x on the 2nd try."},{"Start":"05:26.295 ","End":"05:29.370","Text":"Not the 1st 1, but the 2nd 1,"},{"Start":"05:29.370 ","End":"05:32.960","Text":"and that\u0027s exactly what this characteristic means."},{"Start":"05:32.960 ","End":"05:38.884","Text":"It\u0027s basically called the characteristic where x is memory less,"},{"Start":"05:38.884 ","End":"05:45.355","Text":"where we disregard the first 3 balls right here where k is greater than 3,"},{"Start":"05:45.355 ","End":"05:49.550","Text":"and we just take a look at these 2 guys,"},{"Start":"05:49.550 ","End":"05:53.630","Text":"the remainder of the balls that we have to take a look where we basically"},{"Start":"05:53.630 ","End":"05:58.600","Text":"forgot about the 3 balls because we know that we didn\u0027t succeed with them."},{"Start":"05:58.600 ","End":"06:04.145","Text":"Now, this is 1 of the major characteristics of that geometric distribution."},{"Start":"06:04.145 ","End":"06:10.580","Text":"But I prefer you doing this the long way around."},{"Start":"06:10.580 ","End":"06:14.840","Text":"Don\u0027t be lazy, do it the long way around so you can understand exactly"},{"Start":"06:14.840 ","End":"06:18.964","Text":"what\u0027s going on instead of using the shortcut right here."},{"Start":"06:18.964 ","End":"06:21.095","Text":"Know that it exists,"},{"Start":"06:21.095 ","End":"06:24.680","Text":"but do it the long way around so you\u0027ll know that"},{"Start":"06:24.680 ","End":"06:28.590","Text":"you\u0027re doing it right and don\u0027t be lazy with this."},{"Start":"06:28.590 ","End":"06:31.970","Text":"Good luck and let\u0027s go on to the next section,"},{"Start":"06:31.970 ","End":"06:36.080","Text":"Section e. In Section e we\u0027re asked,"},{"Start":"06:36.080 ","End":"06:41.250","Text":"what is the expectation and standard deviation of the number of balls removed?"},{"Start":"06:41.740 ","End":"06:51.680","Text":"Let\u0027s just remember our equations for the expectation and the variance of x."},{"Start":"06:51.680 ","End":"06:55.700","Text":"If x is distributed with the geometric distribution,"},{"Start":"06:55.700 ","End":"07:02.190","Text":"then the expectation of x would be 1 over p,"},{"Start":"07:02.190 ","End":"07:09.730","Text":"and the variance of x would be q over p squared."},{"Start":"07:09.730 ","End":"07:11.960","Text":"Now, for the expectation,"},{"Start":"07:11.960 ","End":"07:13.970","Text":"we know that p equals 0.3,"},{"Start":"07:13.970 ","End":"07:22.900","Text":"so the expectation would be 1 over 0.3 and that equals to 3 and 1/3."},{"Start":"07:22.900 ","End":"07:27.500","Text":"In our little game of removing balls from a basket,"},{"Start":"07:27.500 ","End":"07:29.270","Text":"until we get a green 1,"},{"Start":"07:29.270 ","End":"07:35.590","Text":"we\u0027re expected to remove 3 and 1/3 balls before we get a green 1."},{"Start":"07:35.590 ","End":"07:38.180","Text":"Now, what\u0027s the variance?"},{"Start":"07:38.180 ","End":"07:40.100","Text":"Well, that\u0027s q over p squared."},{"Start":"07:40.100 ","End":"07:45.660","Text":"Now, q is 0.7 over 0.3"},{"Start":"07:45.660 ","End":"07:53.815","Text":"squared and that equals to 7 and 7/9."},{"Start":"07:53.815 ","End":"07:56.855","Text":"The standard deviation of x,"},{"Start":"07:56.855 ","End":"08:00.380","Text":"well that equals to the square root of the variance,"},{"Start":"08:00.380 ","End":"08:02.660","Text":"which is 7 and 7/9,"},{"Start":"08:02.660 ","End":"08:07.720","Text":"and that equals to 2.79."},{"Start":"08:07.720 ","End":"08:10.040","Text":"Now, what does that mean?"},{"Start":"08:10.040 ","End":"08:18.590","Text":"That means that although we\u0027re expected to remove 3 in 1/3 balls until we get a green 1,"},{"Start":"08:18.590 ","End":"08:26.005","Text":"here we\u0027re expected to be 2.79 balls away from the expectation."},{"Start":"08:26.005 ","End":"08:28.580","Text":"Here we answered the question,"},{"Start":"08:28.580 ","End":"08:34.350","Text":"what\u0027s the expectation and what\u0027s the variance of x?"}],"ID":13014},{"Watched":false,"Name":"Exercise 1","Duration":"5m 31s","ChapterTopicVideoID":12536,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"In this question, we\u0027ll be asked about quality assurance in a manufacturing line."},{"Start":"00:05.535 ","End":"00:08.925","Text":"Now, a mass production line manufacturers products,"},{"Start":"00:08.925 ","End":"00:11.160","Text":"5 percent of which are faulty."},{"Start":"00:11.160 ","End":"00:14.310","Text":"A quality assurance experts samples products from"},{"Start":"00:14.310 ","End":"00:17.110","Text":"the production line until he finds a faulty product,"},{"Start":"00:17.110 ","End":"00:20.670","Text":"and we\u0027re asked to calculate the following probabilities where the"},{"Start":"00:20.670 ","End":"00:24.555","Text":"QA experts samples 3 products or 4 or 5 products,"},{"Start":"00:24.555 ","End":"00:27.885","Text":"or when he samples more than 7 products,"},{"Start":"00:27.885 ","End":"00:29.820","Text":"or at least 8 products."},{"Start":"00:29.820 ","End":"00:35.685","Text":"Now, here we have a geometric probability distribution, and why is that?"},{"Start":"00:35.685 ","End":"00:39.045","Text":"Well, let\u0027s take a look at the characteristics."},{"Start":"00:39.045 ","End":"00:41.150","Text":"Here are the characteristics."},{"Start":"00:41.150 ","End":"00:49.625","Text":"The characteristics are that we have the same Bernoulli trial that\u0027s repeated n times,"},{"Start":"00:49.625 ","End":"00:53.185","Text":"where each trial is independent of each other."},{"Start":"00:53.185 ","End":"00:57.245","Text":"The 2nd characteristic is that x is defined as the number of"},{"Start":"00:57.245 ","End":"01:01.465","Text":"trials carried out until the first success."},{"Start":"01:01.465 ","End":"01:05.750","Text":"What do we have here in our case?"},{"Start":"01:05.750 ","End":"01:11.060","Text":"Well, each Bernoulli trial is basically a product that\u0027s sampled."},{"Start":"01:11.310 ","End":"01:21.870","Text":"Let\u0027s define our success as equal to a faulty product."},{"Start":"01:22.480 ","End":"01:26.224","Text":"Now, what\u0027s the probability of success?"},{"Start":"01:26.224 ","End":"01:30.500","Text":"Well, that\u0027s 5 percent, 0.05."},{"Start":"01:30.500 ","End":"01:36.294","Text":"Now each time we remove a product,"},{"Start":"01:36.294 ","End":"01:41.270","Text":"it\u0027s independent of any other time that we remove a product."},{"Start":"01:41.270 ","End":"01:43.100","Text":"We have independence,"},{"Start":"01:43.100 ","End":"01:44.735","Text":"we have a Bernoulli trial,"},{"Start":"01:44.735 ","End":"01:46.100","Text":"and what do we want to know?"},{"Start":"01:46.100 ","End":"01:50.220","Text":"We want to define a random variable."},{"Start":"01:50.220 ","End":"01:52.370","Text":"What\u0027s a random variable?"},{"Start":"01:52.370 ","End":"01:56.525","Text":"That would be the number of products"},{"Start":"01:56.525 ","End":"02:04.200","Text":"sampled until we find a faulty 1."},{"Start":"02:04.200 ","End":"02:09.620","Text":"We know here that x is distributed geometrically"},{"Start":"02:09.620 ","End":"02:15.335","Text":"with a probability of 0.05 percent."},{"Start":"02:15.335 ","End":"02:21.030","Text":"Now, what\u0027s the equation for geometric probability?"},{"Start":"02:21.030 ","End":"02:26.785","Text":"Well, that\u0027s q^k minus 1 times p,"},{"Start":"02:26.785 ","End":"02:31.765","Text":"or the other way around, p times q^x minus 1."},{"Start":"02:31.765 ","End":"02:33.515","Text":"Once we have that,"},{"Start":"02:33.515 ","End":"02:37.400","Text":"then let\u0027s take a look at the sections here."},{"Start":"02:37.400 ","End":"02:39.980","Text":"Section 1 as the QA expert,"},{"Start":"02:39.980 ","End":"02:43.580","Text":"what\u0027s the probability that the QA experts samples 3 products?"},{"Start":"02:43.580 ","End":"02:46.100","Text":"Well, in our case,"},{"Start":"02:46.100 ","End":"02:51.725","Text":"p, where x equals 3, what\u0027s q?"},{"Start":"02:51.725 ","End":"02:53.360","Text":"Q is 1 minus p,"},{"Start":"02:53.360 ","End":"03:02.160","Text":"which is 0.95^2 times 0.05,"},{"Start":"03:02.390 ","End":"03:09.555","Text":"and what\u0027s the probability of x being equal to 4?"},{"Start":"03:09.555 ","End":"03:14.655","Text":"Well, 0.95^k minus 1, that\u0027s 3,"},{"Start":"03:14.655 ","End":"03:21.045","Text":"4 minus 1 is 3 times 0.05."},{"Start":"03:21.045 ","End":"03:26.685","Text":"What\u0027s the probability of x being equal to 5?"},{"Start":"03:26.685 ","End":"03:30.030","Text":"Well again, this is simple, 0.95^4,"},{"Start":"03:30.030 ","End":"03:37.360","Text":"5 minus 1 times 0.05."},{"Start":"03:38.150 ","End":"03:46.790","Text":"Let\u0027s continue. What\u0027s the probability that the QA samples more than 7 products?"},{"Start":"03:46.790 ","End":"03:51.890","Text":"That means that the probability of x is greater than 7."},{"Start":"03:51.890 ","End":"03:53.630","Text":"Now, if you remember,"},{"Start":"03:53.630 ","End":"04:00.185","Text":"1 of the special characteristics of a geometric probability is that when p,"},{"Start":"04:00.185 ","End":"04:03.980","Text":"the probability of x being greater than k,"},{"Start":"04:03.980 ","End":"04:07.360","Text":"that equals to q^k."},{"Start":"04:07.360 ","End":"04:10.260","Text":"Now, in our case,"},{"Start":"04:10.260 ","End":"04:16.470","Text":"what\u0027s q? That\u0027s 0.95^7."},{"Start":"04:16.470 ","End":"04:23.535","Text":"Now let\u0027s take a look at e. The QA expert samples at least 8 products."},{"Start":"04:23.535 ","End":"04:29.765","Text":"The probability of x being greater or equal to 8."},{"Start":"04:29.765 ","End":"04:36.470","Text":"Now, here we have a problem because we want to use this characteristic,"},{"Start":"04:36.470 ","End":"04:42.770","Text":"but we\u0027re told that the probability of x being greater than k,"},{"Start":"04:42.770 ","End":"04:44.585","Text":"not greater or equal to."},{"Start":"04:44.585 ","End":"04:46.895","Text":"So here we have a problem,"},{"Start":"04:46.895 ","End":"04:52.780","Text":"but we still can use this because if x is greater or equal to 8,"},{"Start":"04:52.780 ","End":"04:56.560","Text":"that means that x is equal to 8,"},{"Start":"04:56.560 ","End":"04:58.330","Text":"or 9, or 10,"},{"Start":"04:58.330 ","End":"05:00.165","Text":"and so on and so forth."},{"Start":"05:00.165 ","End":"05:07.475","Text":"But that\u0027s comparable to saying that x is greater than 7,"},{"Start":"05:07.475 ","End":"05:11.010","Text":"because if x is greater than 7 then it equals to 8,"},{"Start":"05:11.010 ","End":"05:13.040","Text":"or 9, or 10, and so on and so forth,"},{"Start":"05:13.040 ","End":"05:20.135","Text":"so t1hat equals to the probability of x being greater than 7,"},{"Start":"05:20.135 ","End":"05:25.670","Text":"but that\u0027s exactly what we calculated here in Section d."},{"Start":"05:25.670 ","End":"05:32.920","Text":"Again, that\u0027s 0.95^7."}],"ID":13015},{"Watched":false,"Name":"Exercise 2","Duration":"9m 49s","ChapterTopicVideoID":12537,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.125","Text":"In this problem, we\u0027ll be talking about X-rays."},{"Start":"00:04.125 ","End":"00:08.970","Text":"The probability of an X-ray negative being in good order is 0.9."},{"Start":"00:08.970 ","End":"00:12.525","Text":"A person goes to the Imaging Institute for an X-ray."},{"Start":"00:12.525 ","End":"00:16.740","Text":"He leaves only when he has an X-ray negative in good order."},{"Start":"00:16.740 ","End":"00:20.040","Text":"We\u0027re asked, what is the probability of being X-rayed"},{"Start":"00:20.040 ","End":"00:24.510","Text":"exactly 3 times or more than 4 times and so on and so forth?"},{"Start":"00:24.510 ","End":"00:30.125","Text":"Well, here we have a geometric distribution. Why is that?"},{"Start":"00:30.125 ","End":"00:36.360","Text":"Well, let\u0027s take a look at the characteristics of a geometric distribution and prove it."},{"Start":"00:36.860 ","End":"00:39.470","Text":"Here are the characteristics."},{"Start":"00:39.470 ","End":"00:45.050","Text":"Now the first 1 says that we need a Bernoulli trial that\u0027s repeated n times,"},{"Start":"00:45.050 ","End":"00:49.220","Text":"where each trial is independent of the other."},{"Start":"00:49.220 ","End":"00:52.940","Text":"Well, here we do have a Bernoulli trial."},{"Start":"00:52.940 ","End":"00:59.030","Text":"What\u0027s the success or how do we define success in a Bernoulli trial?"},{"Start":"00:59.030 ","End":"01:08.620","Text":"Here we define success as a good photo."},{"Start":"01:09.230 ","End":"01:12.015","Text":"A good X-ray."},{"Start":"01:12.015 ","End":"01:16.870","Text":"What\u0027s the probability of this good photo?"},{"Start":"01:16.870 ","End":"01:20.885","Text":"That would be 0.9."},{"Start":"01:20.885 ","End":"01:23.145","Text":"Yeah, let\u0027s just write this out,"},{"Start":"01:23.145 ","End":"01:25.715","Text":"that equals to 0.9."},{"Start":"01:25.715 ","End":"01:27.865","Text":"Now, what else do we need to know?"},{"Start":"01:27.865 ","End":"01:31.975","Text":"Well, the second characteristics is that"},{"Start":"01:31.975 ","End":"01:37.690","Text":"the X is being defined as the number of trials carried out until the first success."},{"Start":"01:37.690 ","End":"01:42.660","Text":"In our case, X would be the number of"},{"Start":"01:42.660 ","End":"01:51.695","Text":"X-rays a person has to undergo until he gets a good 1 and then he can go home."},{"Start":"01:51.695 ","End":"01:58.625","Text":"In that case, now we know that X is distributed with the geometric distribution,"},{"Start":"01:58.625 ","End":"02:03.185","Text":"where p equals 0.9."},{"Start":"02:03.185 ","End":"02:07.100","Text":"Now, just to make things clear,"},{"Start":"02:07.100 ","End":"02:11.045","Text":"the probability of X being equal to k. Well,"},{"Start":"02:11.045 ","End":"02:14.150","Text":"what\u0027s the equation for a geometric distribution?"},{"Start":"02:14.150 ","End":"02:17.720","Text":"Well, that\u0027s q to the power of k minus 1"},{"Start":"02:17.720 ","End":"02:24.860","Text":"times p. Just to make things a little bit more clear,"},{"Start":"02:24.860 ","End":"02:28.010","Text":"q here would be equal to 0.1,"},{"Start":"02:28.010 ","End":"02:34.100","Text":"that\u0027s 1 minus p. Let\u0027s answer the first section here."},{"Start":"02:34.100 ","End":"02:38.300","Text":"What\u0027s the probability of being X-rayed exactly 3 times?"},{"Start":"02:38.300 ","End":"02:43.000","Text":"That means the probability of X being equal to 3."},{"Start":"02:43.000 ","End":"02:45.735","Text":"Now, let\u0027s plug in the numbers. What\u0027s q?"},{"Start":"02:45.735 ","End":"02:47.130","Text":"q, that\u0027s 0.1,"},{"Start":"02:47.130 ","End":"02:50.450","Text":"0.1 to the power of k minus 1."},{"Start":"02:50.450 ","End":"02:52.640","Text":"Well that\u0027s 3 minus 1, that\u0027s 2,"},{"Start":"02:52.640 ","End":"03:03.970","Text":"times p, and that\u0027s 0.9 and that equals to 0.009."},{"Start":"03:04.370 ","End":"03:07.610","Text":"Let\u0027s go to Section B."},{"Start":"03:07.610 ","End":"03:13.135","Text":"Section B asks what\u0027s the probability of being X-rayed more than 4 times?"},{"Start":"03:13.135 ","End":"03:19.905","Text":"That means that we\u0027re looking at the probability of X being more than 4."},{"Start":"03:19.905 ","End":"03:22.600","Text":"Now, if we remember one of the characteristics of"},{"Start":"03:22.600 ","End":"03:29.915","Text":"the geometric distribution is that the probability where X is greater than k,"},{"Start":"03:29.915 ","End":"03:36.405","Text":"that equals to q to the power of k. In our case, what\u0027s q?"},{"Start":"03:36.405 ","End":"03:41.775","Text":"q, that\u0027s 0.1 to the power of 4. k equals 4."},{"Start":"03:41.775 ","End":"03:46.510","Text":"Now that equals to 0.0001."},{"Start":"03:49.860 ","End":"03:56.600","Text":"Section C asks what are the expectation and variance of the number of X-rays carried out?"},{"Start":"03:57.680 ","End":"04:01.750","Text":"Let\u0027s see the expectation of X,"},{"Start":"04:01.750 ","End":"04:08.910","Text":"where X is defined with the geometric distribution that equals to 1 over p. In our case,"},{"Start":"04:08.910 ","End":"04:16.270","Text":"that\u0027s 1 over 0.9 and that equals to 1.11 photos."},{"Start":"04:16.890 ","End":"04:24.580","Text":"That means that a person takes 1.11 X-rays before he\u0027s allowed to go home."},{"Start":"04:24.580 ","End":"04:29.615","Text":"Now, what\u0027s the variance of X?"},{"Start":"04:29.615 ","End":"04:36.275","Text":"Well, the variance of X is defined as q over p squared,"},{"Start":"04:36.275 ","End":"04:41.195","Text":"and that equals to 0.1 over 0.9 squared."},{"Start":"04:41.195 ","End":"04:48.680","Text":"Now, that equals to 0.123 photos squared."},{"Start":"04:48.680 ","End":"04:51.410","Text":"Now, that\u0027s not an intuitive unit,"},{"Start":"04:51.410 ","End":"04:54.170","Text":"but that\u0027s the units of the variance."},{"Start":"04:54.170 ","End":"05:00.050","Text":"Here we\u0027ve answered the question for the variance and the expectation of X."},{"Start":"05:00.050 ","End":"05:06.260","Text":"In section D, we\u0027re given every X-ray costs the Imaging Institute $50."},{"Start":"05:06.260 ","End":"05:09.970","Text":"A person pays the Institute a $100 for an X-ray in good order."},{"Start":"05:09.970 ","End":"05:12.920","Text":"We\u0027re asked, what\u0027s the expectation and variance of"},{"Start":"05:12.920 ","End":"05:17.150","Text":"the Institute\u0027s profit from a person who goes there for an X-ray?"},{"Start":"05:17.150 ","End":"05:20.750","Text":"Well, whenever somebody asks me for the expectation variance,"},{"Start":"05:20.750 ","End":"05:22.205","Text":"I have to ask, what?"},{"Start":"05:22.205 ","End":"05:24.710","Text":"Let\u0027s just read it out."},{"Start":"05:24.710 ","End":"05:26.585","Text":"This is the profits,"},{"Start":"05:26.585 ","End":"05:31.800","Text":"were looking for the profits of a person who goes for an X-ray."},{"Start":"05:32.390 ","End":"05:38.750","Text":"I think we pretty sure that we have a linear transformation, and why is that?"},{"Start":"05:38.750 ","End":"05:43.470","Text":"Because we\u0027re going from X which is the number of X-rays."},{"Start":"05:43.940 ","End":"05:48.270","Text":"We need to find out what is the profits."},{"Start":"05:48.270 ","End":"05:50.920","Text":"That\u0027s right here, that\u0027s the profit."},{"Start":"05:51.190 ","End":"05:54.275","Text":"Now, the minute we have a linear transformation,"},{"Start":"05:54.275 ","End":"06:00.785","Text":"let\u0027s just look at the steps associated with a linear transformation. Here they are."},{"Start":"06:00.785 ","End":"06:03.695","Text":"Now, let\u0027s look at the first step."},{"Start":"06:03.695 ","End":"06:05.990","Text":"First step is to recognize that you\u0027re"},{"Start":"06:05.990 ","End":"06:08.210","Text":"dealing with a linear transformation. Well, we are."},{"Start":"06:08.210 ","End":"06:10.700","Text":"We\u0027re taking the X-rays,"},{"Start":"06:10.700 ","End":"06:11.780","Text":"the number of X-rays,"},{"Start":"06:11.780 ","End":"06:16.350","Text":"and want to transform them into profits."},{"Start":"06:17.210 ","End":"06:22.595","Text":"What\u0027s the next step? Let\u0027s write the transformation rule according to the data."},{"Start":"06:22.595 ","End":"06:24.109","Text":"Well, what are profits?"},{"Start":"06:24.109 ","End":"06:28.400","Text":"Profits are basically income minus expenses."},{"Start":"06:28.400 ","End":"06:30.830","Text":"Why the profits?"},{"Start":"06:30.830 ","End":"06:35.000","Text":"That\u0027s our income minus expenses. What are our income?"},{"Start":"06:35.000 ","End":"06:39.605","Text":"A person pays the institute a $100 for an X-ray in good order."},{"Start":"06:39.605 ","End":"06:44.220","Text":"Our income is a $100 minus expenses, minus what?"},{"Start":"06:44.220 ","End":"06:50.210","Text":"Well, how much does it cost the Imaging Institute for an X-ray?"},{"Start":"06:50.210 ","End":"06:55.035","Text":"Well, each X-ray cost $50, right?"},{"Start":"06:55.035 ","End":"06:59.075","Text":"We have to multiply that by the number of X-rays taken."},{"Start":"06:59.075 ","End":"07:05.390","Text":"Now, let\u0027s take a look at the general form of a linear transformation."},{"Start":"07:05.390 ","End":"07:10.770","Text":"That\u0027s y equals ax plus b."},{"Start":"07:10.770 ","End":"07:13.320","Text":"That\u0027s basically step number 3,"},{"Start":"07:13.320 ","End":"07:16.890","Text":"we have to simplify the rule and identify the values of a and b."},{"Start":"07:16.890 ","End":"07:21.700","Text":"In order to bring this expression into this form,"},{"Start":"07:21.700 ","End":"07:24.535","Text":"then let\u0027s just read this out again."},{"Start":"07:24.535 ","End":"07:29.050","Text":"That\u0027ll be minus 50x plus a 100."},{"Start":"07:29.050 ","End":"07:32.950","Text":"Now in this form, we can easily identify that a equals minus"},{"Start":"07:32.950 ","End":"07:38.740","Text":"50 and b equals 100. a minus 50,"},{"Start":"07:38.740 ","End":"07:40.090","Text":"that\u0027s the multiplier,"},{"Start":"07:40.090 ","End":"07:42.220","Text":"that\u0027s right here, the multiplier of x,"},{"Start":"07:42.220 ","End":"07:47.520","Text":"and b is this guy right here, that\u0027s 100."},{"Start":"07:47.520 ","End":"07:50.485","Text":"Now, what else do we need to know?"},{"Start":"07:50.485 ","End":"07:54.560","Text":"Well, we need to know what the expectation of X is."},{"Start":"07:54.560 ","End":"07:57.020","Text":"Now we calculated then the previous section,"},{"Start":"07:57.020 ","End":"08:01.355","Text":"that\u0027s 1.11 and the variance of X."},{"Start":"08:01.355 ","End":"08:06.415","Text":"Well that equals 0.123."},{"Start":"08:06.415 ","End":"08:08.820","Text":"Now I think we\u0027re ready."},{"Start":"08:08.820 ","End":"08:11.940","Text":"The expectation of what?"},{"Start":"08:11.940 ","End":"08:16.070","Text":"Of a linear transformation that equals to a times"},{"Start":"08:16.070 ","End":"08:21.080","Text":"the expectation of X plus b. You remember that?"},{"Start":"08:21.080 ","End":"08:23.690","Text":"Now, in our case,"},{"Start":"08:23.690 ","End":"08:28.160","Text":"a is minus 50, that\u0027s minus 50 times the expectation of X."},{"Start":"08:28.160 ","End":"08:33.140","Text":"Well that\u0027s 1.11 plus b."},{"Start":"08:33.140 ","End":"08:37.325","Text":"What\u0027s b? b is a 100, plus 100."},{"Start":"08:37.325 ","End":"08:45.330","Text":"That comes out to 44.5."},{"Start":"08:45.330 ","End":"08:47.510","Text":"From every person that comes in,"},{"Start":"08:47.510 ","End":"08:54.065","Text":"the Imaging Institute expects to earn a profit of $44.5."},{"Start":"08:54.065 ","End":"08:56.980","Text":"Let\u0027s take a look at the variance of y."},{"Start":"08:56.980 ","End":"08:58.790","Text":"Again, if we recall,"},{"Start":"08:58.790 ","End":"09:00.860","Text":"the variance of a linear transformation,"},{"Start":"09:00.860 ","End":"09:04.715","Text":"that\u0027s a squared times the variance of X."},{"Start":"09:04.715 ","End":"09:08.390","Text":"In our case, a squared is minus 50,"},{"Start":"09:08.390 ","End":"09:13.970","Text":"that\u0027s minus 50 squared times the variance of X."},{"Start":"09:13.970 ","End":"09:17.910","Text":"The variance of X is 0.123,"},{"Start":"09:19.750 ","End":"09:28.370","Text":"and that comes out to $307.50."},{"Start":"09:28.370 ","End":"09:32.210","Text":"Now the units are dollars, that\u0027s dollars squared."},{"Start":"09:32.210 ","End":"09:34.490","Text":"Those are the units of the variance."},{"Start":"09:34.490 ","End":"09:37.400","Text":"Not very intuitive, but there we have it."},{"Start":"09:37.400 ","End":"09:41.570","Text":"We\u0027ve calculated the variance of Y,"},{"Start":"09:41.570 ","End":"09:45.710","Text":"and the variance and the expectation of Y right here."},{"Start":"09:45.710 ","End":"09:49.260","Text":"Answering section D here."}],"ID":13016},{"Watched":false,"Name":"Exercise 3 Parts a-b","Duration":"7m 47s","ChapterTopicVideoID":12538,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this problem, we\u0027ll be talking about tossing a coin."},{"Start":"00:04.050 ","End":"00:07.710","Text":"Now, a coin is tossed until tails is obtained."},{"Start":"00:07.710 ","End":"00:13.350","Text":"We\u0027re asked, what\u0027s the probability of tossing the coin at most 10 times?"},{"Start":"00:13.350 ","End":"00:17.265","Text":"Now here we\u0027re dealing with the geometric distribution."},{"Start":"00:17.265 ","End":"00:20.100","Text":"Why is that? Well, let\u0027s take a look at"},{"Start":"00:20.100 ","End":"00:24.540","Text":"the characteristics of the geometric distribution."},{"Start":"00:24.540 ","End":"00:26.760","Text":"Here are the characteristics."},{"Start":"00:26.760 ","End":"00:29.280","Text":"Well, let\u0027s look at the first 1."},{"Start":"00:29.280 ","End":"00:32.355","Text":"The same Bernoulli trial is repeated n times,"},{"Start":"00:32.355 ","End":"00:35.490","Text":"where each trial is independent of each other."},{"Start":"00:35.490 ","End":"00:39.866","Text":"Well, we know that each toss of the coin is independent of each other,"},{"Start":"00:39.866 ","End":"00:42.070","Text":"so do we have a Bernoulli trial or not?"},{"Start":"00:42.070 ","End":"00:43.760","Text":"Well, yes, we do."},{"Start":"00:43.760 ","End":"00:47.810","Text":"Because a Bernoulli trial deals with either success or"},{"Start":"00:47.810 ","End":"00:52.505","Text":"failure to trial that has success or failure with the appropriate probability."},{"Start":"00:52.505 ","End":"00:59.300","Text":"Now let\u0027s define success here as getting a tail, heads or tails."},{"Start":"00:59.300 ","End":"01:01.400","Text":"Now we\u0027re getting a tail, that\u0027s the success."},{"Start":"01:01.400 ","End":"01:04.265","Text":"What\u0027s the probability of getting a tail?"},{"Start":"01:04.265 ","End":"01:06.710","Text":"Well, let\u0027s put it this way."},{"Start":"01:06.710 ","End":"01:08.885","Text":"If the coin is a fair coin,"},{"Start":"01:08.885 ","End":"01:15.350","Text":"then the probability of getting tails is 0.5 or 50 percent."},{"Start":"01:15.350 ","End":"01:18.450","Text":"Now, what else do we know?"},{"Start":"01:18.450 ","End":"01:24.380","Text":"We define X as the number of trials carried out until the first success,"},{"Start":"01:24.380 ","End":"01:26.060","Text":"and this is what we\u0027re asked."},{"Start":"01:26.060 ","End":"01:29.645","Text":"A coin is tossed until tails is obtained."},{"Start":"01:29.645 ","End":"01:32.480","Text":"So X would be the number"},{"Start":"01:32.480 ","End":"01:41.955","Text":"of tosses until what?"},{"Start":"01:41.955 ","End":"01:46.120","Text":"Until tails, the first tail."},{"Start":"01:47.000 ","End":"01:52.010","Text":"If that\u0027s the case, then we know that X is distributed with"},{"Start":"01:52.010 ","End":"01:57.995","Text":"the geometric distribution where the probability is 0.5."},{"Start":"01:57.995 ","End":"02:00.830","Text":"Now, just to remind ourselves,"},{"Start":"02:00.830 ","End":"02:05.765","Text":"what\u0027s the probability of X equaling k in a geometric distribution?"},{"Start":"02:05.765 ","End":"02:11.870","Text":"Well, that\u0027s q to the power of k minus 1 times"},{"Start":"02:11.870 ","End":"02:18.170","Text":"p. Let\u0027s start with our first section."},{"Start":"02:18.170 ","End":"02:25.385","Text":"What\u0027s the probability of tossing a coin at most 10 times? Let\u0027s go."},{"Start":"02:25.385 ","End":"02:32.180","Text":"The probability where X is less than or equal to 10,"},{"Start":"02:32.180 ","End":"02:34.420","Text":"at most 10 times."},{"Start":"02:34.420 ","End":"02:41.600","Text":"Now, we could go and answer and calculate all the probabilities where X equals 1,"},{"Start":"02:41.600 ","End":"02:43.190","Text":"plug that in, X equals 2,"},{"Start":"02:43.190 ","End":"02:44.503","Text":"and so on and so forth,"},{"Start":"02:44.503 ","End":"02:47.045","Text":"but let\u0027s take the complimentary set,"},{"Start":"02:47.045 ","End":"02:48.895","Text":"I think it would be easier."},{"Start":"02:48.895 ","End":"02:56.365","Text":"Now that would be 1 minus the probability of X being greater than 10."},{"Start":"02:56.365 ","End":"03:00.800","Text":"Now, here we have a probability that we"},{"Start":"03:00.800 ","End":"03:06.605","Text":"know and we can use the special characteristics of a geometric distribution."},{"Start":"03:06.605 ","End":"03:11.665","Text":"We know that the probability where X is greater than k,"},{"Start":"03:11.665 ","End":"03:15.780","Text":"that equals to q^k."},{"Start":"03:15.780 ","End":"03:18.660","Text":"Great. If that\u0027s the case,"},{"Start":"03:18.660 ","End":"03:23.100","Text":"this equals 1 minus 0.5."},{"Start":"03:23.100 ","End":"03:26.085","Text":"Because if p is 0.5,"},{"Start":"03:26.085 ","End":"03:28.070","Text":"then 1 minus p,"},{"Start":"03:28.070 ","End":"03:33.170","Text":"which is equal to q, and that equals to 0.5 as well."},{"Start":"03:33.170 ","End":"03:36.620","Text":"That\u0027s 1 minus q^k."},{"Start":"03:36.620 ","End":"03:40.540","Text":"Well, that\u0027s 0.5^10,"},{"Start":"03:40.540 ","End":"03:46.720","Text":"and that equals to 0.999."},{"Start":"03:48.110 ","End":"03:50.495","Text":"In Section B, we\u0027re asked,"},{"Start":"03:50.495 ","End":"03:53.930","Text":"what\u0027s the probability of tossing the coin at most 5"},{"Start":"03:53.930 ","End":"03:58.655","Text":"times if it\u0027s known that the coin was tossed at least 3 times?"},{"Start":"03:58.655 ","End":"04:01.580","Text":"Well, here we have a conditional probability."},{"Start":"04:01.580 ","End":"04:05.000","Text":"What I\u0027ve written here is basically what we\u0027ve learned from"},{"Start":"04:05.000 ","End":"04:10.615","Text":"the question and from answering question A or section A."},{"Start":"04:10.615 ","End":"04:13.685","Text":"Let\u0027s get to work on section B."},{"Start":"04:13.685 ","End":"04:16.940","Text":"Now, since we have a conditional probability,"},{"Start":"04:16.940 ","End":"04:19.040","Text":"basically what we\u0027re asked is this,"},{"Start":"04:19.040 ","End":"04:24.950","Text":"what\u0027s the probability of X being less than or equal to 5?"},{"Start":"04:24.950 ","End":"04:30.035","Text":"That means the probability of tossing a coin at most 5 times,"},{"Start":"04:30.035 ","End":"04:34.985","Text":"given that it\u0027s known that the coin was tossed at least 3 times,"},{"Start":"04:34.985 ","End":"04:40.545","Text":"that means that X is greater or equal to 3."},{"Start":"04:40.545 ","End":"04:45.215","Text":"This is a conditional probability that we have to calculate."},{"Start":"04:45.215 ","End":"04:48.615","Text":"Well, we know how to do that."},{"Start":"04:48.615 ","End":"04:52.213","Text":"That\u0027s basically in the denominator,"},{"Start":"04:52.213 ","End":"04:58.220","Text":"we have the probability of what\u0027s given that X is greater or equal to 3."},{"Start":"04:58.220 ","End":"05:00.260","Text":"In the numerator, well,"},{"Start":"05:00.260 ","End":"05:02.480","Text":"that\u0027s the intersection of these guys."},{"Start":"05:02.480 ","End":"05:04.865","Text":"That\u0027s the probability."},{"Start":"05:04.865 ","End":"05:10.220","Text":"Now here, that\u0027s X is greater or equal to"},{"Start":"05:10.220 ","End":"05:15.920","Text":"3 and X is less than or equal to 5."},{"Start":"05:15.920 ","End":"05:18.485","Text":"Now we can write this out better."},{"Start":"05:18.485 ","End":"05:21.770","Text":"We\u0027ll say that this in the numerator,"},{"Start":"05:21.770 ","End":"05:27.275","Text":"that\u0027s the probability of 3 less than or equal to X,"},{"Start":"05:27.275 ","End":"05:28.864","Text":"less than or equal to 5,"},{"Start":"05:28.864 ","End":"05:38.920","Text":"divided by the probability of X being greater or equal to 3."},{"Start":"05:38.930 ","End":"05:44.525","Text":"Now, from our previous questions,"},{"Start":"05:44.525 ","End":"05:49.625","Text":"we know that the probability of X being greater or equal to 3,"},{"Start":"05:49.625 ","End":"05:55.855","Text":"that\u0027s the probability of X being greater than 2."},{"Start":"05:55.855 ","End":"06:02.570","Text":"Do you remember that? Wherever we see that X is greater or"},{"Start":"06:02.570 ","End":"06:08.480","Text":"equal to something in a discrete distribution where X is only whole numbers,"},{"Start":"06:08.480 ","End":"06:13.805","Text":"then that is comparable to saying that X is greater than 2."},{"Start":"06:13.805 ","End":"06:20.465","Text":"Now, what\u0027s the probability of X being between 3 and 5?"},{"Start":"06:20.465 ","End":"06:23.900","Text":"Well, that\u0027s the probability of X being equal to"},{"Start":"06:23.900 ","End":"06:29.510","Text":"3 plus the probability of X being equal to 4,"},{"Start":"06:29.510 ","End":"06:34.500","Text":"plus the probability of X being equal to 5."},{"Start":"06:35.390 ","End":"06:39.055","Text":"Now, that equals to,"},{"Start":"06:39.055 ","End":"06:44.110","Text":"what\u0027s the probability of X being equal to 3?"},{"Start":"06:44.110 ","End":"06:46.250","Text":"Now let\u0027s use this guy right here."},{"Start":"06:46.250 ","End":"06:52.560","Text":"That\u0027s 0.5. Now k equals 3,"},{"Start":"06:52.560 ","End":"06:54.510","Text":"so this is 3 minus 1, that\u0027s 2,"},{"Start":"06:54.510 ","End":"06:59.385","Text":"that\u0027s 0.5 squared times 0.5,"},{"Start":"06:59.385 ","End":"07:02.900","Text":"plus what\u0027s the probability of X being equal to 4?"},{"Start":"07:02.900 ","End":"07:11.360","Text":"Well, that\u0027s 0.5 cubed times 0.5 plus the probability of X being equal to 5."},{"Start":"07:11.360 ","End":"07:13.580","Text":"Well, that\u0027s 0.5^4,"},{"Start":"07:13.580 ","End":"07:17.720","Text":"5 minus 1 is 4 times 0.5,"},{"Start":"07:17.720 ","End":"07:22.056","Text":"that\u0027s our p, divided by,"},{"Start":"07:22.056 ","End":"07:26.555","Text":"now, what\u0027s the probability of X greater than 2?"},{"Start":"07:26.555 ","End":"07:28.760","Text":"Well, here we\u0027ll use this guy right here,"},{"Start":"07:28.760 ","End":"07:31.220","Text":"this equation right here, that\u0027s q,"},{"Start":"07:31.220 ","End":"07:33.935","Text":"0.5, so that\u0027s 0.5 squared. 0.5 squared."},{"Start":"07:33.935 ","End":"07:40.040","Text":"Now, when you calculate everything out,"},{"Start":"07:40.040 ","End":"07:45.480","Text":"this turns out to be 0.875."}],"ID":13017},{"Watched":false,"Name":"Exercise 3 Parts c-d","Duration":"9m 2s","ChapterTopicVideoID":12539,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.575","Text":"In Section C, we\u0027re asked if it\u0027s known that heads was obtained on the first 2 tosses,"},{"Start":"00:07.575 ","End":"00:11.835","Text":"what\u0027s the probability that a person tossed the coin 7 times?"},{"Start":"00:11.835 ","End":"00:14.865","Text":"Well again, here we have a conditional probability,"},{"Start":"00:14.865 ","End":"00:16.380","Text":"but with a little twist."},{"Start":"00:16.380 ","End":"00:19.200","Text":"Let\u0027s just write this out."},{"Start":"00:19.200 ","End":"00:21.180","Text":"What\u0027s the probability?"},{"Start":"00:21.180 ","End":"00:22.800","Text":"Now, what\u0027s given to us?"},{"Start":"00:22.800 ","End":"00:29.595","Text":"It\u0027s known that heads was obtained on the first 2 tosses but x was defined,"},{"Start":"00:29.595 ","End":"00:32.925","Text":"or the success was defined as getting tails."},{"Start":"00:32.925 ","End":"00:36.660","Text":"If heads was obtained in the first 2 tosses,"},{"Start":"00:36.660 ","End":"00:40.260","Text":"we know that x must have been tossed with the number of tosses"},{"Start":"00:40.260 ","End":"00:45.220","Text":"must have been greater than 2."},{"Start":"00:45.980 ","End":"00:49.710","Text":"Because on the first 2 tosses, we got heads,"},{"Start":"00:49.710 ","End":"00:52.560","Text":"we didn\u0027t get tails, and tails is our success,"},{"Start":"00:52.560 ","End":"00:54.380","Text":"that\u0027s our definition of success."},{"Start":"00:54.380 ","End":"00:59.285","Text":"It\u0027s known that we toss the coin more than twice."},{"Start":"00:59.285 ","End":"01:01.790","Text":"Now, what do we want to know?"},{"Start":"01:01.790 ","End":"01:05.546","Text":"What\u0027s the probability that a person tossed the coin 7 times."},{"Start":"01:05.546 ","End":"01:08.615","Text":"That means that x equals to 7."},{"Start":"01:08.615 ","End":"01:10.805","Text":"Now, once we have this,"},{"Start":"01:10.805 ","End":"01:12.410","Text":"then we know how to solve it."},{"Start":"01:12.410 ","End":"01:15.470","Text":"This is a straightforward conditional probability,"},{"Start":"01:15.470 ","End":"01:18.305","Text":"where in the denominator,"},{"Start":"01:18.305 ","End":"01:22.970","Text":"we have a probability of what\u0027s known."},{"Start":"01:22.970 ","End":"01:26.635","Text":"Probability x being greater than 2."},{"Start":"01:26.635 ","End":"01:30.620","Text":"Now, what\u0027s the probability in the numerator?"},{"Start":"01:30.620 ","End":"01:32.915","Text":"Well, that\u0027s the intersection of these guys."},{"Start":"01:32.915 ","End":"01:41.390","Text":"That\u0027s the probability of x being equal to 7 and that x is greater than 2."},{"Start":"01:41.390 ","End":"01:46.070","Text":"Now, if we have x being greater than 2,"},{"Start":"01:46.070 ","End":"01:47.480","Text":"then x can be 3,"},{"Start":"01:47.480 ","End":"01:49.085","Text":"4, 5,"},{"Start":"01:49.085 ","End":"01:51.095","Text":"6, 7,8,"},{"Start":"01:51.095 ","End":"01:53.165","Text":"and so on and so forth."},{"Start":"01:53.165 ","End":"01:57.875","Text":"But what\u0027s the intersection between this and x equals 7?"},{"Start":"01:57.875 ","End":"02:00.070","Text":"Well, that\u0027s right here."},{"Start":"02:00.070 ","End":"02:03.635","Text":"x equals to 7, that\u0027s the intercept."},{"Start":"02:03.635 ","End":"02:08.915","Text":"The numerator just boils down to the probability"},{"Start":"02:08.915 ","End":"02:15.215","Text":"of x being equal to 7 divided by what\u0027s in the denominator,"},{"Start":"02:15.215 ","End":"02:18.590","Text":"the probability of x is greater than 2."},{"Start":"02:18.590 ","End":"02:22.430","Text":"Now, let\u0027s calculate this guy right here."},{"Start":"02:22.430 ","End":"02:25.265","Text":"What\u0027s the probability of x being greater than 2?"},{"Start":"02:25.265 ","End":"02:27.965","Text":"Well, let\u0027s use this formula right here."},{"Start":"02:27.965 ","End":"02:31.790","Text":"Now we know that q equals 0.5 and k equals 2,"},{"Start":"02:31.790 ","End":"02:33.560","Text":"so in the denominator,"},{"Start":"02:33.560 ","End":"02:36.230","Text":"we\u0027ll have 0.5 squared."},{"Start":"02:36.230 ","End":"02:39.905","Text":"What\u0027s the probability of x being equal to 7?"},{"Start":"02:39.905 ","End":"02:42.950","Text":"Well, let\u0027s use this formula right here,"},{"Start":"02:42.950 ","End":"02:46.355","Text":"that\u0027s 0.5 to the power of 6,"},{"Start":"02:46.355 ","End":"02:48.845","Text":"7 minus 1, k minus 1,"},{"Start":"02:48.845 ","End":"02:53.615","Text":"times 0.5, that\u0027s our p. Now,"},{"Start":"02:53.615 ","End":"02:55.970","Text":"when we calculate this all out,"},{"Start":"02:55.970 ","End":"03:03.540","Text":"this comes to 0.03125."},{"Start":"03:04.640 ","End":"03:07.695","Text":"As an addendum to this section,"},{"Start":"03:07.695 ","End":"03:12.065","Text":"let\u0027s recall one of the characteristics of the geometric distribution,"},{"Start":"03:12.065 ","End":"03:18.925","Text":"where it says that the probability of x being equal to n plus k,"},{"Start":"03:18.925 ","End":"03:23.425","Text":"given that x is greater than k,"},{"Start":"03:23.425 ","End":"03:29.600","Text":"that would equal to the probability of x being equal to n. Now,"},{"Start":"03:29.600 ","End":"03:34.190","Text":"if we define k as being equal to 2,"},{"Start":"03:34.190 ","End":"03:37.550","Text":"that means that n equals what?"},{"Start":"03:37.550 ","End":"03:43.835","Text":"Equals to 5, because 5 plus 2 equals 7, that\u0027s right here."},{"Start":"03:43.835 ","End":"03:48.375","Text":"Now, if that\u0027s the case,"},{"Start":"03:48.375 ","End":"03:50.090","Text":"then if we plug this in,"},{"Start":"03:50.090 ","End":"03:54.800","Text":"the probability of x being equal to 7,"},{"Start":"03:54.800 ","End":"03:58.715","Text":"given that x is greater than 2,"},{"Start":"03:58.715 ","End":"04:02.690","Text":"that equals to the probability of x being equal to n."},{"Start":"04:02.690 ","End":"04:06.650","Text":"Now what\u0027s the probability of x being equal to n, n equals 5."},{"Start":"04:06.650 ","End":"04:09.545","Text":"This guy right here, let\u0027s plug in the numbers,"},{"Start":"04:09.545 ","End":"04:14.820","Text":"that\u0027ll be 0.5,"},{"Start":"04:14.820 ","End":"04:18.945","Text":"q to the power of k minus 1,"},{"Start":"04:18.945 ","End":"04:21.900","Text":"that\u0027s n minus 1, n is 5,"},{"Start":"04:21.900 ","End":"04:26.835","Text":"so that\u0027ll be 4 times 0.5."},{"Start":"04:26.835 ","End":"04:33.470","Text":"Now that also equals 0.03125."},{"Start":"04:33.470 ","End":"04:41.315","Text":"Again, this is called the characteristic where x is memory-less."},{"Start":"04:41.315 ","End":"04:48.755","Text":"Again, we have to understand that this characteristic exists in this distribution,"},{"Start":"04:48.755 ","End":"04:53.465","Text":"but it\u0027s much better to go about this the long way,"},{"Start":"04:53.465 ","End":"04:57.860","Text":"that means calculate this thing out by hand the long way."},{"Start":"04:57.860 ","End":"04:59.480","Text":"Don\u0027t take the shortcuts,"},{"Start":"04:59.480 ","End":"05:03.350","Text":"this will always get you into trouble, you\u0027ll get confused."},{"Start":"05:03.350 ","End":"05:05.990","Text":"Try to do this the long way round,"},{"Start":"05:05.990 ","End":"05:10.470","Text":"do it step-by-step until you get to the right answer,"},{"Start":"05:10.470 ","End":"05:14.975","Text":"that\u0027ll just minimize your risk of making a mistake."},{"Start":"05:14.975 ","End":"05:20.345","Text":"A suggestion. In Section D we\u0027re asked,"},{"Start":"05:20.345 ","End":"05:24.380","Text":"what\u0027s the expectation of the number of heads received?"},{"Start":"05:24.380 ","End":"05:31.610","Text":"Now don\u0027t forget the success was the number of tails."},{"Start":"05:31.610 ","End":"05:35.925","Text":"Getting a tail was the success and x was"},{"Start":"05:35.925 ","End":"05:40.660","Text":"a random variable of the number of tosses until the first tail."},{"Start":"05:40.660 ","End":"05:44.660","Text":"Now the rest of the stuff that\u0027s written here is what"},{"Start":"05:44.660 ","End":"05:48.755","Text":"we got from the previous questions in sections."},{"Start":"05:48.755 ","End":"05:50.810","Text":"Now, let\u0027s get to work."},{"Start":"05:50.810 ","End":"05:53.465","Text":"What\u0027s the expectation or number of heads we see?"},{"Start":"05:53.465 ","End":"05:58.415","Text":"Well, if in a geometric distribution,"},{"Start":"05:58.415 ","End":"06:04.130","Text":"if x is a number of tosses until my first tail,"},{"Start":"06:04.130 ","End":"06:08.929","Text":"then all the previous tosses were heads."},{"Start":"06:08.929 ","End":"06:13.780","Text":"That means that if I toss x times,"},{"Start":"06:13.780 ","End":"06:15.690","Text":"in order to get a tail,"},{"Start":"06:15.690 ","End":"06:21.090","Text":"then I tossed x minus 1 time in order to get heads,"},{"Start":"06:21.090 ","End":"06:28.690","Text":"and so y is the number of heads that I received."},{"Start":"06:28.690 ","End":"06:35.045","Text":"Now, this very much looks like a linear transformation. Why is that?"},{"Start":"06:35.045 ","End":"06:42.190","Text":"Well, the general form of a linear transformation is ax plus b."},{"Start":"06:42.190 ","End":"06:45.650","Text":"Let\u0027s just go over the steps for"},{"Start":"06:45.650 ","End":"06:49.500","Text":"linear transformation to make sure that we\u0027re on the right track."},{"Start":"06:51.170 ","End":"06:54.605","Text":"These are the steps, let\u0027s take a look at the first step."},{"Start":"06:54.605 ","End":"06:58.310","Text":"Recognize that we\u0027re dealing with a linear transformation where we are."},{"Start":"06:58.310 ","End":"07:01.355","Text":"We\u0027re taking the number of tails,"},{"Start":"07:01.355 ","End":"07:04.849","Text":"and we\u0027re calculating the number of heads."},{"Start":"07:04.849 ","End":"07:08.635","Text":"Now, write the transformation rule where we did that."},{"Start":"07:08.635 ","End":"07:12.750","Text":"3, simplify the rule and identify the values of a and b."},{"Start":"07:12.750 ","End":"07:17.285","Text":"The general form of a linear transformation is this."},{"Start":"07:17.285 ","End":"07:20.750","Text":"Let\u0027s just write this out like this."},{"Start":"07:20.750 ","End":"07:25.300","Text":"This is 1 times x minus 1."},{"Start":"07:25.300 ","End":"07:28.069","Text":"Now, having written it like this,"},{"Start":"07:28.069 ","End":"07:32.870","Text":"we can easily identify that a equals 1 and b equals"},{"Start":"07:32.870 ","End":"07:37.470","Text":"minus 1. a would be the multiplier of x,"},{"Start":"07:37.470 ","End":"07:41.380","Text":"that\u0027s right here, and b is this guy right here."},{"Start":"07:41.380 ","End":"07:43.700","Text":"Now, having done that,"},{"Start":"07:43.700 ","End":"07:46.850","Text":"we\u0027re asked what\u0027s the expectation of the number of heads?"},{"Start":"07:46.850 ","End":"07:51.805","Text":"That means, what\u0027s the expectation of y?"},{"Start":"07:51.805 ","End":"07:59.810","Text":"Now that equals if we remember to the a times the expectation of x plus b."},{"Start":"07:59.810 ","End":"08:02.600","Text":"But hold on, what\u0027s the expectation of x?"},{"Start":"08:02.600 ","End":"08:08.555","Text":"Well, we know that the expectation of x in a geometric distribution,"},{"Start":"08:08.555 ","End":"08:10.655","Text":"that equals 1 over p,"},{"Start":"08:10.655 ","End":"08:13.910","Text":"and that equals to 1 over 0.5,"},{"Start":"08:13.910 ","End":"08:15.650","Text":"and that equals to 2."},{"Start":"08:15.650 ","End":"08:18.710","Text":"In our case, p is 0.5."},{"Start":"08:18.710 ","End":"08:21.020","Text":"Let\u0027s just plug in the numbers,"},{"Start":"08:21.020 ","End":"08:22.850","Text":"a equals 1, that\u0027s 1,"},{"Start":"08:22.850 ","End":"08:24.800","Text":"times the expectation of x,"},{"Start":"08:24.800 ","End":"08:28.370","Text":"that\u0027s 2, plus b. b is minus 1,"},{"Start":"08:28.370 ","End":"08:29.675","Text":"so that\u0027s minus 1,"},{"Start":"08:29.675 ","End":"08:31.825","Text":"and that equals to 1."},{"Start":"08:31.825 ","End":"08:34.200","Text":"This makes sense."},{"Start":"08:34.200 ","End":"08:37.855","Text":"So let\u0027s talk about y because if we\u0027re"},{"Start":"08:37.855 ","End":"08:43.385","Text":"expected to toss a coin twice in order to get a tail,"},{"Start":"08:43.385 ","End":"08:47.240","Text":"that means that we have to go through heads first."},{"Start":"08:47.240 ","End":"08:50.434","Text":"We have to toss the coin once,"},{"Start":"08:50.434 ","End":"08:54.680","Text":"and the expectation of the first toss would be a head,"},{"Start":"08:54.680 ","End":"08:58.025","Text":"and then the second one would be a tail."},{"Start":"08:58.025 ","End":"09:01.630","Text":"That pretty much makes sense."}],"ID":13018},{"Watched":false,"Name":"Exercise 4","Duration":"7m 12s","ChapterTopicVideoID":12540,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this question, we\u0027ll be talking about parked cars."},{"Start":"00:03.750 ","End":"00:06.450","Text":"30 percent of the cars in Ohio are white."},{"Start":"00:06.450 ","End":"00:11.745","Text":"10 random cars enter a parking lot every day, and we\u0027re asked,"},{"Start":"00:11.745 ","End":"00:14.535","Text":"what\u0027s the probability that at a given day,"},{"Start":"00:14.535 ","End":"00:18.225","Text":"exactly half of the cars in the parking lot are white?"},{"Start":"00:18.225 ","End":"00:21.180","Text":"Well, if we notice,"},{"Start":"00:21.180 ","End":"00:26.835","Text":"we see that although we\u0027re in the chapter dealing with the geometric probability,"},{"Start":"00:26.835 ","End":"00:31.260","Text":"this question has nothing to do with a geometric distribution."},{"Start":"00:31.260 ","End":"00:34.995","Text":"It has everything to do with a binomial distribution."},{"Start":"00:34.995 ","End":"00:37.600","Text":"Let\u0027s take a look why."},{"Start":"00:39.020 ","End":"00:44.475","Text":"These are the characteristics of a binomial distribution."},{"Start":"00:44.475 ","End":"00:50.220","Text":"The first one is that we have the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:50.220 ","End":"00:54.525","Text":"A Bernoulli trial is a trial where we have success or failure."},{"Start":"00:54.525 ","End":"00:56.370","Text":"What\u0027s our success here?"},{"Start":"00:56.370 ","End":"01:02.500","Text":"Let\u0027s define our success as a white car."},{"Start":"01:04.220 ","End":"01:08.505","Text":"Now, are the cars independent of each other?"},{"Start":"01:08.505 ","End":"01:15.690","Text":"Well, we\u0027re looking at 10 cars out of the whole population of cars in the State of Ohio."},{"Start":"01:15.690 ","End":"01:18.690","Text":"That means we have a small sample size from"},{"Start":"01:18.690 ","End":"01:22.650","Text":"a large population so we can assume independence."},{"Start":"01:22.650 ","End":"01:24.735","Text":"Yes, they are independent."},{"Start":"01:24.735 ","End":"01:27.270","Text":"The trial is repeated n times,"},{"Start":"01:27.270 ","End":"01:31.930","Text":"n equals 10, given right here."},{"Start":"01:32.750 ","End":"01:37.890","Text":"The third characteristic is the most important one, and why is that?"},{"Start":"01:37.890 ","End":"01:43.870","Text":"Here it says, X is defined as the total number of successes obtained."},{"Start":"01:44.480 ","End":"01:48.270","Text":"What does it say in the geometric distribution?"},{"Start":"01:48.270 ","End":"01:50.130","Text":"Not the total number of successes,"},{"Start":"01:50.130 ","End":"01:52.470","Text":"but the first success."},{"Start":"01:52.470 ","End":"01:55.350","Text":"That\u0027s what differentiates between"},{"Start":"01:55.350 ","End":"01:58.710","Text":"a geometric distribution and the binomial distribution."},{"Start":"01:58.710 ","End":"02:02.925","Text":"A binomial distribution counts the total number of successes."},{"Start":"02:02.925 ","End":"02:09.520","Text":"Let\u0027s define X as the number of white cars."},{"Start":"02:11.030 ","End":"02:13.230","Text":"Once we\u0027ve done that,"},{"Start":"02:13.230 ","End":"02:16.905","Text":"we know that X is distributed"},{"Start":"02:16.905 ","End":"02:23.430","Text":"binomially with n equals 10 and p being equal to what?"},{"Start":"02:23.430 ","End":"02:26.550","Text":"Well, 30 percent of the cars in Ohio are white,"},{"Start":"02:26.550 ","End":"02:31.000","Text":"so p equals 0.3, right here."},{"Start":"02:35.240 ","End":"02:37.320","Text":"X equals k. Well,"},{"Start":"02:37.320 ","End":"02:40.875","Text":"let\u0027s remember the equation for binomial distribution."},{"Start":"02:40.875 ","End":"02:43.755","Text":"The probability of X being equal to k,"},{"Start":"02:43.755 ","End":"02:48.538","Text":"that equals to n/k, p^k,"},{"Start":"02:48.538 ","End":"02:52.935","Text":"and q to the power of n minus k,"},{"Start":"02:52.935 ","End":"02:58.770","Text":"where q is 1 minus p. So if we plug in q here,"},{"Start":"02:58.770 ","End":"03:02.170","Text":"that\u0027s equals to 0.7."},{"Start":"03:02.480 ","End":"03:05.265","Text":"Having done all this work,"},{"Start":"03:05.265 ","End":"03:07.470","Text":"let\u0027s get back to our question."},{"Start":"03:07.470 ","End":"03:09.900","Text":"What\u0027s the probability that on a given day,"},{"Start":"03:09.900 ","End":"03:13.290","Text":"exactly half of the cars in the parking lot are white?"},{"Start":"03:13.290 ","End":"03:18.240","Text":"That means we\u0027re looking at k being equal to 5. Why 5?"},{"Start":"03:18.240 ","End":"03:20.010","Text":"Because 5 is half of 10,"},{"Start":"03:20.010 ","End":"03:27.690","Text":"so the probability of x being equal to 5 equals n/k,"},{"Start":"03:27.690 ","End":"03:33.580","Text":"that\u0027s 10 over 5. p,"},{"Start":"03:33.710 ","End":"03:36.570","Text":"0.3^5 times, q,"},{"Start":"03:36.570 ","End":"03:41.040","Text":"that\u0027s 0.7 to the power of n minus k,"},{"Start":"03:41.040 ","End":"03:43.050","Text":"that\u0027s 10 minus 5, that\u0027s 5,"},{"Start":"03:43.050 ","End":"03:48.040","Text":"and that equals to 0.1029."},{"Start":"03:49.790 ","End":"03:52.410","Text":"In this section, we\u0027re asked,"},{"Start":"03:52.410 ","End":"03:56.490","Text":"what\u0027s the expectation of the number of days one has to wait"},{"Start":"03:56.490 ","End":"04:01.815","Text":"until the first day that half of the cars in the parking lot will be white?"},{"Start":"04:01.815 ","End":"04:06.930","Text":"Here we have a geometric distribution, and here\u0027s why."},{"Start":"04:06.930 ","End":"04:09.675","Text":"If we look at the characteristics,"},{"Start":"04:09.675 ","End":"04:11.445","Text":"we\u0027ll see that we have the following."},{"Start":"04:11.445 ","End":"04:15.930","Text":"First of all, we have the same Bernoulli trial that\u0027s repeated n times,"},{"Start":"04:15.930 ","End":"04:18.495","Text":"where each trial is independent of each other."},{"Start":"04:18.495 ","End":"04:20.550","Text":"If we have a Bernoulli trial,"},{"Start":"04:20.550 ","End":"04:22.815","Text":"then let\u0027s define our success."},{"Start":"04:22.815 ","End":"04:27.945","Text":"Success would be defined as"},{"Start":"04:27.945 ","End":"04:37.120","Text":"half the cars are white."},{"Start":"04:38.960 ","End":"04:45.315","Text":"Half the cars would be white in the parking lot. What else do we know?"},{"Start":"04:45.315 ","End":"04:52.440","Text":"Well, what\u0027s the probability that half of the cars would be white?"},{"Start":"04:52.440 ","End":"04:58.470","Text":"The probability is exactly what we calculated in Section A,"},{"Start":"04:58.470 ","End":"05:01.450","Text":"and that was 0.1029."},{"Start":"05:03.050 ","End":"05:07.515","Text":"That\u0027s what we did. Here,"},{"Start":"05:07.515 ","End":"05:09.660","Text":"we\u0027re not talking about the number of cars,"},{"Start":"05:09.660 ","End":"05:11.925","Text":"but we\u0027re talking about the number of days"},{"Start":"05:11.925 ","End":"05:18.135","Text":"until we get a situation where half the cars in the parking lot are white."},{"Start":"05:18.135 ","End":"05:20.340","Text":"Let\u0027s define y,"},{"Start":"05:20.340 ","End":"05:22.635","Text":"not X, we don\u0027t want to get confused."},{"Start":"05:22.635 ","End":"05:29.880","Text":"That\u0027s the number of days I have to wait until"},{"Start":"05:29.880 ","End":"05:39.610","Text":"half of the cars are white."},{"Start":"05:41.870 ","End":"05:48.840","Text":"When we defined all these guys, in essence,"},{"Start":"05:48.840 ","End":"05:54.390","Text":"what we\u0027ve done is we\u0027ve made sure that y would be"},{"Start":"05:54.390 ","End":"06:02.730","Text":"distributed with a geometric distribution where p equals to 0.1029."},{"Start":"06:02.730 ","End":"06:08.175","Text":"That means that the probability of y being equal to k"},{"Start":"06:08.175 ","End":"06:16.890","Text":"equals q to the power of k minus 1 times p. In our case,"},{"Start":"06:16.890 ","End":"06:19.860","Text":"what do we want to know?"},{"Start":"06:19.860 ","End":"06:26.610","Text":"We want to know what\u0027s the expectation of the number of days one has to wait."},{"Start":"06:26.610 ","End":"06:32.550","Text":"That means that we\u0027re looking at the expectation of y. y is the number of days,"},{"Start":"06:32.550 ","End":"06:35.100","Text":"so what\u0027s the expected number of days?"},{"Start":"06:35.100 ","End":"06:38.970","Text":"If y is distributed with a geometric distribution,"},{"Start":"06:38.970 ","End":"06:42.420","Text":"then the expectation is 1/p,"},{"Start":"06:42.420 ","End":"06:47.580","Text":"which in our case is 1 over 0.1029,"},{"Start":"06:47.580 ","End":"06:50.955","Text":"and that equals to 9.72."},{"Start":"06:50.955 ","End":"06:52.155","Text":"What are the units?"},{"Start":"06:52.155 ","End":"06:55.000","Text":"The units are days."},{"Start":"06:55.220 ","End":"07:03.195","Text":"In essence, what we\u0027re looking at is waiting 9.72 days"},{"Start":"07:03.195 ","End":"07:11.710","Text":"until we get a situation where half the parking lot is filled with cars that are white."}],"ID":13019},{"Watched":false,"Name":"Exercise 5 Parts a-b","Duration":"5m 20s","ChapterTopicVideoID":12541,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this question, we\u0027ll be talking about games of chance."},{"Start":"00:03.840 ","End":"00:07.125","Text":"Now, a person plays a game of chance until he loses."},{"Start":"00:07.125 ","End":"00:08.610","Text":"The expectation is that,"},{"Start":"00:08.610 ","End":"00:11.085","Text":"he will play the game 10 times,"},{"Start":"00:11.085 ","End":"00:14.820","Text":"and we\u0027re asked what are his chances of losing the individual game?"},{"Start":"00:14.820 ","End":"00:18.240","Text":"Well, this looks like a geometric distribution,"},{"Start":"00:18.240 ","End":"00:21.645","Text":"but let\u0027s just look at the characteristics to make sure."},{"Start":"00:21.645 ","End":"00:24.345","Text":"So these are the characteristics."},{"Start":"00:24.345 ","End":"00:26.460","Text":"We have the first 1 being that"},{"Start":"00:26.460 ","End":"00:31.590","Text":"the same Bernoulli trials repeated n times where each trial is independent of each other."},{"Start":"00:31.590 ","End":"00:36.475","Text":"Well, we know that we have Bernoulli trials here and"},{"Start":"00:36.475 ","End":"00:41.715","Text":"we\u0027ll define the Bernoulli trial as a game that the person play."},{"Start":"00:41.715 ","End":"00:44.375","Text":"In the Bernoulli trial or in the game,"},{"Start":"00:44.375 ","End":"00:47.695","Text":"there are chances of success and failure."},{"Start":"00:47.695 ","End":"00:49.400","Text":"Now, first of all,"},{"Start":"00:49.400 ","End":"00:51.545","Text":"let\u0027s define what success is."},{"Start":"00:51.545 ","End":"00:54.290","Text":"Success for us right now,"},{"Start":"00:54.290 ","End":"01:03.470","Text":"we\u0027ll define that as losing a game and we have the probability of losing a game,"},{"Start":"01:03.470 ","End":"01:05.224","Text":"the probability of success."},{"Start":"01:05.224 ","End":"01:07.130","Text":"We don\u0027t know what that is right now,"},{"Start":"01:07.130 ","End":"01:09.020","Text":"but we\u0027ll get to that in a bit."},{"Start":"01:09.020 ","End":"01:12.290","Text":"The next characteristic is that X is defined as"},{"Start":"01:12.290 ","End":"01:15.350","Text":"the number of trials carried out until the first success."},{"Start":"01:15.350 ","End":"01:16.790","Text":"Well, that\u0027s what the person\u0027s doing."},{"Start":"01:16.790 ","End":"01:19.760","Text":"He\u0027s playing a game until he loses,"},{"Start":"01:19.760 ","End":"01:21.590","Text":"until the first loss."},{"Start":"01:21.590 ","End":"01:26.420","Text":"We\u0027ll define X as the number of games"},{"Start":"01:26.420 ","End":"01:36.450","Text":"played until the first loss."},{"Start":"01:37.690 ","End":"01:41.735","Text":"Well, that seems fine."},{"Start":"01:41.735 ","End":"01:44.825","Text":"Once we\u0027ve defined these things,"},{"Start":"01:44.825 ","End":"01:51.140","Text":"then we can say that X is distributed with the geometric distribution,"},{"Start":"01:51.140 ","End":"02:01.025","Text":"with the parameter P. We know that the equation for the geometric distribution is this,"},{"Start":"02:01.025 ","End":"02:03.890","Text":"that the probability of X being equal to k,"},{"Start":"02:03.890 ","End":"02:13.090","Text":"that\u0027s q to the power of k minus 1 times P. Let\u0027s now try to figure out what P is."},{"Start":"02:13.090 ","End":"02:15.110","Text":"Well, on 1 hand,"},{"Start":"02:15.110 ","End":"02:23.765","Text":"we\u0027re given that the expectation that the person will play 10 games."},{"Start":"02:23.765 ","End":"02:26.690","Text":"The expectation of X,"},{"Start":"02:26.690 ","End":"02:30.530","Text":"X is the number of games played while the expectation is 10. That\u0027s given right here."},{"Start":"02:30.530 ","End":"02:33.335","Text":"But we also know that the expectation of"},{"Start":"02:33.335 ","End":"02:36.890","Text":"X when it\u0027s distributed with the geometric distribution,"},{"Start":"02:36.890 ","End":"02:41.290","Text":"well that\u0027s 1 over P. This equals right here,"},{"Start":"02:41.290 ","End":"02:43.190","Text":"1 over P,"},{"Start":"02:43.190 ","End":"02:44.845","Text":"and if that\u0027s the case,"},{"Start":"02:44.845 ","End":"02:47.709","Text":"then we have the following."},{"Start":"02:47.709 ","End":"02:51.660","Text":"We can say that\u0027s 10p, that equals to 1."},{"Start":"02:51.660 ","End":"02:54.825","Text":"That means P equals to 1 over 10."},{"Start":"02:54.825 ","End":"02:56.370","Text":"If that\u0027s the case,"},{"Start":"02:56.370 ","End":"02:59.575","Text":"then q is equal to 1 minus p,"},{"Start":"02:59.575 ","End":"03:02.665","Text":"that equals to 9 over 10."},{"Start":"03:02.665 ","End":"03:07.675","Text":"Excellent. Now that we have these guys,"},{"Start":"03:07.675 ","End":"03:10.870","Text":"then we can go ahead and answer the questions here,"},{"Start":"03:10.870 ","End":"03:14.705","Text":"section A to D. In section A,"},{"Start":"03:14.705 ","End":"03:19.030","Text":"we\u0027re asked what\u0027s the probability of playing the game exactly 6 times?"},{"Start":"03:19.030 ","End":"03:20.985","Text":"Well, let\u0027s take a look."},{"Start":"03:20.985 ","End":"03:28.330","Text":"The probability that we play the game exactly 6 times, well,"},{"Start":"03:28.330 ","End":"03:33.140","Text":"let\u0027s use this expression right here, q,"},{"Start":"03:33.140 ","End":"03:38.630","Text":"which is 9/10s is 0.9 to the power of k minus 1,"},{"Start":"03:38.630 ","End":"03:44.390","Text":"that\u0027s 5, times 0.1 times p,"},{"Start":"03:44.390 ","End":"03:50.190","Text":"and that equals to 0.06 approximately."},{"Start":"03:50.680 ","End":"03:53.390","Text":"Now in section B, we\u0027re asked what\u0027s"},{"Start":"03:53.390 ","End":"03:57.290","Text":"the probability of playing the game at most 12 times?"},{"Start":"03:57.290 ","End":"04:05.405","Text":"B is the probability that X is less than or equal to 12."},{"Start":"04:05.405 ","End":"04:09.980","Text":"Now, this seems to be work intensive because we"},{"Start":"04:09.980 ","End":"04:14.690","Text":"have to calculate what\u0027s the probability of X when it\u0027s 12 and then when it\u0027s 11,"},{"Start":"04:14.690 ","End":"04:16.475","Text":"then when it\u0027s 10 until 1,"},{"Start":"04:16.475 ","End":"04:18.635","Text":"then add up all these guys up,"},{"Start":"04:18.635 ","End":"04:22.940","Text":"and then we\u0027ll have the probability of this event."},{"Start":"04:22.940 ","End":"04:26.270","Text":"But on the other hand, let\u0027s be a little bit smarter."},{"Start":"04:26.270 ","End":"04:30.235","Text":"Let\u0027s take the complimentary set of this,"},{"Start":"04:30.235 ","End":"04:36.830","Text":"that would equal to 1 minus the probability of X being greater than 12."},{"Start":"04:36.830 ","End":"04:40.895","Text":"This is the complimentary set of this."},{"Start":"04:40.895 ","End":"04:43.205","Text":"Also, what did we learn?"},{"Start":"04:43.205 ","End":"04:50.510","Text":"We learned that when X is distributed with a geometric distribution,"},{"Start":"04:50.510 ","End":"04:55.760","Text":"that means that the probability of X being greater than k, well,"},{"Start":"04:55.760 ","End":"05:00.975","Text":"that equals to q to the power of k. In our case,"},{"Start":"05:00.975 ","End":"05:03.000","Text":"there\u0027ll be 1 minus."},{"Start":"05:03.000 ","End":"05:05.310","Text":"Now, what\u0027s q? q is 0.9,"},{"Start":"05:05.310 ","End":"05:09.480","Text":"0.9 to the power of 12."},{"Start":"05:09.480 ","End":"05:16.810","Text":"That equals to 0.7176."}],"ID":13020},{"Watched":false,"Name":"Exercise 5 Parts c-d","Duration":"5m 45s","ChapterTopicVideoID":12542,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In Section C, we\u0027re given that the person played over 6 games and"},{"Start":"00:04.560 ","End":"00:09.105","Text":"we\u0027re asked what\u0027s the probability that he played the game exactly 10 times?"},{"Start":"00:09.105 ","End":"00:15.470","Text":"Now, here\u0027s the information that we\u0027ve received from answering the previous questions."},{"Start":"00:15.470 ","End":"00:20.875","Text":"Now, this looks like a conditional probability."},{"Start":"00:20.875 ","End":"00:25.610","Text":"What we need to do is to write this down,"},{"Start":"00:25.610 ","End":"00:33.915","Text":"probability of now what\u0027s given to us that X is being played over 6 times."},{"Start":"00:33.915 ","End":"00:40.834","Text":"We want to know what\u0027s the probability of playing the game 10 times."},{"Start":"00:40.834 ","End":"00:48.605","Text":"Now, we can use 1 of the special characteristics of the geometric distribution,"},{"Start":"00:48.605 ","End":"00:51.230","Text":"where X is memoryless."},{"Start":"00:51.230 ","End":"00:53.075","Text":"Now, if you remember,"},{"Start":"00:53.075 ","End":"01:01.680","Text":"that\u0027s the probability of X being equal to n plus k given that X"},{"Start":"01:01.680 ","End":"01:05.330","Text":"is greater than k. That means that equals to"},{"Start":"01:05.330 ","End":"01:10.490","Text":"the probability of X being equal to n. Now in our case,"},{"Start":"01:10.490 ","End":"01:17.680","Text":"k would be equal to 6 and n would be equal to 4."},{"Start":"01:17.680 ","End":"01:21.035","Text":"That means now we can plug this in."},{"Start":"01:21.035 ","End":"01:26.150","Text":"The probability of X being equal to 4,"},{"Start":"01:26.150 ","End":"01:28.730","Text":"that would be equal to 0.9,"},{"Start":"01:28.730 ","End":"01:32.940","Text":"that\u0027s q to the power of 3,"},{"Start":"01:32.940 ","End":"01:36.570","Text":"k minus 1 times 0.1,"},{"Start":"01:36.570 ","End":"01:41.535","Text":"times p. Let\u0027s put that aside right now."},{"Start":"01:41.535 ","End":"01:43.910","Text":"We can use this,"},{"Start":"01:43.910 ","End":"01:46.910","Text":"but I suggest strongly that you don\u0027t."},{"Start":"01:46.910 ","End":"01:48.215","Text":"Why is that?"},{"Start":"01:48.215 ","End":"01:51.440","Text":"Because experience shows that students"},{"Start":"01:51.440 ","End":"01:54.980","Text":"really get confused when using this and they make lots of mistakes."},{"Start":"01:54.980 ","End":"01:59.600","Text":"It\u0027s best for you guys just to do it the long way."},{"Start":"01:59.600 ","End":"02:02.690","Text":"Just calculate things out step-by-step,"},{"Start":"02:02.690 ","End":"02:08.780","Text":"and then we\u0027ll figure things out much better with less mistakes. Let\u0027s do that."},{"Start":"02:08.780 ","End":"02:15.365","Text":"Now, the probability of X equaling 10 given that X is greater than 6."},{"Start":"02:15.365 ","End":"02:20.119","Text":"Well, what do we have in the denominator and what do we have in the numerator?"},{"Start":"02:20.119 ","End":"02:23.300","Text":"Well, we know that in the denominator we have"},{"Start":"02:23.300 ","End":"02:27.295","Text":"the probability of what\u0027s given that X greater than 6."},{"Start":"02:27.295 ","End":"02:29.615","Text":"What do we have in the numerator?"},{"Start":"02:29.615 ","End":"02:30.950","Text":"Well, that\u0027s the intersection."},{"Start":"02:30.950 ","End":"02:38.850","Text":"That\u0027s the probability of X equaling 10 intersect X being greater than 6."},{"Start":"02:39.950 ","End":"02:43.740","Text":"Let\u0027s see how we can continue."},{"Start":"02:43.740 ","End":"02:49.025","Text":"Now, let\u0027s look at the numerator right here,"},{"Start":"02:49.025 ","End":"02:51.860","Text":"we have X being greater than 6,"},{"Start":"02:51.860 ","End":"02:54.770","Text":"and that means that X equals 7, 8,"},{"Start":"02:54.770 ","End":"02:59.974","Text":"9, 10, 11, and so on and so forth till infinity."},{"Start":"02:59.974 ","End":"03:06.020","Text":"Now, what\u0027s the intersect between this set and X equals 10?"},{"Start":"03:06.020 ","End":"03:09.780","Text":"Well, obviously is just X equals 10."},{"Start":"03:09.780 ","End":"03:14.900","Text":"The numerator just boils down to the probability of X being equal to 10."},{"Start":"03:14.900 ","End":"03:21.340","Text":"Now, what\u0027s the probability of X being greater than 6?"},{"Start":"03:21.340 ","End":"03:25.190","Text":"Well, we know 1 of the characteristics of"},{"Start":"03:25.190 ","End":"03:30.200","Text":"the geometric distribution is that when X is greater than k,"},{"Start":"03:30.200 ","End":"03:38.850","Text":"what that equals to q to the power of k. That means that we have X being greater than 6."},{"Start":"03:38.850 ","End":"03:42.780","Text":"We have here q. What\u0027s q?"},{"Start":"03:42.780 ","End":"03:48.195","Text":"Q is 0.9 to the power of 6."},{"Start":"03:48.195 ","End":"03:52.610","Text":"Now, what\u0027s the probability of X equaling 10,"},{"Start":"03:52.610 ","End":"03:53.825","Text":"but that\u0027s q,"},{"Start":"03:53.825 ","End":"03:57.925","Text":"that\u0027s 0.9 to the power of 9."},{"Start":"03:57.925 ","End":"04:03.350","Text":"That\u0027s 10 minus 1 times 0.1 and that"},{"Start":"04:03.350 ","End":"04:10.145","Text":"equals to 0.9 to the power of 3 times 0.1."},{"Start":"04:10.145 ","End":"04:16.370","Text":"Again, we can see that we receive the same answer both here and here."},{"Start":"04:16.370 ","End":"04:21.110","Text":"Here when we calculated this the long way, step-by-step,"},{"Start":"04:21.110 ","End":"04:28.040","Text":"being very careful and using the short equation or the shortcut right here."},{"Start":"04:28.040 ","End":"04:34.085","Text":"But again, I highly recommend that you use this way,"},{"Start":"04:34.085 ","End":"04:38.960","Text":"this method, instead of using this method right here."},{"Start":"04:38.960 ","End":"04:42.540","Text":"Just take it under consideration."},{"Start":"04:43.310 ","End":"04:45.930","Text":"In Section D, we\u0027re asked,"},{"Start":"04:45.930 ","End":"04:48.310","Text":"what\u0027s the standard deviation of X?"},{"Start":"04:48.310 ","End":"04:55.130","Text":"Well, we know that X is distributed geometrically with a probability of 0.1."},{"Start":"04:55.130 ","End":"04:58.985","Text":"Now, let\u0027s calculate the variance of X."},{"Start":"04:58.985 ","End":"05:04.294","Text":"The variance of X is defined as q divided by p squared."},{"Start":"05:04.294 ","End":"05:06.050","Text":"What\u0027s q in our case,"},{"Start":"05:06.050 ","End":"05:10.460","Text":"that\u0027s 0.9 divided by 0.1,"},{"Start":"05:10.460 ","End":"05:17.570","Text":"that\u0027s p squared, and that equals to 90 games."},{"Start":"05:17.570 ","End":"05:20.045","Text":"Let\u0027s see units game squared."},{"Start":"05:20.045 ","End":"05:23.960","Text":"Not very intuitive, but we\u0027re asked about the standard deviation."},{"Start":"05:23.960 ","End":"05:29.900","Text":"Well, we know that the standard deviation of X is the square root of the variance."},{"Start":"05:29.900 ","End":"05:34.070","Text":"That means that that\u0027s the square root of 90 and that equals to"},{"Start":"05:34.070 ","End":"05:40.950","Text":"9.487 and the units, games."},{"Start":"05:41.120 ","End":"05:45.940","Text":"This is a standard deviation of X."}],"ID":13021},{"Watched":false,"Name":"Exercise 6 Parts a-b","Duration":"6m 35s","ChapterTopicVideoID":12543,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this question, we\u0027ll be talking about cakes."},{"Start":"00:02.550 ","End":"00:07.950","Text":"A bakery makes cheesecakes and chocolate cakes that are packed in sealed packages."},{"Start":"00:07.950 ","End":"00:11.940","Text":"40 percent of the cakes are cheesecakes and the rest are chocolate cakes."},{"Start":"00:11.940 ","End":"00:15.765","Text":"Labels are glued on the packages at a later stage of production."},{"Start":"00:15.765 ","End":"00:19.725","Text":"A person enters the bakery and randomly selects a cake."},{"Start":"00:19.725 ","End":"00:23.340","Text":"We\u0027re asked, what\u0027s the probability that he will have to"},{"Start":"00:23.340 ","End":"00:27.015","Text":"select 5 cakes before getting a chocolate cake?"},{"Start":"00:27.015 ","End":"00:31.410","Text":"This is obviously a geometric distribution. Why is that?"},{"Start":"00:31.410 ","End":"00:35.715","Text":"Again, let\u0027s recall a geometric distribution deals with"},{"Start":"00:35.715 ","End":"00:42.340","Text":"Bernoulli trials and Bernoulli trials that are independent of each other."},{"Start":"00:42.340 ","End":"00:44.870","Text":"On the other hand, we have"},{"Start":"00:44.870 ","End":"00:50.525","Text":"a random variable that counts the number of trials until the first success."},{"Start":"00:50.525 ","End":"00:54.695","Text":"In a Bernoulli trial in our case, what success?"},{"Start":"00:54.695 ","End":"01:02.440","Text":"Let\u0027s define success as being the chocolate cake."},{"Start":"01:03.560 ","End":"01:08.480","Text":"What\u0027s the probability of getting a chocolate cake?"},{"Start":"01:08.480 ","End":"01:11.585","Text":"If 40 percent of the cakes are cheesecakes,"},{"Start":"01:11.585 ","End":"01:13.715","Text":"that means that 60 percent,"},{"Start":"01:13.715 ","End":"01:23.390","Text":"1 minus p, 1 minus 40 percent is chocolate cake and that\u0027s right here."},{"Start":"01:23.390 ","End":"01:28.685","Text":"What else do we have to find out whether the trials are independent or not?"},{"Start":"01:28.685 ","End":"01:34.145","Text":"We\u0027re assuming that this is a massive production line."},{"Start":"01:34.145 ","End":"01:40.769","Text":"That means that we\u0027re sampling a very small number from a very large population,"},{"Start":"01:40.769 ","End":"01:42.335","Text":"and whenever we do that,"},{"Start":"01:42.335 ","End":"01:44.330","Text":"we can assume independence."},{"Start":"01:44.330 ","End":"01:46.850","Text":"What\u0027s the other thing, the random variable."},{"Start":"01:46.850 ","End":"01:51.545","Text":"We\u0027re counting here the number of trials until we get a success."},{"Start":"01:51.545 ","End":"01:53.660","Text":"Here that\u0027s exactly what we\u0027re doing."},{"Start":"01:53.660 ","End":"01:59.355","Text":"We define X as the number of"},{"Start":"01:59.355 ","End":"02:06.029","Text":"selections until we get a chocolate cake."},{"Start":"02:06.029 ","End":"02:08.975","Text":"In question a,"},{"Start":"02:08.975 ","End":"02:13.840","Text":"we\u0027re asked what\u0027s the probability that we will have to select 5 cakes?"},{"Start":"02:13.840 ","End":"02:16.360","Text":"Again, X, we said,"},{"Start":"02:16.360 ","End":"02:21.770","Text":"is distributed geometrically with p equaling 0.6."},{"Start":"02:21.770 ","End":"02:25.270","Text":"The probability of X being equal to k,"},{"Start":"02:25.270 ","End":"02:33.420","Text":"that equals to q to the power of k minus 1 times p. If p equals 0.6,"},{"Start":"02:33.420 ","End":"02:36.660","Text":"then q equals 0.4."},{"Start":"02:36.660 ","End":"02:40.610","Text":"That\u0027s 1 minus p. In question a,"},{"Start":"02:40.610 ","End":"02:44.020","Text":"we\u0027re looking at k being equal to 5."},{"Start":"02:44.020 ","End":"02:47.975","Text":"What\u0027s the probability that he will have to select 5 cakes?"},{"Start":"02:47.975 ","End":"02:54.020","Text":"That means we\u0027re looking at the probability where X equals 5."},{"Start":"02:54.020 ","End":"02:56.480","Text":"Let\u0027s plug in the numbers."},{"Start":"02:56.480 ","End":"03:01.520","Text":"Q is 0.4 to the power of k minus 1,"},{"Start":"03:01.520 ","End":"03:04.130","Text":"5 minus 1 that\u0027s 4, times p,"},{"Start":"03:04.130 ","End":"03:10.740","Text":"that\u0027s 0.6, and that equals to 0.015."},{"Start":"03:11.330 ","End":"03:15.440","Text":"That\u0027s the probability that we will have to select"},{"Start":"03:15.440 ","End":"03:19.740","Text":"5 cakes before getting a chocolate cake. It\u0027s right here."},{"Start":"03:19.760 ","End":"03:22.280","Text":"In section B we\u0027re asked,"},{"Start":"03:22.280 ","End":"03:26.330","Text":"if a person samples less than 7 cakes before getting a chocolate cake,"},{"Start":"03:26.330 ","End":"03:30.680","Text":"what\u0027s the probability that he actually samples more than 4 cakes?"},{"Start":"03:30.680 ","End":"03:36.840","Text":"This looks like a conditional probability. Let\u0027s set it up."},{"Start":"03:37.330 ","End":"03:40.685","Text":"We want the probability,"},{"Start":"03:40.685 ","End":"03:42.740","Text":"what\u0027s given to us,"},{"Start":"03:42.740 ","End":"03:47.920","Text":"that a person samples less than 7. What do we want to know?"},{"Start":"03:47.920 ","End":"03:51.250","Text":"What\u0027s the probability that the actual sample\u0027s more than 4 cakes?"},{"Start":"03:51.250 ","End":"03:54.160","Text":"That means that X is greater than 4."},{"Start":"03:54.160 ","End":"03:58.386","Text":"In a conditional probability,"},{"Start":"03:58.386 ","End":"04:02.620","Text":"in the denominator, we have the probability of this guy,"},{"Start":"04:02.620 ","End":"04:05.440","Text":"X being less than 7."},{"Start":"04:05.440 ","End":"04:09.190","Text":"In the numerator, that\u0027s the intersection of this guy,"},{"Start":"04:09.190 ","End":"04:13.410","Text":"X has to be greater than 4 and less than 7,"},{"Start":"04:13.410 ","End":"04:16.845","Text":"not equaling to the 4 and 7."},{"Start":"04:16.845 ","End":"04:18.275","Text":"That means that,"},{"Start":"04:18.275 ","End":"04:24.590","Text":"that\u0027s the probability of X being greater than 4 and less than 7."},{"Start":"04:24.590 ","End":"04:31.845","Text":"Let\u0027s take a look at the denominator here."},{"Start":"04:31.845 ","End":"04:35.780","Text":"The probability of X being less than 7,"},{"Start":"04:35.780 ","End":"04:38.090","Text":"that\u0027s 1 minus,"},{"Start":"04:38.090 ","End":"04:40.880","Text":"we\u0027re taking the complimentary set right here,"},{"Start":"04:40.880 ","End":"04:43.070","Text":"the probability of X."},{"Start":"04:43.070 ","End":"04:47.220","Text":"What\u0027s the complimentary set of X being less than 7,"},{"Start":"04:47.220 ","End":"04:51.210","Text":"that\u0027s X being greater than 6."},{"Start":"04:51.210 ","End":"04:59.425","Text":"What\u0027s the probability of X being less than 7 and greater than 4,"},{"Start":"04:59.425 ","End":"05:02.180","Text":"but not equaling 4 and 7?"},{"Start":"05:02.180 ","End":"05:11.400","Text":"That\u0027s the probability where X equals 5 plus the probability where X equals 6."},{"Start":"05:11.400 ","End":"05:15.020","Text":"Let\u0027s just remind ourselves of"},{"Start":"05:15.020 ","End":"05:21.260","Text":"the geometric probability distribution. What\u0027s the function?"},{"Start":"05:21.260 ","End":"05:24.245","Text":"The probability of X being equal to k,"},{"Start":"05:24.245 ","End":"05:28.220","Text":"that equals to q to the power of k minus 1 times p,"},{"Start":"05:28.220 ","End":"05:31.340","Text":"and let\u0027s also remember"},{"Start":"05:31.340 ","End":"05:36.665","Text":"the characteristic where the probability of X being greater than k,"},{"Start":"05:36.665 ","End":"05:41.795","Text":"that equals to q to the power of k. Having remembered these,"},{"Start":"05:41.795 ","End":"05:45.990","Text":"let\u0027s just calculate these guys right here."},{"Start":"05:46.450 ","End":"05:50.555","Text":"Let\u0027s look at the probability of X being equal to 5."},{"Start":"05:50.555 ","End":"05:55.100","Text":"Q, that\u0027s 0.4 to the power of 4."},{"Start":"05:55.100 ","End":"06:00.095","Text":"That\u0027s 5 minus 1 times p, which is 0.6."},{"Start":"06:00.095 ","End":"06:03.690","Text":"Plus, the probability of X equaling 6,"},{"Start":"06:03.690 ","End":"06:06.000","Text":"that\u0027s 0.4 to the power of 5."},{"Start":"06:06.000 ","End":"06:13.830","Text":"That\u0027s 6 minus 1 times 0.6 divided by 1 minus,"},{"Start":"06:13.830 ","End":"06:18.555","Text":"what\u0027s the probability of X being greater than 6?"},{"Start":"06:18.555 ","End":"06:22.975","Text":"That\u0027s 0.4 to the power of 6."},{"Start":"06:22.975 ","End":"06:26.420","Text":"Once we\u0027ve calculated this,"},{"Start":"06:26.420 ","End":"06:35.370","Text":"this comes out to 0.0215. That\u0027s our answer."}],"ID":13022},{"Watched":false,"Name":"Exercise 6 Parts c-d","Duration":"10m 24s","ChapterTopicVideoID":12544,"CourseChapterTopicPlaylistID":64486,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In Section C, we\u0027re given that a person samples cakes until he finds a chocolate cake."},{"Start":"00:05.175 ","End":"00:09.930","Text":"Now it\u0027s known that the cost of making a cheesecake is $50,"},{"Start":"00:09.930 ","End":"00:12.810","Text":"and a chocolate cake $30, and we\u0027re asked,"},{"Start":"00:12.810 ","End":"00:14.730","Text":"what are the expectation and variance of"},{"Start":"00:14.730 ","End":"00:17.715","Text":"the total cost of producing the cakes in the sample?"},{"Start":"00:17.715 ","End":"00:20.055","Text":"Well, that\u0027s just recall,"},{"Start":"00:20.055 ","End":"00:27.225","Text":"X is distributed with the geometric distribution with a probability of 0.6."},{"Start":"00:27.225 ","End":"00:33.780","Text":"X is defined also as the number of"},{"Start":"00:33.780 ","End":"00:35.830","Text":"samples"},{"Start":"00:37.580 ","End":"00:45.430","Text":"until the first chocolate cake."},{"Start":"00:47.540 ","End":"00:51.780","Text":"What\u0027s the probability where X equals k?"},{"Start":"00:51.780 ","End":"01:01.670","Text":"Well, that equals to q^k minus 1 times p. What else do we know?"},{"Start":"01:01.670 ","End":"01:07.460","Text":"We can know that the expectation of X equals 1/p."},{"Start":"01:07.460 ","End":"01:17.460","Text":"That means that that\u0027s 1/0.6 and that equals to 1 and 2/3."},{"Start":"01:17.460 ","End":"01:19.950","Text":"What\u0027s the variance of X?"},{"Start":"01:19.950 ","End":"01:23.239","Text":"That\u0027s q over squared,"},{"Start":"01:23.239 ","End":"01:28.280","Text":"and that means that it\u0027s 0.4/0.6 squared and"},{"Start":"01:28.280 ","End":"01:34.360","Text":"that equals to 1 and 1/9."},{"Start":"01:34.360 ","End":"01:39.950","Text":"Having calculated that, let\u0027s take a look at what we\u0027re asked,"},{"Start":"01:39.950 ","End":"01:42.890","Text":"were asked not for X,"},{"Start":"01:42.890 ","End":"01:47.375","Text":"but for the total cost of producing the cakes."},{"Start":"01:47.375 ","End":"01:51.260","Text":"That looks like a linear transformation, and why is that?"},{"Start":"01:51.260 ","End":"01:53.860","Text":"Because we\u0027re moving from X,"},{"Start":"01:53.860 ","End":"01:57.400","Text":"which is the number of samples,"},{"Start":"01:57.710 ","End":"02:04.120","Text":"to y, which is the total cost."},{"Start":"02:05.540 ","End":"02:11.400","Text":"Let\u0027s see how we can figure out what the total cost is."},{"Start":"02:11.600 ","End":"02:16.020","Text":"We know that we\u0027ve sampled X,"},{"Start":"02:16.020 ","End":"02:20.355","Text":"that\u0027s the number of samples until we ate a chocolate cake."},{"Start":"02:20.355 ","End":"02:24.860","Text":"How many cheese cakes did we have to sample before that?"},{"Start":"02:24.860 ","End":"02:30.580","Text":"We had to sample X minus 1 cheesecakes."},{"Start":"02:30.980 ","End":"02:36.020","Text":"How much does a cheesecake cost to make,"},{"Start":"02:36.020 ","End":"02:42.260","Text":"that\u0027s $50, plus the last sample,"},{"Start":"02:42.260 ","End":"02:46.550","Text":"which is a chocolate cake, times $30."},{"Start":"02:46.550 ","End":"02:48.995","Text":"That would be our y,"},{"Start":"02:48.995 ","End":"02:51.020","Text":"that would be the total cost."},{"Start":"02:51.020 ","End":"03:00.185","Text":"This is the linear transformation of taking the number of samples,"},{"Start":"03:00.185 ","End":"03:05.180","Text":"and transforming them into costs."},{"Start":"03:05.180 ","End":"03:09.110","Text":"But we\u0027re not through with this yet. Why is that?"},{"Start":"03:09.110 ","End":"03:12.005","Text":"Because it\u0027s not in the generic form."},{"Start":"03:12.005 ","End":"03:16.080","Text":"So let\u0027s put this in a generic form."},{"Start":"03:16.330 ","End":"03:19.895","Text":"If y equals this,"},{"Start":"03:19.895 ","End":"03:27.045","Text":"then y equals 50X minus 50 plus 30,"},{"Start":"03:27.045 ","End":"03:32.775","Text":"and that equals to 50X minus 20."},{"Start":"03:32.775 ","End":"03:37.730","Text":"The generic form for a linear transformation,"},{"Start":"03:37.730 ","End":"03:41.145","Text":"that\u0027s aX plus b."},{"Start":"03:41.145 ","End":"03:48.360","Text":"That means that a in our case is 50 and b equals minus 20,"},{"Start":"03:48.360 ","End":"03:51.000","Text":"a is the multiplier of X, that\u0027s 50,"},{"Start":"03:51.000 ","End":"03:56.080","Text":"that\u0027s right here, and B is minus 20, that\u0027s right here."},{"Start":"03:57.170 ","End":"04:02.540","Text":"Again, we\u0027re not really done yet because we\u0027re asked for"},{"Start":"04:02.540 ","End":"04:07.490","Text":"the expectation of y and the variance of y."},{"Start":"04:07.490 ","End":"04:09.530","Text":"Now, if we recall,"},{"Start":"04:09.530 ","End":"04:17.705","Text":"the expectation of Y is a times the expectation of X plus b."},{"Start":"04:17.705 ","End":"04:20.337","Text":"Now let\u0027s plug in our numbers,"},{"Start":"04:20.337 ","End":"04:22.495","Text":"a is 50,"},{"Start":"04:22.495 ","End":"04:26.130","Text":"times the expectation of X, that\u0027s right here."},{"Start":"04:26.130 ","End":"04:31.740","Text":"That\u0027s 1 and 2/3 minus 20."},{"Start":"04:31.740 ","End":"04:41.970","Text":"That comes out to 63 and 1/3."},{"Start":"04:41.970 ","End":"04:43.370","Text":"What\u0027s the variance of y?"},{"Start":"04:43.370 ","End":"04:47.340","Text":"Well, that\u0027s a squared times the variance of X."},{"Start":"04:47.930 ","End":"04:51.975","Text":"That equals to, that\u0027s 50 squared,"},{"Start":"04:51.975 ","End":"04:54.395","Text":"that\u0027s a is 50."},{"Start":"04:54.395 ","End":"04:56.150","Text":"Now what\u0027s the variance of X?"},{"Start":"04:56.150 ","End":"05:00.545","Text":"The variance of X is 1 and 1/9 times 1 and 1/9,"},{"Start":"05:00.545 ","End":"05:02.900","Text":"and that comes out to"},{"Start":"05:02.900 ","End":"05:12.210","Text":"be 2,777 and 7/9."},{"Start":"05:13.310 ","End":"05:19.730","Text":"Here we figured out the expectation of y and the variance of y,"},{"Start":"05:19.730 ","End":"05:23.830","Text":"and variance is the total cost of the cakes in the sample."},{"Start":"05:23.830 ","End":"05:27.430","Text":"So the total cost is $63 and 1/3,"},{"Start":"05:27.430 ","End":"05:29.355","Text":"that\u0027s the units here,"},{"Start":"05:29.355 ","End":"05:37.005","Text":"and here the variance is $2,777 and 7/9 squared."},{"Start":"05:37.005 ","End":"05:44.205","Text":"These are the answers for Section C. In Section D we\u0027re asked,"},{"Start":"05:44.205 ","End":"05:46.470","Text":"following the previous question,"},{"Start":"05:46.470 ","End":"05:47.870","Text":"what are the expectations,"},{"Start":"05:47.870 ","End":"05:53.100","Text":"standard deviation of the number of cheesecakes sampled by the person?"},{"Start":"05:53.180 ","End":"06:03.885","Text":"If we recall, X was the number of selections in the sample,"},{"Start":"06:03.885 ","End":"06:09.820","Text":"until the first chocolate cake,"},{"Start":"06:10.850 ","End":"06:22.535","Text":"where X is distributed with a geometric distribution where p equals 0.6."},{"Start":"06:22.535 ","End":"06:26.960","Text":"What are we asked? Were not asked about X,"},{"Start":"06:26.960 ","End":"06:31.140","Text":"but we\u0027re asked about the cheesecakes,"},{"Start":"06:31.140 ","End":"06:32.660","Text":"and what do we want to know,"},{"Start":"06:32.660 ","End":"06:37.860","Text":"the expectation and standard deviation of the cheesecakes."},{"Start":"06:38.600 ","End":"06:43.475","Text":"If X is a number of selections until the first chocolate cake,"},{"Start":"06:43.475 ","End":"06:50.215","Text":"then how many cheesecakes did we select before we got chocolate cake?"},{"Start":"06:50.215 ","End":"06:54.225","Text":"That would have been X minus 1."},{"Start":"06:54.225 ","End":"07:04.280","Text":"Because we had to sample x minus 1 cheesecake until we got the first chocolate cake."},{"Start":"07:04.280 ","End":"07:09.000","Text":"So y is X minus 1,"},{"Start":"07:09.000 ","End":"07:13.690","Text":"which would be the number of cheesecakes."},{"Start":"07:17.450 ","End":"07:22.515","Text":"Now, this really looks like a linear transformation,"},{"Start":"07:22.515 ","End":"07:27.530","Text":"because it answers the generic form of a linear transformation,"},{"Start":"07:27.530 ","End":"07:31.730","Text":"y equals aX plus b."},{"Start":"07:31.730 ","End":"07:37.150","Text":"In our case, a would be equal to 1,"},{"Start":"07:37.150 ","End":"07:41.680","Text":"and b would be equal to minus 1."},{"Start":"07:42.380 ","End":"07:45.965","Text":"What are we asked? We\u0027re asked basically,"},{"Start":"07:45.965 ","End":"07:49.355","Text":"what is the expectation of y,"},{"Start":"07:49.355 ","End":"07:52.470","Text":"and what\u0027s the variance of y?"},{"Start":"07:52.640 ","End":"08:02.255","Text":"The expectation of y as a times the expectation of X plus b."},{"Start":"08:02.255 ","End":"08:07.750","Text":"The variance is a squared times the variance of X."},{"Start":"08:07.750 ","End":"08:10.775","Text":"If we recall from the last section,"},{"Start":"08:10.775 ","End":"08:14.690","Text":"the expectation of X was what?"},{"Start":"08:14.690 ","End":"08:16.980","Text":"That was 1 and 2/3."},{"Start":"08:17.810 ","End":"08:20.805","Text":"The variance of X,"},{"Start":"08:20.805 ","End":"08:24.370","Text":"that was 1 and 1/9."},{"Start":"08:24.370 ","End":"08:30.860","Text":"So let\u0027s just plug in the numbers and see what the expectation of y is,"},{"Start":"08:30.860 ","End":"08:34.175","Text":"where y is a linear transformation of X,"},{"Start":"08:34.175 ","End":"08:37.655","Text":"which counts the number of cheesecakes,"},{"Start":"08:37.655 ","End":"08:40.255","Text":"not the chocolate cakes, but the cheesecakes."},{"Start":"08:40.255 ","End":"08:43.726","Text":"So the expectation of y,"},{"Start":"08:43.726 ","End":"08:47.865","Text":"that equals to a, a is what?"},{"Start":"08:47.865 ","End":"08:49.830","Text":"a we said is 1,"},{"Start":"08:49.830 ","End":"08:52.910","Text":"1 times the expectation of X, that\u0027s right here,"},{"Start":"08:52.910 ","End":"08:57.245","Text":"1 times 1 and 2/3 plus b,"},{"Start":"08:57.245 ","End":"08:59.580","Text":"b is minus 1."},{"Start":"08:59.830 ","End":"09:05.350","Text":"That equals to 2/3."},{"Start":"09:05.350 ","End":"09:08.325","Text":"This is the expectation of y,"},{"Start":"09:08.325 ","End":"09:16.990","Text":"the number of cheesecakes selections would be 2/3 cakes."},{"Start":"09:17.150 ","End":"09:20.700","Text":"Now what\u0027s the variance of y?"},{"Start":"09:20.700 ","End":"09:24.015","Text":"Well, the variance of y is a squared,"},{"Start":"09:24.015 ","End":"09:28.970","Text":"a is 1, so that\u0027s 1 squared times the variance of X."},{"Start":"09:28.970 ","End":"09:34.220","Text":"Well, the variance of X is 1 and 1/9."},{"Start":"09:34.220 ","End":"09:43.295","Text":"That equals to 1 and 1/9 cheesecakes squared."},{"Start":"09:43.295 ","End":"09:47.090","Text":"But again, we\u0027re asked for the standard deviation, not the variance."},{"Start":"09:47.090 ","End":"09:53.870","Text":"Now we know that the standard deviation of y equals the square root of the variance of y."},{"Start":"09:53.870 ","End":"09:58.940","Text":"Now, that equals to the square root of 1 and 1/9, well,"},{"Start":"09:58.940 ","End":"10:03.110","Text":"that equals to 1.054,"},{"Start":"10:03.110 ","End":"10:07.200","Text":"the units are cakes."},{"Start":"10:08.140 ","End":"10:15.275","Text":"This is our answer for the standard deviation and the expectation of y,"},{"Start":"10:15.275 ","End":"10:23.860","Text":"which is the amount of cheesecakes sampled before we sampled a chocolate cake."}],"ID":13023}],"Thumbnail":null,"ID":64486},{"Name":"Special Discrete Probability Distributions - Uniform Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"4m 44s","ChapterTopicVideoID":12545,"CourseChapterTopicPlaylistID":64487,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.060 ","End":"00:04.990","Text":"In this chapter, we\u0027ll be discussing special discrete probabilities,"},{"Start":"00:04.990 ","End":"00:08.065","Text":"specifically the uniform distribution."},{"Start":"00:08.065 ","End":"00:11.095","Text":"Now, in this distribution,"},{"Start":"00:11.095 ","End":"00:14.140","Text":"each result has the same probability"},{"Start":"00:14.140 ","End":"00:18.635","Text":"and the values obtained in this distribution are between a and b,"},{"Start":"00:18.635 ","End":"00:21.315","Text":"with increasing increments of 1."},{"Start":"00:21.315 ","End":"00:27.085","Text":"What that means is that the distribution has a lower limit and an upper limit."},{"Start":"00:27.085 ","End":"00:28.930","Text":"The lower limit being a,"},{"Start":"00:28.930 ","End":"00:30.940","Text":"the upper limit being b,"},{"Start":"00:30.940 ","End":"00:36.370","Text":"and X takes on the values between a and b with increments of 1."},{"Start":"00:36.370 ","End":"00:38.245","Text":"That means that it starts with a,"},{"Start":"00:38.245 ","End":"00:40.360","Text":"then a plus 1, a plus 2,"},{"Start":"00:40.360 ","End":"00:42.785","Text":"and so on and so forth until we get to b."},{"Start":"00:42.785 ","End":"00:45.725","Text":"Now, whenever we have this scenario,"},{"Start":"00:45.725 ","End":"00:51.250","Text":"then we say that X is distributed with a uniform distribution,"},{"Start":"00:51.250 ","End":"00:52.905","Text":"that\u0027s U for uniform,"},{"Start":"00:52.905 ","End":"00:54.950","Text":"with 2 parameters, a and b,"},{"Start":"00:54.950 ","End":"00:56.480","Text":"where a is the lower limit of"},{"Start":"00:56.480 ","End":"01:00.365","Text":"the distribution and b is the upper limit of the distribution."},{"Start":"01:00.365 ","End":"01:08.265","Text":"Now, the probability of X where X equals K,"},{"Start":"01:08.265 ","End":"01:12.860","Text":"that would equal 1 over b minus a plus 1."},{"Start":"01:12.860 ","End":"01:19.335","Text":"That\u0027s what makes this a uniform distribution for every K,"},{"Start":"01:19.335 ","End":"01:22.380","Text":"where K gets a value of a, a plus 1,"},{"Start":"01:22.380 ","End":"01:27.295","Text":"a plus 2 until b for every K. Then"},{"Start":"01:27.295 ","End":"01:32.445","Text":"the probability of X equaling K is a constant,"},{"Start":"01:32.445 ","End":"01:36.060","Text":"it\u0027s 1 over b minus a plus 1."},{"Start":"01:36.060 ","End":"01:41.070","Text":"Now, let\u0027s go to the expectation and variance."},{"Start":"01:41.090 ","End":"01:45.900","Text":"The equation for the expectation is as follows."},{"Start":"01:45.900 ","End":"01:50.690","Text":"The expectation of X equals a plus b over 2,"},{"Start":"01:50.690 ","End":"01:57.725","Text":"and the variance of x is b minus a plus 1 squared minus 1 over 12."},{"Start":"01:57.725 ","End":"02:04.715","Text":"Now, these are the equations that we\u0027re"},{"Start":"02:04.715 ","End":"02:12.200","Text":"given when X is distributed with a uniform distribution with parameters a and b."},{"Start":"02:12.200 ","End":"02:17.940","Text":"Now, let\u0027s take a look at an example and see how we can work with these things."},{"Start":"02:18.170 ","End":"02:24.320","Text":"In our example, a person randomly selects a number between 1 and 100"},{"Start":"02:24.320 ","End":"02:26.810","Text":"inclusively and we\u0027re asked what\u0027s"},{"Start":"02:26.810 ","End":"02:31.910","Text":"the probability function of the number and what\u0027s its expectation?"},{"Start":"02:31.910 ","End":"02:38.915","Text":"Well, here, we obviously have a uniform distribution. Why is that?"},{"Start":"02:38.915 ","End":"02:46.130","Text":"Because for every number that\u0027s selected randomly between 1 and 100,"},{"Start":"02:46.130 ","End":"02:49.080","Text":"the probability would be the same probability."},{"Start":"02:49.640 ","End":"02:55.250","Text":"Let\u0027s just use our equations for the uniform probability."},{"Start":"02:55.250 ","End":"03:00.125","Text":"The probability of X being equal to K,"},{"Start":"03:00.125 ","End":"03:02.475","Text":"that would equal to what?"},{"Start":"03:02.475 ","End":"03:10.590","Text":"1 over b minus a plus 1, that\u0027s the probability."},{"Start":"03:10.590 ","End":"03:17.285","Text":"Again, we have a uniform probability and what\u0027s our lower limit and upper limit?"},{"Start":"03:17.285 ","End":"03:20.730","Text":"We have the lower limit being 1,"},{"Start":"03:20.730 ","End":"03:22.160","Text":"it\u0027s given right here,"},{"Start":"03:22.160 ","End":"03:25.595","Text":"and the upper limit being 100."},{"Start":"03:25.595 ","End":"03:28.580","Text":"This is our X,"},{"Start":"03:28.580 ","End":"03:31.980","Text":"X is a randomly selected number."},{"Start":"03:33.860 ","End":"03:39.410","Text":"This is our probability function and when we just plug in the numbers,"},{"Start":"03:39.410 ","End":"03:44.730","Text":"we have 1 over now b is 100 minus a,"},{"Start":"03:44.730 ","End":"03:47.350","Text":"that\u0027s 1 plus 1,"},{"Start":"03:47.480 ","End":"03:52.905","Text":"and that equals to 1 over 100."},{"Start":"03:52.905 ","End":"03:59.405","Text":"That\u0027s the probability where X equals any number between 1 and 100."},{"Start":"03:59.405 ","End":"04:04.080","Text":"Now, we\u0027re also asked about the expectation."},{"Start":"04:04.310 ","End":"04:10.295","Text":"Now, what\u0027s the equation for the expectation in a uniform distribution?"},{"Start":"04:10.295 ","End":"04:14.360","Text":"That\u0027s b plus a over 2."},{"Start":"04:14.360 ","End":"04:16.460","Text":"Again, let\u0027s plug in the numbers,"},{"Start":"04:16.460 ","End":"04:19.700","Text":"b is 100 plus a,"},{"Start":"04:19.700 ","End":"04:23.120","Text":"a is 1 divided by 2,"},{"Start":"04:23.120 ","End":"04:27.120","Text":"and that equals 50.5."},{"Start":"04:27.190 ","End":"04:33.690","Text":"This is basically the middle of the distribution between 1 and 100,"},{"Start":"04:33.690 ","End":"04:41.840","Text":"and this is the probability of landing on any random number between a and b,"},{"Start":"04:41.840 ","End":"04:44.250","Text":"between 1 and 100."}],"ID":13024},{"Watched":false,"Name":"Exercise 1","Duration":"5m 32s","ChapterTopicVideoID":12546,"CourseChapterTopicPlaylistID":64487,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"This question deals with a game of chance."},{"Start":"00:02.415 ","End":"00:03.780","Text":"Now, in a lotto game,"},{"Start":"00:03.780 ","End":"00:07.035","Text":"there are 45 balls numbered from 1-45,"},{"Start":"00:07.035 ","End":"00:11.055","Text":"and x is the number of the first ball drawn by the machine."},{"Start":"00:11.055 ","End":"00:18.255","Text":"Now, we can see that x is distributed with a uniform distribution. Why is that?"},{"Start":"00:18.255 ","End":"00:23.305","Text":"Because it really doesn\u0027t matter what number was chosen."},{"Start":"00:23.305 ","End":"00:28.100","Text":"The probability of getting that number is a constant."},{"Start":"00:28.100 ","End":"00:31.490","Text":"It doesn\u0027t matter if it was number 1,"},{"Start":"00:31.490 ","End":"00:33.590","Text":"or number 10, or number 15,"},{"Start":"00:33.590 ","End":"00:34.805","Text":"and so on and so forth."},{"Start":"00:34.805 ","End":"00:37.190","Text":"For each value of x,"},{"Start":"00:37.190 ","End":"00:39.940","Text":"we have the same probability."},{"Start":"00:39.940 ","End":"00:43.400","Text":"This points to a uniform distribution."},{"Start":"00:43.400 ","End":"00:47.795","Text":"Now, what\u0027s the parameters of a uniform distribution?"},{"Start":"00:47.795 ","End":"00:50.500","Text":"Well, we have a,"},{"Start":"00:50.500 ","End":"00:53.510","Text":"which is the lower bound and that equals to 1,"},{"Start":"00:53.510 ","End":"00:55.010","Text":"that\u0027s given right here."},{"Start":"00:55.010 ","End":"00:56.480","Text":"And the upper bound,"},{"Start":"00:56.480 ","End":"00:57.750","Text":"that\u0027s b, well,"},{"Start":"00:57.750 ","End":"00:59.950","Text":"that equals to 45."},{"Start":"00:59.950 ","End":"01:06.810","Text":"The probability of x being equal to k,"},{"Start":"01:06.810 ","End":"01:11.795","Text":"well that equals to 1 over b minus a plus 1,"},{"Start":"01:11.795 ","End":"01:14.030","Text":"where k equals 1,"},{"Start":"01:14.030 ","End":"01:17.105","Text":"2, and so on and so forth until 45."},{"Start":"01:17.105 ","End":"01:19.580","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"01:19.580 ","End":"01:22.020","Text":"1 over, b is 45,"},{"Start":"01:22.020 ","End":"01:27.555","Text":"1 over 45 minus 1 plus 1,"},{"Start":"01:27.555 ","End":"01:33.105","Text":"that equals to 1 over 45."},{"Start":"01:33.105 ","End":"01:36.900","Text":"So having said that,"},{"Start":"01:36.900 ","End":"01:43.090","Text":"let\u0027s calculate the first section right here, a."},{"Start":"01:43.090 ","End":"01:50.645","Text":"Now a asks us to calculate the probability where x equals 2 and as we said,"},{"Start":"01:50.645 ","End":"01:53.870","Text":"it really doesn\u0027t matter what value x has,"},{"Start":"01:53.870 ","End":"01:58.775","Text":"because x is distributed with the uniform distribution,"},{"Start":"01:58.775 ","End":"02:07.200","Text":"therefore, the probability of any value of x would be 1 over 45."},{"Start":"02:07.930 ","End":"02:14.525","Text":"Section b asks us to calculate the probability of x being less than or equal to 30."},{"Start":"02:14.525 ","End":"02:20.195","Text":"What\u0027s the probability of x being less than or equal to 30?"},{"Start":"02:20.195 ","End":"02:23.195","Text":"Well, let\u0027s take a look."},{"Start":"02:23.195 ","End":"02:25.790","Text":"The probability of x being equal to 1,"},{"Start":"02:25.790 ","End":"02:28.540","Text":"that equals 1 over 45,"},{"Start":"02:28.540 ","End":"02:31.515","Text":"and the probability of x equaling 2,"},{"Start":"02:31.515 ","End":"02:33.740","Text":"that\u0027s also 1 over 45,"},{"Start":"02:33.740 ","End":"02:38.555","Text":"and so on and so forth until we get to x being equal to 30."},{"Start":"02:38.555 ","End":"02:40.970","Text":"That\u0027s also 1 over 45."},{"Start":"02:40.970 ","End":"02:46.490","Text":"Now, how many values of x are there between 1 and 30?"},{"Start":"02:46.490 ","End":"02:55.170","Text":"Well, that\u0027s 30 minus 1 plus 1 times the probability of each 1 of these values."},{"Start":"02:55.170 ","End":"03:00.275","Text":"Well, each 1 of these values have a probability of 1 over 45,"},{"Start":"03:00.275 ","End":"03:05.190","Text":"that equals to 30 over 45."},{"Start":"03:05.720 ","End":"03:10.690","Text":"In section c we\u0027re asked to calculate the probability of x being"},{"Start":"03:10.690 ","End":"03:15.130","Text":"greater than 4 given that x is less than or equal to 10."},{"Start":"03:15.130 ","End":"03:19.570","Text":"That\u0027s again, probability of x being greater than"},{"Start":"03:19.570 ","End":"03:23.950","Text":"4 given that x is less than or equal to 10."},{"Start":"03:23.950 ","End":"03:28.920","Text":"Well, that\u0027s a conditional probability and we do know how to solve that."},{"Start":"03:28.920 ","End":"03:33.370","Text":"In the denominator, we\u0027ll have the probability of what\u0027s given,"},{"Start":"03:33.370 ","End":"03:36.220","Text":"x being less than or equal to 10."},{"Start":"03:36.220 ","End":"03:38.620","Text":"Now, what\u0027s the numerator?"},{"Start":"03:38.620 ","End":"03:43.390","Text":"The numerator is the probability of the intersection of these 2 guys."},{"Start":"03:43.390 ","End":"03:48.800","Text":"So that\u0027s the probability where x is greater than 4,"},{"Start":"03:48.800 ","End":"03:51.285","Text":"not equaling 4, but greater than 4,"},{"Start":"03:51.285 ","End":"03:54.120","Text":"but less than or equal to 10."},{"Start":"03:54.120 ","End":"03:59.030","Text":"Let\u0027s just see what\u0027s going on here."},{"Start":"03:59.030 ","End":"04:03.215","Text":"The probability of x being less than or equal to 10,"},{"Start":"04:03.215 ","End":"04:07.120","Text":"well, that\u0027s 10 over 45."},{"Start":"04:07.120 ","End":"04:11.240","Text":"What\u0027s the probability of x being greater than 4,"},{"Start":"04:11.240 ","End":"04:16.070","Text":"not equaling 4 but it\u0027s greater than 4 and less than or equal to 10?"},{"Start":"04:16.070 ","End":"04:20.210","Text":"Well, what are the values of x which are greater than 4 and less than or equal to 10?"},{"Start":"04:20.210 ","End":"04:22.385","Text":"That\u0027s 5, 6, 7,"},{"Start":"04:22.385 ","End":"04:24.560","Text":"8, 9, and 10. That\u0027s 6."},{"Start":"04:24.560 ","End":"04:28.790","Text":"We have 6 values of x in this range,"},{"Start":"04:28.790 ","End":"04:31.925","Text":"so that\u0027s 6 over 45,"},{"Start":"04:31.925 ","End":"04:38.350","Text":"and that comes out to 0.6."},{"Start":"04:38.350 ","End":"04:43.760","Text":"In section d we\u0027re asked to calculate the probability where x equals k."},{"Start":"04:43.760 ","End":"04:49.100","Text":"The probability of x being equal to k. Well,"},{"Start":"04:49.100 ","End":"04:53.235","Text":"in our situation here in this question,"},{"Start":"04:53.235 ","End":"04:56.225","Text":"x is distributed uniformly."},{"Start":"04:56.225 ","End":"05:05.475","Text":"With the ranges where a equals 1 and b equals 45, so that\u0027s easy."},{"Start":"05:05.475 ","End":"05:09.380","Text":"That equals to 1 over 45 because what does this say?"},{"Start":"05:09.380 ","End":"05:11.210","Text":"This basically asks us,"},{"Start":"05:11.210 ","End":"05:18.760","Text":"what\u0027s the probability of x being equal to any value within the range between 1 and 45?"},{"Start":"05:18.760 ","End":"05:22.465","Text":"Now, we know since x is distributed uniformly,"},{"Start":"05:22.465 ","End":"05:25.820","Text":"it\u0027s a constant probability of 1 over 45."},{"Start":"05:25.820 ","End":"05:27.755","Text":"We\u0027ve calculated that right here,"},{"Start":"05:27.755 ","End":"05:32.100","Text":"so this is our answer for d."}],"ID":13025},{"Watched":false,"Name":"Exercise 2","Duration":"12m 14s","ChapterTopicVideoID":12547,"CourseChapterTopicPlaylistID":64487,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this question, we\u0027ll be talking about magic."},{"Start":"00:02.940 ","End":"00:07.301","Text":"A magician wishes to select a random number between 1 and a 100."},{"Start":"00:07.301 ","End":"00:11.580","Text":"Assume that there\u0027s no information about the magician and we\u0027re asked,"},{"Start":"00:11.580 ","End":"00:15.555","Text":"what\u0027s the expectation and standard deviation of the number selected?"},{"Start":"00:15.555 ","End":"00:19.605","Text":"Well, we know that we have a uniform distribution."},{"Start":"00:19.605 ","End":"00:22.965","Text":"That is that x is being distributed"},{"Start":"00:22.965 ","End":"00:27.816","Text":"uniformly and we need to know what the parameters are."},{"Start":"00:27.816 ","End":"00:30.970","Text":"Now, why is it a uniform distribution?"},{"Start":"00:30.970 ","End":"00:34.939","Text":"Because no matter what number a person selects,"},{"Start":"00:34.939 ","End":"00:38.479","Text":"the probability of getting that number is uniform."},{"Start":"00:38.479 ","End":"00:40.175","Text":"It\u0027s a constant."},{"Start":"00:40.175 ","End":"00:46.175","Text":"Now, let\u0027s look at the parameters of this uniform distribution."},{"Start":"00:46.175 ","End":"00:47.360","Text":"Well, the lower bound,"},{"Start":"00:47.360 ","End":"00:49.610","Text":"that\u0027s a, that equals to 1,"},{"Start":"00:49.610 ","End":"00:52.490","Text":"that\u0027s given right here and what\u0027s the upper bound,"},{"Start":"00:52.490 ","End":"00:55.350","Text":"that\u0027s b? That\u0027s a 100."},{"Start":"00:56.030 ","End":"01:02.420","Text":"We know that the probability of x being equal to k,"},{"Start":"01:02.420 ","End":"01:09.275","Text":"well, that equals to 1 over b minus a plus 1."},{"Start":"01:09.275 ","End":"01:11.525","Text":"Now in our case,"},{"Start":"01:11.525 ","End":"01:19.495","Text":"probability of x being equal to k,"},{"Start":"01:19.495 ","End":"01:22.740","Text":"that equals to 1 over what\u0027s b?"},{"Start":"01:22.740 ","End":"01:25.665","Text":"B is a 100 minus 1 plus 1,"},{"Start":"01:25.665 ","End":"01:28.290","Text":"a 100 minus 1 plus 1,"},{"Start":"01:28.290 ","End":"01:32.470","Text":"and that equals to 1 over 100."},{"Start":"01:33.200 ","End":"01:35.600","Text":"If that\u0027s the case,"},{"Start":"01:35.600 ","End":"01:37.370","Text":"what do we asked?"},{"Start":"01:37.370 ","End":"01:40.400","Text":"Well, we\u0027re asked to figure out what\u0027s"},{"Start":"01:40.400 ","End":"01:43.955","Text":"the expectation is standard deviation of the number."},{"Start":"01:43.955 ","End":"01:48.750","Text":"Fine. Let\u0027s get to it."},{"Start":"01:51.340 ","End":"01:56.995","Text":"Now, the expectation of x,"},{"Start":"01:56.995 ","End":"02:02.470","Text":"well, that\u0027s given to us as b plus a over 2."},{"Start":"02:02.470 ","End":"02:06.550","Text":"Plugging in the numbers these a 100 plus a,"},{"Start":"02:06.550 ","End":"02:12.515","Text":"that\u0027s 1 over 2 and that equals to 50.5."},{"Start":"02:12.515 ","End":"02:15.555","Text":"What\u0027s the variance of x?"},{"Start":"02:15.555 ","End":"02:21.100","Text":"Well, that\u0027s b minus a plus 1 squared,"},{"Start":"02:21.100 ","End":"02:25.220","Text":"minus 1 over 12."},{"Start":"02:25.220 ","End":"02:27.355","Text":"There\u0027s nothing we can do about it."},{"Start":"02:27.355 ","End":"02:29.929","Text":"This is the equation for the variance,"},{"Start":"02:29.929 ","End":"02:31.540","Text":"so let\u0027s just use it."},{"Start":"02:31.540 ","End":"02:33.475","Text":"Let\u0027s plug in the numbers right now."},{"Start":"02:33.475 ","End":"02:36.640","Text":"B is a 100 minus a,"},{"Start":"02:36.640 ","End":"02:44.525","Text":"that\u0027s minus 1 plus 1 squared minus 1 over 12,"},{"Start":"02:44.525 ","End":"02:51.740","Text":"and that equals to 833.25."},{"Start":"02:51.740 ","End":"02:54.170","Text":"But hold on, we\u0027re not asked about the variance,"},{"Start":"02:54.170 ","End":"02:57.635","Text":"we\u0027re asked about the standard deviation."},{"Start":"02:57.635 ","End":"03:00.140","Text":"The standard deviation of x, well,"},{"Start":"03:00.140 ","End":"03:03.109","Text":"that equals to the square root of the variance,"},{"Start":"03:03.109 ","End":"03:06.365","Text":"which is 833.25,"},{"Start":"03:06.365 ","End":"03:14.900","Text":"and that equals to 28.87."},{"Start":"03:14.900 ","End":"03:17.810","Text":"In this section, we\u0027re giving it to the magician asks"},{"Start":"03:17.810 ","End":"03:20.720","Text":"6 people to select a number and we\u0027re asked,"},{"Start":"03:20.720 ","End":"03:24.665","Text":"what\u0027s the probability the 3 of them will select a number larger than 80."},{"Start":"03:24.665 ","End":"03:30.230","Text":"Now, here, this reminds me more of a binomial distribution,"},{"Start":"03:30.230 ","End":"03:32.270","Text":"more than a uniform distribution."},{"Start":"03:32.270 ","End":"03:36.125","Text":"But let\u0027s see how everything just gels together."},{"Start":"03:36.125 ","End":"03:42.170","Text":"First of all, let\u0027s remember that x was defined as"},{"Start":"03:42.170 ","End":"03:48.360","Text":"any number chosen randomly between 1 and a 100."},{"Start":"03:48.360 ","End":"03:54.855","Text":"The probability was 1 over a 100 to select any number."},{"Start":"03:54.855 ","End":"03:58.730","Text":"Now, what are we looking for right now?"},{"Start":"03:58.730 ","End":"04:02.285","Text":"We\u0027re looking at a new variable,"},{"Start":"04:02.285 ","End":"04:08.515","Text":"y and we\u0027ll define that is the number that\u0027s chosen,"},{"Start":"04:08.515 ","End":"04:11.715","Text":"which is over 80."},{"Start":"04:11.715 ","End":"04:14.010","Text":"That would be our success."},{"Start":"04:14.010 ","End":"04:20.960","Text":"That would be the random variable that we\u0027ll have to deal with."},{"Start":"04:20.960 ","End":"04:27.080","Text":"Now this random variable is binomially distributed. Why is that?"},{"Start":"04:27.080 ","End":"04:30.560","Text":"Well, because we\u0027re taking 3 out of"},{"Start":"04:30.560 ","End":"04:33.410","Text":"the 6 people and we want to know what\u0027s"},{"Start":"04:33.410 ","End":"04:38.150","Text":"the probability that 3 of them will select a number larger than 80."},{"Start":"04:38.150 ","End":"04:43.290","Text":"First of all, n is equal to 6,"},{"Start":"04:43.290 ","End":"04:49.230","Text":"the magician asked 6 people and k would be equal to 3."},{"Start":"04:49.230 ","End":"04:53.045","Text":"3 of them will select a number larger than 80."},{"Start":"04:53.045 ","End":"04:57.415","Text":"Now we need to figure out what\u0027s the probability"},{"Start":"04:57.415 ","End":"05:03.170","Text":"of a number being chosen that\u0027s greater than 80."},{"Start":"05:03.170 ","End":"05:05.510","Text":"Well, let\u0027s go back here."},{"Start":"05:05.510 ","End":"05:09.760","Text":"X is any number chosen with a probability of 1 over a 100,"},{"Start":"05:09.760 ","End":"05:15.014","Text":"so what\u0027s the probability of x being greater than 80?"},{"Start":"05:15.014 ","End":"05:17.795","Text":"Well, that equals to,"},{"Start":"05:17.795 ","End":"05:20.175","Text":"first of all, something over a 100."},{"Start":"05:20.175 ","End":"05:24.515","Text":"Now, how many values of x do we have,"},{"Start":"05:24.515 ","End":"05:25.925","Text":"which are over 80?"},{"Start":"05:25.925 ","End":"05:27.680","Text":"Well, we have 81, 82,"},{"Start":"05:27.680 ","End":"05:29.720","Text":"83, until a 100."},{"Start":"05:29.720 ","End":"05:31.070","Text":"Now if you count them up,"},{"Start":"05:31.070 ","End":"05:33.315","Text":"we\u0027ll see that we have 20 of them."},{"Start":"05:33.315 ","End":"05:35.660","Text":"That equals to 0.2."},{"Start":"05:35.660 ","End":"05:43.985","Text":"The probability, of a number that\u0027s chosen to be over 80,"},{"Start":"05:43.985 ","End":"05:46.890","Text":"well, that would be 0.2."},{"Start":"05:47.090 ","End":"05:52.175","Text":"If that\u0027s the case and Y is now a random variable,"},{"Start":"05:52.175 ","End":"05:58.790","Text":"then we know that y is being distributed binomially,"},{"Start":"05:58.790 ","End":"06:04.550","Text":"where n equals 6 and p equals 0.2."},{"Start":"06:04.550 ","End":"06:09.590","Text":"Now if we remember the probability of Y being equal to k,"},{"Start":"06:09.590 ","End":"06:12.560","Text":"well, that equals to n over k,"},{"Start":"06:12.560 ","End":"06:14.840","Text":"p to the power of k,"},{"Start":"06:14.840 ","End":"06:18.740","Text":"q to the power of n minus k,"},{"Start":"06:18.740 ","End":"06:24.305","Text":"where we remember q is 1 minus p. Let\u0027s just plug in the numbers,"},{"Start":"06:24.305 ","End":"06:26.270","Text":"K here is equal to 3,"},{"Start":"06:26.270 ","End":"06:31.700","Text":"so the probability of Y being equal to 3 and that\u0027s what we\u0027re looking for."},{"Start":"06:31.700 ","End":"06:36.560","Text":"We\u0027re looking for 3 people who selected a number larger than 80."},{"Start":"06:36.560 ","End":"06:40.415","Text":"Well, that\u0027s the probability of Y being equal to 3."},{"Start":"06:40.415 ","End":"06:42.440","Text":"That equals to n over k,"},{"Start":"06:42.440 ","End":"06:45.875","Text":"n is 6, k is 3,"},{"Start":"06:45.875 ","End":"06:56.060","Text":"p is equal to 0.20 to the power of 3 to the power of k times q."},{"Start":"06:56.060 ","End":"07:00.215","Text":"That\u0027s 0.8 to the power of n minus k, 6 minus 3."},{"Start":"07:00.215 ","End":"07:05.620","Text":"That also equals 3 and this turns out to be 0.08192."},{"Start":"07:11.680 ","End":"07:13.700","Text":"In this next section,"},{"Start":"07:13.700 ","End":"07:17.690","Text":"we\u0027re asked what the expectation and standard deviation of the sum of"},{"Start":"07:17.690 ","End":"07:22.070","Text":"the numbers selected by the people. Let\u0027s get started."},{"Start":"07:22.070 ","End":"07:29.015","Text":"We know from the previous section that x was distributed with a uniform distribution."},{"Start":"07:29.015 ","End":"07:38.255","Text":"X was distributed with a uniform distribution where a equals 1 and b equals 100."},{"Start":"07:38.255 ","End":"07:43.935","Text":"Now, let\u0027s define xi. What\u0027s that?"},{"Start":"07:43.935 ","End":"07:47.970","Text":"That would be the number that was"},{"Start":"07:47.970 ","End":"07:55.550","Text":"selected by the i^th person."},{"Start":"07:55.550 ","End":"08:00.040","Text":"Now, this has the same distribution as,"},{"Start":"08:00.040 ","End":"08:02.455","Text":"we figured out previously,"},{"Start":"08:02.455 ","End":"08:07.570","Text":"this xi is distributed also uniformly,"},{"Start":"08:07.570 ","End":"08:13.435","Text":"where a equals 1 and b equals 100."},{"Start":"08:13.435 ","End":"08:15.625","Text":"What do we need to do right now?"},{"Start":"08:15.625 ","End":"08:22.120","Text":"Well, first of all, let\u0027s also recall that the expectation of x was"},{"Start":"08:22.120 ","End":"08:31.430","Text":"50.5 and the variance of x was 833.25."},{"Start":"08:31.430 ","End":"08:38.050","Text":"Now, that means that the expectation of the number of"},{"Start":"08:38.050 ","End":"08:45.915","Text":"the i^th person is also 50.5 and the variance of x,"},{"Start":"08:45.915 ","End":"08:51.180","Text":"of the i^th person is also 833.25."},{"Start":"08:51.180 ","End":"08:54.115","Text":"Let\u0027s see how we can continue."},{"Start":"08:54.115 ","End":"08:56.485","Text":"Now, what are we asked?"},{"Start":"08:56.485 ","End":"09:03.730","Text":"We\u0027re asked to find the expectation and standard deviation of the sum of the numbers."},{"Start":"09:03.730 ","End":"09:05.170","Text":"The sum of the numbers,"},{"Start":"09:05.170 ","End":"09:09.500","Text":"that\u0027s the total so let\u0027s just write the sum of the total."},{"Start":"09:09.500 ","End":"09:13.535","Text":"Let\u0027s write this out as the sum of xi,"},{"Start":"09:13.535 ","End":"09:21.280","Text":"where i equals 1 to 6 and we\u0027re looking for the expectation of the total."},{"Start":"09:21.280 ","End":"09:24.605","Text":"Now, from previous lessons,"},{"Start":"09:24.605 ","End":"09:31.490","Text":"we know that the expectation of the sum equals the sum of the expectations,"},{"Start":"09:31.490 ","End":"09:34.670","Text":"then that\u0027s always, so that equals to"},{"Start":"09:34.670 ","End":"09:40.460","Text":"the expectation of x_1 plus the expectation of x_2,"},{"Start":"09:40.460 ","End":"09:43.265","Text":"and so on and so forth until when?"},{"Start":"09:43.265 ","End":"09:46.460","Text":"Until the expectation of x_6,"},{"Start":"09:46.460 ","End":"09:49.680","Text":"we have 6 people."},{"Start":"09:49.680 ","End":"09:52.240","Text":"What is the expectation of x_1?"},{"Start":"09:52.240 ","End":"09:55.540","Text":"Well, that\u0027s 50.5."},{"Start":"09:55.540 ","End":"09:59.125","Text":"What about the expectation of x_2?"},{"Start":"09:59.125 ","End":"10:04.580","Text":"Again, that\u0027s 50.5 and so on and so forth 6 times."},{"Start":"10:04.580 ","End":"10:09.975","Text":"We have 6 times 50.5 and that equals to"},{"Start":"10:09.975 ","End":"10:14.380","Text":"303 so this is the expectation of"},{"Start":"10:14.380 ","End":"10:20.720","Text":"the sum of the numbers of these 6 people selected by these 6 people."},{"Start":"10:21.600 ","End":"10:24.790","Text":"Let\u0027s take a look at the variance."},{"Start":"10:24.790 ","End":"10:32.130","Text":"What\u0027s the variance of the total or the sum of the number selected by the 6 people."},{"Start":"10:32.130 ","End":"10:35.180","Text":"Well, that\u0027s the variance of the sum."},{"Start":"10:35.180 ","End":"10:40.460","Text":"Now, when the z equal to the sum of the variance,"},{"Start":"10:40.460 ","End":"10:44.750","Text":"that means that we\u0027re looking at the variance of x_1 plus"},{"Start":"10:44.750 ","End":"10:50.630","Text":"the variance of x_2 and so on and so forth until the variance of x_6."},{"Start":"10:50.630 ","End":"10:53.290","Text":"Now, when is that equal to this?"},{"Start":"10:53.290 ","End":"10:57.410","Text":"Under the condition that x_1 and x_2 and so forth."},{"Start":"10:57.410 ","End":"11:01.610","Text":"All the 6 people independent of each other and they are,"},{"Start":"11:01.610 ","End":"11:08.044","Text":"they\u0027re selected randomly and they choose their numbers independently of each other."},{"Start":"11:08.044 ","End":"11:12.620","Text":"The variance of the sum here equals the sum of the variance."},{"Start":"11:12.620 ","End":"11:15.926","Text":"Again, just like in the expectation,"},{"Start":"11:15.926 ","End":"11:18.364","Text":"the variance of each xi,"},{"Start":"11:18.364 ","End":"11:21.150","Text":"x_1, x_2,"},{"Start":"11:21.150 ","End":"11:30.875","Text":"x_3 is the same and that\u0027s 833.25 plus 833.25,"},{"Start":"11:30.875 ","End":"11:33.760","Text":"and so on and so forth 6 times,"},{"Start":"11:33.760 ","End":"11:39.360","Text":"6 times 833.25,"},{"Start":"11:39.360 ","End":"11:48.865","Text":"and that turns out to be 4,999.5."},{"Start":"11:48.865 ","End":"11:52.190","Text":"But again, we\u0027re not asked for the variance,"},{"Start":"11:52.190 ","End":"11:56.600","Text":"we\u0027re asked for the standard deviation so let\u0027s figure that out."},{"Start":"11:56.600 ","End":"12:00.095","Text":"Now the standard deviation of the total, well,"},{"Start":"12:00.095 ","End":"12:03.290","Text":"that equals to the square root of the variance and that equals to"},{"Start":"12:03.290 ","End":"12:08.540","Text":"the square root of 4999.5,"},{"Start":"12:08.540 ","End":"12:14.340","Text":"which equals to 70.71."}],"ID":13026},{"Watched":false,"Name":"Exercise 3","Duration":"7m 50s","ChapterTopicVideoID":12548,"CourseChapterTopicPlaylistID":64487,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.820","Text":"This question will be talking about throwing a die."},{"Start":"00:03.820 ","End":"00:06.745","Text":"Now, X is a result of throwing 1 die."},{"Start":"00:06.745 ","End":"00:10.075","Text":"We\u0027re asked, what\u0027s the probability distribution of X?"},{"Start":"00:10.075 ","End":"00:15.565","Text":"Well, we know that X is uniformly distributed,"},{"Start":"00:15.565 ","End":"00:19.945","Text":"where a equals 1 and b equals 6."},{"Start":"00:19.945 ","End":"00:21.535","Text":"Now, why is that?"},{"Start":"00:21.535 ","End":"00:25.060","Text":"Well, what is the values of a die?"},{"Start":"00:25.060 ","End":"00:27.820","Text":"Well, a die can have a value of 1 or 2,"},{"Start":"00:27.820 ","End":"00:28.900","Text":"or 3, or 4,"},{"Start":"00:28.900 ","End":"00:30.775","Text":"or 5, or 6."},{"Start":"00:30.775 ","End":"00:34.585","Text":"What are the probabilities of getting any 1 of the values?"},{"Start":"00:34.585 ","End":"00:37.150","Text":"Well, the probabilities are the same."},{"Start":"00:37.150 ","End":"00:38.770","Text":"It\u0027s a constant."},{"Start":"00:38.770 ","End":"00:42.870","Text":"Now, if X is uniformly distributed,"},{"Start":"00:42.870 ","End":"00:47.430","Text":"then the probability of X equaling k. Well,"},{"Start":"00:47.430 ","End":"00:49.350","Text":"that equals to 1 over b,"},{"Start":"00:49.350 ","End":"00:51.240","Text":"minus a plus 1."},{"Start":"00:51.240 ","End":"00:54.660","Text":"In our case, that\u0027s 1 over 6,"},{"Start":"00:54.660 ","End":"00:56.235","Text":"minus 1 plus 1."},{"Start":"00:56.235 ","End":"00:59.320","Text":"That equals to 1 over 6."},{"Start":"01:00.050 ","End":"01:04.580","Text":"In section b, we\u0027re asked what\u0027s the expectation of X?"},{"Start":"01:04.580 ","End":"01:11.785","Text":"Well, if X is distributed uniformly then the expectation of X,"},{"Start":"01:11.785 ","End":"01:14.270","Text":"we have a formula for that."},{"Start":"01:14.270 ","End":"01:17.660","Text":"That\u0027s b plus a over 2."},{"Start":"01:17.660 ","End":"01:19.325","Text":"Now, let\u0027s plug that in."},{"Start":"01:19.325 ","End":"01:25.210","Text":"That\u0027s 6 plus 1 over 2 and that equals to 3.5."},{"Start":"01:25.210 ","End":"01:27.900","Text":"Now, having done that,"},{"Start":"01:27.900 ","End":"01:32.730","Text":"we calculated the expectation of X in a various number of ways."},{"Start":"01:32.730 ","End":"01:40.865","Text":"One of them would be the sum of X times the probability of X."},{"Start":"01:40.865 ","End":"01:43.925","Text":"Another technique would be to use a table."},{"Start":"01:43.925 ","End":"01:48.560","Text":"We have X and we have the probability of X,"},{"Start":"01:48.560 ","End":"01:51.140","Text":"where X would be 1 or 2,"},{"Start":"01:51.140 ","End":"01:53.570","Text":"or 3, 4, or 5,"},{"Start":"01:53.570 ","End":"02:00.560","Text":"or 6 with a probability of a 6th for each one of the values."},{"Start":"02:00.560 ","End":"02:05.675","Text":"Now, I\u0027m not saying that these techniques are not valid,"},{"Start":"02:05.675 ","End":"02:09.890","Text":"but in this case the redundant. Why is that?"},{"Start":"02:09.890 ","End":"02:15.335","Text":"Because we have a shortened version to calculate the expectation of X."},{"Start":"02:15.335 ","End":"02:20.540","Text":"In the case of a uniform distribution that short form"},{"Start":"02:20.540 ","End":"02:26.720","Text":"that equation to calculate the expectation of X is this guy right here."},{"Start":"02:26.720 ","End":"02:29.467","Text":"So these guys are good,"},{"Start":"02:29.467 ","End":"02:33.600","Text":"but in this case, they\u0027re redundant."},{"Start":"02:34.860 ","End":"02:39.685","Text":"In this section, we\u0027re given that the die is thrown 4 times."},{"Start":"02:39.685 ","End":"02:42.370","Text":"We\u0027re asked what are the expectation and variance of"},{"Start":"02:42.370 ","End":"02:45.980","Text":"the sum of the results of the four throws."},{"Start":"02:46.010 ","End":"02:49.800","Text":"Let\u0027s define Xi as"},{"Start":"02:49.800 ","End":"02:58.360","Text":"the result of the throw."},{"Start":"02:59.810 ","End":"03:04.980","Text":"In essence, what we\u0027re looking at is throwing a dice"},{"Start":"03:04.980 ","End":"03:09.820","Text":"4 times and x1 would be the result in the first throw,"},{"Start":"03:09.820 ","End":"03:15.190","Text":"x2, the result of the second throw and so on and so forth for the fourth throw,"},{"Start":"03:15.190 ","End":"03:19.630","Text":"so i equals 1, 2, 4."},{"Start":"03:19.640 ","End":"03:23.220","Text":"Now, what do we know about Xi?"},{"Start":"03:23.220 ","End":"03:25.725","Text":"Well, we know that Xi,"},{"Start":"03:25.725 ","End":"03:29.190","Text":"all the X\u0027s are independent of each other."},{"Start":"03:29.190 ","End":"03:32.535","Text":"Each throw is independent of the other."},{"Start":"03:32.535 ","End":"03:37.950","Text":"We also know that they have the same distribution."},{"Start":"03:38.780 ","End":"03:41.565","Text":"Now, what does that mean?"},{"Start":"03:41.565 ","End":"03:47.255","Text":"That means that each Xi is distributed uniformly,"},{"Start":"03:47.255 ","End":"03:49.100","Text":"where a equals 1,"},{"Start":"03:49.100 ","End":"03:51.905","Text":"and b equals 6."},{"Start":"03:51.905 ","End":"03:58.810","Text":"Now, we know also that since X was distributed,"},{"Start":"03:58.810 ","End":"04:03.515","Text":"only 1 throw was distributed with a uniform distribution,"},{"Start":"04:03.515 ","End":"04:06.365","Text":"where a equals 1 and b equals 6."},{"Start":"04:06.365 ","End":"04:10.165","Text":"We know that the expectation of X,"},{"Start":"04:10.165 ","End":"04:12.950","Text":"that equal to 3.5."},{"Start":"04:12.950 ","End":"04:16.650","Text":"We calculated that in the last section."},{"Start":"04:16.750 ","End":"04:23.975","Text":"Now, what we\u0027re looking at is the expectation right here,"},{"Start":"04:23.975 ","End":"04:29.960","Text":"and variance of the sum of Xi\u0027s."},{"Start":"04:29.960 ","End":"04:32.870","Text":"That means that we\u0027re looking now at T,"},{"Start":"04:32.870 ","End":"04:36.860","Text":"total, is equal to the sum of Xi,"},{"Start":"04:36.860 ","End":"04:40.710","Text":"where Xi equals 1 to 4,"},{"Start":"04:40.710 ","End":"04:44.460","Text":"or X1, plus X2,"},{"Start":"04:44.460 ","End":"04:48.910","Text":"plus X3, plus X4."},{"Start":"04:49.190 ","End":"04:51.270","Text":"Now, what we want to know,"},{"Start":"04:51.270 ","End":"04:55.835","Text":"is we want to know what\u0027s the expectation of the sum of the total."},{"Start":"04:55.835 ","End":"04:59.735","Text":"Well, we know that the expectation of the sum"},{"Start":"04:59.735 ","End":"05:04.550","Text":"is always equals to the sum of the expectation."},{"Start":"05:04.550 ","End":"05:07.765","Text":"That means that we\u0027re looking at the expectation of X1,"},{"Start":"05:07.765 ","End":"05:11.065","Text":"plus the expectation of X2,"},{"Start":"05:11.065 ","End":"05:17.835","Text":"plus the expectation of X3 plus the expectation of X4."},{"Start":"05:17.835 ","End":"05:22.285","Text":"Now, what did we say about the individual Xi\u0027s?"},{"Start":"05:22.285 ","End":"05:25.230","Text":"They have the same distribution."},{"Start":"05:25.230 ","End":"05:30.260","Text":"That means that the expectation is the same for each exercise."},{"Start":"05:30.260 ","End":"05:33.155","Text":"That means that\u0027s 3.5,"},{"Start":"05:33.155 ","End":"05:40.110","Text":"plus 3.5, plus 3.5, plus 3.5."},{"Start":"05:41.150 ","End":"05:46.260","Text":"That equals to 14."},{"Start":"05:46.260 ","End":"05:53.990","Text":"Which means that if we sum up all the results of the 4th rows when throwing a die,"},{"Start":"05:53.990 ","End":"05:58.620","Text":"we\u0027re expected to get a value of 14."},{"Start":"06:00.170 ","End":"06:05.055","Text":"Let\u0027s take a look now at the variance."},{"Start":"06:05.055 ","End":"06:07.270","Text":"Now, the variance of x,"},{"Start":"06:07.270 ","End":"06:12.620","Text":"which is an individual throw that equals to b minus a,"},{"Start":"06:12.620 ","End":"06:15.060","Text":"plus 1 squared,"},{"Start":"06:15.060 ","End":"06:18.150","Text":"minus 1 over 12."},{"Start":"06:18.150 ","End":"06:20.280","Text":"When we plug in, the numbers,"},{"Start":"06:20.280 ","End":"06:22.050","Text":"we\u0027ll get 6 minus 1,"},{"Start":"06:22.050 ","End":"06:23.910","Text":"plus 1 squared,"},{"Start":"06:23.910 ","End":"06:26.490","Text":"minus 1 over 12."},{"Start":"06:26.490 ","End":"06:31.845","Text":"That equals to 2.917."},{"Start":"06:31.845 ","End":"06:33.995","Text":"Having said that, now,"},{"Start":"06:33.995 ","End":"06:36.785","Text":"what\u0027s the variance of the total?"},{"Start":"06:36.785 ","End":"06:41.540","Text":"That means that the variance of the sum of the Xi\u0027s."},{"Start":"06:41.540 ","End":"06:47.615","Text":"Well, that equals to the sum of the variances."},{"Start":"06:47.615 ","End":"06:52.485","Text":"The variance of X1 plus the variance of X2,"},{"Start":"06:52.485 ","End":"06:55.650","Text":"plus the variance X3,"},{"Start":"06:55.650 ","End":"06:58.320","Text":"plus the variance of X4."},{"Start":"06:58.320 ","End":"07:02.060","Text":"Now, that\u0027s true under which condition?"},{"Start":"07:02.060 ","End":"07:05.255","Text":"Well, the condition is that each X,"},{"Start":"07:05.255 ","End":"07:09.290","Text":"all the Xi\u0027s have to be independent."},{"Start":"07:09.290 ","End":"07:13.250","Text":"Now, because they\u0027re not only independent,"},{"Start":"07:13.250 ","End":"07:15.860","Text":"but they have the same distribution,"},{"Start":"07:15.860 ","End":"07:19.180","Text":"then they also have the same variance."},{"Start":"07:19.180 ","End":"07:28.370","Text":"The variance for X1 equals 2.917 and that equals to the variance of X2 and X3 and X4,"},{"Start":"07:28.370 ","End":"07:33.490","Text":"so we have to multiply this 4 times."},{"Start":"07:33.490 ","End":"07:40.610","Text":"That equals to 11.668. Here we go."},{"Start":"07:40.610 ","End":"07:42.035","Text":"We\u0027ve answered the question."},{"Start":"07:42.035 ","End":"07:50.110","Text":"We\u0027ve calculated the variance of the total and the expectation of the total."}],"ID":13027},{"Watched":false,"Name":"Exercise 4","Duration":"6m 4s","ChapterTopicVideoID":12549,"CourseChapterTopicPlaylistID":64487,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this question, we\u0027ll be talking about removing balls from a basket."},{"Start":"00:04.140 ","End":"00:06.585","Text":"Now, there are 10 balls in a basket,"},{"Start":"00:06.585 ","End":"00:08.400","Text":"only 1 of which is red."},{"Start":"00:08.400 ","End":"00:10.320","Text":"A person takes out the balls,"},{"Start":"00:10.320 ","End":"00:14.145","Text":"1 after the other without returns until he removes the red ball,"},{"Start":"00:14.145 ","End":"00:15.525","Text":"and then he stops."},{"Start":"00:15.525 ","End":"00:21.360","Text":"Now, what are the expectation and variance of the number of balls that were removed?"},{"Start":"00:21.360 ","End":"00:26.199","Text":"Now, this isn\u0027t a trivial question,"},{"Start":"00:26.199 ","End":"00:31.050","Text":"it requires a little bit of thought. Let\u0027s get to it."},{"Start":"00:31.050 ","End":"00:32.505","Text":"Well, first of all,"},{"Start":"00:32.505 ","End":"00:33.675","Text":"what are we looking at?"},{"Start":"00:33.675 ","End":"00:35.595","Text":"What\u0027s a random variable?"},{"Start":"00:35.595 ","End":"00:41.170","Text":"Well, our random variable is the number of balls that were removed,"},{"Start":"00:41.170 ","End":"00:47.930","Text":"and we\u0027re looking to calculate the expectation and variance of this random variable."},{"Start":"00:47.930 ","End":"00:50.795","Text":"Let\u0027s just write this out."},{"Start":"00:50.795 ","End":"00:57.370","Text":"X equals the number of balls that were removed."},{"Start":"00:57.770 ","End":"01:02.495","Text":"Now, I maintain that X has"},{"Start":"01:02.495 ","End":"01:09.350","Text":"a uniform distribution where a equals 1 and b equals 10."},{"Start":"01:09.350 ","End":"01:12.080","Text":"Now, to prove this,"},{"Start":"01:12.080 ","End":"01:14.990","Text":"I\u0027ll need the help of the probability tree."},{"Start":"01:14.990 ","End":"01:21.630","Text":"Let\u0027s construct this tree and see why this is so."},{"Start":"01:22.430 ","End":"01:25.410","Text":"This is basically the first step."},{"Start":"01:25.410 ","End":"01:27.944","Text":"When we get to the basket,"},{"Start":"01:27.944 ","End":"01:30.770","Text":"we initially have 10 balls,"},{"Start":"01:30.770 ","End":"01:33.180","Text":"and we only have 1 red ball."},{"Start":"01:33.180 ","End":"01:37.340","Text":"The probability of getting a red ball is 1 over 10"},{"Start":"01:37.340 ","End":"01:42.125","Text":"and the probability of getting a ball that\u0027s not red is 9 over 10."},{"Start":"01:42.125 ","End":"01:46.350","Text":"Now, if we got the red ball right here, we stop."},{"Start":"01:47.200 ","End":"01:51.780","Text":"If we didn\u0027t get a red ball, we continue."},{"Start":"01:51.910 ","End":"01:55.085","Text":"Now, this is the second step."},{"Start":"01:55.085 ","End":"01:58.055","Text":"In the second step, we have 9 balls."},{"Start":"01:58.055 ","End":"02:02.345","Text":"Again, if we didn\u0027t get a red ball in the first step,"},{"Start":"02:02.345 ","End":"02:08.035","Text":"then the probability of getting a red ball on the second step is 1 over 9."},{"Start":"02:08.035 ","End":"02:09.990","Text":"If we got a red ball there,"},{"Start":"02:09.990 ","End":"02:13.890","Text":"then we stop, and if not, then we continue."},{"Start":"02:13.890 ","End":"02:18.840","Text":"Because the probability of not getting a red ball right here is 8/9."},{"Start":"02:18.840 ","End":"02:23.985","Text":"Let\u0027s continue to the next step. Here\u0027s the next step."},{"Start":"02:23.985 ","End":"02:26.420","Text":"The next step, we have 8 balls and"},{"Start":"02:26.420 ","End":"02:30.380","Text":"the probability of getting a red ball would be 1 over 8."},{"Start":"02:30.380 ","End":"02:33.230","Text":"If we get that red ball, then we stop."},{"Start":"02:33.230 ","End":"02:36.865","Text":"If not, then the probability of getting a not red,"},{"Start":"02:36.865 ","End":"02:40.635","Text":"is 7 over 8 and so on and so forth."},{"Start":"02:40.635 ","End":"02:47.400","Text":"Now, let\u0027s calculate the probability of X being equal to 1."},{"Start":"02:47.400 ","End":"02:49.910","Text":"Well, when X equals to 1,"},{"Start":"02:49.910 ","End":"02:53.615","Text":"then we\u0027re looking at this branch right here."},{"Start":"02:53.615 ","End":"02:58.225","Text":"Now, that would mean that we got the ball,"},{"Start":"02:58.225 ","End":"03:01.390","Text":"we got the red ball on the first try."},{"Start":"03:01.390 ","End":"03:05.140","Text":"The probability of that happening is 1 over 10."},{"Start":"03:05.140 ","End":"03:11.425","Text":"Now, let\u0027s take a look at the probability where X equals 2."},{"Start":"03:11.425 ","End":"03:16.300","Text":"X equals 2 means that we didn\u0027t get the ball in the first trial,"},{"Start":"03:16.300 ","End":"03:18.760","Text":"but we did get the red ball in the second trial."},{"Start":"03:18.760 ","End":"03:22.020","Text":"So that\u0027s the probability of X equals 2."},{"Start":"03:22.020 ","End":"03:29.160","Text":"That\u0027s 9 over 10 times 1 over 9."},{"Start":"03:29.160 ","End":"03:30.885","Text":"We didn\u0027t get on the first try,"},{"Start":"03:30.885 ","End":"03:34.080","Text":"and we did get it on the second try."},{"Start":"03:34.080 ","End":"03:36.210","Text":"That\u0027s the multiplication these probabilities."},{"Start":"03:36.210 ","End":"03:41.500","Text":"Now, these 9s cancel out and we get a probability of 1 over 10."},{"Start":"03:41.500 ","End":"03:45.940","Text":"Well, let\u0027s take a look at what happens when X equals 3."},{"Start":"03:45.940 ","End":"03:48.280","Text":"Again, that means that we didn\u0027t get it in"},{"Start":"03:48.280 ","End":"03:51.220","Text":"the first trial we didn\u0027t get the ball on the second try,"},{"Start":"03:51.220 ","End":"03:53.575","Text":"but in the third try we did get the ball."},{"Start":"03:53.575 ","End":"03:56.530","Text":"So let\u0027s multiply the probabilities."},{"Start":"03:56.530 ","End":"04:04.160","Text":"That\u0027s 9 over 10 times 8 over 9 times 1 over 8."},{"Start":"04:04.680 ","End":"04:08.470","Text":"That\u0027s 9 over 10 times 8 over 9,"},{"Start":"04:08.470 ","End":"04:10.350","Text":"and we did get it in the third try,"},{"Start":"04:10.350 ","End":"04:11.490","Text":"so that\u0027s 1 over 8."},{"Start":"04:11.490 ","End":"04:13.545","Text":"This is this branch right here."},{"Start":"04:13.545 ","End":"04:15.855","Text":"Again, the 9s cancel out,"},{"Start":"04:15.855 ","End":"04:18.630","Text":"the 8 cancel out, and we get 1 over 10."},{"Start":"04:18.630 ","End":"04:25.220","Text":"The probability of X being equal to k, well,"},{"Start":"04:25.220 ","End":"04:27.800","Text":"that equals to 1 over 10,"},{"Start":"04:27.800 ","End":"04:30.110","Text":"where k equals 1, 2,"},{"Start":"04:30.110 ","End":"04:33.420","Text":"3, and so forth until 10."},{"Start":"04:34.360 ","End":"04:39.060","Text":"This is our probability tree,"},{"Start":"04:39.060 ","End":"04:43.090","Text":"and we saw that for every value of X,"},{"Start":"04:43.090 ","End":"04:46.270","Text":"the probability was 1 over 10."},{"Start":"04:46.270 ","End":"04:50.060","Text":"That makes this a uniform distribution."},{"Start":"04:50.060 ","End":"04:56.435","Text":"Now we know for sure we proved that X is distributed with a uniform distribution,"},{"Start":"04:56.435 ","End":"05:00.560","Text":"where a equals 1 and b equals 10."},{"Start":"05:00.560 ","End":"05:02.420","Text":"Now, what are we asked?"},{"Start":"05:02.420 ","End":"05:06.580","Text":"We\u0027re asked for the expectation and variance of X."},{"Start":"05:06.580 ","End":"05:08.670","Text":"Let\u0027s figure that out."},{"Start":"05:08.670 ","End":"05:12.755","Text":"Well, the expectation of X,"},{"Start":"05:12.755 ","End":"05:19.295","Text":"where X is distributed with the uniform distribution is b plus a over 2."},{"Start":"05:19.295 ","End":"05:21.830","Text":"Well, that\u0027s just plug in the numbers."},{"Start":"05:21.830 ","End":"05:27.395","Text":"That\u0027s 10 plus 1 over 2 and that\u0027s 5.5."},{"Start":"05:27.395 ","End":"05:30.050","Text":"What\u0027s the variance of X?"},{"Start":"05:30.050 ","End":"05:38.170","Text":"Well, that\u0027s B minus a plus 1 squared minus 1 over 12."},{"Start":"05:38.170 ","End":"05:40.535","Text":"Again, plugging in the numbers,"},{"Start":"05:40.535 ","End":"05:45.410","Text":"that\u0027s 10 minus 1 plus 1 squared,"},{"Start":"05:45.410 ","End":"05:48.185","Text":"minus 1 over 12,"},{"Start":"05:48.185 ","End":"05:53.305","Text":"and that equals to 8.25."},{"Start":"05:53.305 ","End":"05:57.285","Text":"This is the variance of X."},{"Start":"05:57.285 ","End":"05:59.160","Text":"That\u0027s what we\u0027re asked for right here,"},{"Start":"05:59.160 ","End":"06:01.434","Text":"and this is the expectation,"},{"Start":"06:01.434 ","End":"06:03.720","Text":"that\u0027s were asked right here."}],"ID":13028},{"Watched":false,"Name":"Exercise 5","Duration":"5m 32s","ChapterTopicVideoID":12550,"CourseChapterTopicPlaylistID":64487,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"In this question, we\u0027ll be talking about randomly selecting numbers."},{"Start":"00:04.575 ","End":"00:09.715","Text":"Now, a number selected randomly between 1 and 50 inclusively."},{"Start":"00:09.715 ","End":"00:14.035","Text":"We\u0027re asked what are the chances that the number 4 will be selected?"},{"Start":"00:14.035 ","End":"00:21.240","Text":"Well, first of all, let\u0027s define X as being the number that was selected."},{"Start":"00:21.520 ","End":"00:30.980","Text":"Now, X has a uniform distribution where a equals 1 and b equals 15. Why is that?"},{"Start":"00:30.980 ","End":"00:36.249","Text":"Well, for every value that X has between 1 and 50,"},{"Start":"00:36.249 ","End":"00:41.255","Text":"each value has the same probability of being selected."},{"Start":"00:41.255 ","End":"00:46.579","Text":"We also know that because X is uniformly distributed,"},{"Start":"00:46.579 ","End":"00:50.570","Text":"that the probability of X being equal to k,"},{"Start":"00:50.570 ","End":"00:56.390","Text":"well, that equals to 1 over b minus a plus 1."},{"Start":"00:56.390 ","End":"01:00.845","Text":"Now, having said that, let\u0027s answer a."},{"Start":"01:00.845 ","End":"01:03.200","Text":"A asks us,"},{"Start":"01:03.200 ","End":"01:09.230","Text":"what\u0027s the probability that the number 4 will be selected X equal to 4."},{"Start":"01:09.230 ","End":"01:12.920","Text":"Well, that equals to 1 over,"},{"Start":"01:12.920 ","End":"01:15.145","Text":"now b is 15."},{"Start":"01:15.145 ","End":"01:18.180","Text":"That\u0027s 50. Let\u0027s just plug in the numbers."},{"Start":"01:18.180 ","End":"01:26.640","Text":"Minus 1, a is 1 plus 1 and that equals to 1 over 50."},{"Start":"01:27.020 ","End":"01:29.570","Text":"In section b, we\u0027re asked what are"},{"Start":"01:29.570 ","End":"01:33.280","Text":"the chances that the number selected is larger than 20?"},{"Start":"01:33.280 ","End":"01:36.495","Text":"Well, b is saying,"},{"Start":"01:36.495 ","End":"01:40.385","Text":"what\u0027s the probability where X is greater than 20?"},{"Start":"01:40.385 ","End":"01:45.335","Text":"Well, if X is greater than 20, non-inclusive,"},{"Start":"01:45.335 ","End":"01:47.870","Text":"then we can write this up,"},{"Start":"01:47.870 ","End":"01:52.815","Text":"that X is greater or equal to 21."},{"Start":"01:52.815 ","End":"01:57.810","Text":"What\u0027s the probability of X being greater or equal to 21?"},{"Start":"01:57.810 ","End":"02:02.570","Text":"That equals to the probability of X being greater than 20. Why is that?"},{"Start":"02:02.570 ","End":"02:09.420","Text":"Well, because X is a discrete random variable with increments of 1."},{"Start":"02:09.830 ","End":"02:13.155","Text":"We can write this so,"},{"Start":"02:13.155 ","End":"02:18.005","Text":"what\u0027s the probability of X equaling any value?"},{"Start":"02:18.005 ","End":"02:19.640","Text":"That\u0027s 1 over 50."},{"Start":"02:19.640 ","End":"02:22.220","Text":"Let\u0027s put 50 in the denominator."},{"Start":"02:22.220 ","End":"02:24.860","Text":"Now, what do we have to do with the numerator?"},{"Start":"02:24.860 ","End":"02:27.935","Text":"Well, we have to know or calculate"},{"Start":"02:27.935 ","End":"02:33.650","Text":"how many values of X there are between 21 and 15 inclusive."},{"Start":"02:33.650 ","End":"02:39.040","Text":"Well, that\u0027s 50 minus 21 plus 1."},{"Start":"02:39.040 ","End":"02:42.250","Text":"That equals to 30 over 50."},{"Start":"02:44.300 ","End":"02:46.815","Text":"Section C here says,"},{"Start":"02:46.815 ","End":"02:49.845","Text":"if a number larger than 20 is selected,"},{"Start":"02:49.845 ","End":"02:55.240","Text":"what\u0027s the probability that the number selected is smaller than 28?"},{"Start":"02:55.240 ","End":"02:58.640","Text":"Well, this is a conditional probability."},{"Start":"02:58.640 ","End":"03:02.060","Text":"Let\u0027s just set it up. What\u0027s the probability now?"},{"Start":"03:02.060 ","End":"03:03.920","Text":"What\u0027s the condition? Right here,"},{"Start":"03:03.920 ","End":"03:05.405","Text":"that\u0027s what\u0027s given to us."},{"Start":"03:05.405 ","End":"03:08.360","Text":"If a number larger than 20 is selected,"},{"Start":"03:08.360 ","End":"03:12.950","Text":"that means X is greater than 20 and what did we ask for?"},{"Start":"03:12.950 ","End":"03:16.280","Text":"The probability that the number selected is smarter than 28."},{"Start":"03:16.280 ","End":"03:20.580","Text":"That means that X is less than 28."},{"Start":"03:22.130 ","End":"03:26.780","Text":"Now, to the probability of what\u0027s given in the denominator,"},{"Start":"03:26.780 ","End":"03:30.960","Text":"that\u0027s X being greater than 20."},{"Start":"03:31.220 ","End":"03:36.545","Text":"In the numerator, the probability of the intersect."},{"Start":"03:36.545 ","End":"03:43.580","Text":"Now, what\u0027s the intersect here that x is greater than 20 and less than 28?"},{"Start":"03:43.580 ","End":"03:48.920","Text":"We\u0027ll write this down, x less 28."},{"Start":"03:48.920 ","End":"03:52.940","Text":"Now, just like we did in section b,"},{"Start":"03:52.940 ","End":"03:56.120","Text":"we can rephrase these guys in order to"},{"Start":"03:56.120 ","End":"04:00.725","Text":"calculate the number of values that we have within a range."},{"Start":"04:00.725 ","End":"04:08.450","Text":"Let\u0027s do that. That would equal the probability of"},{"Start":"04:08.450 ","End":"04:18.760","Text":"X being greater or equal to 21 and less than and equal to 27,"},{"Start":"04:18.760 ","End":"04:26.830","Text":"divided by the probability of X being greater or equal to 21."},{"Start":"04:27.650 ","End":"04:30.695","Text":"Let\u0027s take a look at the denominator."},{"Start":"04:30.695 ","End":"04:32.629","Text":"Well, in the denominator,"},{"Start":"04:32.629 ","End":"04:35.510","Text":"that\u0027s what we calculated in section b."},{"Start":"04:35.510 ","End":"04:37.895","Text":"That equals to 30 over 50."},{"Start":"04:37.895 ","End":"04:40.675","Text":"That\u0027s right here. We\u0027ll put that down here."},{"Start":"04:40.675 ","End":"04:46.075","Text":"Now, what is the probability of X,"},{"Start":"04:46.075 ","End":"04:49.865","Text":"X being between 21 and 27 inclusive?"},{"Start":"04:49.865 ","End":"04:55.970","Text":"Well, let\u0027s figure out how many values of X are there between 21 and 27 inclusive."},{"Start":"04:55.970 ","End":"05:03.010","Text":"Well, that\u0027s 27 minus 21 plus 1."},{"Start":"05:03.010 ","End":"05:08.965","Text":"Again, that\u0027s a problem, because it\u0027s a probability that has to be over 50."},{"Start":"05:08.965 ","End":"05:18.490","Text":"Now, that equals to 7 over 50 divided by 30 over 50."},{"Start":"05:18.490 ","End":"05:22.620","Text":"That comes out to 7/30."},{"Start":"05:22.620 ","End":"05:25.470","Text":"This is the answer for section C,"},{"Start":"05:25.470 ","End":"05:28.185","Text":"this is the answer for section B,"},{"Start":"05:28.185 ","End":"05:31.870","Text":"and this is the answer for section A."}],"ID":13029},{"Watched":false,"Name":"Exercise 6","Duration":"4m 17s","ChapterTopicVideoID":12551,"CourseChapterTopicPlaylistID":64487,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this question, we\u0027re asked to prove that if X is"},{"Start":"00:03.270 ","End":"00:07.365","Text":"distributed with a uniform distribution with parameters a and b,"},{"Start":"00:07.365 ","End":"00:12.130","Text":"then the expectation of X equals a plus b over 2."},{"Start":"00:12.500 ","End":"00:20.880","Text":"Let\u0027s get at it. We know that the general formula to calculate the expectation of X,"},{"Start":"00:20.880 ","End":"00:26.880","Text":"well that equals to the sum of x times the probability of x."},{"Start":"00:26.880 ","End":"00:33.275","Text":"Because we know that X is distributed with a uniform distribution with these parameters,"},{"Start":"00:33.275 ","End":"00:42.770","Text":"we know that the probability of x being equal to k equals 1 over b minus a plus 1,"},{"Start":"00:42.770 ","End":"00:46.190","Text":"where k equals a,"},{"Start":"00:46.190 ","End":"00:47.945","Text":"a plus 1,"},{"Start":"00:47.945 ","End":"00:50.910","Text":"and so on and so forth until b."},{"Start":"00:51.590 ","End":"00:56.415","Text":"Let\u0027s just plug this back into here."},{"Start":"00:56.415 ","End":"01:01.250","Text":"Now, that equals to the sum of x times the probability of x."},{"Start":"01:01.250 ","End":"01:02.480","Text":"Now the probability of x,"},{"Start":"01:02.480 ","End":"01:04.925","Text":"we said that\u0027s 1 over b minus a plus 1."},{"Start":"01:04.925 ","End":"01:09.630","Text":"That\u0027s 1 over b minus a plus 1."},{"Start":"01:10.850 ","End":"01:13.399","Text":"This is a constant."},{"Start":"01:13.399 ","End":"01:18.020","Text":"It has nothing to do with the sum operator right here."},{"Start":"01:18.020 ","End":"01:20.630","Text":"We can take this out and write it like this,"},{"Start":"01:20.630 ","End":"01:28.085","Text":"1 over b minus a plus 1 times the sum of x,"},{"Start":"01:28.085 ","End":"01:34.270","Text":"where x starts with a and ends up with b."},{"Start":"01:35.810 ","End":"01:37.910","Text":"What do we have here?"},{"Start":"01:37.910 ","End":"01:45.585","Text":"We have a constant times the sum of a series. What is that?"},{"Start":"01:45.585 ","End":"01:47.595","Text":"Why is this a series?"},{"Start":"01:47.595 ","End":"01:49.680","Text":"Well, let\u0027s take a look at this."},{"Start":"01:49.680 ","End":"01:55.665","Text":"That\u0027s 1 over b minus a plus 1 times,"},{"Start":"01:55.665 ","End":"01:57.320","Text":"let\u0027s take a look at this."},{"Start":"01:57.320 ","End":"02:02.030","Text":"That\u0027s a plus a plus 1,"},{"Start":"02:02.030 ","End":"02:04.850","Text":"plus a plus 2,"},{"Start":"02:04.850 ","End":"02:08.610","Text":"and so on and so forth, plus b."},{"Start":"02:09.170 ","End":"02:12.435","Text":"This is a series."},{"Start":"02:12.435 ","End":"02:19.880","Text":"We have to remember what the formulas for the sum of a series."},{"Start":"02:19.880 ","End":"02:22.565","Text":"Let\u0027s just go over here to the side."},{"Start":"02:22.565 ","End":"02:24.800","Text":"The sum of a series,"},{"Start":"02:24.800 ","End":"02:31.460","Text":"that equals to the first member in the series plus"},{"Start":"02:31.460 ","End":"02:40.610","Text":"the last member of the series times the number of members in the series divided by 2."},{"Start":"02:40.610 ","End":"02:47.225","Text":"In our case a_1 will equal to"},{"Start":"02:47.225 ","End":"02:56.819","Text":"a and a_n will equal to b. A_1 that\u0027s the first value in the series,"},{"Start":"02:56.819 ","End":"03:01.080","Text":"a_n that\u0027s the last value in the series."},{"Start":"03:01.080 ","End":"03:09.335","Text":"Now, what\u0027s n? N is the number of values or the number of members in the series."},{"Start":"03:09.335 ","End":"03:11.420","Text":"How many members in the series do we have?"},{"Start":"03:11.420 ","End":"03:13.895","Text":"We have 1, 2, 3, and so on and so forth."},{"Start":"03:13.895 ","End":"03:18.940","Text":"But we have b minus a plus 1."},{"Start":"03:18.940 ","End":"03:22.830","Text":"Let\u0027s plug all this back into here."},{"Start":"03:22.830 ","End":"03:26.780","Text":"That\u0027s 1 over b minus a plus 1."},{"Start":"03:26.780 ","End":"03:29.950","Text":"That\u0027s this guy right here times, now,"},{"Start":"03:29.950 ","End":"03:35.355","Text":"instead of this let\u0027s plug in these guys right here."},{"Start":"03:35.355 ","End":"03:40.560","Text":"A_1, that\u0027s a plus a_n,"},{"Start":"03:40.560 ","End":"03:42.609","Text":"well, a_n is b,"},{"Start":"03:42.609 ","End":"03:49.515","Text":"divide by 2 times"},{"Start":"03:49.515 ","End":"03:55.330","Text":"n. Now n is b minus a plus 1."},{"Start":"03:57.770 ","End":"04:00.450","Text":"This cancels this out,"},{"Start":"04:00.450 ","End":"04:02.295","Text":"this cancels this out."},{"Start":"04:02.295 ","End":"04:06.465","Text":"We have a plus b over 2."},{"Start":"04:06.465 ","End":"04:10.480","Text":"That is the expectation of X,"},{"Start":"04:10.480 ","End":"04:17.790","Text":"where X is distributed with a uniform distribution with parameters a and b."}],"ID":13030}],"Thumbnail":null,"ID":64487},{"Name":"Special Discrete Probability Distributions - Hypergeometric Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"6m 2s","ChapterTopicVideoID":12552,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:04.410","Text":"In this chapter we\u0027ll be talking about special discrete probability,"},{"Start":"00:04.410 ","End":"00:07.320","Text":"specifically the hypergeometric distribution."},{"Start":"00:07.320 ","End":"00:10.770","Text":"Now again, the special discrete probabilities,"},{"Start":"00:10.770 ","End":"00:15.870","Text":"we\u0027re given the probability function of the distribution in 1 hand and"},{"Start":"00:15.870 ","End":"00:21.270","Text":"we\u0027re also given the equations for the expectation and variance."},{"Start":"00:21.270 ","End":"00:26.280","Text":"Now, also we\u0027re going to have to understand that we\u0027re dealing with"},{"Start":"00:26.280 ","End":"00:28.830","Text":"the hypergeometric distribution from"},{"Start":"00:28.830 ","End":"00:32.280","Text":"the context of the question that we\u0027ll be trying to solve,"},{"Start":"00:32.280 ","End":"00:36.050","Text":"so we have to understand the characteristics of this distribution."},{"Start":"00:36.050 ","End":"00:39.185","Text":"Now, let\u0027s take a look specifically"},{"Start":"00:39.185 ","End":"00:42.830","Text":"at this type of distribution, the hypergeometric distribution."},{"Start":"00:42.830 ","End":"00:47.755","Text":"Now, given a population with N individuals,"},{"Start":"00:47.755 ","End":"00:51.590","Text":"of which D individuals have a given characteristic."},{"Start":"00:51.590 ","End":"00:55.255","Text":"These D individuals are called special."},{"Start":"00:55.255 ","End":"01:02.285","Text":"Now, n individuals are selected from the population with no returns."},{"Start":"01:02.285 ","End":"01:07.610","Text":"X is defined as the number of special individuals in the sample."},{"Start":"01:07.610 ","End":"01:12.640","Text":"Now, hypergeometric random variable with parameters N,"},{"Start":"01:12.640 ","End":"01:16.190","Text":"D, and n would be denoted by this,"},{"Start":"01:16.190 ","End":"01:20.735","Text":"X is distributed with the hypergeometric distribution,"},{"Start":"01:20.735 ","End":"01:24.815","Text":"H for hypergeometric, with parameters N, D,"},{"Start":"01:24.815 ","End":"01:30.820","Text":"and n. The probability function of X,"},{"Start":"01:30.820 ","End":"01:34.750","Text":"where X has a hypergeometric distribution is this,"},{"Start":"01:34.750 ","End":"01:39.340","Text":"is the probability of X being equal to sum value k and that would equal d over"},{"Start":"01:39.340 ","End":"01:46.505","Text":"k times N minus D over n minus k divided by big N over n. Now,"},{"Start":"01:46.505 ","End":"01:49.975","Text":"this looks complicated, so let\u0027s simplify this."},{"Start":"01:49.975 ","End":"01:52.375","Text":"Well, let\u0027s take a look at the denominator."},{"Start":"01:52.375 ","End":"01:55.600","Text":"That\u0027s N over n. Now,"},{"Start":"01:55.600 ","End":"01:59.095","Text":"N is defined as the size of the population,"},{"Start":"01:59.095 ","End":"02:03.264","Text":"and small n is defined as the size of our sample."},{"Start":"02:03.264 ","End":"02:10.280","Text":"So N over n would give me the number of samples,"},{"Start":"02:10.280 ","End":"02:17.030","Text":"of size n from a population of N. For example,"},{"Start":"02:17.030 ","End":"02:21.875","Text":"if we had 100 people and we wanted to sample size of 10,"},{"Start":"02:21.875 ","End":"02:26.480","Text":"we wanted to sample 10 individuals from a population of a 100."},{"Start":"02:26.480 ","End":"02:33.380","Text":"Then how many samples of size 10 could we get out of a population of size 100?"},{"Start":"02:33.380 ","End":"02:35.800","Text":"Well, that\u0027s this expression right here."},{"Start":"02:35.800 ","End":"02:39.125","Text":"Let\u0027s take a look at the numerator."},{"Start":"02:39.125 ","End":"02:42.215","Text":"Well, D is defined as the number of"},{"Start":"02:42.215 ","End":"02:47.310","Text":"individuals in the population that have a special characteristic."},{"Start":"02:47.390 ","End":"02:54.755","Text":"N minus D is the number of people that do not have the special characteristic."},{"Start":"02:54.755 ","End":"03:03.395","Text":"The numerator gives us the number of samples of size n,"},{"Start":"03:03.395 ","End":"03:07.055","Text":"where k individuals in our sample,"},{"Start":"03:07.055 ","End":"03:13.880","Text":"have a special characteristic and that means that n minus"},{"Start":"03:13.880 ","End":"03:21.050","Text":"k individuals do not have a special characteristic."},{"Start":"03:21.050 ","End":"03:28.805","Text":"Now, this ratio right here basically describes inwards what this thing means."},{"Start":"03:28.805 ","End":"03:32.335","Text":"That\u0027s the ratio of the number of samples,"},{"Start":"03:32.335 ","End":"03:36.050","Text":"where we have k special individuals and n minus"},{"Start":"03:36.050 ","End":"03:41.450","Text":"k individuals that are not special over the total number"},{"Start":"03:41.450 ","End":"03:43.445","Text":"of samples of size"},{"Start":"03:43.445 ","End":"03:51.035","Text":"n. Here are the equations for the expectation of X and the variance of X."},{"Start":"03:51.035 ","End":"03:54.395","Text":"Now, let\u0027s take a closer look at the expectation of X."},{"Start":"03:54.395 ","End":"03:56.600","Text":"Well, that\u0027s n times D over n,"},{"Start":"03:56.600 ","End":"04:03.170","Text":"the sample size times the proportion of individuals that are special in the population."},{"Start":"04:03.170 ","End":"04:11.540","Text":"Well, let\u0027s define p probability as D over N. When we write this out,"},{"Start":"04:11.540 ","End":"04:15.035","Text":"this equals to n times p. Now,"},{"Start":"04:15.035 ","End":"04:19.250","Text":"which distribution has an expectation of n times p?"},{"Start":"04:19.250 ","End":"04:22.175","Text":"Well, that\u0027s a binomial distribution."},{"Start":"04:22.175 ","End":"04:28.790","Text":"Let\u0027s assume that Y has a binomial distribution with parameters n and"},{"Start":"04:28.790 ","End":"04:35.855","Text":"p with the expectation of Y would be n times p. Let\u0027s recall that."},{"Start":"04:35.855 ","End":"04:38.750","Text":"Now we see here that that\u0027s the same,"},{"Start":"04:38.750 ","End":"04:40.265","Text":"the expectation here is the same,"},{"Start":"04:40.265 ","End":"04:43.040","Text":"but there\u0027s a big difference between"},{"Start":"04:43.040 ","End":"04:48.505","Text":"the binomial distribution and the hypergeometric distribution, and what\u0027s that?"},{"Start":"04:48.505 ","End":"04:50.615","Text":"In the binomial distributions,"},{"Start":"04:50.615 ","End":"04:52.700","Text":"we\u0027re sampling with returns."},{"Start":"04:52.700 ","End":"04:57.290","Text":"The observations are independent of each other,"},{"Start":"04:57.290 ","End":"05:00.320","Text":"and in the hypergeometric distribution,"},{"Start":"05:00.320 ","End":"05:02.700","Text":"we\u0027re sampling without returns,"},{"Start":"05:02.700 ","End":"05:07.290","Text":"so the observations here are dependent on each other."},{"Start":"05:07.600 ","End":"05:13.100","Text":"Again, the expectation of Y is n times p,"},{"Start":"05:13.100 ","End":"05:16.070","Text":"where Y is distributed with a binomial distribution,"},{"Start":"05:16.070 ","End":"05:18.500","Text":"and the variance of Y, well,"},{"Start":"05:18.500 ","End":"05:25.580","Text":"that was n times p times 1 minus p. Let\u0027s recall that."},{"Start":"05:25.580 ","End":"05:30.800","Text":"Here we have something similar in the hypergeometric distribution."},{"Start":"05:30.800 ","End":"05:32.990","Text":"We have n times D over N,"},{"Start":"05:32.990 ","End":"05:35.960","Text":"that\u0027s p times 1 minus D over N,"},{"Start":"05:35.960 ","End":"05:40.640","Text":"that\u0027s 1 minus p times this factor here right here,"},{"Start":"05:40.640 ","End":"05:45.830","Text":"which is N minus n over N minus 1."},{"Start":"05:45.830 ","End":"05:51.500","Text":"This is the factor that differentiates between"},{"Start":"05:51.500 ","End":"05:57.709","Text":"the variances of the binomial distribution and the hypergeometric distribution."},{"Start":"05:57.709 ","End":"06:01.920","Text":"Let\u0027s go to an example right now."}],"ID":13031},{"Watched":false,"Name":"Example","Duration":"4m 46s","ChapterTopicVideoID":12553,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"In our example there are 40 children in a class,"},{"Start":"00:03.120 ","End":"00:05.700","Text":"10 of whom are girls in the rest of boys."},{"Start":"00:05.700 ","End":"00:09.630","Text":"A group of 4 students are selected to take part in a delegation."},{"Start":"00:09.630 ","End":"00:11.355","Text":"In section a, we\u0027re asked,"},{"Start":"00:11.355 ","End":"00:16.080","Text":"what\u0027s the probability distribution of the number of boys in the delegation?"},{"Start":"00:16.080 ","End":"00:19.605","Text":"Well, this is a Hyper Geometric Distribution."},{"Start":"00:19.605 ","End":"00:22.770","Text":"But let\u0027s take a look at the characteristics"},{"Start":"00:22.770 ","End":"00:27.120","Text":"of this distribution to make sure that we\u0027re correct."},{"Start":"00:27.120 ","End":"00:30.180","Text":"Here are the characteristics right here."},{"Start":"00:30.180 ","End":"00:32.985","Text":"The first 1 being that we have N individuals,"},{"Start":"00:32.985 ","End":"00:35.295","Text":"D which are special."},{"Start":"00:35.295 ","End":"00:37.340","Text":"What\u0027s our big N?"},{"Start":"00:37.340 ","End":"00:38.810","Text":"What\u0027s their population size?"},{"Start":"00:38.810 ","End":"00:41.525","Text":"Well, we have 40 children in a class,"},{"Start":"00:41.525 ","End":"00:45.005","Text":"and D of which are special."},{"Start":"00:45.005 ","End":"00:46.730","Text":"What are we looking for?"},{"Start":"00:46.730 ","End":"00:53.030","Text":"We\u0027re looking for the number of boys because that\u0027s what we\u0027re being asked about."},{"Start":"00:53.030 ","End":"00:56.750","Text":"We\u0027re asked about the probability distribution of the number of boys."},{"Start":"00:56.750 ","End":"00:59.690","Text":"D equals to 30,"},{"Start":"00:59.690 ","End":"01:03.270","Text":"40 minus 10 girls is 30."},{"Start":"01:03.290 ","End":"01:05.760","Text":"What about our sample size?"},{"Start":"01:05.760 ","End":"01:09.390","Text":"Well, our sample size is 4 students that\u0027s right here."},{"Start":"01:09.390 ","End":"01:13.070","Text":"Now, X is the number of special individuals who are in our case,"},{"Start":"01:13.070 ","End":"01:17.180","Text":"as we said, X equals the number of boys."},{"Start":"01:17.180 ","End":"01:21.430","Text":"Now, what else do we have to look at?"},{"Start":"01:21.430 ","End":"01:24.530","Text":"Well, we\u0027re going to have to look at how we sampled."},{"Start":"01:24.530 ","End":"01:27.340","Text":"Are we sampling with or without returns?"},{"Start":"01:27.340 ","End":"01:31.955","Text":"Here we\u0027re sampling without returns because once a student,"},{"Start":"01:31.955 ","End":"01:35.885","Text":"a child has been selected for the delegation, well,"},{"Start":"01:35.885 ","End":"01:37.805","Text":"he\u0027s put aside,"},{"Start":"01:37.805 ","End":"01:42.290","Text":"and then another selection is made from all the other children."},{"Start":"01:42.290 ","End":"01:44.090","Text":"We\u0027re not selecting him again."},{"Start":"01:44.090 ","End":"01:49.100","Text":"We\u0027re not putting him back into the pool of children to be selected again."},{"Start":"01:49.100 ","End":"01:53.520","Text":"Here we\u0027re dealing with sampling without returns."},{"Start":"01:53.520 ","End":"01:55.100","Text":"If that\u0027s the case,"},{"Start":"01:55.100 ","End":"01:58.215","Text":"to answer section a, X,"},{"Start":"01:58.215 ","End":"02:03.105","Text":"the number of boys is distributed with the Hyper Geometric Distribution,"},{"Start":"02:03.105 ","End":"02:05.625","Text":"where big N equals 40,"},{"Start":"02:05.625 ","End":"02:11.085","Text":"D equals 30, and small n equals 4."},{"Start":"02:11.085 ","End":"02:14.419","Text":"In section b, we\u0027re asked what are the expectation"},{"Start":"02:14.419 ","End":"02:17.765","Text":"and variance of the number of boys in the delegation?"},{"Start":"02:17.765 ","End":"02:22.400","Text":"Well, the expectation of X equals to"},{"Start":"02:22.400 ","End":"02:28.025","Text":"small n times D over big N. Now let\u0027s plug in the numbers."},{"Start":"02:28.025 ","End":"02:32.615","Text":"Small n is 4 times D is 30,"},{"Start":"02:32.615 ","End":"02:34.940","Text":"and big N is 40."},{"Start":"02:34.940 ","End":"02:39.615","Text":"That means that this equals to 3 boys."},{"Start":"02:39.615 ","End":"02:42.100","Text":"What\u0027s the variance of X?"},{"Start":"02:42.100 ","End":"02:48.470","Text":"Well, the variance is defined as n times D over big N times"},{"Start":"02:48.470 ","End":"02:56.870","Text":"1 minus D over N times big N minus small n divided by big N minus 1."},{"Start":"02:56.870 ","End":"02:59.000","Text":"Now, let\u0027s plug in the numbers."},{"Start":"02:59.000 ","End":"03:03.995","Text":"Small n is 4 times 30 over 40."},{"Start":"03:03.995 ","End":"03:09.075","Text":"D over N times 1 minus"},{"Start":"03:09.075 ","End":"03:15.656","Text":"30 over 40 time big N minus small n,"},{"Start":"03:15.656 ","End":"03:21.180","Text":"that\u0027s 40 minus 4 divided by 40 minus 1."},{"Start":"03:21.180 ","End":"03:25.510","Text":"That equals to 0.692."},{"Start":"03:28.340 ","End":"03:35.305","Text":"In section c, we\u0027re asked what are the chances of there being 3 boys in the delegation?"},{"Start":"03:35.305 ","End":"03:41.275","Text":"c asks, what\u0027s the probability of X being equal to 3?"},{"Start":"03:41.275 ","End":"03:47.065","Text":"Now, let\u0027s just remind ourselves of the probability function,"},{"Start":"03:47.065 ","End":"03:55.860","Text":"probability of X being equal to K. That equals to D over K times big N"},{"Start":"03:55.860 ","End":"04:04.950","Text":"minus D over small n minus K divided by big N over small n. In our case,"},{"Start":"04:04.950 ","End":"04:06.810","Text":"let\u0027s just plug in the numbers."},{"Start":"04:06.810 ","End":"04:15.540","Text":"D is 30 and K is 3 times N is 40, and D is 30."},{"Start":"04:15.540 ","End":"04:19.830","Text":"That\u0027s 40 minus 30 over,"},{"Start":"04:19.830 ","End":"04:27.540","Text":"that\u0027s 4 minus 3 divided by big N over small n. Well,"},{"Start":"04:27.540 ","End":"04:36.825","Text":"that\u0027s 40 over 4 and that would equal 30 over 3,"},{"Start":"04:36.825 ","End":"04:40.590","Text":"times 10 over 1,"},{"Start":"04:40.590 ","End":"04:46.420","Text":"divided by 40 over 4. That\u0027s our answer."}],"ID":13032},{"Watched":false,"Name":"Exercise 1 Part a","Duration":"5m 12s","ChapterTopicVideoID":12554,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"This question will be talking about sampling balls from a basket."},{"Start":"00:04.980 ","End":"00:09.085","Text":"Now, a basket has 5 red balls and 4 green balls."},{"Start":"00:09.085 ","End":"00:13.560","Text":"3 balls are randomly removed from the basket with no returns."},{"Start":"00:13.560 ","End":"00:17.340","Text":"In section a we\u0027re asked to use a table to construct"},{"Start":"00:17.340 ","End":"00:22.095","Text":"the probability function of the number of red balls removed from the container."},{"Start":"00:22.095 ","End":"00:25.770","Text":"Here we have a hyper geometric distribution."},{"Start":"00:25.770 ","End":"00:30.940","Text":"But let\u0027s look at the characteristics of this distribution."},{"Start":"00:31.370 ","End":"00:33.675","Text":"This is a list."},{"Start":"00:33.675 ","End":"00:36.170","Text":"Let\u0027s take a look at the first characteristic."},{"Start":"00:36.170 ","End":"00:40.030","Text":"We need to have N individuals D of which are special."},{"Start":"00:40.030 ","End":"00:44.195","Text":"N would be the population size."},{"Start":"00:44.195 ","End":"00:49.080","Text":"Here we have 5 red balls and 4 green balls in a basket"},{"Start":"00:49.080 ","End":"00:53.560","Text":"so our population of balls would equal to 9."},{"Start":"00:53.560 ","End":"00:55.745","Text":"That\u0027s the population size."},{"Start":"00:55.745 ","End":"00:59.075","Text":"Now, what about the special balls here?"},{"Start":"00:59.075 ","End":"01:01.790","Text":"We\u0027re asked about the red balls."},{"Start":"01:01.790 ","End":"01:06.770","Text":"D would be the number of red balls in our population, and that\u0027s 5."},{"Start":"01:06.770 ","End":"01:10.490","Text":"That\u0027s right here. Now, what about the sample?"},{"Start":"01:10.490 ","End":"01:16.800","Text":"Well, 3 balls are sampled and we\u0027re sampling with no returns."},{"Start":"01:16.800 ","End":"01:19.890","Text":"So n equals 3."},{"Start":"01:19.890 ","End":"01:22.410","Text":"Now, what about X?"},{"Start":"01:22.410 ","End":"01:31.540","Text":"X would be the number of red balls in our sample."},{"Start":"01:31.840 ","End":"01:36.425","Text":"Now, having met all these criteria,"},{"Start":"01:36.425 ","End":"01:41.690","Text":"then we know that X is distributed with a hyper geometric distribution,"},{"Start":"01:41.690 ","End":"01:43.445","Text":"where N equals 9,"},{"Start":"01:43.445 ","End":"01:48.445","Text":"D equals 5, and small n equals 3."},{"Start":"01:48.445 ","End":"01:53.510","Text":"The probability of X being equal to"},{"Start":"01:53.510 ","End":"02:01.850","Text":"K value would equal to D over K,"},{"Start":"02:01.850 ","End":"02:06.970","Text":"N minus D over n minus k"},{"Start":"02:06.970 ","End":"02:13.190","Text":"divided by big N over small n. Now,"},{"Start":"02:13.190 ","End":"02:16.385","Text":"let\u0027s take a look at the values of X."},{"Start":"02:16.385 ","End":"02:20.810","Text":"Remember we have to use a table to construct a probability function."},{"Start":"02:20.810 ","End":"02:27.875","Text":"What values of X do we have?"},{"Start":"02:27.875 ","End":"02:34.340","Text":"X can be 0 where we have no red balls in our sample."},{"Start":"02:34.340 ","End":"02:38.530","Text":"We could have 1 ball in our sample 2 or 3 balls in the sample."},{"Start":"02:38.530 ","End":"02:42.295","Text":"3 being the maximum number because that\u0027s the sample size."},{"Start":"02:42.295 ","End":"02:48.810","Text":"We have either 0,1,2 or 3."},{"Start":"02:48.810 ","End":"02:57.080","Text":"Now we want to calculate the probability of X for each 1 of the values of X."},{"Start":"02:57.080 ","End":"02:59.635","Text":"Let\u0027s get started."},{"Start":"02:59.635 ","End":"03:06.255","Text":"The probability of X being equal to 0,"},{"Start":"03:06.255 ","End":"03:08.360","Text":"let\u0027s just plug in the numbers here."},{"Start":"03:08.360 ","End":"03:16.065","Text":"That\u0027s D over k. That\u0027s 5 over 0 times big N minus D,"},{"Start":"03:16.065 ","End":"03:20.265","Text":"that\u0027s 9 minus 5, that\u0027s 4."},{"Start":"03:20.265 ","End":"03:23.070","Text":"Small n minus k, that\u0027s 3 minus 0,"},{"Start":"03:23.070 ","End":"03:27.050","Text":"that\u0027s 3 divided by big N over small n,"},{"Start":"03:27.050 ","End":"03:29.340","Text":"that\u0027s 9 over 3."},{"Start":"03:29.870 ","End":"03:35.670","Text":"That would equal 4 divided by 84."},{"Start":"03:35.670 ","End":"03:41.765","Text":"Now, what about the probability of X being equal to 1?"},{"Start":"03:41.765 ","End":"03:44.614","Text":"Well again, let\u0027s plug in the numbers."},{"Start":"03:44.614 ","End":"03:48.935","Text":"That\u0027s 5 over 1 times"},{"Start":"03:48.935 ","End":"03:55.455","Text":"4 over 2 divided by 9 over 3."},{"Start":"03:55.455 ","End":"04:00.605","Text":"That comes out to 30 divided by 84."},{"Start":"04:00.605 ","End":"04:03.350","Text":"Now, let\u0027s first plug in the numbers in the table."},{"Start":"04:03.350 ","End":"04:05.960","Text":"That\u0027s 4 over 84,"},{"Start":"04:05.960 ","End":"04:09.990","Text":"that\u0027s 30 over 84."},{"Start":"04:09.990 ","End":"04:19.070","Text":"Now we have to figure out what the probability is for X equals 2. Here we go."},{"Start":"04:19.070 ","End":"04:22.565","Text":"That\u0027s the probability of X equaling 2."},{"Start":"04:22.565 ","End":"04:26.815","Text":"That equals to 5 over 2"},{"Start":"04:26.815 ","End":"04:33.975","Text":"times 4 over 1 divided by 9 over 3."},{"Start":"04:33.975 ","End":"04:40.590","Text":"That comes out to 40 divided by 84."},{"Start":"04:40.590 ","End":"04:44.690","Text":"Let\u0027s plug that in. That\u0027s 40 divided by 84."},{"Start":"04:44.690 ","End":"04:50.915","Text":"Now, we can plug in 3 into the equation here."},{"Start":"04:50.915 ","End":"04:52.370","Text":"But why do that?"},{"Start":"04:52.370 ","End":"04:56.250","Text":"We know that the sum of the probabilities equal 1."},{"Start":"04:56.320 ","End":"05:01.075","Text":"When we calculate the sum of these guys,"},{"Start":"05:01.075 ","End":"05:03.834","Text":"we take 1 minus the sum of these guys,"},{"Start":"05:03.834 ","End":"05:07.225","Text":"that\u0027ll be 10 over 84."},{"Start":"05:07.225 ","End":"05:12.050","Text":"This would be the answer for section a."}],"ID":13033},{"Watched":false,"Name":"Exercise 1 Parts b-c","Duration":"7m 16s","ChapterTopicVideoID":12555,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this section, we\u0027re asked to calculate"},{"Start":"00:01.980 ","End":"00:05.160","Text":"the expectation and variance of the number of red balls removed,"},{"Start":"00:05.160 ","End":"00:08.310","Text":"once using the probability function and once"},{"Start":"00:08.310 ","End":"00:12.045","Text":"using the formulas for the hyper-geometric probability."},{"Start":"00:12.045 ","End":"00:18.870","Text":"Let\u0027s take a look at the table that we constructed in section A. Here it is."},{"Start":"00:18.870 ","End":"00:20.760","Text":"Now, if that\u0027s the case,"},{"Start":"00:20.760 ","End":"00:24.945","Text":"let\u0027s start calculating the expectation of x."},{"Start":"00:24.945 ","End":"00:30.990","Text":"Well, that equals to the sum of x times its probability."},{"Start":"00:30.990 ","End":"00:34.070","Text":"Let\u0027s start calculating this."},{"Start":"00:34.070 ","End":"00:36.140","Text":"That\u0027s 0 times 4 over 84."},{"Start":"00:36.140 ","End":"00:40.145","Text":"We don\u0027t need to write this down because it\u0027s a multiplication of 0. Let\u0027s start from 1."},{"Start":"00:40.145 ","End":"00:49.755","Text":"That\u0027s 1 times 30 over 84 plus 2 times 40 over 84,"},{"Start":"00:49.755 ","End":"00:54.405","Text":"plus 3 times 10 over"},{"Start":"00:54.405 ","End":"01:00.805","Text":"84 and that comes out to 1 and 2/3s."},{"Start":"01:00.805 ","End":"01:03.935","Text":"Now, what about the variance of X?"},{"Start":"01:03.935 ","End":"01:08.240","Text":"Well, that equals to the sum of x squared times"},{"Start":"01:08.240 ","End":"01:13.675","Text":"its probability minus the expectation squared of X."},{"Start":"01:13.675 ","End":"01:15.520","Text":"That equals 2."},{"Start":"01:15.520 ","End":"01:17.390","Text":"Again, we don\u0027t have to start here."},{"Start":"01:17.390 ","End":"01:19.280","Text":"This is a multiplication by 0,"},{"Start":"01:19.280 ","End":"01:22.895","Text":"so it\u0027s 1 squared times 30 over"},{"Start":"01:22.895 ","End":"01:30.735","Text":"84 plus 2 squared times 40 over 84,"},{"Start":"01:30.735 ","End":"01:35.595","Text":"plus 3 squared times 10 over 84"},{"Start":"01:35.595 ","End":"01:41.565","Text":"minus 1 and 2/3s squared."},{"Start":"01:41.565 ","End":"01:47.410","Text":"That comes out to 5 over 9."},{"Start":"01:48.500 ","End":"01:51.470","Text":"Let\u0027s use the formulas for"},{"Start":"01:51.470 ","End":"01:56.960","Text":"the hyper-geometric probability distribution to calculate the expectation variance."},{"Start":"01:56.960 ","End":"02:02.145","Text":"Now, the expectation of x according to the formulas,"},{"Start":"02:02.145 ","End":"02:07.580","Text":"is n times D over big N. Now,"},{"Start":"02:07.580 ","End":"02:10.010","Text":"in our case, n is equals to 3."},{"Start":"02:10.010 ","End":"02:11.960","Text":"So 3 times,"},{"Start":"02:11.960 ","End":"02:13.535","Text":"what\u0027s D over N?"},{"Start":"02:13.535 ","End":"02:19.310","Text":"Well D is 5 and big N is 9."},{"Start":"02:19.310 ","End":"02:23.195","Text":"Now, that equals to 15 over 9,"},{"Start":"02:23.195 ","End":"02:32.550","Text":"and that equals to 5 over 3 and that basically equals to 1 and 2/3s."},{"Start":"02:32.550 ","End":"02:37.410","Text":"As we can see, this is equal to this."},{"Start":"02:38.280 ","End":"02:41.260","Text":"Now, what about the variance?"},{"Start":"02:41.260 ","End":"02:49.360","Text":"Well, the variance of x according to the formulas that equals to n times D over"},{"Start":"02:49.360 ","End":"02:53.650","Text":"N times 1 minus D over"},{"Start":"02:53.650 ","End":"03:00.895","Text":"N times big N minus small n,"},{"Start":"03:00.895 ","End":"03:05.305","Text":"but it\u0027s a minus divided by big N minus 1."},{"Start":"03:05.305 ","End":"03:07.345","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"03:07.345 ","End":"03:13.900","Text":"That\u0027s 3 times D over N. That\u0027ll be 5 over 9"},{"Start":"03:13.900 ","End":"03:21.170","Text":"times 1 minus 5 over 9 times 9 minus 3,"},{"Start":"03:21.170 ","End":"03:25.740","Text":"that\u0027s 9 minus 3 over 9 minus 1."},{"Start":"03:25.760 ","End":"03:30.495","Text":"This comes out to 5 over 9."},{"Start":"03:30.495 ","End":"03:34.365","Text":"Again, we can see that this equals this."},{"Start":"03:34.365 ","End":"03:38.290","Text":"There it is we answered section B."},{"Start":"03:38.920 ","End":"03:43.580","Text":"Section C, we\u0027re asked what is the expectation and variance of the number of"},{"Start":"03:43.580 ","End":"03:47.945","Text":"red balls if the red balls removed are returned each time."},{"Start":"03:47.945 ","End":"03:54.035","Text":"Now, the key phrase here is that the balls are returned each time before the next sample."},{"Start":"03:54.035 ","End":"03:56.675","Text":"Now, if that\u0027s the case."},{"Start":"03:56.675 ","End":"04:00.350","Text":"Then we\u0027re not talking about a hyper-geometric distribution because"},{"Start":"04:00.350 ","End":"04:06.355","Text":"the hyper-geometric distribution talks about sampling without returns."},{"Start":"04:06.355 ","End":"04:08.645","Text":"Let\u0027s see what we have here."},{"Start":"04:08.645 ","End":"04:17.340","Text":"Let\u0027s define success as the number or the red balls,"},{"Start":"04:18.760 ","End":"04:21.755","Text":"not the number of red balls but red balls."},{"Start":"04:21.755 ","End":"04:24.515","Text":"Now getting a red ball is a success for us."},{"Start":"04:24.515 ","End":"04:27.350","Text":"Now, what\u0027s the probability of a success?"},{"Start":"04:27.350 ","End":"04:30.575","Text":"Well, we had 5 red balls in the sample out of a total of 9."},{"Start":"04:30.575 ","End":"04:33.265","Text":"The probability is 5 over 9."},{"Start":"04:33.265 ","End":"04:35.630","Text":"How many did we sample?"},{"Start":"04:35.630 ","End":"04:36.880","Text":"Well, we sampled 3,"},{"Start":"04:36.880 ","End":"04:39.590","Text":"so, n here equals 3."},{"Start":"04:39.590 ","End":"04:42.585","Text":"Now, remember what we said about"},{"Start":"04:42.585 ","End":"04:49.075","Text":"the hyper-geometric distribution versus the binomial distribution."},{"Start":"04:49.075 ","End":"04:58.340","Text":"With the binomial distribution has the same characteristics except that the samples,"},{"Start":"04:58.340 ","End":"05:00.080","Text":"each time you sample something,"},{"Start":"05:00.080 ","End":"05:01.825","Text":"you have to return it."},{"Start":"05:01.825 ","End":"05:05.810","Text":"That\u0027s the binomial distribution with the hyper-geometric distribution,"},{"Start":"05:05.810 ","End":"05:08.300","Text":"you don\u0027t return the balls."},{"Start":"05:08.300 ","End":"05:13.055","Text":"Now, the minute you return the ball each time before you sample the next ball,"},{"Start":"05:13.055 ","End":"05:15.910","Text":"well, then you have independence."},{"Start":"05:15.910 ","End":"05:24.560","Text":"In essence, we have 3 Bernoulli trials where success is defined as getting a red ball and"},{"Start":"05:24.560 ","End":"05:29.855","Text":"the probability of getting a red ball is always 5 over 9 because we\u0027re returning"},{"Start":"05:29.855 ","End":"05:36.240","Text":"a ball back into the pool or the population."},{"Start":"05:36.240 ","End":"05:39.920","Text":"The trials are independent."},{"Start":"05:39.920 ","End":"05:44.120","Text":"They have the same distribution and that reminds us,"},{"Start":"05:44.120 ","End":"05:47.615","Text":"as we said, of a binomial distribution."},{"Start":"05:47.615 ","End":"05:54.625","Text":"If we define x as the number of successes,"},{"Start":"05:54.625 ","End":"06:05.305","Text":"then x has a binomial distribution where n equals 3 and p equals 5 over 9."},{"Start":"06:05.305 ","End":"06:07.295","Text":"If that\u0027s the case,"},{"Start":"06:07.295 ","End":"06:13.340","Text":"then the expectation of x equals n times p and a binomial distribution,"},{"Start":"06:13.340 ","End":"06:15.905","Text":"that\u0027s the definition of the expectation of x."},{"Start":"06:15.905 ","End":"06:17.690","Text":"Now, plugging in the numbers,"},{"Start":"06:17.690 ","End":"06:26.495","Text":"that\u0027s 3 times 5 over 9 that equals to 1 and 2/3s."},{"Start":"06:26.495 ","End":"06:27.995","Text":"Now as we can see,"},{"Start":"06:27.995 ","End":"06:31.505","Text":"the expectation in the binomial distribution is"},{"Start":"06:31.505 ","End":"06:35.405","Text":"equal to the expectation and hyper-geometric distribution."},{"Start":"06:35.405 ","End":"06:38.180","Text":"We saw that in the previous section."},{"Start":"06:38.180 ","End":"06:40.550","Text":"Now what about the variance of x?"},{"Start":"06:40.550 ","End":"06:46.365","Text":"Well, the variance of x, let\u0027s define it that\u0027s n times p"},{"Start":"06:46.365 ","End":"06:52.985","Text":"times 1 minus p. That\u0027s the definition of the variance in a binomial distribution."},{"Start":"06:52.985 ","End":"06:58.625","Text":"Now let\u0027s plug in the numbers that\u0027s 3 times 5 over 9 times"},{"Start":"06:58.625 ","End":"07:04.830","Text":"1 minus 5 over 9 and that equals to 20 over 27."},{"Start":"07:04.830 ","End":"07:12.660","Text":"If we recall, this does not equal the variance in a hyper-geometric distribution."},{"Start":"07:12.660 ","End":"07:15.870","Text":"This is a binomial distribution."}],"ID":13034},{"Watched":false,"Name":"Exercise 2 Part a","Duration":"5m 24s","ChapterTopicVideoID":12556,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.555","Text":"In this question, we\u0027ll be talking about questions in a quiz."},{"Start":"00:03.555 ","End":"00:08.670","Text":"There are 10 questions on 3 different subjects in a quiz: 3 about sports,"},{"Start":"00:08.670 ","End":"00:10.110","Text":"4 about entertainment,"},{"Start":"00:10.110 ","End":"00:11.820","Text":"and the rest about science."},{"Start":"00:11.820 ","End":"00:15.345","Text":"A participant in the quiz randomly picks 4 questions."},{"Start":"00:15.345 ","End":"00:18.735","Text":"We define X as the number of sports questions picked."},{"Start":"00:18.735 ","End":"00:20.970","Text":"We\u0027re asked in section a to construct"},{"Start":"00:20.970 ","End":"00:25.065","Text":"a probability function of X using the formula, not the table."},{"Start":"00:25.065 ","End":"00:29.970","Text":"Here we have a hypergeometric distribution."},{"Start":"00:29.970 ","End":"00:32.160","Text":"Why is that? First of all,"},{"Start":"00:32.160 ","End":"00:34.845","Text":"let\u0027s look at the population size."},{"Start":"00:34.845 ","End":"00:37.410","Text":"We have 10 questions in the quiz,"},{"Start":"00:37.410 ","End":"00:40.275","Text":"so big N equals 10."},{"Start":"00:40.275 ","End":"00:42.460","Text":"That\u0027s the population size."},{"Start":"00:42.460 ","End":"00:47.030","Text":"Now let\u0027s take a look at the questions that have a special characteristic."},{"Start":"00:47.030 ","End":"00:52.550","Text":"Now for us, a question that has a special characteristic is the sports questions."},{"Start":"00:52.550 ","End":"00:59.030","Text":"So D means that we\u0027re counting the sports questions."},{"Start":"00:59.030 ","End":"01:01.310","Text":"How many sports questions are there?"},{"Start":"01:01.310 ","End":"01:04.385","Text":"There are 3 sports questions."},{"Start":"01:04.385 ","End":"01:07.025","Text":"What about the sample size?"},{"Start":"01:07.025 ","End":"01:13.310","Text":"Well, it says that the participants in the quiz randomly picks 4 questions."},{"Start":"01:13.310 ","End":"01:15.620","Text":"So n equals 4."},{"Start":"01:15.620 ","End":"01:19.010","Text":"The sampling is done randomly,"},{"Start":"01:19.010 ","End":"01:21.905","Text":"and obviously it\u0027s done without returns."},{"Start":"01:21.905 ","End":"01:24.495","Text":"How\u0027s X defined?"},{"Start":"01:24.495 ","End":"01:28.280","Text":"X is defined as the number of sports questions."},{"Start":"01:28.280 ","End":"01:33.760","Text":"Sports questions."},{"Start":"01:33.760 ","End":"01:42.575","Text":"That means that X now has a hypergeometric distribution where big N equals 10,"},{"Start":"01:42.575 ","End":"01:45.035","Text":"D equals 3,"},{"Start":"01:45.035 ","End":"01:48.035","Text":"and small n equals 4."},{"Start":"01:48.035 ","End":"01:50.675","Text":"Now in section a,"},{"Start":"01:50.675 ","End":"01:56.000","Text":"we\u0027re asked to construct a probability function of X using the formula, not the table."},{"Start":"01:56.000 ","End":"02:00.540","Text":"What\u0027s the probability of X being equal to k?"},{"Start":"02:00.620 ","End":"02:06.170","Text":"In a generic equation of the hypergeometric distribution,"},{"Start":"02:06.170 ","End":"02:12.500","Text":"that\u0027s d over K times big N minus d over small n"},{"Start":"02:12.500 ","End":"02:18.770","Text":"minus k divided by big N over small n. So in our case,"},{"Start":"02:18.770 ","End":"02:21.805","Text":"let\u0027s just plug in the numbers: d equals 3,"},{"Start":"02:21.805 ","End":"02:26.915","Text":"K we leave it k because we want this to be a formula,"},{"Start":"02:26.915 ","End":"02:29.675","Text":"times n minus D,"},{"Start":"02:29.675 ","End":"02:31.235","Text":"that\u0027s 10 minus 3,"},{"Start":"02:31.235 ","End":"02:34.220","Text":"that\u0027s 7, over small n minus K,"},{"Start":"02:34.220 ","End":"02:40.568","Text":"that\u0027s 4 minus K divided by big N over small n,"},{"Start":"02:40.568 ","End":"02:43.600","Text":"that\u0027s 10 over 4."},{"Start":"02:44.180 ","End":"02:48.770","Text":"Let\u0027s take a look at the values that X can have."},{"Start":"02:48.770 ","End":"02:54.600","Text":"In our case, we\u0027re looking at X being equal to 0."},{"Start":"02:54.600 ","End":"02:56.415","Text":"In our sample of 4,"},{"Start":"02:56.415 ","End":"03:03.170","Text":"we can have a situation where we don\u0027t have any sports questions."},{"Start":"03:03.170 ","End":"03:05.205","Text":"We can have 1 sport question,"},{"Start":"03:05.205 ","End":"03:08.240","Text":"or 2, or 3 sports questions."},{"Start":"03:08.240 ","End":"03:11.300","Text":"Now, the question is because we took a sample of 4,"},{"Start":"03:11.300 ","End":"03:13.580","Text":"can we have 4 sports questions?"},{"Start":"03:13.580 ","End":"03:18.320","Text":"Obviously not because we only have 3 sports questions in the whole population."},{"Start":"03:18.320 ","End":"03:21.870","Text":"So X can have only these values: 0,"},{"Start":"03:21.870 ","End":"03:23.910","Text":"1, 2, and 3."},{"Start":"03:23.910 ","End":"03:28.760","Text":"Let\u0027s just look at the generic formula for"},{"Start":"03:28.760 ","End":"03:34.590","Text":"calculating the values of X. K,"},{"Start":"03:34.590 ","End":"03:36.315","Text":"which is the value of X,"},{"Start":"03:36.315 ","End":"03:40.680","Text":"can have at its minimum value;"},{"Start":"03:40.680 ","End":"03:45.740","Text":"the minimum value is defined as the maximum between"},{"Start":"03:45.740 ","End":"03:52.710","Text":"0 and small n plus D minus big N,"},{"Start":"03:52.710 ","End":"03:57.905","Text":"and it could have a maximum value defined as"},{"Start":"03:57.905 ","End":"04:04.510","Text":"minimum between small n and D. Now,"},{"Start":"04:04.510 ","End":"04:07.010","Text":"let\u0027s just take a look at this."},{"Start":"04:07.010 ","End":"04:10.760","Text":"Let\u0027s just plug in the numbers and see where we\u0027re at."},{"Start":"04:10.760 ","End":"04:21.005","Text":"K equals the maximum between 0 and small n plus d minus big N. Small n is 4,"},{"Start":"04:21.005 ","End":"04:22.655","Text":"D is 3,"},{"Start":"04:22.655 ","End":"04:24.770","Text":"and big N is 10."},{"Start":"04:24.770 ","End":"04:28.050","Text":"So that\u0027s 4 plus 3,"},{"Start":"04:28.050 ","End":"04:30.420","Text":"that\u0027s 7 minus 10."},{"Start":"04:30.420 ","End":"04:32.670","Text":"That\u0027s minus 3."},{"Start":"04:32.670 ","End":"04:36.495","Text":"So it\u0027s maximum of 0 and minus 3."},{"Start":"04:36.495 ","End":"04:44.835","Text":"What\u0027s the maximum value that X can have or the K can have?"},{"Start":"04:44.835 ","End":"04:49.860","Text":"That\u0027ll be the minimum between n. That\u0027s 4,"},{"Start":"04:49.860 ","End":"04:52.275","Text":"and D is 3."},{"Start":"04:52.275 ","End":"05:01.780","Text":"So K can have the maximum of 0 minus 3: that\u0027s 0."},{"Start":"05:02.630 ","End":"05:10.295","Text":"The minimum value of maximum value would be the minimum of 4 and 3. Well that\u0027s 3."},{"Start":"05:10.295 ","End":"05:14.150","Text":"So K can have the values between 0 and 3."},{"Start":"05:14.150 ","End":"05:15.995","Text":"That\u0027s what we got right here."},{"Start":"05:15.995 ","End":"05:23.509","Text":"So this is how you calculate the range of the values that X can have."}],"ID":13035},{"Watched":false,"Name":"Exercise 2 Parts b-c","Duration":"4m 34s","ChapterTopicVideoID":12557,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.545","Text":"In section B, we\u0027re asked,"},{"Start":"00:01.545 ","End":"00:06.105","Text":"what is the expectation and standard deviation of x?"},{"Start":"00:06.105 ","End":"00:08.055","Text":"Let\u0027s start calculating."},{"Start":"00:08.055 ","End":"00:15.945","Text":"The expectation of X is defined as small n times D divided by big N. In our case,"},{"Start":"00:15.945 ","End":"00:20.220","Text":"that\u0027ll be 4 times 3 divided by 10 and that"},{"Start":"00:20.220 ","End":"00:27.580","Text":"equals to 1.2 sports questions."},{"Start":"00:28.730 ","End":"00:31.905","Text":"Now, what about the variance of X?"},{"Start":"00:31.905 ","End":"00:38.790","Text":"Well, that equals to n times D divided by big N times 1 minus D"},{"Start":"00:38.790 ","End":"00:46.445","Text":"divided by big N times big N minus small n divided by big N minus 1."},{"Start":"00:46.445 ","End":"00:49.249","Text":"Now, let\u0027s plug in the numbers."},{"Start":"00:49.249 ","End":"00:58.860","Text":"That\u0027s 4 times 3 over 10 times 1 minus 3 over 10,"},{"Start":"00:58.860 ","End":"01:04.820","Text":"times 10 minus 4 divided by 10 minus 1,"},{"Start":"01:04.820 ","End":"01:08.870","Text":"and that equals to 0.56."},{"Start":"01:08.870 ","End":"01:13.700","Text":"Now, the standard deviation of x is the square root of the variance,"},{"Start":"01:13.700 ","End":"01:16.550","Text":"that\u0027s the square root of 0.56,"},{"Start":"01:16.550 ","End":"01:22.595","Text":"and that equals to 0.748."},{"Start":"01:22.595 ","End":"01:26.855","Text":"In Section C, we\u0027re asked to calculate the following probability."},{"Start":"01:26.855 ","End":"01:32.485","Text":"The probability of X equaling 2 given that X is greater than 1."},{"Start":"01:32.485 ","End":"01:37.735","Text":"That\u0027s the probability of X being equal to 2,"},{"Start":"01:37.735 ","End":"01:40.875","Text":"given that X is greater than 1."},{"Start":"01:40.875 ","End":"01:44.360","Text":"Well, we know how to do that."},{"Start":"01:44.360 ","End":"01:46.265","Text":"That\u0027s in the denominator."},{"Start":"01:46.265 ","End":"01:48.000","Text":"That\u0027s the probability of what\u0027s given."},{"Start":"01:48.000 ","End":"01:50.975","Text":"That\u0027s the probability of X being greater than 1."},{"Start":"01:50.975 ","End":"01:54.815","Text":"In the numerator, that\u0027s the probability of the intersects."},{"Start":"01:54.815 ","End":"02:01.985","Text":"That\u0027s the probability of X equaling 2 and X being greater than 1."},{"Start":"02:01.985 ","End":"02:07.130","Text":"Now, let\u0027s make this a little bit more simple."},{"Start":"02:07.130 ","End":"02:12.785","Text":"Now, what\u0027s the probability of X equaling 2 and X being greater than 1."},{"Start":"02:12.785 ","End":"02:15.050","Text":"What\u0027s the intersect of these events?"},{"Start":"02:15.050 ","End":"02:23.600","Text":"That\u0027s the probability of X being equal to 2 divided by the probability of"},{"Start":"02:23.600 ","End":"02:31.520","Text":"X being"},{"Start":"02:31.520 ","End":"02:34.505","Text":"greater than 1."},{"Start":"02:34.505 ","End":"02:39.965","Text":"Let\u0027s first recall what we did in the previous sections."},{"Start":"02:39.965 ","End":"02:44.689","Text":"We said that the probability of X being equal to k,"},{"Start":"02:44.689 ","End":"02:50.140","Text":"well, that equals to 3 over k,"},{"Start":"02:50.140 ","End":"02:57.275","Text":"7 over 4 minus k divided by 10 over 4,"},{"Start":"02:57.275 ","End":"03:00.620","Text":"where k equals 0,"},{"Start":"03:00.620 ","End":"03:03.505","Text":"1, 2, and 3."},{"Start":"03:03.505 ","End":"03:07.815","Text":"In our case right here,"},{"Start":"03:07.815 ","End":"03:12.185","Text":"let\u0027s calculate this expression right here."},{"Start":"03:12.185 ","End":"03:16.765","Text":"Now, what\u0027s this expression right here?"},{"Start":"03:16.765 ","End":"03:20.810","Text":"What\u0027s the probability of X being equal to 2?"},{"Start":"03:20.810 ","End":"03:28.485","Text":"Well, that\u0027s 3 over 2 times 7 over 4 minus 2, that\u0027s 2."},{"Start":"03:28.485 ","End":"03:31.900","Text":"Divided by 10 over 4."},{"Start":"03:31.900 ","End":"03:34.370","Text":"Now that was the numerator."},{"Start":"03:34.370 ","End":"03:36.875","Text":"Let\u0027s take a look at the denominator."},{"Start":"03:36.875 ","End":"03:41.060","Text":"Well, the probability of x being greater than 1,"},{"Start":"03:41.060 ","End":"03:44.440","Text":"that means that x is 2 and x equals 3."},{"Start":"03:44.440 ","End":"03:47.130","Text":"Let\u0027s plug in these numbers."},{"Start":"03:47.130 ","End":"03:48.761","Text":"Now, when X equals 2,"},{"Start":"03:48.761 ","End":"03:50.450","Text":"that\u0027s just like the numerator."},{"Start":"03:50.450 ","End":"03:52.190","Text":"Let\u0027s just copy this cell."},{"Start":"03:52.190 ","End":"04:00.420","Text":"That\u0027s 3 over 2 times 7 over 2 divided by 10 over 4,"},{"Start":"04:00.420 ","End":"04:04.570","Text":"plus, now x equals 3."},{"Start":"04:04.570 ","End":"04:06.260","Text":"That\u0027s this guy right here."},{"Start":"04:06.260 ","End":"04:13.980","Text":"So that\u0027s 3 over 3 times 7 over 4 minus 3, that\u0027s 1."},{"Start":"04:13.980 ","End":"04:17.475","Text":"Divided by 10 over 4."},{"Start":"04:17.475 ","End":"04:23.985","Text":"Now, all this expression turns out to be 0.9."},{"Start":"04:23.985 ","End":"04:34.080","Text":"So, 0.9 is the probability that X equals 2 given that X is greater than 1."}],"ID":13036},{"Watched":false,"Name":"Exercise 3","Duration":"6m 9s","ChapterTopicVideoID":12558,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.454","Text":"In this question we\u0027ll be dealing with people who have a driving license."},{"Start":"00:04.454 ","End":"00:08.040","Text":"Now, 6 people are sampled from a population in"},{"Start":"00:08.040 ","End":"00:11.730","Text":"which 60 percent of the population have a driving license."},{"Start":"00:11.730 ","End":"00:16.125","Text":"Now, let X be the number of people sampled who have a driving license."},{"Start":"00:16.125 ","End":"00:19.800","Text":"We\u0027re asked to identify the probability distribution and"},{"Start":"00:19.800 ","End":"00:23.265","Text":"to calculate the expectation of variants of the following,"},{"Start":"00:23.265 ","End":"00:25.650","Text":"where the following are 2 special cases."},{"Start":"00:25.650 ","End":"00:28.530","Text":"1 is where we sample from a large population,"},{"Start":"00:28.530 ","End":"00:33.345","Text":"and secondly, where we sample from a population of 10 people."},{"Start":"00:33.345 ","End":"00:35.400","Text":"Let\u0027s start with Section A,"},{"Start":"00:35.400 ","End":"00:37.630","Text":"sampling from a large population."},{"Start":"00:37.630 ","End":"00:43.730","Text":"Now, whenever we sample a small number of individuals from a very large population,"},{"Start":"00:43.730 ","End":"00:46.610","Text":"then we can assume independence."},{"Start":"00:46.610 ","End":"00:48.400","Text":"Now, having said that,"},{"Start":"00:48.400 ","End":"00:54.275","Text":"then what we have here is a binomial distribution, and why is that?"},{"Start":"00:54.275 ","End":"00:57.125","Text":"Well, what is a binomial distribution?"},{"Start":"00:57.125 ","End":"01:01.490","Text":"This is where we have Bernoulli trials which defines"},{"Start":"01:01.490 ","End":"01:06.350","Text":"success and failure that are independent of each other,"},{"Start":"01:06.350 ","End":"01:10.400","Text":"and they have obviously the same distribution with"},{"Start":"01:10.400 ","End":"01:15.395","Text":"the same probability for success on each sample because they\u0027re independent."},{"Start":"01:15.395 ","End":"01:19.655","Text":"Now, what\u0027s the definition of success here?"},{"Start":"01:19.655 ","End":"01:26.430","Text":"A definition of success is where we have a driving license."},{"Start":"01:28.460 ","End":"01:31.640","Text":"What\u0027s the probability of success?"},{"Start":"01:31.640 ","End":"01:33.860","Text":"Well, that\u0027s 60 percent."},{"Start":"01:33.860 ","End":"01:37.280","Text":"Now, how many trials are we doing?"},{"Start":"01:37.280 ","End":"01:39.320","Text":"Well, we\u0027re doing 6 trials,"},{"Start":"01:39.320 ","End":"01:42.765","Text":"we\u0027re sampling 6 people."},{"Start":"01:42.765 ","End":"01:44.390","Text":"What are we doing here?"},{"Start":"01:44.390 ","End":"01:46.580","Text":"What are we defining as X?"},{"Start":"01:46.580 ","End":"01:57.115","Text":"X is the number of people with a driving license."},{"Start":"01:57.115 ","End":"02:06.665","Text":"X is distributed with a binomial distribution where n equals 6 and p equals 0.6."},{"Start":"02:06.665 ","End":"02:09.560","Text":"Now, the minute we identify this,"},{"Start":"02:09.560 ","End":"02:12.830","Text":"then we can identify the expectation of X."},{"Start":"02:12.830 ","End":"02:17.465","Text":"Well, if X is defined with a binomial distribution,"},{"Start":"02:17.465 ","End":"02:22.440","Text":"then the expectation of X is n times p. In our case,"},{"Start":"02:22.440 ","End":"02:25.200","Text":"that\u0027ll be 6, n is 6,"},{"Start":"02:25.200 ","End":"02:27.660","Text":"times p which is 0.6,"},{"Start":"02:27.660 ","End":"02:30.760","Text":"and that equals to 3.6."},{"Start":"02:30.760 ","End":"02:33.605","Text":"What about the variance of X?"},{"Start":"02:33.605 ","End":"02:43.920","Text":"Well, that\u0027s n times p times 1 minus p. That equals 2."},{"Start":"02:43.920 ","End":"02:49.065","Text":"As we said, n is 6 p is 0.6,"},{"Start":"02:49.065 ","End":"02:52.995","Text":"and 1 minus p is 0.4."},{"Start":"02:52.995 ","End":"02:58.630","Text":"Now, that equals to 1.46."},{"Start":"02:59.210 ","End":"03:02.830","Text":"Now, let\u0027s see about Section B."},{"Start":"03:02.830 ","End":"03:07.325","Text":"Section B asks us about a population of 10 people."},{"Start":"03:07.325 ","End":"03:11.615","Text":"Here we\u0027re dealing with a small population."},{"Start":"03:11.615 ","End":"03:16.450","Text":"We have a sample taken from a small populations,"},{"Start":"03:16.450 ","End":"03:20.720","Text":"so we cannot assume independence."},{"Start":"03:20.720 ","End":"03:25.550","Text":"As such, we\u0027re dealing with the hypergeometric distribution,"},{"Start":"03:25.550 ","End":"03:30.020","Text":"where N equals 10,"},{"Start":"03:30.020 ","End":"03:32.180","Text":"that\u0027s the size of the population,"},{"Start":"03:32.180 ","End":"03:33.845","Text":"that\u0027s given to us."},{"Start":"03:33.845 ","End":"03:36.020","Text":"Now, what\u0027s D? Well,"},{"Start":"03:36.020 ","End":"03:39.860","Text":"D are the people that have a driving license,"},{"Start":"03:39.860 ","End":"03:42.230","Text":"and what are we looking for?"},{"Start":"03:42.230 ","End":"03:44.955","Text":"We\u0027re looking at 6."},{"Start":"03:44.955 ","End":"03:47.400","Text":"Why do we say D is equal to 6?"},{"Start":"03:47.400 ","End":"03:55.475","Text":"Well, that\u0027s the proportion of the people who have a driving license in the population,"},{"Start":"03:55.475 ","End":"03:56.570","Text":"that\u0027s 60 percent,"},{"Start":"03:56.570 ","End":"03:59.180","Text":"that\u0027s given to us."},{"Start":"03:59.180 ","End":"04:04.025","Text":"6 over 10, that\u0027s the proportion of people who have a driving license."},{"Start":"04:04.025 ","End":"04:06.115","Text":"Now, what\u0027s N?"},{"Start":"04:06.115 ","End":"04:15.000","Text":"N is the amount of people that we sampled or the sample size and again, that\u0027s 6."},{"Start":"04:15.000 ","End":"04:16.830","Text":"If that\u0027s the case,"},{"Start":"04:16.830 ","End":"04:23.945","Text":"then X has a hypergeometric distribution where N equals 10,"},{"Start":"04:23.945 ","End":"04:29.190","Text":"D equals 6, and n equals 6."},{"Start":"04:29.290 ","End":"04:31.850","Text":"Now, if that\u0027s the case,"},{"Start":"04:31.850 ","End":"04:39.485","Text":"then the expectation of X equals to n times D over N. Now,"},{"Start":"04:39.485 ","End":"04:40.730","Text":"let\u0027s plug in the numbers."},{"Start":"04:40.730 ","End":"04:44.050","Text":"That\u0027s 6 times 6 over 0,"},{"Start":"04:44.050 ","End":"04:47.730","Text":"and that equals to 3.6."},{"Start":"04:47.730 ","End":"04:49.235","Text":"As we can see,"},{"Start":"04:49.235 ","End":"04:53.920","Text":"the expectation of the hypergeometric distribution"},{"Start":"04:53.920 ","End":"04:59.180","Text":"is the same as that of the binomial distribution."},{"Start":"04:59.180 ","End":"05:05.360","Text":"Let\u0027s take a look now at the variance of the hypergeometric distribution."},{"Start":"05:05.360 ","End":"05:13.640","Text":"Well, the variance is defined as n times D over"},{"Start":"05:13.640 ","End":"05:18.270","Text":"N times 1 minus D over"},{"Start":"05:18.270 ","End":"05:24.810","Text":"N times N minus small n divided by N minus 1."},{"Start":"05:24.810 ","End":"05:26.490","Text":"Now, let\u0027s plug in the numbers."},{"Start":"05:26.490 ","End":"05:33.150","Text":"That\u0027s 6 times 6 over 10 times"},{"Start":"05:33.150 ","End":"05:40.695","Text":"4 over 10 times 10 minus 6,"},{"Start":"05:40.695 ","End":"05:45.090","Text":"that\u0027s 4 over 10 minus 1 that\u0027s 9,"},{"Start":"05:45.090 ","End":"05:48.890","Text":"and that comes out to 0.64."},{"Start":"05:48.890 ","End":"05:50.335","Text":"Now as you can see,"},{"Start":"05:50.335 ","End":"05:56.695","Text":"this variance does not equal this variance right here in the binomial distribution,"},{"Start":"05:56.695 ","End":"06:01.210","Text":"and that\u0027s really what the difference between the binomial distribution and"},{"Start":"06:01.210 ","End":"06:05.885","Text":"the hypergeometric distribution with respect to the variance."},{"Start":"06:05.885 ","End":"06:09.390","Text":"It\u0027s this expression right here."}],"ID":13037},{"Watched":false,"Name":"Exercise 4","Duration":"5m ","ChapterTopicVideoID":12559,"CourseChapterTopicPlaylistID":64488,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this question would be dealing with choosing a delegation."},{"Start":"00:03.840 ","End":"00:07.230","Text":"Now an organization has 7 academic engineers,"},{"Start":"00:07.230 ","End":"00:10.170","Text":"3 technicians, and 5 practical engineers."},{"Start":"00:10.170 ","End":"00:15.630","Text":"4 employees are randomly selected for a delegation to attend the conference in Madrid."},{"Start":"00:15.630 ","End":"00:17.460","Text":"In section a, we\u0027re asked what\u0027s"},{"Start":"00:17.460 ","End":"00:21.780","Text":"the probability that only academic engineers will be selected?"},{"Start":"00:21.780 ","End":"00:25.230","Text":"Well, this seems to be a hypergeometric distribution."},{"Start":"00:25.230 ","End":"00:27.060","Text":"Let\u0027s take a look at it."},{"Start":"00:27.060 ","End":"00:29.820","Text":"What\u0027s the total population that we have?"},{"Start":"00:29.820 ","End":"00:32.865","Text":"What\u0027s the total population size that we\u0027re dealing with?"},{"Start":"00:32.865 ","End":"00:35.430","Text":"Well, we have 7,"},{"Start":"00:35.430 ","End":"00:39.735","Text":"that\u0027s big N. We have 7 academic engineers,"},{"Start":"00:39.735 ","End":"00:42.495","Text":"and 3 technicians,"},{"Start":"00:42.495 ","End":"00:45.690","Text":"and 5 practical engineers."},{"Start":"00:45.690 ","End":"00:49.940","Text":"That means we\u0027re dealing with 15 employees in total,"},{"Start":"00:49.940 ","End":"00:55.530","Text":"so our total population size is 15 employees. Now what\u0027s D?"},{"Start":"00:55.530 ","End":"00:58.895","Text":"D are the employees that have a special characteristic."},{"Start":"00:58.895 ","End":"01:03.820","Text":"But we\u0027re dealing here with only academic engineers in section a,"},{"Start":"01:03.820 ","End":"01:06.320","Text":"and how many academic engineers do we have?"},{"Start":"01:06.320 ","End":"01:08.110","Text":"Well, we have 7,"},{"Start":"01:08.110 ","End":"01:16.605","Text":"and we\u0027re dealing with academic engineers."},{"Start":"01:16.605 ","End":"01:19.880","Text":"What about our sample size?"},{"Start":"01:19.880 ","End":"01:23.630","Text":"Well, we\u0027re asking for 4 employees to be in the delegation."},{"Start":"01:23.630 ","End":"01:26.165","Text":"So n equals 4."},{"Start":"01:26.165 ","End":"01:33.560","Text":"If we define x as the number of academic engineers,"},{"Start":"01:33.560 ","End":"01:41.080","Text":"then x has a hypergeometric distribution where big N equals 15,"},{"Start":"01:41.080 ","End":"01:44.065","Text":"D equals 7,"},{"Start":"01:44.065 ","End":"01:47.525","Text":"and small n equals 4."},{"Start":"01:47.525 ","End":"01:52.670","Text":"Now let\u0027s remind ourselves of the equation for the hypergeometric distribution."},{"Start":"01:52.670 ","End":"01:56.915","Text":"That\u0027s the probability of x being equal to k. Well,"},{"Start":"01:56.915 ","End":"02:00.240","Text":"that equals to D/k,"},{"Start":"02:00.240 ","End":"02:09.950","Text":"N minus D over small n minus k divided by big N over small n. Now,"},{"Start":"02:09.950 ","End":"02:14.375","Text":"in our case, the probability of x being equal to what?"},{"Start":"02:14.375 ","End":"02:20.405","Text":"Well, we\u0027re asked, what\u0027s the probability that only academic engineers will be selected?"},{"Start":"02:20.405 ","End":"02:28.320","Text":"That means that all the 4 people in the delegation must be academic engineer,"},{"Start":"02:28.320 ","End":"02:31.710","Text":"so we\u0027re looking at x being equal to 4,"},{"Start":"02:31.710 ","End":"02:33.960","Text":"and that equals to,"},{"Start":"02:33.960 ","End":"02:35.220","Text":"well, let\u0027s plug in the numbers."},{"Start":"02:35.220 ","End":"02:38.970","Text":"D equals 7, k equals 4."},{"Start":"02:38.970 ","End":"02:41.400","Text":"Right now, big N and minus D,"},{"Start":"02:41.400 ","End":"02:43.440","Text":"well that\u0027s 15 minus 7,"},{"Start":"02:43.440 ","End":"02:46.280","Text":"that\u0027s 8, and n minus k,"},{"Start":"02:46.280 ","End":"02:47.630","Text":"well that\u0027s 4 minus 4,"},{"Start":"02:47.630 ","End":"02:55.840","Text":"that\u0027s 0 divided by big N, that\u0027s 15/4."},{"Start":"02:56.140 ","End":"03:04.559","Text":"That equals to 0.0256."},{"Start":"03:04.559 ","End":"03:07.310","Text":"In section b, we\u0027re asked what is the expectation of"},{"Start":"03:07.310 ","End":"03:10.580","Text":"the number of technicians that will be selected?"},{"Start":"03:10.580 ","End":"03:14.420","Text":"In section a, remember that we\u0027re dealing with academic engineers,"},{"Start":"03:14.420 ","End":"03:17.540","Text":"but here we\u0027re dealing with technicians."},{"Start":"03:17.540 ","End":"03:20.700","Text":"Again, N is still 15."},{"Start":"03:20.700 ","End":"03:21.910","Text":"That\u0027s our population,"},{"Start":"03:21.910 ","End":"03:24.020","Text":"that\u0027s the size of our population."},{"Start":"03:24.020 ","End":"03:29.255","Text":"D, now, what would that equal to?"},{"Start":"03:29.255 ","End":"03:35.075","Text":"That would equal to 3 because now we\u0027re dealing with technicians,"},{"Start":"03:35.075 ","End":"03:40.250","Text":"not academic engineers, but technicians and how many technicians do we have?"},{"Start":"03:40.250 ","End":"03:43.640","Text":"We have 3. Small n is still"},{"Start":"03:43.640 ","End":"03:49.325","Text":"4 because we\u0027re still looking at 4 employees for the delegation."},{"Start":"03:49.325 ","End":"03:57.990","Text":"Let\u0027s now define y as the number of technicians in the sample."},{"Start":"03:58.610 ","End":"04:02.510","Text":"That\u0027s the case then y is distributed with"},{"Start":"04:02.510 ","End":"04:06.770","Text":"the hypergeometric distribution where big N equals 15,"},{"Start":"04:06.770 ","End":"04:09.470","Text":"D now equals 3 because again,"},{"Start":"04:09.470 ","End":"04:10.925","Text":"we\u0027re dealing with technicians,"},{"Start":"04:10.925 ","End":"04:14.780","Text":"not with academic engineers like we did in section a,"},{"Start":"04:14.780 ","End":"04:22.040","Text":"and small n equals 4 because we\u0027re still choosing 4 people for the delegation."},{"Start":"04:22.040 ","End":"04:29.000","Text":"What\u0027s the expectation of a random variable having a hypergeometric distribution?"},{"Start":"04:29.000 ","End":"04:36.380","Text":"Well, that\u0027s n times D divided by big N. Let\u0027s plug in the numbers."},{"Start":"04:36.380 ","End":"04:42.755","Text":"Small n is 4 times D. Well that\u0027s 3 divided by"},{"Start":"04:42.755 ","End":"04:50.790","Text":"big N. Well that big N is15,and that comes out to 0.8."},{"Start":"04:50.790 ","End":"05:00.600","Text":"We can expect 0.8 technicians to be included in our delegation of 4 employees."}],"ID":13038}],"Thumbnail":null,"ID":64488},{"Name":"Special Discrete Probability Distributions - Negative Binomial Distribution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"1m 59s","ChapterTopicVideoID":12560,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this chapter we\u0027ll be talking about special discrete probability,"},{"Start":"00:03.390 ","End":"00:06.840","Text":"specifically the negative binomial distribution."},{"Start":"00:06.840 ","End":"00:10.185","Text":"Now first of all, why special discrete probabilities?"},{"Start":"00:10.185 ","End":"00:15.120","Text":"Well, and that\u0027s because we\u0027re given the probability function and we\u0027re also"},{"Start":"00:15.120 ","End":"00:21.330","Text":"given the equations to calculate the expectation and variance of these distributions."},{"Start":"00:21.330 ","End":"00:24.930","Text":"Let\u0027s get into the negative binomial distribution."},{"Start":"00:24.930 ","End":"00:27.915","Text":"Now, in this probability distribution,"},{"Start":"00:27.915 ","End":"00:34.350","Text":"the same Bernoulli trial is repeated independently until there are successes."},{"Start":"00:34.350 ","End":"00:38.220","Text":"Now, these successes don\u0027t have to be consecutive successes,"},{"Start":"00:38.220 ","End":"00:40.725","Text":"but we do need to have r of them,"},{"Start":"00:40.725 ","End":"00:43.880","Text":"r successes. Then we stop."},{"Start":"00:43.880 ","End":"00:49.340","Text":"Now, X is the number of repetitions until our successes are obtained."},{"Start":"00:49.340 ","End":"00:53.200","Text":"Again, after we have our successes, we stop."},{"Start":"00:53.200 ","End":"00:56.120","Text":"When we have these conditions,"},{"Start":"00:56.120 ","End":"01:01.370","Text":"then we say that X is distributed with a negative binomial distribution"},{"Start":"01:01.370 ","End":"01:06.880","Text":"with 2 parameters r and the number of successes that we want,"},{"Start":"01:06.880 ","End":"01:10.925","Text":"and p the probability of a success."},{"Start":"01:10.925 ","End":"01:13.760","Text":"Now, what\u0027s the probability function here?"},{"Start":"01:13.760 ","End":"01:17.315","Text":"Well, that\u0027s the probability of X being equal to k."},{"Start":"01:17.315 ","End":"01:21.440","Text":"That would equal to k minus 1 over r minus 1,"},{"Start":"01:21.440 ","End":"01:27.740","Text":"times p to the power of r times 1 minus p to the power of k minus r,"},{"Start":"01:27.740 ","End":"01:30.575","Text":"where k runs from r,"},{"Start":"01:30.575 ","End":"01:33.680","Text":"r plus 1 until infinity."},{"Start":"01:33.680 ","End":"01:39.979","Text":"The expectation of X is given to us as r divided"},{"Start":"01:39.979 ","End":"01:45.770","Text":"by p. The number of successes divided by the probability."},{"Start":"01:45.770 ","End":"01:55.230","Text":"The variance of X is given to us as r times 1 minus p divided by p squared."},{"Start":"01:55.360 ","End":"01:59.880","Text":"Enough of the theory. Let\u0027s get to the example."}],"ID":13039},{"Watched":false,"Name":"Example","Duration":"6m 38s","ChapterTopicVideoID":12561,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.640","Text":"In our example, a dice thrown until the result larger than 4 is obtained 3 times."},{"Start":"00:05.640 ","End":"00:07.110","Text":"We\u0027re asked in section A,"},{"Start":"00:07.110 ","End":"00:10.935","Text":"what are the chances that the die will be thrown 6 times?"},{"Start":"00:10.935 ","End":"00:15.465","Text":"Well, here we have other conditions for negative binomial distribution."},{"Start":"00:15.465 ","End":"00:22.380","Text":"Why is that? Well, we have a Bernoulli trial where we have a success or a failure."},{"Start":"00:22.380 ","End":"00:26.670","Text":"Now, the success is"},{"Start":"00:26.670 ","End":"00:33.225","Text":"defined as a result of being greater than 4."},{"Start":"00:33.225 ","End":"00:38.580","Text":"That means that in the roll of the die we get either a 5 or a 6."},{"Start":"00:38.580 ","End":"00:42.235","Text":"Now, what\u0027s the probability of getting the success,"},{"Start":"00:42.235 ","End":"00:44.175","Text":"getting either 5 or 6?"},{"Start":"00:44.175 ","End":"00:45.855","Text":"Well that\u0027s 2 over 6."},{"Start":"00:45.855 ","End":"00:53.475","Text":"We get the probability of a 5 is 1/6 and the probability of a 6 is another 1/6."},{"Start":"00:53.475 ","End":"00:55.080","Text":"We have 2/6,"},{"Start":"00:55.080 ","End":"00:57.490","Text":"and then just boils down to 1/3,"},{"Start":"00:57.490 ","End":"01:00.590","Text":"that\u0027s the probability of our success."},{"Start":"01:00.590 ","End":"01:03.185","Text":"Now, what else do we have?"},{"Start":"01:03.185 ","End":"01:10.205","Text":"We have to roll the die until we get 3 successes."},{"Start":"01:10.205 ","End":"01:12.470","Text":"That means that again,"},{"Start":"01:12.470 ","End":"01:16.595","Text":"result larger than 4 is obtained 3 times."},{"Start":"01:16.595 ","End":"01:19.910","Text":"The number of successes,"},{"Start":"01:19.910 ","End":"01:24.890","Text":"the number of results greater than 4 we want to have 3 of them."},{"Start":"01:24.890 ","End":"01:32.360","Text":"Now, let\u0027s define x as the number of times we roll"},{"Start":"01:32.360 ","End":"01:41.250","Text":"the die until we get 3 successes."},{"Start":"01:43.400 ","End":"01:47.795","Text":"That means that x now has"},{"Start":"01:47.795 ","End":"01:55.860","Text":"a negative binomial distribution where r equals 3 and p equals 1/3."},{"Start":"01:56.960 ","End":"02:00.650","Text":"Now if x is a negative binomial distribution,"},{"Start":"02:00.650 ","End":"02:04.340","Text":"let\u0027s just remind ourselves of the equation here."},{"Start":"02:04.340 ","End":"02:08.270","Text":"That\u0027s the probability of x being equal to k. Now,"},{"Start":"02:08.270 ","End":"02:17.720","Text":"that equals to k minus 1 over r minus 1 times p^r"},{"Start":"02:17.720 ","End":"02:27.685","Text":"times 1 minus p^k minus r. Let\u0027s now go to section A and answer that 1."},{"Start":"02:27.685 ","End":"02:33.490","Text":"A asks us, what\u0027s the probability of X being equal to 6?"},{"Start":"02:33.490 ","End":"02:37.220","Text":"What are the chances that the die will be thrown 6 times?"},{"Start":"02:37.220 ","End":"02:39.260","Text":"Well, let\u0027s plug in the numbers here."},{"Start":"02:39.260 ","End":"02:42.440","Text":"K is 6, so that\u0027s 6 minus 1."},{"Start":"02:42.440 ","End":"02:45.880","Text":"R is 3, so that\u0027s 3 minus 1."},{"Start":"02:45.880 ","End":"02:54.645","Text":"That\u0027s 3 minus 1 times p is 1/3,"},{"Start":"02:54.645 ","End":"02:58.530","Text":"so that\u0027s 1/3^3,"},{"Start":"02:58.530 ","End":"03:03.510","Text":"times 1 minus p,"},{"Start":"03:03.510 ","End":"03:06.330","Text":"that\u0027s 1 minus 1/3^k,"},{"Start":"03:06.330 ","End":"03:10.080","Text":"so that\u0027s 6 minus 3."},{"Start":"03:10.080 ","End":"03:13.575","Text":"Now that equals to 5 over 2,"},{"Start":"03:13.575 ","End":"03:20.090","Text":"1/3^3, 2/3^3 as well."},{"Start":"03:20.090 ","End":"03:24.335","Text":"Now, that\u0027s the answer to section A."},{"Start":"03:24.335 ","End":"03:29.140","Text":"Now, let\u0027s take a look at the rationale behind this answer right here."},{"Start":"03:29.140 ","End":"03:33.950","Text":"Basically, that\u0027s the rationale behind the negative binomial distribution."},{"Start":"03:33.950 ","End":"03:36.125","Text":"Now, in our case,"},{"Start":"03:36.125 ","End":"03:38.645","Text":"we had 6 roll of the dice."},{"Start":"03:38.645 ","End":"03:39.890","Text":"That\u0027s the first row,"},{"Start":"03:39.890 ","End":"03:42.460","Text":"that\u0027s the second, that\u0027s third,"},{"Start":"03:42.460 ","End":"03:47.095","Text":"fourth, fifth, and sixth roll of the die."},{"Start":"03:47.095 ","End":"03:49.350","Text":"Now, on the sixth row,"},{"Start":"03:49.350 ","End":"03:51.270","Text":"we had a success."},{"Start":"03:51.270 ","End":"03:56.270","Text":"That was part of the 3 successes that we were looking for,"},{"Start":"03:56.270 ","End":"03:57.890","Text":"and after the third success,"},{"Start":"03:57.890 ","End":"04:00.600","Text":"we just stopped rolling the die."},{"Start":"04:00.680 ","End":"04:05.265","Text":"That means that in the previous 5 throws,"},{"Start":"04:05.265 ","End":"04:07.830","Text":"in these throws right here,"},{"Start":"04:07.830 ","End":"04:10.905","Text":"we had 2 other successes."},{"Start":"04:10.905 ","End":"04:16.550","Text":"Assume for now that the successes were on the first row and on the second row."},{"Start":"04:16.550 ","End":"04:21.265","Text":"That means that under 3 other throws we had failures."},{"Start":"04:21.265 ","End":"04:25.790","Text":"Now, what\u0027s the probability of getting a success? That\u0027s 1/3."},{"Start":"04:25.790 ","End":"04:27.890","Text":"On the sixth row,"},{"Start":"04:27.890 ","End":"04:31.970","Text":"the probability of getting a success was 1/3."},{"Start":"04:31.970 ","End":"04:34.264","Text":"What about the other 2 successes?"},{"Start":"04:34.264 ","End":"04:38.520","Text":"Well, that\u0027s 1/3 squared."},{"Start":"04:38.520 ","End":"04:43.010","Text":"Now, we also had 3 failures, so that\u0027s 2/3."},{"Start":"04:43.010 ","End":"04:45.610","Text":"The probability of a failure is 2/3,"},{"Start":"04:45.610 ","End":"04:47.080","Text":"and we had 3 of them."},{"Start":"04:47.080 ","End":"04:50.990","Text":"That\u0027s to the power of 3."},{"Start":"04:50.990 ","End":"04:57.760","Text":"Now, we didn\u0027t say that the successes had to be under first and second role."},{"Start":"04:57.760 ","End":"05:01.675","Text":"They could have been anywhere within the 5 throws."},{"Start":"05:01.675 ","End":"05:03.630","Text":"Out of the 5 throws,"},{"Start":"05:03.630 ","End":"05:05.895","Text":"we needed 2 successes."},{"Start":"05:05.895 ","End":"05:08.040","Text":"They could be anywhere there."},{"Start":"05:08.040 ","End":"05:10.330","Text":"That\u0027s this expression right here."},{"Start":"05:10.330 ","End":"05:11.980","Text":"Now, let\u0027s simplify this."},{"Start":"05:11.980 ","End":"05:14.930","Text":"That\u0027s 5 over 2."},{"Start":"05:15.570 ","End":"05:21.065","Text":"1/3^3 times 2/3 to the power of 3."},{"Start":"05:21.065 ","End":"05:24.170","Text":"Now, we can see that this expression right here"},{"Start":"05:24.170 ","End":"05:28.210","Text":"is exactly the same as this expression right here."},{"Start":"05:28.210 ","End":"05:36.270","Text":"Here I\u0027ve explained the rationale behind the negative binomial distribution."},{"Start":"05:36.320 ","End":"05:39.860","Text":"In section b, we\u0027re asked what is the expectation and"},{"Start":"05:39.860 ","End":"05:43.430","Text":"variance of the number of times the die was thrown?"},{"Start":"05:43.430 ","End":"05:47.810","Text":"Well, again, the expectation of x,"},{"Start":"05:47.810 ","End":"05:51.395","Text":"where x has a negative binomial distribution."},{"Start":"05:51.395 ","End":"05:55.980","Text":"Well, that equals to r divided by p. Now,"},{"Start":"05:55.980 ","End":"06:01.440","Text":"in our case, r is 3 divided by p. Now, p is 1/3."},{"Start":"06:01.510 ","End":"06:05.320","Text":"That equals to 9."},{"Start":"06:05.320 ","End":"06:09.255","Text":"What about the variance of x?"},{"Start":"06:09.255 ","End":"06:17.345","Text":"The variance of x is defined as r times 1 minus p divided by p squared."},{"Start":"06:17.345 ","End":"06:26.600","Text":"Now, that equals to 3 times 2/3 divided by 1/3 squared,"},{"Start":"06:26.600 ","End":"06:29.715","Text":"and that equals to 18."},{"Start":"06:29.715 ","End":"06:38.110","Text":"This is the expectation right here and this is the variance of x."}],"ID":13040},{"Watched":false,"Name":"Exercise 1 Parts a-b","Duration":"8m 4s","ChapterTopicVideoID":12562,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this question, we\u0027ll be talking about moving balls from a container."},{"Start":"00:03.930 ","End":"00:07.980","Text":"A container has 4 black balls and 6 white balls."},{"Start":"00:07.980 ","End":"00:11.460","Text":"A person removes 1 ball at a time with returns."},{"Start":"00:11.460 ","End":"00:15.750","Text":"X is the number of times that a ball is removed until we get 2 white balls,"},{"Start":"00:15.750 ","End":"00:18.210","Text":"although not necessarily consecutively."},{"Start":"00:18.210 ","End":"00:23.610","Text":"We\u0027re asked to calculate the probability of X equaling 2, equaling 3, 4,"},{"Start":"00:23.610 ","End":"00:30.380","Text":"and k. Here we have a negative binomial distribution."},{"Start":"00:30.380 ","End":"00:34.610","Text":"Why is that? Well, let\u0027s take a look at the conditions of"},{"Start":"00:34.610 ","End":"00:39.395","Text":"a negative binomial distribution and see whether we meet them or not."},{"Start":"00:39.395 ","End":"00:41.675","Text":"Here are the conditions,"},{"Start":"00:41.675 ","End":"00:43.370","Text":"where the first 1 states that,"},{"Start":"00:43.370 ","End":"00:49.770","Text":"the same Bernoulli trial is repeated independently until there are successes, okay."},{"Start":"00:49.770 ","End":"00:51.830","Text":"What about a Bernoulli trial?"},{"Start":"00:51.830 ","End":"00:57.160","Text":"But we have a Bernoulli trial every time we sample a ball from the container."},{"Start":"00:57.160 ","End":"00:59.150","Text":"In a Bernoulli trial,"},{"Start":"00:59.150 ","End":"01:01.745","Text":"we\u0027re looking at successes and failures."},{"Start":"01:01.745 ","End":"01:03.695","Text":"What\u0027s our success?"},{"Start":"01:03.695 ","End":"01:09.540","Text":"Our success is defined as removing a white ball."},{"Start":"01:10.490 ","End":"01:15.090","Text":"What\u0027s the probability of our success."},{"Start":"01:15.090 ","End":"01:17.890","Text":"The probability of removing a white ball?"},{"Start":"01:17.890 ","End":"01:20.380","Text":"Well, how many balls in total do we have in the container?"},{"Start":"01:20.380 ","End":"01:21.990","Text":"We have 10 balls, right,"},{"Start":"01:21.990 ","End":"01:24.735","Text":"4 black balls and 6 white balls."},{"Start":"01:24.735 ","End":"01:29.095","Text":"How many white balls do we have in the container? We got 6."},{"Start":"01:29.095 ","End":"01:36.620","Text":"The probability of removing a white ball that\u0027s 6 over 10 or 0.6."},{"Start":"01:36.620 ","End":"01:41.260","Text":"Now, what about the Bernoulli trial being independent?"},{"Start":"01:41.260 ","End":"01:44.485","Text":"They are independent of each other. Why is that?"},{"Start":"01:44.485 ","End":"01:46.360","Text":"Because every time we sample a ball,"},{"Start":"01:46.360 ","End":"01:49.015","Text":"we return it back into the container,"},{"Start":"01:49.015 ","End":"01:54.410","Text":"which makes the probability of each success the same."},{"Start":"01:54.410 ","End":"01:59.450","Text":"That means that the Bernoulli trials are independent of each other."},{"Start":"01:59.450 ","End":"02:02.900","Text":"Now, how many successes are we interested in?"},{"Start":"02:02.900 ","End":"02:08.880","Text":"Well, we want to get 2 white balls so r in our case equals 2."},{"Start":"02:08.880 ","End":"02:11.610","Text":"We keep on sampling the container,"},{"Start":"02:11.610 ","End":"02:17.890","Text":"sampling the balls until we get 2 white balls, and then we stop."},{"Start":"02:18.230 ","End":"02:28.385","Text":"What we have here is the conditions associated with the negative binomial distribution."},{"Start":"02:28.385 ","End":"02:34.920","Text":"X is the number of times that we have to"},{"Start":"02:34.920 ","End":"02:43.815","Text":"remove balls from the container until we get 2 white balls."},{"Start":"02:43.815 ","End":"02:49.285","Text":"That means that x is distributed with a negative binomial distribution,"},{"Start":"02:49.285 ","End":"02:55.000","Text":"where r equals 2 and p equals 0.6."},{"Start":"02:55.000 ","End":"02:56.650","Text":"Now, having said that,"},{"Start":"02:56.650 ","End":"03:01.285","Text":"what\u0027s the general equation for negative binomial distribution?"},{"Start":"03:01.285 ","End":"03:07.660","Text":"Well, that\u0027s the probability of X being equal to k. That equals"},{"Start":"03:07.660 ","End":"03:15.505","Text":"to k minus 1 over r minus 1 times p to the power of r,"},{"Start":"03:15.505 ","End":"03:20.395","Text":"and times 1 minus p to the power of"},{"Start":"03:20.395 ","End":"03:26.715","Text":"k minus r. Where k has the values of r,"},{"Start":"03:26.715 ","End":"03:28.230","Text":"r plus 1,"},{"Start":"03:28.230 ","End":"03:32.145","Text":"and so on and so forth until infinity."},{"Start":"03:32.145 ","End":"03:36.065","Text":"Let\u0027s answer Section A right here."},{"Start":"03:36.065 ","End":"03:41.755","Text":"Section A asks, what\u0027s the probability of X being equal to 2?"},{"Start":"03:41.755 ","End":"03:50.680","Text":"Lets just plug in the numbers into this equation right here where k equals 2."},{"Start":"03:50.680 ","End":"03:57.085","Text":"We have 2 minus 1 over r minus 1, now r is 2."},{"Start":"03:57.085 ","End":"04:01.995","Text":"That\u0027s also 2 minus 1 times the probability."},{"Start":"04:01.995 ","End":"04:03.210","Text":"What\u0027s the probability?"},{"Start":"04:03.210 ","End":"04:07.845","Text":"That\u0027s 0.6 to the power of r. Now r equals 2"},{"Start":"04:07.845 ","End":"04:15.750","Text":"times 1 minus 0.6 to the power of 2 minus 2,"},{"Start":"04:15.750 ","End":"04:19.830","Text":"That\u0027s k minus r. k is 2 and r is 2,"},{"Start":"04:19.830 ","End":"04:21.865","Text":"so that\u0027s 2 minus 2."},{"Start":"04:21.865 ","End":"04:24.988","Text":"Now, what is that equal to?"},{"Start":"04:24.988 ","End":"04:35.809","Text":"That\u0027s 1 over 1 times 0.6 squared times 0.4 to the power of 0."},{"Start":"04:35.809 ","End":"04:39.180","Text":"Now, 1 over 1,"},{"Start":"04:39.180 ","End":"04:43.669","Text":"that equals 1 and 0.4 to the power of 0,"},{"Start":"04:43.669 ","End":"04:45.035","Text":"that equals 1 also."},{"Start":"04:45.035 ","End":"04:48.425","Text":"That equals to 0.6 squared,"},{"Start":"04:48.425 ","End":"04:52.260","Text":"which equals to 0.36."},{"Start":"04:53.020 ","End":"04:59.885","Text":"Section B, we\u0027re asked to calculate the probability of X being equal to 3."},{"Start":"04:59.885 ","End":"05:07.170","Text":"Let\u0027s get to b equals the probability of X equaling 3. What does that mean?"},{"Start":"05:07.170 ","End":"05:16.970","Text":"That the probability that we\u0027ll get 2 white balls when we sample only 3 balls."},{"Start":"05:16.970 ","End":"05:21.020","Text":"Now, that equals, let\u0027s plug in the numbers k equals 3."},{"Start":"05:21.020 ","End":"05:25.100","Text":"That\u0027s 3 minus 1 over r minus 1,"},{"Start":"05:25.100 ","End":"05:28.940","Text":"that\u0027s 2 minus 1 times 0.6,"},{"Start":"05:28.940 ","End":"05:32.060","Text":"p, to the power of r. r is 2."},{"Start":"05:32.060 ","End":"05:39.095","Text":"Times 1 minus 0.6 to the power of k minus r. Now k is 3,"},{"Start":"05:39.095 ","End":"05:42.065","Text":"r is 2, that\u0027s 3 minus 2."},{"Start":"05:42.065 ","End":"05:45.200","Text":"Now, that equals to 2 over 1,"},{"Start":"05:45.200 ","End":"05:53.325","Text":"0.6 squared times 0.4 to the power of 1."},{"Start":"05:53.325 ","End":"05:58.810","Text":"Now that equals to 0.288."},{"Start":"05:59.200 ","End":"06:03.290","Text":"Now let\u0027s take a look at the rationale behind this number."},{"Start":"06:03.290 ","End":"06:09.230","Text":"We defined x as a number of times we have to remove balls until we get 2 white ones."},{"Start":"06:09.230 ","End":"06:14.210","Text":"Well, here we\u0027re given the probability of X being equal to 3."},{"Start":"06:14.210 ","End":"06:19.610","Text":"Let\u0027s say we\u0027re sampling 3 balls right here."},{"Start":"06:19.610 ","End":"06:25.535","Text":"Now, we know that the last ball is a success, and then we stop."},{"Start":"06:25.535 ","End":"06:28.445","Text":"Now, how many successes are we looking at?"},{"Start":"06:28.445 ","End":"06:30.290","Text":"Well, we\u0027re looking at 2 successes."},{"Start":"06:30.290 ","End":"06:32.610","Text":"We want 2 successes."},{"Start":"06:33.160 ","End":"06:38.280","Text":"Here in the last sample."},{"Start":"06:38.280 ","End":"06:40.770","Text":"Then we have a success and then we stop."},{"Start":"06:40.770 ","End":"06:46.035","Text":"That means that here we have to have at least 1 success"},{"Start":"06:46.035 ","End":"06:52.505","Text":"and 1 failure in order for us to have 2 successes in 3 samples."},{"Start":"06:52.505 ","End":"06:56.630","Text":"Now, what does this mean with respect to probabilities?"},{"Start":"06:56.630 ","End":"07:00.619","Text":"Well, the probability of the last success,"},{"Start":"07:00.619 ","End":"07:03.365","Text":"that equals to 0.6."},{"Start":"07:03.365 ","End":"07:08.265","Text":"Now, in the previous 2 samples,"},{"Start":"07:08.265 ","End":"07:11.235","Text":"we need 1 success and 1 failure."},{"Start":"07:11.235 ","End":"07:17.150","Text":"That equals to 1, 0.6 times 0.4."},{"Start":"07:17.150 ","End":"07:24.020","Text":"Now, who said anything about success being had on the first sample?"},{"Start":"07:24.020 ","End":"07:25.565","Text":"It could be in the second sample."},{"Start":"07:25.565 ","End":"07:28.250","Text":"Out of the 2 places here,"},{"Start":"07:28.250 ","End":"07:31.380","Text":"we\u0027re looking for 1 success."},{"Start":"07:31.380 ","End":"07:33.185","Text":"If we simplify that,"},{"Start":"07:33.185 ","End":"07:35.780","Text":"that would be 2 over 1,"},{"Start":"07:35.780 ","End":"07:40.575","Text":"0.6 squared times 0.4."},{"Start":"07:40.575 ","End":"07:44.565","Text":"This is exactly equal to this."},{"Start":"07:44.565 ","End":"07:46.970","Text":"This is the rationale,"},{"Start":"07:46.970 ","End":"07:51.905","Text":"the logic behind this probability right here."},{"Start":"07:51.905 ","End":"07:55.265","Text":"This is how we figure it out from a logical perspective,"},{"Start":"07:55.265 ","End":"07:57.395","Text":"from a combinatorial perspective,"},{"Start":"07:57.395 ","End":"08:04.080","Text":"instead of just plugging in the number into the negative binomial distribution."}],"ID":13041},{"Watched":false,"Name":"Exercise 1 Parts c-d","Duration":"5m 30s","ChapterTopicVideoID":12563,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"In Section c, we\u0027re asked to calculate the probability where X equals 4."},{"Start":"00:04.995 ","End":"00:09.195","Text":"That\u0027s the probability where X equals 4."},{"Start":"00:09.195 ","End":"00:11.640","Text":"Now again, let\u0027s plug in the numbers."},{"Start":"00:11.640 ","End":"00:14.970","Text":"Case 4, so that\u0027s 4 minus 1,"},{"Start":"00:14.970 ","End":"00:17.850","Text":"r is still 2 so that\u0027s 2 minus 1,"},{"Start":"00:17.850 ","End":"00:20.595","Text":"so that\u0027s 4 minus 1 over 2 minus 1,"},{"Start":"00:20.595 ","End":"00:24.870","Text":"times 0.6 to the power of"},{"Start":"00:24.870 ","End":"00:32.745","Text":"2 times 1 minus 0.6^k minus r, that\u0027s 4 minus 2."},{"Start":"00:32.745 ","End":"00:34.110","Text":"Let\u0027s simplify that."},{"Start":"00:34.110 ","End":"00:37.515","Text":"That\u0027s 3/1,"},{"Start":"00:37.515 ","End":"00:48.870","Text":"0.6^2, 0.6 squared times 0.4 to the power 2, that\u0027s also squared."},{"Start":"00:48.870 ","End":"00:54.740","Text":"That\u0027s the answer for Section c. In Section d,"},{"Start":"00:54.740 ","End":"00:57.800","Text":"we\u0027re asked to calculate the probability where X equals"},{"Start":"00:57.800 ","End":"01:03.785","Text":"k. That\u0027s the probability of X being equal to k. Now,"},{"Start":"01:03.785 ","End":"01:06.815","Text":"let\u0027s plug in the numbers right here."},{"Start":"01:06.815 ","End":"01:12.210","Text":"Well, that equals to k minus 1 over r minus 1."},{"Start":"01:12.210 ","End":"01:15.285","Text":"Now, r is 2, so that\u0027s 2 minus 1."},{"Start":"01:15.285 ","End":"01:19.500","Text":"Now, 0.6^r and r is 2,"},{"Start":"01:19.500 ","End":"01:21.225","Text":"so that\u0027s 2 right here,"},{"Start":"01:21.225 ","End":"01:29.575","Text":"times 1 minus 0.6 to the power of k minus r, minus 2."},{"Start":"01:29.575 ","End":"01:31.955","Text":"Let\u0027s just simplify that."},{"Start":"01:31.955 ","End":"01:36.800","Text":"That would be k minus 1 over 1,"},{"Start":"01:36.800 ","End":"01:47.080","Text":"times 0.6 squared times 0.4 to the power of k minus 2."},{"Start":"01:47.080 ","End":"01:51.075","Text":"This is the answer for Section d. Now again,"},{"Start":"01:51.075 ","End":"01:57.180","Text":"let\u0027s look at the rationale behind this expression right here."},{"Start":"01:57.230 ","End":"02:01.565","Text":"When we\u0027re talking about X being equal to k,"},{"Start":"02:01.565 ","End":"02:06.110","Text":"that means that we\u0027ve sampled k balls from"},{"Start":"02:06.110 ","End":"02:10.670","Text":"the container where the last sample is a success,"},{"Start":"02:10.670 ","End":"02:11.980","Text":"being a white ball,"},{"Start":"02:11.980 ","End":"02:17.705","Text":"and then we stop because we\u0027ve already had 1 previous success."},{"Start":"02:17.705 ","End":"02:19.490","Text":"We\u0027re looking for 2 successes,"},{"Start":"02:19.490 ","End":"02:21.455","Text":"we\u0027re looking for 2 white balls."},{"Start":"02:21.455 ","End":"02:24.290","Text":"Let\u0027s take a look at this from a graphical perspective."},{"Start":"02:24.290 ","End":"02:26.615","Text":"Here we have the first ball sample,"},{"Start":"02:26.615 ","End":"02:28.399","Text":"the second ball sample,"},{"Start":"02:28.399 ","End":"02:30.050","Text":"the third ball sample,"},{"Start":"02:30.050 ","End":"02:33.425","Text":"and so on and so forth until the kth ball."},{"Start":"02:33.425 ","End":"02:36.950","Text":"We know that the last ball sampled is a success,"},{"Start":"02:36.950 ","End":"02:38.495","Text":"that\u0027s a white ball."},{"Start":"02:38.495 ","End":"02:44.960","Text":"That means that in all the previous samples,"},{"Start":"02:44.960 ","End":"02:48.530","Text":"we have k minus 1 previous samples."},{"Start":"02:48.530 ","End":"02:52.265","Text":"We\u0027re looking at 1 success and all the rest failures"},{"Start":"02:52.265 ","End":"02:56.790","Text":"because we\u0027ve stopped on the kth sample,"},{"Start":"02:56.790 ","End":"02:58.755","Text":"the kth removal of a ball."},{"Start":"02:58.755 ","End":"03:02.295","Text":"Here we have k minus 1 sample,"},{"Start":"03:02.295 ","End":"03:04.470","Text":"here we have the kth sample."},{"Start":"03:04.470 ","End":"03:06.780","Text":"That\u0027s a success."},{"Start":"03:06.780 ","End":"03:09.590","Text":"How many successes do we have in this area?"},{"Start":"03:09.590 ","End":"03:11.980","Text":"We have 1 success."},{"Start":"03:11.980 ","End":"03:16.235","Text":"Let\u0027s assume it\u0027s on the first sample for now, let\u0027s just assume it."},{"Start":"03:16.235 ","End":"03:20.240","Text":"All the rest of the samples are failures."},{"Start":"03:20.240 ","End":"03:25.070","Text":"Now, what\u0027s the probability of getting a white ball?"},{"Start":"03:25.070 ","End":"03:30.380","Text":"Well, that\u0027s 0.6 with the last sample here getting a white ball,"},{"Start":"03:30.380 ","End":"03:34.835","Text":"that\u0027s the probability of getting a white ball, that\u0027s 0.6."},{"Start":"03:34.835 ","End":"03:41.720","Text":"What\u0027s the probability of getting 1 white ball in all the other guys right there,"},{"Start":"03:41.720 ","End":"03:45.680","Text":"in all the other areas right here? Well, that\u0027s also 0.6."},{"Start":"03:45.680 ","End":"03:49.475","Text":"Now, how many failures do we have?"},{"Start":"03:49.475 ","End":"03:55.610","Text":"Well, we have basically k minus 2 failures."},{"Start":"03:55.610 ","End":"04:00.800","Text":"Because this is k. We have k samples."},{"Start":"04:00.800 ","End":"04:02.944","Text":"This is a success,"},{"Start":"04:02.944 ","End":"04:04.745","Text":"and this is a success."},{"Start":"04:04.745 ","End":"04:07.415","Text":"That\u0027s k minus 2 failures."},{"Start":"04:07.415 ","End":"04:09.725","Text":"Now, what\u0027s the probability of a failure,"},{"Start":"04:09.725 ","End":"04:14.965","Text":"that\u0027s 0.4 to the power of k minus 2."},{"Start":"04:14.965 ","End":"04:20.990","Text":"Now again, who says that the success has to be in the first place?"},{"Start":"04:20.990 ","End":"04:24.095","Text":"The success can be in the second place, the third place,"},{"Start":"04:24.095 ","End":"04:29.665","Text":"or anywhere in the k minus 1 samples here."},{"Start":"04:29.665 ","End":"04:36.260","Text":"Let\u0027s just translate this graphical representation into probabilities."},{"Start":"04:36.260 ","End":"04:41.235","Text":"In essence, we have 0.6 squared,"},{"Start":"04:41.235 ","End":"04:43.460","Text":"because we have 2 successes,"},{"Start":"04:43.460 ","End":"04:47.980","Text":"times 0.4 to the power of k minus 2."},{"Start":"04:47.980 ","End":"04:50.875","Text":"That\u0027s the k minus 2 failures,"},{"Start":"04:50.875 ","End":"04:55.850","Text":"and we\u0027re looking at the success not being in the first place,"},{"Start":"04:55.850 ","End":"04:59.570","Text":"but being anywhere in the k minus 1 samples."},{"Start":"04:59.570 ","End":"05:02.075","Text":"That\u0027s k minus 1,"},{"Start":"05:02.075 ","End":"05:03.800","Text":"and we\u0027re looking for 1 success,"},{"Start":"05:03.800 ","End":"05:05.480","Text":"so that\u0027s over 1."},{"Start":"05:05.480 ","End":"05:09.110","Text":"As we can see, this equals this."},{"Start":"05:09.110 ","End":"05:12.950","Text":"Here basically is the logic,"},{"Start":"05:12.950 ","End":"05:19.425","Text":"the probabilistic logic of this probability right here."},{"Start":"05:19.425 ","End":"05:21.950","Text":"We can either calculate it like this,"},{"Start":"05:21.950 ","End":"05:25.400","Text":"using combinatorics or you can just plug in"},{"Start":"05:25.400 ","End":"05:30.440","Text":"the numbers into the negative binomial distribution, and we\u0027ll get that."}],"ID":13042},{"Watched":false,"Name":"Exercise 2","Duration":"6m 10s","ChapterTopicVideoID":12564,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this question, we\u0027ll be talking about a game of chance."},{"Start":"00:03.000 ","End":"00:06.675","Text":"Now, the chances of winning a game of chance is 0.4."},{"Start":"00:06.675 ","End":"00:10.440","Text":"A person plays the game and stops playing after he wins twice,"},{"Start":"00:10.440 ","End":"00:12.240","Text":"not necessarily in a row."},{"Start":"00:12.240 ","End":"00:15.990","Text":"We\u0027re asked; what are the chances that he played twice or 3 times,"},{"Start":"00:15.990 ","End":"00:17.445","Text":"and so on and so forth?"},{"Start":"00:17.445 ","End":"00:21.975","Text":"Well, here we have a negative binomial distribution."},{"Start":"00:21.975 ","End":"00:27.789","Text":"Let\u0027s see if we meet all the conditions of this distribution."},{"Start":"00:28.730 ","End":"00:33.950","Text":"Here are the conditions, the first 1 being that we have the same Bernoulli trial"},{"Start":"00:33.950 ","End":"00:38.615","Text":"that\u0027s repeated independently until there are successes."},{"Start":"00:38.615 ","End":"00:44.960","Text":"Well, each game that the person wins is a Bernoulli trial."},{"Start":"00:44.960 ","End":"00:47.390","Text":"In the game, you either win or lose."},{"Start":"00:47.390 ","End":"00:52.940","Text":"Let\u0027s define our success as a win."},{"Start":"00:52.940 ","End":"00:55.340","Text":"Now, what\u0027s the probability of a win?"},{"Start":"00:55.340 ","End":"00:58.750","Text":"Well, it\u0027s given to us, that\u0027s 0.4."},{"Start":"00:58.750 ","End":"01:02.360","Text":"Now, each game is independent of each other,"},{"Start":"01:02.360 ","End":"01:08.525","Text":"and therefore the probability of a win is the same for each game that\u0027s played."},{"Start":"01:08.525 ","End":"01:11.360","Text":"Now, what about r?"},{"Start":"01:11.360 ","End":"01:14.575","Text":"R is the number of wins that we need,"},{"Start":"01:14.575 ","End":"01:17.420","Text":"and we need 2."},{"Start":"01:17.490 ","End":"01:29.320","Text":"Let\u0027s define X as the number of games played until we have 2 wins."},{"Start":"01:29.630 ","End":"01:35.305","Text":"X is distributed with a negative binomial distribution,"},{"Start":"01:35.305 ","End":"01:40.640","Text":"where r equals 2 and p equals 0.4."},{"Start":"01:41.090 ","End":"01:48.025","Text":"If that\u0027s the case, then the probability of X being equal to k,"},{"Start":"01:48.025 ","End":"01:52.175","Text":"well, that equals to k minus 1 over r minus 1,"},{"Start":"01:52.175 ","End":"01:58.390","Text":"P^r, and 1 minus P to the power of"},{"Start":"01:58.390 ","End":"02:05.150","Text":"k minus r. Having said that,"},{"Start":"02:05.150 ","End":"02:10.405","Text":"let\u0027s start calculating the probabilities of these sections."},{"Start":"02:10.405 ","End":"02:12.545","Text":"In Section A, we\u0027re asked,"},{"Start":"02:12.545 ","End":"02:15.140","Text":"what are the chances that he played twice?"},{"Start":"02:15.140 ","End":"02:21.245","Text":"Well, that\u0027s saying, what\u0027s the probability of X being equal to 2?"},{"Start":"02:21.245 ","End":"02:23.645","Text":"Well, let\u0027s plug in the numbers here."},{"Start":"02:23.645 ","End":"02:28.160","Text":"K equals 2, so that\u0027s 2 minus 1 over r minus 1,"},{"Start":"02:28.160 ","End":"02:33.448","Text":"r is also 2, so that\u0027s over 2 minus 1 times,"},{"Start":"02:33.448 ","End":"02:35.330","Text":"now P is 0.4,"},{"Start":"02:35.330 ","End":"02:37.780","Text":"so that\u0027s 0.4^r;"},{"Start":"02:37.780 ","End":"02:40.500","Text":"that\u0027s 2, times 1 minus p;"},{"Start":"02:40.500 ","End":"02:41.969","Text":"that\u0027s 1 minus 0.4,"},{"Start":"02:41.969 ","End":"02:49.230","Text":"that\u0027s 0.6 to the power of k minus r. K is 2 and r is 2."},{"Start":"02:49.230 ","End":"02:51.185","Text":"So that\u0027s 2 minus 2."},{"Start":"02:51.185 ","End":"02:52.880","Text":"Now, let\u0027s simplify that."},{"Start":"02:52.880 ","End":"03:01.985","Text":"That would be 1 over 1 times 0.4 squared times 0.6^0."},{"Start":"03:01.985 ","End":"03:04.490","Text":"Now 1 over 1, that equals 1;"},{"Start":"03:04.490 ","End":"03:08.760","Text":"and 0.6^0, that also equals 1."},{"Start":"03:08.760 ","End":"03:11.540","Text":"We\u0027re left with 0.4 squared."},{"Start":"03:11.540 ","End":"03:19.405","Text":"That\u0027s 0.4 squared and that equals to 0.16."},{"Start":"03:19.405 ","End":"03:21.320","Text":"Section B asks us,"},{"Start":"03:21.320 ","End":"03:24.515","Text":"what are the chances that he played 3 times?"},{"Start":"03:24.515 ","End":"03:29.434","Text":"B, the probability that x equals 3."},{"Start":"03:29.434 ","End":"03:31.910","Text":"Well, again, let\u0027s just plug in the numbers."},{"Start":"03:31.910 ","End":"03:35.930","Text":"That\u0027s 3 minus 1 over 2 minus 1"},{"Start":"03:35.930 ","End":"03:44.100","Text":"times 0.4^2 times 0.6 to the power of 3 minus 2."},{"Start":"03:44.100 ","End":"03:46.160","Text":"Again, let\u0027s simplify that."},{"Start":"03:46.160 ","End":"03:48.140","Text":"That\u0027s 2 over 1,"},{"Start":"03:48.140 ","End":"03:54.605","Text":"0.4 squared times 0.6^1,"},{"Start":"03:54.605 ","End":"04:00.655","Text":"and that comes out to 0.192."},{"Start":"04:00.655 ","End":"04:02.630","Text":"Sections C asks us,"},{"Start":"04:02.630 ","End":"04:05.555","Text":"what are the chances that he played 4 games?"},{"Start":"04:05.555 ","End":"04:09.785","Text":"That\u0027s the probability that x equals 4 now."},{"Start":"04:09.785 ","End":"04:13.880","Text":"That\u0027ll be 4 minus 1 over 2 minus 1,"},{"Start":"04:13.880 ","End":"04:20.775","Text":"0.4 squared times 0.6 to the power of 4 minus 2."},{"Start":"04:20.775 ","End":"04:22.589","Text":"Again, let\u0027s simplify that,"},{"Start":"04:22.589 ","End":"04:25.435","Text":"that\u0027s 3 over 1."},{"Start":"04:25.435 ","End":"04:32.075","Text":"0.4 squared times 0.6 squared,"},{"Start":"04:32.075 ","End":"04:37.825","Text":"and that equals to 0.1728."},{"Start":"04:37.825 ","End":"04:39.740","Text":"Section D asks,"},{"Start":"04:39.740 ","End":"04:42.755","Text":"what are the chances that he played 5 times?"},{"Start":"04:42.755 ","End":"04:47.150","Text":"Well, that\u0027s the probability of x being equal to 5."},{"Start":"04:47.150 ","End":"04:48.620","Text":"But this is getting a little bit boring,"},{"Start":"04:48.620 ","End":"04:49.670","Text":"but let\u0027s do that anyway,"},{"Start":"04:49.670 ","End":"04:53.465","Text":"that\u0027s 5 minus 1 over 2 minus 1,"},{"Start":"04:53.465 ","End":"04:59.790","Text":"0.4 squared times 0.6 to the power of 5 minus 2."},{"Start":"04:59.790 ","End":"05:01.177","Text":"Again, let\u0027s simplify,"},{"Start":"05:01.177 ","End":"05:04.175","Text":"that\u0027ll be 4 over 1,"},{"Start":"05:04.175 ","End":"05:08.900","Text":"0.4 squared times 0.6 cube now,"},{"Start":"05:08.900 ","End":"05:13.290","Text":"and that will equal to 0.13824."},{"Start":"05:17.900 ","End":"05:22.460","Text":"Section E asks, what are the chances that he played k times?"},{"Start":"05:22.460 ","End":"05:27.640","Text":"Well, this is the general equation for this question."},{"Start":"05:27.640 ","End":"05:29.235","Text":"Let\u0027s just write it out."},{"Start":"05:29.235 ","End":"05:33.634","Text":"That\u0027s the probability of x being equal to k."},{"Start":"05:33.634 ","End":"05:40.000","Text":"That will be k minus 1 over 2 minus 1,"},{"Start":"05:40.000 ","End":"05:50.230","Text":"0.4^2 times 0.6 to the power of k minus 2."},{"Start":"05:50.230 ","End":"05:56.695","Text":"Simplifying that, that\u0027ll be k minus 1 over 1,"},{"Start":"05:56.695 ","End":"06:04.520","Text":"0.4 squared times 0.6 to the power of k minus 2."},{"Start":"06:04.520 ","End":"06:10.740","Text":"This is the general equation for our question here."}],"ID":13043},{"Watched":false,"Name":"Exercise 3","Duration":"3m 55s","ChapterTopicVideoID":12565,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.250","Text":"In this question, we\u0027re asked to show that"},{"Start":"00:02.250 ","End":"00:05.040","Text":"the geometric probability distribution is"},{"Start":"00:05.040 ","End":"00:09.555","Text":"a special case of the negative binomial probability distribution."},{"Start":"00:09.555 ","End":"00:16.890","Text":"Well, let\u0027s define X is having a geometric distribution with parameter p. Now,"},{"Start":"00:16.890 ","End":"00:19.305","Text":"how did we define this random variable?"},{"Start":"00:19.305 ","End":"00:24.750","Text":"That was the number of trials we had to"},{"Start":"00:24.750 ","End":"00:32.960","Text":"repeat until we had the first success."},{"Start":"00:32.960 ","End":"00:40.100","Text":"Now, we said that the probability of x being equal to k,"},{"Start":"00:40.100 ","End":"00:46.460","Text":"that would equal to p times 1 minus p to the power of k minus 1,"},{"Start":"00:46.460 ","End":"00:49.130","Text":"where k equals 1,"},{"Start":"00:49.130 ","End":"00:52.535","Text":"2, and so on and so forth until infinity."},{"Start":"00:52.535 ","End":"00:56.510","Text":"Now, let\u0027s define another random variable,"},{"Start":"00:56.510 ","End":"01:05.165","Text":"Y having a negative binomial distribution with parameters r and p. Now,"},{"Start":"01:05.165 ","End":"01:07.550","Text":"how did we define this random variable?"},{"Start":"01:07.550 ","End":"01:12.440","Text":"Well, that was the number of trials we had to"},{"Start":"01:12.440 ","End":"01:20.760","Text":"repeat until the rth success."},{"Start":"01:20.980 ","End":"01:26.150","Text":"Now, the probability of Y being equal to k,"},{"Start":"01:26.150 ","End":"01:30.505","Text":"but that was k minus 1 over r minus 1,"},{"Start":"01:30.505 ","End":"01:32.640","Text":"p to the power of r,"},{"Start":"01:32.640 ","End":"01:36.215","Text":"1 minus p to the power of k minus r,"},{"Start":"01:36.215 ","End":"01:38.840","Text":"where k equals r,"},{"Start":"01:38.840 ","End":"01:40.475","Text":"r plus 1,"},{"Start":"01:40.475 ","End":"01:43.975","Text":"and so on and so forth until infinity."},{"Start":"01:43.975 ","End":"01:51.125","Text":"Now, let\u0027s look at the definitions of both random variables here."},{"Start":"01:51.125 ","End":"01:56.978","Text":"In X, we\u0027re talking about repeating the trials until the first success."},{"Start":"01:56.978 ","End":"02:01.145","Text":"In Y, we have to repeat the trials until the rth success."},{"Start":"02:01.145 ","End":"02:06.160","Text":"We can see that if r equals 1,"},{"Start":"02:06.160 ","End":"02:09.080","Text":"then the geometric distribution would be"},{"Start":"02:09.080 ","End":"02:12.760","Text":"a special case of the negative binomial distribution."},{"Start":"02:12.760 ","End":"02:14.720","Text":"Let\u0027s just prove that."},{"Start":"02:14.720 ","End":"02:17.300","Text":"We said that if r equals 1,"},{"Start":"02:17.300 ","End":"02:20.767","Text":"then let\u0027s plug that into this equation right here,"},{"Start":"02:20.767 ","End":"02:22.595","Text":"the negative binomial distribution,"},{"Start":"02:22.595 ","End":"02:27.260","Text":"then the probability of Y equaling k, well,"},{"Start":"02:27.260 ","End":"02:31.805","Text":"that\u0027ll equal to k minus 1 over 1 minus 1,"},{"Start":"02:31.805 ","End":"02:38.570","Text":"p to the power of 1 times 1 minus p to the power of k minus 1."},{"Start":"02:38.570 ","End":"02:41.030","Text":"Now, let\u0027s simplify that,"},{"Start":"02:41.030 ","End":"02:43.564","Text":"that\u0027s k minus 1 over 0,"},{"Start":"02:43.564 ","End":"02:48.755","Text":"p times 1 minus p to the power of k minus 1."},{"Start":"02:48.755 ","End":"02:51.500","Text":"Now, what\u0027s this expression right here,"},{"Start":"02:51.500 ","End":"02:54.185","Text":"k minus 1 over 0? Well, that equals 1."},{"Start":"02:54.185 ","End":"03:02.300","Text":"This whole thing goes down to p times 1 minus p to the power of k minus 1."},{"Start":"03:02.300 ","End":"03:06.765","Text":"Now, this looks exactly like this."},{"Start":"03:06.765 ","End":"03:12.095","Text":"We see that if r equals 1 in the negative binomial distribution,"},{"Start":"03:12.095 ","End":"03:16.265","Text":"it becomes the geometric distribution."},{"Start":"03:16.265 ","End":"03:20.510","Text":"We can say that if Y is distributed with"},{"Start":"03:20.510 ","End":"03:25.880","Text":"a negative binomial distribution where r equals 1 and p, well,"},{"Start":"03:25.880 ","End":"03:35.540","Text":"that\u0027s congruent to saying that X has a geometric distribution with"},{"Start":"03:35.540 ","End":"03:37.910","Text":"parameter p. We can say that"},{"Start":"03:37.910 ","End":"03:42.770","Text":"the random variables are the same and the distributions are the same"},{"Start":"03:42.770 ","End":"03:46.460","Text":"again where the geometric distribution is"},{"Start":"03:46.460 ","End":"03:50.420","Text":"a special case of the negative binomial distribution,"},{"Start":"03:50.420 ","End":"03:55.110","Text":"and the special case being that r has to equal 1."}],"ID":13044},{"Watched":false,"Name":"Exercise 4","Duration":"8m 52s","ChapterTopicVideoID":12566,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this question, we\u0027ll be dealing with tossing a coin."},{"Start":"00:03.510 ","End":"00:09.330","Text":"Now, a coin is tossed again and again until tails is obtained 3 times and we\u0027re"},{"Start":"00:09.330 ","End":"00:15.615","Text":"asked in Section A to construct the probability function of the total number of tosses."},{"Start":"00:15.615 ","End":"00:20.130","Text":"Here we have a negative binomial distribution. Why is that?"},{"Start":"00:20.130 ","End":"00:24.960","Text":"Well, we have a Bernoulli trial that\u0027s done over and over again."},{"Start":"00:24.960 ","End":"00:26.940","Text":"What\u0027s a Bernoulli trial?"},{"Start":"00:26.940 ","End":"00:28.500","Text":"That\u0027s a coin toss."},{"Start":"00:28.500 ","End":"00:31.350","Text":"We have success and failure here,"},{"Start":"00:31.350 ","End":"00:34.440","Text":"where success is getting a tails."},{"Start":"00:34.440 ","End":"00:42.730","Text":"Let\u0027s define success as getting tails in the coin toss."},{"Start":"00:42.730 ","End":"00:45.680","Text":"Now, what\u0027s the probability of getting a tail?"},{"Start":"00:45.680 ","End":"00:47.900","Text":"But that equals to 0.5."},{"Start":"00:47.900 ","End":"00:51.320","Text":"We assume that the coin is a fair 1,"},{"Start":"00:51.320 ","End":"00:58.655","Text":"so the probability of getting tails is 0.5 and each toss is independent of the other."},{"Start":"00:58.655 ","End":"01:01.055","Text":"Now, what else do we know?"},{"Start":"01:01.055 ","End":"01:04.280","Text":"We want to obtain tails 3 times."},{"Start":"01:04.280 ","End":"01:06.785","Text":"That means, we want r,"},{"Start":"01:06.785 ","End":"01:08.300","Text":"the number of successes,"},{"Start":"01:08.300 ","End":"01:10.430","Text":"we want that to be 3."},{"Start":"01:10.430 ","End":"01:14.720","Text":"Let\u0027s define a random variable as the number of"},{"Start":"01:14.720 ","End":"01:23.470","Text":"tosses until we have 3 successes."},{"Start":"01:23.960 ","End":"01:27.360","Text":"In our case, that\u0027d be 3 tails."},{"Start":"01:27.360 ","End":"01:31.610","Text":"X would have a negative binomial distribution where"},{"Start":"01:31.610 ","End":"01:37.655","Text":"r equals 3 and p would be equal to 0.5."},{"Start":"01:37.655 ","End":"01:42.110","Text":"Now let\u0027s just remember what the equation is for"},{"Start":"01:42.110 ","End":"01:46.970","Text":"the probability function of the negative binomial distribution,"},{"Start":"01:46.970 ","End":"01:49.856","Text":"that\u0027s the probability of x being equal to k,"},{"Start":"01:49.856 ","End":"01:54.890","Text":"well that equals to k minus 1 over r minus 1,"},{"Start":"01:54.890 ","End":"02:00.110","Text":"p to the power of r times 1 minus p to"},{"Start":"02:00.110 ","End":"02:05.660","Text":"the power of k minus r. In our case in Section A,"},{"Start":"02:05.660 ","End":"02:08.630","Text":"we\u0027re asked to construct a probability function, well,"},{"Start":"02:08.630 ","End":"02:12.905","Text":"that\u0027s the probability of x being equal to k. Now,"},{"Start":"02:12.905 ","End":"02:15.470","Text":"let\u0027s just plug in the numbers here."},{"Start":"02:15.470 ","End":"02:17.945","Text":"That\u0027d be k minus 1."},{"Start":"02:17.945 ","End":"02:20.700","Text":"Now r here is 3."},{"Start":"02:20.700 ","End":"02:23.390","Text":"They\u0027d be 3 minus 1."},{"Start":"02:23.390 ","End":"02:25.970","Text":"Now, p is 0.5,"},{"Start":"02:25.970 ","End":"02:32.300","Text":"so that\u0027d be 0.5 to the power of r times 1"},{"Start":"02:32.300 ","End":"02:39.975","Text":"minus p. That\u0027s again 0.5 times k minus r. That\u0027d be k minus 3."},{"Start":"02:39.975 ","End":"02:41.960","Text":"Let\u0027s simplify that."},{"Start":"02:41.960 ","End":"02:52.115","Text":"That would be k minus 1 over 2 times 0.5 to the power of k. Where again,"},{"Start":"02:52.115 ","End":"02:56.080","Text":"k would be equal to 3,"},{"Start":"02:56.080 ","End":"03:01.670","Text":"4, 5, and so on and so forth until infinity."},{"Start":"03:01.760 ","End":"03:04.045","Text":"In Section B we\u0027re asked,"},{"Start":"03:04.045 ","End":"03:08.245","Text":"what are the expectation and variance of the total number of tosses?"},{"Start":"03:08.245 ","End":"03:15.780","Text":"In B, we want to know what\u0027s the expectation of x and the variance of x,"},{"Start":"03:15.780 ","End":"03:19.600","Text":"what the expectation of x in a negative binomial distribution is"},{"Start":"03:19.600 ","End":"03:24.505","Text":"defined as r over p. In our case,"},{"Start":"03:24.505 ","End":"03:28.590","Text":"r is 3 and p equals 0.5,"},{"Start":"03:28.590 ","End":"03:31.140","Text":"that means that that equals 6."},{"Start":"03:31.140 ","End":"03:34.050","Text":"Now, what\u0027s the variance of x?"},{"Start":"03:34.050 ","End":"03:42.115","Text":"The variance of x is defined as r times 1 minus p divided by p squared,"},{"Start":"03:42.115 ","End":"03:51.255","Text":"that\u0027d be 3 times 0.5 divided by 0.5 squared and that also comes out to 6."},{"Start":"03:51.255 ","End":"03:56.605","Text":"This is the variance and this is the expectation of x."},{"Start":"03:56.605 ","End":"04:02.090","Text":"In Section C, we\u0027re given that the above process is repeated 5 times."},{"Start":"04:02.090 ","End":"04:05.660","Text":"We\u0027re asked, what\u0027s the probability that the coin is tossed"},{"Start":"04:05.660 ","End":"04:09.775","Text":"exactly 4 times on 2 of the 5 repetitions?"},{"Start":"04:09.775 ","End":"04:12.020","Text":"Well, this is a little bit complicated,"},{"Start":"04:12.020 ","End":"04:13.280","Text":"so just bear with me."},{"Start":"04:13.280 ","End":"04:15.110","Text":"We\u0027re going to do this in stages."},{"Start":"04:15.110 ","End":"04:18.410","Text":"The first stage is this 1."},{"Start":"04:18.410 ","End":"04:20.014","Text":"We\u0027re going to answer the question,"},{"Start":"04:20.014 ","End":"04:25.930","Text":"what is the probability that the coin is tossed exactly 4 times?"},{"Start":"04:25.930 ","End":"04:28.400","Text":"Well, in essence,"},{"Start":"04:28.400 ","End":"04:35.390","Text":"what we\u0027re looking for is the probability that x equals 4."},{"Start":"04:35.390 ","End":"04:43.760","Text":"Now, from Section A we\u0027ve calculated the probability function for x."},{"Start":"04:43.760 ","End":"04:46.370","Text":"Let\u0027s just plug in the numbers here."},{"Start":"04:46.370 ","End":"04:54.720","Text":"That\u0027d be 4 minus 1 over 2 times 0.5 to the power of 4."},{"Start":"04:54.720 ","End":"04:59.770","Text":"Now, that comes out to 0.1875."},{"Start":"05:02.080 ","End":"05:06.170","Text":"Now the next stage is to understand what we mean"},{"Start":"05:06.170 ","End":"05:09.950","Text":"when we say the process is repeated 5 times."},{"Start":"05:09.950 ","End":"05:15.290","Text":"We want to know about 2 out of the 5 repetitions."},{"Start":"05:15.290 ","End":"05:18.440","Text":"Well, when we\u0027re talking about the process is repeated 5 times,"},{"Start":"05:18.440 ","End":"05:19.460","Text":"what are we talking about?"},{"Start":"05:19.460 ","End":"05:20.705","Text":"Which process?"},{"Start":"05:20.705 ","End":"05:26.570","Text":"Well, that\u0027s the process of tossing the coin until we have 3 successes,"},{"Start":"05:26.570 ","End":"05:28.810","Text":"until we have 3 tails."},{"Start":"05:28.810 ","End":"05:35.664","Text":"The probability of tossing a coin 4 times to get 3 successes,"},{"Start":"05:35.664 ","End":"05:38.240","Text":"that would be this guy right here."},{"Start":"05:38.240 ","End":"05:44.150","Text":"Now, if we repeat this process 5 times,"},{"Start":"05:44.150 ","End":"05:47.750","Text":"we want to know that out of the 5 repetitions,"},{"Start":"05:47.750 ","End":"05:52.700","Text":"what\u0027s the probability that 2 of"},{"Start":"05:52.700 ","End":"05:58.320","Text":"the repetitions will have 4 tosses with getting 3 tails?"},{"Start":"05:58.320 ","End":"06:00.390","Text":"That\u0027s the 3 successes here."},{"Start":"06:00.390 ","End":"06:03.290","Text":"Here we\u0027re not talking now about the number of tosses,"},{"Start":"06:03.290 ","End":"06:05.885","Text":"we\u0027re talking about the number of repetitions."},{"Start":"06:05.885 ","End":"06:12.135","Text":"Let\u0027s define y as the number of repetitions"},{"Start":"06:12.135 ","End":"06:21.350","Text":"with or having 4 tosses."},{"Start":"06:21.350 ","End":"06:26.685","Text":"Now, what\u0027s the probability of y,"},{"Start":"06:26.685 ","End":"06:28.850","Text":"of getting 4 tosses?"},{"Start":"06:28.850 ","End":"06:30.410","Text":"Well, the probability is here,"},{"Start":"06:30.410 ","End":"06:34.385","Text":"we\u0027ve calculated that, that\u0027s 0.1875."},{"Start":"06:34.385 ","End":"06:37.040","Text":"Then how many repetitions do we have?"},{"Start":"06:37.040 ","End":"06:42.140","Text":"Well, we have 5 repetitions that\u0027s given to us right here."},{"Start":"06:42.140 ","End":"06:45.350","Text":"Whenever we have this configuration,"},{"Start":"06:45.350 ","End":"06:51.290","Text":"we\u0027re talking about y having a binomial distribution,"},{"Start":"06:51.290 ","End":"06:53.224","Text":"not a negative binomial,"},{"Start":"06:53.224 ","End":"07:03.780","Text":"but a binomial distribution where n equals 5 and p equals 0.1875."},{"Start":"07:04.250 ","End":"07:11.460","Text":"Let\u0027s just recall that the probability of y being equal to k,"},{"Start":"07:11.460 ","End":"07:14.270","Text":"that\u0027d be n over k,"},{"Start":"07:14.270 ","End":"07:16.550","Text":"p to the power of k,"},{"Start":"07:16.550 ","End":"07:23.435","Text":"and 1 minus p to the power of n minus k. In our case,"},{"Start":"07:23.435 ","End":"07:29.430","Text":"the probability of y being equal to 2,"},{"Start":"07:29.430 ","End":"07:32.300","Text":"we want 2 out of the 5 repetitions."},{"Start":"07:32.300 ","End":"07:34.010","Text":"That\u0027s what we\u0027re looking at."},{"Start":"07:34.010 ","End":"07:36.650","Text":"The probability of y equaling 2."},{"Start":"07:36.650 ","End":"07:38.030","Text":"Let\u0027s plug in the numbers."},{"Start":"07:38.030 ","End":"07:41.730","Text":"N equals 5 over 2,"},{"Start":"07:41.730 ","End":"07:50.340","Text":"p is 0.1875 to the power of 2 times 1"},{"Start":"07:50.340 ","End":"07:59.980","Text":"minus 0.1875 to the power of 5 minus 2, that\u0027s 3."},{"Start":"08:00.680 ","End":"08:09.000","Text":"That comes out to 0.1886."},{"Start":"08:09.000 ","End":"08:11.395","Text":"Now again, this is what we\u0027ve done."},{"Start":"08:11.395 ","End":"08:16.010","Text":"We\u0027ve calculated the probability of"},{"Start":"08:16.010 ","End":"08:23.420","Text":"a repetition having 4 tosses as being 0.1875."},{"Start":"08:23.420 ","End":"08:29.190","Text":"We use this probability in the binomial distribution where,"},{"Start":"08:29.190 ","End":"08:35.375","Text":"y now, the new random variable was the number of repetitions with the 4 tosses."},{"Start":"08:35.375 ","End":"08:40.430","Text":"We could calculate the probability of 2 of"},{"Start":"08:40.430 ","End":"08:46.190","Text":"the 5 repetitions having 4 tosses where in each toss,"},{"Start":"08:46.190 ","End":"08:49.250","Text":"we had 3 successes,"},{"Start":"08:49.250 ","End":"08:51.750","Text":"that means 3 tails."}],"ID":13045},{"Watched":false,"Name":"Exercise 5","Duration":"4m 46s","ChapterTopicVideoID":12567,"CourseChapterTopicPlaylistID":64489,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"Let X_i be the number of repetitions until the first success in"},{"Start":"00:04.200 ","End":"00:09.030","Text":"independent Bernoulli trials where i equals 1 to n. We\u0027re asked to"},{"Start":"00:09.030 ","End":"00:11.580","Text":"prove that the expectation and variance of the sum of"},{"Start":"00:11.580 ","End":"00:14.590","Text":"X_i are the same as the expectation and variance of"},{"Start":"00:14.590 ","End":"00:21.950","Text":"the negative binomial probability distribution with parameters n and p. Whenever"},{"Start":"00:21.950 ","End":"00:25.250","Text":"we\u0027re dealing with a random variable"},{"Start":"00:25.250 ","End":"00:29.701","Text":"that describes the number of repetitions until the first success,"},{"Start":"00:29.701 ","End":"00:32.270","Text":"well, that\u0027s a geometric distribution,"},{"Start":"00:32.270 ","End":"00:37.120","Text":"so each X_i is"},{"Start":"00:37.120 ","End":"00:43.070","Text":"distributed with the geometric distribution with parameter p or the same probability."},{"Start":"00:43.070 ","End":"00:47.600","Text":"Where i equals 1 until n. Now,"},{"Start":"00:47.600 ","End":"00:49.205","Text":"if that\u0027s the case,"},{"Start":"00:49.205 ","End":"00:52.355","Text":"let\u0027s now look at the sum of X_i."},{"Start":"00:52.355 ","End":"00:57.065","Text":"We\u0027re looking basically at the expectation of the sum of X_i,"},{"Start":"00:57.065 ","End":"01:02.105","Text":"where i equals 1 to n. Well, from previous chapters,"},{"Start":"01:02.105 ","End":"01:07.310","Text":"we know that the expectation of the sum is the sum of the expectation."},{"Start":"01:07.310 ","End":"01:13.595","Text":"So we\u0027re dealing with the expectation of X_1 plus the expectation of X_2,"},{"Start":"01:13.595 ","End":"01:18.640","Text":"and so on and so forth until the expectation of X_n."},{"Start":"01:18.640 ","End":"01:21.230","Text":"In a geometric distribution,"},{"Start":"01:21.230 ","End":"01:29.670","Text":"the expectation of a random variable is 1 over p. That\u0027ll be 1 over p plus 1 over p,"},{"Start":"01:29.670 ","End":"01:32.060","Text":"and they\u0027re the same because each X_i has"},{"Start":"01:32.060 ","End":"01:37.610","Text":"the same geometric distribution and the same probability."},{"Start":"01:37.610 ","End":"01:40.165","Text":"We have that."},{"Start":"01:40.165 ","End":"01:42.980","Text":"How many expressions do we have here?"},{"Start":"01:42.980 ","End":"01:44.810","Text":"Well, we have n expressions."},{"Start":"01:44.810 ","End":"01:47.270","Text":"That\u0027s n times 1 over p,"},{"Start":"01:47.270 ","End":"01:50.405","Text":"and that equals to n over p. Now,"},{"Start":"01:50.405 ","End":"01:55.470","Text":"let\u0027s take a look at the variance of the sum of X_i."},{"Start":"01:55.960 ","End":"02:01.310","Text":"The variance of the sum equals the sum of the variance when,"},{"Start":"02:01.310 ","End":"02:04.755","Text":"only where we have independence."},{"Start":"02:04.755 ","End":"02:09.680","Text":"We\u0027re given that each X_i is independent of each other right here,"},{"Start":"02:09.680 ","End":"02:13.580","Text":"in independent Bernoulli trials that\u0027s given to us."},{"Start":"02:13.580 ","End":"02:18.450","Text":"That means we\u0027re dealing with the variance of X_1 plus"},{"Start":"02:18.450 ","End":"02:25.705","Text":"the variance of X_2 and so on and so forth until the variance of X_n."},{"Start":"02:25.705 ","End":"02:29.540","Text":"Again, since each X_i is"},{"Start":"02:29.540 ","End":"02:33.845","Text":"distributed with the geometric distribution that has the same probability,"},{"Start":"02:33.845 ","End":"02:36.060","Text":"then they all have the same variance."},{"Start":"02:36.060 ","End":"02:41.060","Text":"Now, what\u0027s the variance of a random variable having a geometric distribution?"},{"Start":"02:41.060 ","End":"02:44.570","Text":"Well, that\u0027s 1 minus p over p squared. Let\u0027s write that down."},{"Start":"02:44.570 ","End":"02:52.560","Text":"That\u0027s 1 minus p over p squared for X_1 plus 1 minus p over p squared for X_2,"},{"Start":"02:52.560 ","End":"02:54.600","Text":"and so on and so forth until X_n,"},{"Start":"02:54.600 ","End":"02:59.835","Text":"that\u0027s 1 minus p over p squared."},{"Start":"02:59.835 ","End":"03:02.010","Text":"We have n expressions like that."},{"Start":"03:02.010 ","End":"03:07.360","Text":"We have n times 1 minus p over p squared."},{"Start":"03:09.080 ","End":"03:14.505","Text":"Now let\u0027s define a new random variable,"},{"Start":"03:14.505 ","End":"03:17.250","Text":"and we\u0027ll call that random variable Y."},{"Start":"03:17.250 ","End":"03:24.755","Text":"It will have a negative binomial distribution with parameters n and p. Well,"},{"Start":"03:24.755 ","End":"03:32.480","Text":"the expectation of Y is defined as n over p or n divided by"},{"Start":"03:32.480 ","End":"03:36.185","Text":"p. We can see that"},{"Start":"03:36.185 ","End":"03:42.140","Text":"this definition right here for the negative binomial distribution is exactly the same."},{"Start":"03:42.140 ","End":"03:46.805","Text":"It\u0027s the expectation of the sum of random variables,"},{"Start":"03:46.805 ","End":"03:51.950","Text":"each 1 having a geometric distribution with the same probability."},{"Start":"03:51.950 ","End":"03:55.060","Text":"What about the variance of Y?"},{"Start":"03:55.060 ","End":"04:02.150","Text":"Well, the variance of Y is defined as n times 1 minus p divided by p squared."},{"Start":"04:02.150 ","End":"04:04.400","Text":"That\u0027s the definition. Again,"},{"Start":"04:04.400 ","End":"04:10.255","Text":"we can see that this expression right here is the same as this expression right here."},{"Start":"04:10.255 ","End":"04:19.935","Text":"In conclusion, whenever we have a sum of random variables X_i,"},{"Start":"04:19.935 ","End":"04:26.750","Text":"where each X_i has a geometric distribution with the same parameter p,"},{"Start":"04:26.750 ","End":"04:33.470","Text":"then the expectation and variance of the sum of these variables equals"},{"Start":"04:33.470 ","End":"04:36.710","Text":"the expectation and variance of"},{"Start":"04:36.710 ","End":"04:42.365","Text":"a random variable that\u0027s distributed with the negative binomial distribution,"},{"Start":"04:42.365 ","End":"04:45.930","Text":"having parameters n and p."}],"ID":13046}],"Thumbnail":null,"ID":64489},{"Name":"Poisson Approximation of the Binomial Probability Distribution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"1m 6s","ChapterTopicVideoID":12568,"CourseChapterTopicPlaylistID":64490,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.920","Text":"In this chapter, we\u0027ll be looking at"},{"Start":"00:01.920 ","End":"00:06.180","Text":"the Poisson approximation for the binomial probability distribution."},{"Start":"00:06.180 ","End":"00:12.165","Text":"Now, if X is distributed with a binomial distribution with parameters n and p,"},{"Start":"00:12.165 ","End":"00:15.105","Text":"where n is large and p is small,"},{"Start":"00:15.105 ","End":"00:20.910","Text":"then this distribution can be approximated using the Poisson probability distribution,"},{"Start":"00:20.910 ","End":"00:24.920","Text":"where we replace Lambda with n times p,"},{"Start":"00:24.920 ","End":"00:30.125","Text":"and the probability function of the Poisson distribution is this."},{"Start":"00:30.125 ","End":"00:32.600","Text":"It\u0027s the probability of X being equal to k,"},{"Start":"00:32.600 ","End":"00:35.630","Text":"that would equal to e to the power of minus Lambda"},{"Start":"00:35.630 ","End":"00:39.290","Text":"times Lambda to the power of k over k factorial."},{"Start":"00:39.290 ","End":"00:42.740","Text":"Now, some claim that a large n and"},{"Start":"00:42.740 ","End":"00:47.450","Text":"a small p means that the probability is less than 10 percent,"},{"Start":"00:47.450 ","End":"00:53.500","Text":"and the multiplication of n times p,"},{"Start":"00:53.500 ","End":"00:55.905","Text":"that has to be greater than 10,"},{"Start":"00:55.905 ","End":"00:59.240","Text":"so it doesn\u0027t have to meet these conditions,"},{"Start":"00:59.240 ","End":"01:01.160","Text":"but it has to come pretty close to them."},{"Start":"01:01.160 ","End":"01:07.020","Text":"So let\u0027s take a look at an example and see how we can work with this."}],"ID":13047},{"Watched":false,"Name":"Example","Duration":"5m 12s","ChapterTopicVideoID":12569,"CourseChapterTopicPlaylistID":64490,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In our example, we\u0027re given the 10 percent"},{"Start":"00:02.670 ","End":"00:05.505","Text":"of the products in a production line are blue,"},{"Start":"00:05.505 ","End":"00:08.265","Text":"20 products are randomly selected."},{"Start":"00:08.265 ","End":"00:13.335","Text":"We\u0027re asked to calculate the probability that exactly 1 of the products selected is blue."},{"Start":"00:13.335 ","End":"00:16.890","Text":"We\u0027re also asked to perform the calculations once using"},{"Start":"00:16.890 ","End":"00:23.035","Text":"the binomial probability distribution and once using the Poisson approximation."},{"Start":"00:23.035 ","End":"00:28.875","Text":"Let\u0027s start with the binomial distribution."},{"Start":"00:28.875 ","End":"00:31.105","Text":"What do we have here?"},{"Start":"00:31.105 ","End":"00:38.110","Text":"Define success as getting a blue product."},{"Start":"00:38.110 ","End":"00:41.810","Text":"What\u0027s the probability of getting a blue product?"},{"Start":"00:41.810 ","End":"00:44.165","Text":"Well, that\u0027s 10 percent."},{"Start":"00:44.165 ","End":"00:46.850","Text":"How many products is sampled?"},{"Start":"00:46.850 ","End":"00:52.210","Text":"20. Each product is independent of each other."},{"Start":"00:52.210 ","End":"00:59.240","Text":"Because we have a small number of the products within a whole massive production line,"},{"Start":"00:59.240 ","End":"01:01.920","Text":"so we assume independence."},{"Start":"01:01.970 ","End":"01:13.260","Text":"That\u0027s the case, x here is defined as the number of blue products in the sample."},{"Start":"01:14.360 ","End":"01:23.990","Text":"X has a binomial distribution where n equals 20 and p equals 0.1."},{"Start":"01:24.020 ","End":"01:29.650","Text":"Let\u0027s just recall that the probability of x being equal to k,"},{"Start":"01:29.650 ","End":"01:32.155","Text":"where x has a binomial distribution."},{"Start":"01:32.155 ","End":"01:35.170","Text":"Well, that equals to n over k,"},{"Start":"01:35.170 ","End":"01:41.755","Text":"p to the power of k times 1 minus p to the power of n minus k."},{"Start":"01:41.755 ","End":"01:49.940","Text":"In our case we\u0027re asked what\u0027s the probability of x being equal to 1."},{"Start":"01:49.940 ","End":"01:55.540","Text":"Calculate the probability that exactly 1 of the products is selected is blue,"},{"Start":"01:55.540 ","End":"01:57.110","Text":"so x equals 1."},{"Start":"01:57.110 ","End":"02:01.930","Text":"Well, let\u0027s just plug in the numbers n equals 20, k equals 1."},{"Start":"02:01.930 ","End":"02:03.395","Text":"That\u0027s 20 over 1."},{"Start":"02:03.395 ","End":"02:05.690","Text":"The probability is 0.1,"},{"Start":"02:05.690 ","End":"02:14.190","Text":"so it\u0027s 0.1 to the power of k to the power of 1 times 1 minus p. That\u0027s 1 minus 0.1,"},{"Start":"02:14.190 ","End":"02:18.140","Text":"so that\u0027s 0.9 to the power of n minus k,"},{"Start":"02:18.140 ","End":"02:20.960","Text":"that\u0027s 20 minus 1, that\u0027s 19."},{"Start":"02:20.960 ","End":"02:25.890","Text":"That equals to 0.2702."},{"Start":"02:30.020 ","End":"02:34.730","Text":"Let\u0027s take a look at things from the Poisson perspective."},{"Start":"02:34.730 ","End":"02:37.990","Text":"We\u0027re looking at the Poisson distribution,"},{"Start":"02:37.990 ","End":"02:40.520","Text":"and we\u0027re saying, first of all,"},{"Start":"02:40.520 ","End":"02:41.720","Text":"let\u0027s do the check."},{"Start":"02:41.720 ","End":"02:44.060","Text":"How much is n times p?"},{"Start":"02:44.060 ","End":"02:45.790","Text":"Well, n is 20,"},{"Start":"02:45.790 ","End":"02:48.830","Text":"times p, that\u0027s 0.1, and that equals to 2."},{"Start":"02:48.830 ","End":"02:50.420","Text":"Now if we recall,"},{"Start":"02:50.420 ","End":"02:55.150","Text":"we\u0027re saying that n times p has to be greater than 10."},{"Start":"02:55.150 ","End":"02:57.304","Text":"This isn\u0027t greater than 10 obviously,"},{"Start":"02:57.304 ","End":"03:02.645","Text":"but let\u0027s use the Poisson approximation in any case."},{"Start":"03:02.645 ","End":"03:04.495","Text":"If we recall,"},{"Start":"03:04.495 ","End":"03:11.965","Text":"if x distributed with a Poisson distribution with the parameter Lambda,"},{"Start":"03:11.965 ","End":"03:17.015","Text":"then the probability distribution is e to the minus Lambda,"},{"Start":"03:17.015 ","End":"03:22.945","Text":"times Lambda to the power of k divided by k factorial."},{"Start":"03:22.945 ","End":"03:31.365","Text":"In our case, Lambda equals to 20 times 0.1."},{"Start":"03:31.365 ","End":"03:33.075","Text":"That\u0027s n times p,"},{"Start":"03:33.075 ","End":"03:35.100","Text":"and that equals to 2,"},{"Start":"03:35.100 ","End":"03:42.585","Text":"so x is distributed approximately."},{"Start":"03:42.585 ","End":"03:47.850","Text":"We can use this sign right here,"},{"Start":"03:47.850 ","End":"03:51.770","Text":"which is a squiggly arrow,"},{"Start":"03:51.770 ","End":"03:53.990","Text":"or we can use this as well."},{"Start":"03:53.990 ","End":"03:59.645","Text":"This is another sign for approximation you can use whichever side you want,"},{"Start":"03:59.645 ","End":"04:07.890","Text":"that is approximately distributed with a Poisson distribution where Lambda equals 2."},{"Start":"04:08.320 ","End":"04:11.380","Text":"If that\u0027s the case,"},{"Start":"04:11.380 ","End":"04:14.220","Text":"let\u0027s just plug in the numbers."},{"Start":"04:14.220 ","End":"04:22.440","Text":"We\u0027re asked, what\u0027s the probability of x being equal to 1?"},{"Start":"04:22.440 ","End":"04:26.645","Text":"X has a Poisson distribution."},{"Start":"04:26.645 ","End":"04:30.410","Text":"Well, let\u0027s plug in the numbers that\u0027s e to the minus Lambda."},{"Start":"04:30.410 ","End":"04:34.500","Text":"Lambda equals 2 times 2 to the power of k,"},{"Start":"04:34.500 ","End":"04:36.345","Text":"2 to the power of 1,"},{"Start":"04:36.345 ","End":"04:40.605","Text":"divided by k factorial, that\u0027s 1 factorial."},{"Start":"04:40.605 ","End":"04:46.905","Text":"That equals to 0.2707."},{"Start":"04:46.905 ","End":"04:54.025","Text":"Look at how close this approximation is to the binomial probability right here."},{"Start":"04:54.025 ","End":"05:00.290","Text":"We can see that even though n times p is not that much greater than 10,"},{"Start":"05:00.290 ","End":"05:02.465","Text":"and even here it\u0027s less than 10,"},{"Start":"05:02.465 ","End":"05:06.460","Text":"we still have a fairly good approximation."},{"Start":"05:06.460 ","End":"05:12.990","Text":"All we have to do now is just go ahead and do the exercises. Good luck."}],"ID":13048},{"Watched":false,"Name":"Exercise 1","Duration":"4m 28s","ChapterTopicVideoID":12570,"CourseChapterTopicPlaylistID":64490,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.009","Text":"This question, 10 percent of the population in Never-Never land are unemployed."},{"Start":"00:05.009 ","End":"00:07.875","Text":"10 people are randomly selected."},{"Start":"00:07.875 ","End":"00:11.070","Text":"We\u0027re asked to calculate the chances that there is"},{"Start":"00:11.070 ","End":"00:14.535","Text":"1 unemployed person at most in the sample."},{"Start":"00:14.535 ","End":"00:18.804","Text":"We\u0027re also asked to compare the results with a Poisson approximation."},{"Start":"00:18.804 ","End":"00:23.375","Text":"Let\u0027s get started with the binomial distribution."},{"Start":"00:23.375 ","End":"00:29.060","Text":"In a binomial distribution,"},{"Start":"00:29.060 ","End":"00:32.030","Text":"how do we define our success?"},{"Start":"00:32.030 ","End":"00:37.530","Text":"Well, we\u0027ll defined it as being unemployed."},{"Start":"00:38.440 ","End":"00:42.016","Text":"The probability of being unemployed,"},{"Start":"00:42.016 ","End":"00:44.375","Text":"well, that\u0027s 10 percent."},{"Start":"00:44.375 ","End":"00:51.065","Text":"Now since we have a small sample size from a very large population,"},{"Start":"00:51.065 ","End":"00:56.400","Text":"we assume that the people sampled are independent of each other."},{"Start":"00:56.620 ","End":"00:59.195","Text":"How many people are sampled?"},{"Start":"00:59.195 ","End":"01:01.580","Text":"10, so n equals to 10."},{"Start":"01:01.580 ","End":"01:12.635","Text":"Now let\u0027s define X as the number of unemployed people that are in our sample."},{"Start":"01:12.635 ","End":"01:21.817","Text":"So that means that x now has a binomial distribution with parameters n,"},{"Start":"01:21.817 ","End":"01:25.985","Text":"that equals to 10, and parameter p that equals to 0.1."},{"Start":"01:25.985 ","End":"01:29.945","Text":"Now first of all let\u0027s recall that the probability of X being equal to"},{"Start":"01:29.945 ","End":"01:34.955","Text":"k in a binomial distribution well that equals to n over k,"},{"Start":"01:34.955 ","End":"01:37.220","Text":"p to the power of k,"},{"Start":"01:37.220 ","End":"01:43.534","Text":"1 minus p to the power of n minus k. In our case,"},{"Start":"01:43.534 ","End":"01:48.230","Text":"the probability of x being less than or equal to 1 that"},{"Start":"01:48.230 ","End":"01:52.520","Text":"means that the number of unemployed people are at most 1."},{"Start":"01:52.520 ","End":"01:56.480","Text":"Well, that is to the probability of X being equal to 0,"},{"Start":"01:56.480 ","End":"02:00.620","Text":"plus the probability of X being equal to 1."},{"Start":"02:00.620 ","End":"02:03.482","Text":"Let\u0027s just plug in the numbers,"},{"Start":"02:03.482 ","End":"02:05.721","Text":"the probability of X equaling 0,"},{"Start":"02:05.721 ","End":"02:09.980","Text":"well, that\u0027s n over 0,"},{"Start":"02:09.980 ","End":"02:16.310","Text":"and that\u0027s 10 over 0, times 0.1^0,"},{"Start":"02:16.310 ","End":"02:22.760","Text":"times 0.9 to the power of 10 plus now x equals 1,"},{"Start":"02:22.760 ","End":"02:26.030","Text":"well that\u0027s 10 over 1,"},{"Start":"02:26.030 ","End":"02:32.013","Text":"0.1^1 times 0.9 to the power of 10 minus 1,"},{"Start":"02:32.013 ","End":"02:39.215","Text":"well that\u0027s 9 and that comes out to 0.7361."},{"Start":"02:39.215 ","End":"02:43.805","Text":"Let\u0027s take a look at the Poisson approximation."},{"Start":"02:43.805 ","End":"02:48.470","Text":"It\u0027s a Poisson approximation."},{"Start":"02:48.470 ","End":"02:54.800","Text":"Now say here that Lambda equals to n times p. In our case,"},{"Start":"02:54.800 ","End":"02:57.902","Text":"n is 10 and p is 0.1,"},{"Start":"02:57.902 ","End":"03:00.605","Text":"so we\u0027re talking about Lambda being equal to 1."},{"Start":"03:00.605 ","End":"03:02.165","Text":"So if that\u0027s the case,"},{"Start":"03:02.165 ","End":"03:10.985","Text":"X has a Poisson approximation where Lambda here equals to 1."},{"Start":"03:10.985 ","End":"03:14.420","Text":"Now let\u0027s just recall what\u0027s the probability"},{"Start":"03:14.420 ","End":"03:17.750","Text":"of X being equal to k in a Poisson distribution?"},{"Start":"03:17.750 ","End":"03:19.999","Text":"That\u0027s e to the minus Lambda,"},{"Start":"03:19.999 ","End":"03:25.055","Text":"times Lambda to the power of k over k factorial."},{"Start":"03:25.055 ","End":"03:27.845","Text":"Now in our case,"},{"Start":"03:27.845 ","End":"03:33.275","Text":"we want to know what\u0027s the probability of x being less than or equal to 1."},{"Start":"03:33.275 ","End":"03:37.550","Text":"Well, that equals to the probability of X being equal to 0,"},{"Start":"03:37.550 ","End":"03:41.483","Text":"plus the probability of X being equal to 1."},{"Start":"03:41.483 ","End":"03:49.230","Text":"Let\u0027s just plug in the numbers and calculate this probability right here."},{"Start":"03:49.460 ","End":"03:52.455","Text":"When x equals 0,"},{"Start":"03:52.455 ","End":"03:58.395","Text":"we\u0027re talking about e to the power of minus 1,"},{"Start":"03:58.395 ","End":"04:03.920","Text":"times 1^0 over 0 factorial,"},{"Start":"04:03.920 ","End":"04:06.440","Text":"plus now x equals 1,"},{"Start":"04:06.440 ","End":"04:10.310","Text":"well that\u0027s e to the power of minus 1,"},{"Start":"04:10.310 ","End":"04:15.450","Text":"times 1^1, over 1 factorial."},{"Start":"04:15.450 ","End":"04:22.699","Text":"That comes out to 0.7358 and as we can see,"},{"Start":"04:22.699 ","End":"04:28.440","Text":"this is a very good approximation of this."}],"ID":13049},{"Watched":false,"Name":"Exercise 2","Duration":"3m 55s","ChapterTopicVideoID":12571,"CourseChapterTopicPlaylistID":64490,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this question, 1,000 products are sampled from a mass production line."},{"Start":"00:04.860 ","End":"00:08.295","Text":"It\u0027s known that 5 percent of the products are faulty."},{"Start":"00:08.295 ","End":"00:13.469","Text":"We were asked, what are the chances that the sample will contain 45 faulty products?"},{"Start":"00:13.469 ","End":"00:17.445","Text":"Well, here we have a binomial distribution. Why is that?"},{"Start":"00:17.445 ","End":"00:25.540","Text":"Well, first let\u0027s define our success as being a faulty product."},{"Start":"00:27.560 ","End":"00:33.020","Text":"What\u0027s the probability of getting a faulty product?"},{"Start":"00:33.020 ","End":"00:36.950","Text":"Well, that\u0027s 5 percent, that\u0027s 0.05."},{"Start":"00:36.950 ","End":"00:45.920","Text":"Now we\u0027re sampling 1,000 products from a massive production line."},{"Start":"00:45.920 ","End":"00:49.400","Text":"So we know that the products are independent of each other."},{"Start":"00:49.400 ","End":"00:51.125","Text":"Now, what are we counting?"},{"Start":"00:51.125 ","End":"01:01.350","Text":"We\u0027re counting the number of faulty products in our sample."},{"Start":"01:01.370 ","End":"01:03.720","Text":"Now if that\u0027s the case,"},{"Start":"01:03.720 ","End":"01:13.295","Text":"then x has a binomial distribution where n equals 1,000 and p equals 0.05."},{"Start":"01:13.295 ","End":"01:16.220","Text":"Now, for all intent and purposes,"},{"Start":"01:16.220 ","End":"01:21.860","Text":"when you plug in these numbers into your calculator in order to"},{"Start":"01:21.860 ","End":"01:27.988","Text":"calculate the probability of x being equal to 45,"},{"Start":"01:27.988 ","End":"01:32.300","Text":"we want to know the probability of having 45 faulty products,"},{"Start":"01:32.300 ","End":"01:35.660","Text":"then the calculator will have a hard time because we\u0027re dealing"},{"Start":"01:35.660 ","End":"01:39.755","Text":"with numbers which are too large."},{"Start":"01:39.755 ","End":"01:44.015","Text":"Let\u0027s just use the Poisson approximation here."},{"Start":"01:44.015 ","End":"01:48.085","Text":"Now, when using the Poisson approximation,"},{"Start":"01:48.085 ","End":"01:50.765","Text":"all we have to do is just say this, well,"},{"Start":"01:50.765 ","End":"01:56.180","Text":"we have that Lambda here equals to n times p."},{"Start":"01:56.180 ","End":"02:02.004","Text":"Now n here is 1,000 and p equals 0.05,"},{"Start":"02:02.004 ","End":"02:06.845","Text":"so that means that we\u0027re dealing with Lambda equal to 50."},{"Start":"02:06.845 ","End":"02:12.830","Text":"Now let\u0027s take a look at the criteria for using the Poisson distribution."},{"Start":"02:12.830 ","End":"02:21.125","Text":"We say that the probability has to be less than 0.1 or 10 percent,"},{"Start":"02:21.125 ","End":"02:26.180","Text":"and n times p has to be greater or equal to 10."},{"Start":"02:26.180 ","End":"02:28.115","Text":"Well, in our case,"},{"Start":"02:28.115 ","End":"02:30.975","Text":"n times p is 50,"},{"Start":"02:30.975 ","End":"02:34.780","Text":"which is greater or equal to 10 by all means."},{"Start":"02:34.780 ","End":"02:39.310","Text":"Our probability is 0.05,"},{"Start":"02:39.310 ","End":"02:43.070","Text":"which is obviously less than 0.1."},{"Start":"02:43.070 ","End":"02:48.250","Text":"We are justified in using the Poisson distribution."},{"Start":"02:48.250 ","End":"02:54.905","Text":"Again, let\u0027s just use that we have here then x,"},{"Start":"02:54.905 ","End":"03:03.100","Text":"which has a Poisson approximation where Lambda equals n times p,"},{"Start":"03:03.100 ","End":"03:06.430","Text":"which is equal to 50."},{"Start":"03:06.430 ","End":"03:12.730","Text":"Now let\u0027s recall the probability of x being equal to k. In a Poisson distribution,"},{"Start":"03:12.730 ","End":"03:21.685","Text":"that\u0027s e to the minus Lambda times Lambda^k over k factorial."},{"Start":"03:21.685 ","End":"03:24.080","Text":"Let\u0027s just plug in the numbers."},{"Start":"03:24.080 ","End":"03:28.885","Text":"What\u0027s the probability of x equaling 45?"},{"Start":"03:28.885 ","End":"03:35.775","Text":"Well, that equals to e to the power of minus 50."},{"Start":"03:35.775 ","End":"03:42.665","Text":"Lambda now is 50 minus 50 times 50^45,"},{"Start":"03:42.665 ","End":"03:48.230","Text":"our k is 45, divided by 45 factorial."},{"Start":"03:48.230 ","End":"03:55.080","Text":"Now that comes out 0.0458."}],"ID":13050},{"Watched":false,"Name":"Exercise 3","Duration":"2m 49s","ChapterTopicVideoID":12572,"CourseChapterTopicPlaylistID":64490,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this question we\u0027re given that 1 percent of"},{"Start":"00:02.460 ","End":"00:05.654","Text":"a large population suffers from a given disease."},{"Start":"00:05.654 ","End":"00:10.140","Text":"Now 2,000 people are randomly sampled at the local health clinic."},{"Start":"00:10.140 ","End":"00:13.320","Text":"We\u0027re asked to use the Poisson approximation to"},{"Start":"00:13.320 ","End":"00:17.190","Text":"calculate the probability that 18 of them have the disease."},{"Start":"00:17.190 ","End":"00:21.225","Text":"Well, obviously we have here a binomial distribution"},{"Start":"00:21.225 ","End":"00:27.510","Text":"where we define success as being sick."},{"Start":"00:27.510 ","End":"00:30.630","Text":"Now what\u0027s the probability of being sick?"},{"Start":"00:30.630 ","End":"00:35.565","Text":"That\u0027s 0.01, that\u0027s 1 percent here."},{"Start":"00:35.565 ","End":"00:38.155","Text":"How many people are we sampling?"},{"Start":"00:38.155 ","End":"00:41.670","Text":"Well, we\u0027re sampling 2,000 people."},{"Start":"00:41.670 ","End":"00:45.920","Text":"Now the people are independent of each other obviously because we\u0027re"},{"Start":"00:45.920 ","End":"00:52.920","Text":"looking at a sample size from a very large population."},{"Start":"00:52.940 ","End":"01:02.080","Text":"Let\u0027s define x as the number of sick people in our sample."},{"Start":"01:03.020 ","End":"01:07.804","Text":"Therefore, we can say that x has"},{"Start":"01:07.804 ","End":"01:16.340","Text":"a binomial distribution where n equals 2,000 and p equals 0.01."},{"Start":"01:16.820 ","End":"01:26.885","Text":"Now, whenever we have a very large sample size with a very small probability of success,"},{"Start":"01:26.885 ","End":"01:31.870","Text":"then we\u0027re justified in using the Poisson approximation."},{"Start":"01:31.870 ","End":"01:34.245","Text":"Let\u0027s just do that."},{"Start":"01:34.245 ","End":"01:39.035","Text":"Now we did define Lambda as being equal to"},{"Start":"01:39.035 ","End":"01:45.350","Text":"n times p so in our case that\u0027s 2,000 times 0.01,"},{"Start":"01:45.350 ","End":"01:48.484","Text":"that equals to 20."},{"Start":"01:48.484 ","End":"01:58.040","Text":"Again, we say that x has a Poisson approximation of Lambda being equal to 20."},{"Start":"01:58.040 ","End":"02:04.730","Text":"Now let\u0027s just recall what\u0027s the probability function for the Poisson distribution?"},{"Start":"02:04.730 ","End":"02:11.760","Text":"That\u0027s e^ minus Lambda times Lambda to the power of k divided by k factorial."},{"Start":"02:12.050 ","End":"02:18.535","Text":"In our case, we\u0027re looking for the probability of x being equal to 18."},{"Start":"02:18.535 ","End":"02:25.700","Text":"We want to know the probability of 18 of the people in the sample being sick."},{"Start":"02:25.700 ","End":"02:27.340","Text":"Well, that equals to,"},{"Start":"02:27.340 ","End":"02:28.985","Text":"let\u0027s plug in the numbers right here."},{"Start":"02:28.985 ","End":"02:31.190","Text":"That\u0027s e^ minus Lambda well,"},{"Start":"02:31.190 ","End":"02:36.770","Text":"that\u0027s minus 20 times 20^k,"},{"Start":"02:36.770 ","End":"02:42.050","Text":"to the power of 18 divided by 18 factorial That\u0027s our k!."},{"Start":"02:42.050 ","End":"02:45.570","Text":"That equals to 0.0844."}],"ID":13051},{"Watched":false,"Name":"Exercise 4","Duration":"3m 48s","ChapterTopicVideoID":12573,"CourseChapterTopicPlaylistID":64490,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.245","Text":"In this question, we\u0027re given that the population of New York is 9 million,"},{"Start":"00:04.245 ","End":"00:07.290","Text":"900,000 of which are Asians."},{"Start":"00:07.290 ","End":"00:11.760","Text":"We\u0027re asked to use the Poisson approximation to calculate the probability"},{"Start":"00:11.760 ","End":"00:16.795","Text":"that at least 2 of the 100 randomly selected New Yorkers are Asian."},{"Start":"00:16.795 ","End":"00:27.090","Text":"Well, here we see that we have a binomial distribution where success is defined as being"},{"Start":"00:27.090 ","End":"00:33.720","Text":"Asia and the probability of being Asian that\u0027s 900,000"},{"Start":"00:33.720 ","End":"00:41.405","Text":"over 9 million and that equals 0.1."},{"Start":"00:41.405 ","End":"00:43.880","Text":"Now, what else do we have here?"},{"Start":"00:43.880 ","End":"00:46.145","Text":"We have a sample."},{"Start":"00:46.145 ","End":"00:50.660","Text":"We\u0027re taking a sample size of 100 people and"},{"Start":"00:50.660 ","End":"00:55.235","Text":"when we have a small sample size from a large population,"},{"Start":"00:55.235 ","End":"00:59.640","Text":"we say that the people are independent to each other."},{"Start":"01:00.050 ","End":"01:09.400","Text":"Let\u0027s define x as the number of Asians in our sample."},{"Start":"01:10.310 ","End":"01:14.330","Text":"When we do that, we can say then that x has"},{"Start":"01:14.330 ","End":"01:22.295","Text":"a binomial distribution where n equals 100 and p equals 0.1."},{"Start":"01:22.295 ","End":"01:27.620","Text":"Now, we\u0027re asked to use the Poisson approximation."},{"Start":"01:27.620 ","End":"01:32.030","Text":"Let\u0027s see if we meet the criteria for the Poisson approximation."},{"Start":"01:32.030 ","End":"01:37.040","Text":"The criteria states that probability has to be less than or equal to"},{"Start":"01:37.040 ","End":"01:43.595","Text":"0.1 and the multiplication of n times p has to be greater or equal to 10."},{"Start":"01:43.595 ","End":"01:45.260","Text":"Now in our case,"},{"Start":"01:45.260 ","End":"01:48.170","Text":"our probability here is 0.1."},{"Start":"01:48.170 ","End":"01:50.345","Text":"We\u0027re right on the edge here."},{"Start":"01:50.345 ","End":"01:52.160","Text":"What about n times p,"},{"Start":"01:52.160 ","End":"01:57.710","Text":"where n is 100 and p is 0.1 and that equals to 10 so again,"},{"Start":"01:57.710 ","End":"02:00.005","Text":"we\u0027re right on the edge here."},{"Start":"02:00.005 ","End":"02:06.365","Text":"We can use the Poisson approximation. Let\u0027s do that."},{"Start":"02:06.365 ","End":"02:11.780","Text":"We define Lambda as n times p and that equals again to 100"},{"Start":"02:11.780 ","End":"02:19.370","Text":"times 0.1 and that equals to 10."},{"Start":"02:19.370 ","End":"02:20.585","Text":"If that\u0027s the case,"},{"Start":"02:20.585 ","End":"02:30.275","Text":"we say then that x has a Poisson approximation with parameter Lambda being equal to 10."},{"Start":"02:30.275 ","End":"02:37.010","Text":"Now let\u0027s just be called the probability distribution of a Poisson distribution"},{"Start":"02:37.010 ","End":"02:44.580","Text":"that\u0027s e to the power of minus Lambda times Lambda to the power of k over k factorial."},{"Start":"02:45.190 ","End":"02:50.195","Text":"Now, we\u0027re asked, what\u0027s the probability"},{"Start":"02:50.195 ","End":"02:56.850","Text":"of x being greater or equal to 2?"},{"Start":"02:56.850 ","End":"02:59.550","Text":"What do we want? We want that"},{"Start":"02:59.550 ","End":"03:03.290","Text":"at least 2 of the 100 randomly selected New Yorkers are"},{"Start":"03:03.290 ","End":"03:07.520","Text":"Asians so x has to be greater or equal to 2."},{"Start":"03:07.520 ","End":"03:11.600","Text":"Well, it\u0027s easier to calculate the complementary set here."},{"Start":"03:11.600 ","End":"03:19.885","Text":"That\u0027ll be equal to1 minus the probability of x being less than or equal to 1."},{"Start":"03:19.885 ","End":"03:28.610","Text":"Now, that equals to 1 minus the probability of x equaling 0 plus the probability of"},{"Start":"03:28.610 ","End":"03:33.270","Text":"x being equal to1 and that"},{"Start":"03:33.270 ","End":"03:39.105","Text":"comes out to 0.9995."},{"Start":"03:39.105 ","End":"03:48.690","Text":"This is the probability of having at least 2 Asians in a sample of 100 people."}],"ID":13052}],"Thumbnail":null,"ID":64490},{"Name":"The Discrete Random Variable - Summary Questions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"6m 19s","ChapterTopicVideoID":12574,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.615","Text":"Given that X has a binomial distribution with n equals 4 and p equals 1/2,"},{"Start":"00:06.615 ","End":"00:12.817","Text":"and Y also has a binomial distribution with n equals 10 and p equals 1/4,"},{"Start":"00:12.817 ","End":"00:19.050","Text":"we\u0027re asked in Section a to calculate the expectation and standard deviation of X."},{"Start":"00:19.050 ","End":"00:25.080","Text":"Well, if X is distributed with a binomial distribution,"},{"Start":"00:25.080 ","End":"00:27.780","Text":"then the expectation of X,"},{"Start":"00:27.780 ","End":"00:33.585","Text":"that would equal to n times p. In our case,"},{"Start":"00:33.585 ","End":"00:36.300","Text":"n equals 4,"},{"Start":"00:36.300 ","End":"00:42.190","Text":"that\u0027s right here, and p would equal to 1/2, that\u0027s right here."},{"Start":"00:42.410 ","End":"00:50.965","Text":"The expectation of X would be 4 times 1/2 and that would equal to 2."},{"Start":"00:50.965 ","End":"00:55.070","Text":"Now, what about the standard deviation of X?"},{"Start":"00:55.070 ","End":"01:01.685","Text":"Well, we know that the variance of X equals n times p times q,"},{"Start":"01:01.685 ","End":"01:05.730","Text":"where q is 1 minus p. Here,"},{"Start":"01:05.730 ","End":"01:07.305","Text":"let\u0027s just plug in the numbers,"},{"Start":"01:07.305 ","End":"01:11.640","Text":"that\u0027s n equals 4 times p is 1/2."},{"Start":"01:11.640 ","End":"01:14.535","Text":"Now, q is also 1/2,"},{"Start":"01:14.535 ","End":"01:20.530","Text":"it\u0027s 1 minus p. That will equal to 1."},{"Start":"01:20.530 ","End":"01:21.830","Text":"Now that\u0027s the variance,"},{"Start":"01:21.830 ","End":"01:23.690","Text":"that\u0027s not the standard deviation,"},{"Start":"01:23.690 ","End":"01:29.605","Text":"but we know that the standard deviation of X is the square root of the variance."},{"Start":"01:29.605 ","End":"01:33.220","Text":"That will equal 1 as well."},{"Start":"01:33.440 ","End":"01:39.650","Text":"In Section b, we\u0027re given that W equals 2X minus 4 and we\u0027re asked to calculate"},{"Start":"01:39.650 ","End":"01:45.395","Text":"the expectation and standard deviation of W. Let\u0027s just write this out again."},{"Start":"01:45.395 ","End":"01:49.595","Text":"W equals 2X minus 4."},{"Start":"01:49.595 ","End":"01:55.610","Text":"Now, just looks like a linear transformation of the random variable X"},{"Start":"01:55.610 ","End":"02:01.565","Text":"into W. Let\u0027s just recall what are the rules for linear transformation."},{"Start":"02:01.565 ","End":"02:08.360","Text":"The standard form for a linear transformation is ax plus b."},{"Start":"02:08.360 ","End":"02:12.035","Text":"Why is that? Because we want to identify a and b."},{"Start":"02:12.035 ","End":"02:15.020","Text":"Well, taking a look at this,"},{"Start":"02:15.020 ","End":"02:21.200","Text":"we can easily identify that a equals 2 and b equals minus 4."},{"Start":"02:21.200 ","End":"02:24.785","Text":"Now, what about the expectation of W?"},{"Start":"02:24.785 ","End":"02:31.570","Text":"Well, that equals to a times the expectation of X plus b."},{"Start":"02:31.570 ","End":"02:35.525","Text":"What about the standard deviation of W?"},{"Start":"02:35.525 ","End":"02:41.515","Text":"Well, that\u0027s the absolute value of a times the standard deviation of X."},{"Start":"02:41.515 ","End":"02:43.670","Text":"Once we recall this,"},{"Start":"02:43.670 ","End":"02:48.110","Text":"we can go ahead and solve for Section b."},{"Start":"02:48.110 ","End":"02:56.945","Text":"The expectation of W would be a times the expectation of X plus b."},{"Start":"02:56.945 ","End":"02:58.945","Text":"Well, a equals 2,"},{"Start":"02:58.945 ","End":"03:01.680","Text":"the expectation of X equals 2,"},{"Start":"03:01.680 ","End":"03:03.255","Text":"we figured that up here,"},{"Start":"03:03.255 ","End":"03:05.640","Text":"and b is minus 4."},{"Start":"03:05.640 ","End":"03:11.955","Text":"The expectation of W is 2 times 2 minus 4 and that equals to 0."},{"Start":"03:11.955 ","End":"03:15.005","Text":"What about the standard deviation of W?"},{"Start":"03:15.005 ","End":"03:17.600","Text":"Well, that\u0027s the absolute value of a."},{"Start":"03:17.600 ","End":"03:19.610","Text":"Again, a is 2,"},{"Start":"03:19.610 ","End":"03:22.795","Text":"so that\u0027s the absolute value of 2,"},{"Start":"03:22.795 ","End":"03:26.220","Text":"times the standard deviation of X."},{"Start":"03:26.220 ","End":"03:28.560","Text":"Well, standard deviation of X is 1,"},{"Start":"03:28.560 ","End":"03:31.215","Text":"so that equals to 2."},{"Start":"03:31.215 ","End":"03:34.425","Text":"This is the standard deviation of W,"},{"Start":"03:34.425 ","End":"03:40.845","Text":"and this is the expectation of W. In Section c,"},{"Start":"03:40.845 ","End":"03:44.660","Text":"we\u0027re given T being equal to X plus Y,"},{"Start":"03:44.660 ","End":"03:48.140","Text":"where X plus Y have the binomial distribution,"},{"Start":"03:48.140 ","End":"03:53.240","Text":"and we\u0027re asked to calculate the expectation of T. Also we\u0027re asked whether it\u0027s"},{"Start":"03:53.240 ","End":"03:58.545","Text":"possible to calculate standard deviation of T. Well, let\u0027s see."},{"Start":"03:58.545 ","End":"04:07.280","Text":"X is distributed with a binomial distribution where n equals 4 and p equals 1/2."},{"Start":"04:07.280 ","End":"04:18.090","Text":"Y is distributed with a binomial distribution where n equals 10 and p equals 1/4."},{"Start":"04:19.280 ","End":"04:23.915","Text":"We know what the expectation of X was,"},{"Start":"04:23.915 ","End":"04:26.860","Text":"that was n times p,"},{"Start":"04:26.860 ","End":"04:30.225","Text":"that equal to 4 times 1/2,"},{"Start":"04:30.225 ","End":"04:32.110","Text":"and that equal to 2."},{"Start":"04:32.110 ","End":"04:34.655","Text":"Now what\u0027s the expectation of Y?"},{"Start":"04:34.655 ","End":"04:37.625","Text":"Well, that\u0027s n times p again,"},{"Start":"04:37.625 ","End":"04:42.915","Text":"but that now n equals 10 and p equals 1/4,"},{"Start":"04:42.915 ","End":"04:48.450","Text":"so the expectation of Y equals 2.5."},{"Start":"04:49.610 ","End":"05:00.265","Text":"We\u0027re asked for the expectation of T. That equals to the expectation of X plus Y."},{"Start":"05:00.265 ","End":"05:03.885","Text":"Now, this is the expectation of a sum."},{"Start":"05:03.885 ","End":"05:05.900","Text":"From previous lessons,"},{"Start":"05:05.900 ","End":"05:11.450","Text":"we know that the expectation of a sum is the sum of the expectations."},{"Start":"05:11.450 ","End":"05:16.100","Text":"We have the expectation of X plus the expectation of Y."},{"Start":"05:16.100 ","End":"05:19.310","Text":"Now the expectation of X equals 2,"},{"Start":"05:19.310 ","End":"05:23.870","Text":"that\u0027s right here, plus the expectation of Y,"},{"Start":"05:23.870 ","End":"05:26.780","Text":"that\u0027s right here, that\u0027s 2.5."},{"Start":"05:26.780 ","End":"05:33.815","Text":"The expectation of the total or the sum of X and Y equals 4.5."},{"Start":"05:33.815 ","End":"05:39.140","Text":"Now, what about the standard deviation of T?"},{"Start":"05:39.140 ","End":"05:41.060","Text":"Can we calculate that?"},{"Start":"05:41.060 ","End":"05:43.430","Text":"Well, no, we can\u0027t."},{"Start":"05:43.430 ","End":"05:49.520","Text":"Why is that? Because we learned that the variance of the sum of"},{"Start":"05:49.520 ","End":"05:58.410","Text":"random variables equals to the sum of the variables under which conditions?"},{"Start":"05:58.410 ","End":"06:02.450","Text":"Where both X and Y are independent of each other,"},{"Start":"06:02.450 ","End":"06:05.380","Text":"and there\u0027s no mention of that right here."},{"Start":"06:05.380 ","End":"06:11.000","Text":"We don\u0027t know if X and Y are independent of each other or not, and as such,"},{"Start":"06:11.000 ","End":"06:17.045","Text":"we can\u0027t use this formula to calculate the variance and therefore,"},{"Start":"06:17.045 ","End":"06:19.740","Text":"the standard deviation of T."}],"ID":13053},{"Watched":false,"Name":"Exercise 2","Duration":"10m 43s","ChapterTopicVideoID":12575,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this question, we\u0027ll be talking about winning at slot machines."},{"Start":"00:04.005 ","End":"00:08.160","Text":"Now, Joe plays on 2 slot machines at a casino,"},{"Start":"00:08.160 ","End":"00:09.630","Text":"1 game on each machine,"},{"Start":"00:09.630 ","End":"00:11.745","Text":"machine A, and machine B."},{"Start":"00:11.745 ","End":"00:15.810","Text":"His chances of winning on machine A are 8 percent and"},{"Start":"00:15.810 ","End":"00:20.265","Text":"his chances of winning only on machine A are 5 percent."},{"Start":"00:20.265 ","End":"00:25.260","Text":"His chances of losing both games on a given day are 88 percent."},{"Start":"00:25.260 ","End":"00:26.715","Text":"Whereas in section A,"},{"Start":"00:26.715 ","End":"00:29.715","Text":"what is the chances of Joe winning both games?"},{"Start":"00:29.715 ","End":"00:31.140","Text":"Well, first of all,"},{"Start":"00:31.140 ","End":"00:35.765","Text":"let\u0027s define some events that event A"},{"Start":"00:35.765 ","End":"00:41.910","Text":"be the event of winning on machine A and event B,"},{"Start":"00:41.910 ","End":"00:45.235","Text":"there\u0027ll be a win on machine B."},{"Start":"00:45.235 ","End":"00:47.480","Text":"Now, what data are we given?"},{"Start":"00:47.480 ","End":"00:52.110","Text":"Well, we\u0027re given that the probability of event A,"},{"Start":"00:52.110 ","End":"00:54.925","Text":"that means a win on machine A,"},{"Start":"00:54.925 ","End":"00:58.025","Text":"well, that equals to 8 percent."},{"Start":"00:58.025 ","End":"01:04.160","Text":"The probability of winning only on machine A,"},{"Start":"01:04.160 ","End":"01:09.090","Text":"that means I win on A and I do not win on B."},{"Start":"01:09.090 ","End":"01:11.960","Text":"That\u0027s the meaning of only right here."},{"Start":"01:11.960 ","End":"01:15.830","Text":"Now, that equals to 0.05, that\u0027s 5 percent."},{"Start":"01:15.830 ","End":"01:19.250","Text":"The probability of losing both games,"},{"Start":"01:19.250 ","End":"01:22.850","Text":"that means that it\u0027s not A and not B."},{"Start":"01:22.850 ","End":"01:26.050","Text":"Well, that equals to 88 percent."},{"Start":"01:26.050 ","End":"01:29.910","Text":"We\u0027re asked, what is the chances of Joe winning both games?"},{"Start":"01:29.910 ","End":"01:36.080","Text":"We\u0027re asked, basically, what\u0027s the probability of A and B?"},{"Start":"01:36.080 ","End":"01:42.230","Text":"Now, let\u0027s see how we can represent all the probabilities,"},{"Start":"01:42.230 ","End":"01:44.030","Text":"not only these 3,"},{"Start":"01:44.030 ","End":"01:50.975","Text":"in such a way that I can answer this question or I can calculate this probability."},{"Start":"01:50.975 ","End":"01:53.210","Text":"The best way that I know how to do this,"},{"Start":"01:53.210 ","End":"01:54.530","Text":"I mean, there are many techniques,"},{"Start":"01:54.530 ","End":"01:59.990","Text":"but the most comfortable for me right now would be to use the probability matrix."},{"Start":"01:59.990 ","End":"02:06.960","Text":"Here it is. We can see that here we have A and not A,"},{"Start":"02:06.960 ","End":"02:10.090","Text":"and here we have B and not B."},{"Start":"02:10.130 ","End":"02:12.770","Text":"In the middle of the matrix, well,"},{"Start":"02:12.770 ","End":"02:17.020","Text":"these are all the combinations of A and not A, B and not B."},{"Start":"02:17.020 ","End":"02:19.500","Text":"Let\u0027s now fill in the matrix."},{"Start":"02:19.500 ","End":"02:23.160","Text":"Well, p of A is 0.8, here\u0027s p of A."},{"Start":"02:23.160 ","End":"02:26.775","Text":"Let\u0027s write this down, 0.08."},{"Start":"02:26.775 ","End":"02:31.620","Text":"What about the probability of A and not B?"},{"Start":"02:31.620 ","End":"02:32.820","Text":"Well, here\u0027s A,"},{"Start":"02:32.820 ","End":"02:34.230","Text":"and here\u0027s not B,"},{"Start":"02:34.230 ","End":"02:37.740","Text":"that\u0027s right here, so that\u0027s 0.05."},{"Start":"02:37.740 ","End":"02:40.440","Text":"What about not A and not B?"},{"Start":"02:40.440 ","End":"02:43.140","Text":"Well, that\u0027s right here, that\u0027s not A and that\u0027s not B,"},{"Start":"02:43.140 ","End":"02:45.050","Text":"so this will be the cell right here."},{"Start":"02:45.050 ","End":"02:47.795","Text":"That\u0027ll be 0.88."},{"Start":"02:47.795 ","End":"02:52.115","Text":"Since we know that other probabilities have to add up to 1,"},{"Start":"02:52.115 ","End":"02:55.740","Text":"let\u0027s fill out the rest of this matrix."},{"Start":"02:55.960 ","End":"03:01.890","Text":"0.88 plus 0.05, that\u0027s 0.93."},{"Start":"03:03.800 ","End":"03:08.165","Text":"Here there\u0027ll be 0.07."},{"Start":"03:08.165 ","End":"03:13.385","Text":"Here we\u0027ll have 0.03,"},{"Start":"03:13.385 ","End":"03:19.975","Text":"and here, we\u0027ll have 0.92."},{"Start":"03:19.975 ","End":"03:24.690","Text":"Here we have 0.04."},{"Start":"03:24.690 ","End":"03:29.715","Text":"Again, 0.04 plus 0.03, that\u0027s 0.07,"},{"Start":"03:29.715 ","End":"03:34.810","Text":"0.88 and 0.05 that equals to 0.93,"},{"Start":"03:34.850 ","End":"03:37.440","Text":"0.04 and 0.88, that\u0027s 0.92,"},{"Start":"03:37.440 ","End":"03:38.805","Text":"and so on and so forth."},{"Start":"03:38.805 ","End":"03:43.290","Text":"Everything adds up and the whole probabilities add up to 1."},{"Start":"03:43.310 ","End":"03:45.560","Text":"Now that we\u0027ve done this,"},{"Start":"03:45.560 ","End":"03:49.805","Text":"we can easily answer what\u0027s the probability of A and B?"},{"Start":"03:49.805 ","End":"03:52.730","Text":"A and B means that he wins on both games."},{"Start":"03:52.730 ","End":"03:56.466","Text":"Well, here\u0027s A and here\u0027s B,"},{"Start":"03:56.466 ","End":"04:01.190","Text":"so the probability of winning A and B,"},{"Start":"04:01.190 ","End":"04:02.630","Text":"but that\u0027s this cell right here,"},{"Start":"04:02.630 ","End":"04:06.965","Text":"and that equals to 0.03."},{"Start":"04:06.965 ","End":"04:12.420","Text":"He has a 3 percent chance of winning on both games."},{"Start":"04:12.550 ","End":"04:16.010","Text":"In section b, we\u0027re asked what is the expectation and"},{"Start":"04:16.010 ","End":"04:19.205","Text":"variance of the number of times Joe wins?"},{"Start":"04:19.205 ","End":"04:22.415","Text":"First of all, let\u0027s define a random variable."},{"Start":"04:22.415 ","End":"04:23.720","Text":"We\u0027ll call it x,"},{"Start":"04:23.720 ","End":"04:27.825","Text":"and that will be the number of wins that Joe has."},{"Start":"04:27.825 ","End":"04:30.330","Text":"Now, how many times can Joe win?"},{"Start":"04:30.330 ","End":"04:32.295","Text":"Well, he can win 0 times,"},{"Start":"04:32.295 ","End":"04:36.015","Text":"he win once or he can win twice on 2 machines."},{"Start":"04:36.015 ","End":"04:39.825","Text":"These are the values that x can have."},{"Start":"04:39.825 ","End":"04:45.670","Text":"Now let\u0027s try to figure out the probabilities for each 1 of the values of x."},{"Start":"04:45.670 ","End":"04:48.575","Text":"Let\u0027s just do this in a table form."},{"Start":"04:48.575 ","End":"04:51.865","Text":"That\u0027s x, that\u0027s the probability of x,"},{"Start":"04:51.865 ","End":"04:54.965","Text":"x can be 0, 1, or 2."},{"Start":"04:54.965 ","End":"04:58.775","Text":"What are the probabilities for each of the values here?"},{"Start":"04:58.775 ","End":"05:02.360","Text":"Well, let\u0027s look at the probability matrix that"},{"Start":"05:02.360 ","End":"05:08.630","Text":"we\u0027ve calculated in section a. Here it is."},{"Start":"05:08.630 ","End":"05:13.100","Text":"What\u0027s the probability of x equaling 0?"},{"Start":"05:13.100 ","End":"05:17.825","Text":"Well, x equaling 0 means that he didn\u0027t win on A and he didn\u0027t win on B."},{"Start":"05:17.825 ","End":"05:21.710","Text":"That\u0027s the probability of not A and not B."},{"Start":"05:21.710 ","End":"05:26.405","Text":"Well, that\u0027s right here. That\u0027s 0.88."},{"Start":"05:26.405 ","End":"05:30.935","Text":"What\u0027s the probability that he won on both machines?"},{"Start":"05:30.935 ","End":"05:34.790","Text":"We\u0027re looking at the probability of A and B."},{"Start":"05:34.790 ","End":"05:36.440","Text":"Well, that\u0027s this guy right here,"},{"Start":"05:36.440 ","End":"05:37.895","Text":"that\u0027s this cell right here."},{"Start":"05:37.895 ","End":"05:40.675","Text":"That\u0027s 0.03."},{"Start":"05:40.675 ","End":"05:46.565","Text":"Since the sum of the probabilities have to add up to 1,"},{"Start":"05:46.565 ","End":"05:49.685","Text":"well, the probability of x being equal to 1,"},{"Start":"05:49.685 ","End":"05:53.190","Text":"well, that\u0027s 0.09."},{"Start":"05:53.390 ","End":"06:02.135","Text":"Now we have basically x and the probabilities of x."},{"Start":"06:02.135 ","End":"06:06.800","Text":"We can start to calculate the expectation variance."},{"Start":"06:06.800 ","End":"06:10.070","Text":"Now, let\u0027s look at the expectation of x."},{"Start":"06:10.070 ","End":"06:16.260","Text":"The hazard defined, that\u0027s defined as the sum of x times the probability of x."},{"Start":"06:16.260 ","End":"06:17.870","Text":"Now, that equals to,"},{"Start":"06:17.870 ","End":"06:21.110","Text":"let\u0027s just plug in the numbers 0 times 0.88."},{"Start":"06:21.110 ","End":"06:22.850","Text":"Well, we don\u0027t have to write that down,"},{"Start":"06:22.850 ","End":"06:24.350","Text":"but we have this,"},{"Start":"06:24.350 ","End":"06:30.695","Text":"1 times 0.09 plus 2 times 0.03,"},{"Start":"06:30.695 ","End":"06:34.175","Text":"and that equals to 0.15."},{"Start":"06:34.175 ","End":"06:35.435","Text":"What are the units?"},{"Start":"06:35.435 ","End":"06:39.260","Text":"Wins. This is 0.15 wins,"},{"Start":"06:39.260 ","End":"06:41.555","Text":"that\u0027s the expectation of x."},{"Start":"06:41.555 ","End":"06:44.300","Text":"What about the variance of x?"},{"Start":"06:44.300 ","End":"06:48.410","Text":"Well, that\u0027s defined as the sum of x squared"},{"Start":"06:48.410 ","End":"06:54.025","Text":"times the probabilities minus the expectation squared of x."},{"Start":"06:54.025 ","End":"06:56.810","Text":"Now again, 0 squared times this."},{"Start":"06:56.810 ","End":"06:57.875","Text":"We won\u0027t write that down,"},{"Start":"06:57.875 ","End":"06:59.030","Text":"but we\u0027ll start here."},{"Start":"06:59.030 ","End":"07:04.934","Text":"There\u0027ll be 1 squared times 0.09 plus"},{"Start":"07:04.934 ","End":"07:11.550","Text":"2 squared times 0.03 minus 0.15 squared."},{"Start":"07:11.550 ","End":"07:17.160","Text":"Now that equals to 0.1875."},{"Start":"07:17.160 ","End":"07:18.530","Text":"What are the units here?"},{"Start":"07:18.530 ","End":"07:22.200","Text":"It\u0027s wins squared."},{"Start":"07:22.700 ","End":"07:25.380","Text":"In Section C we\u0027re asked,"},{"Start":"07:25.380 ","End":"07:32.040","Text":"if Joe enters the casino 5 times and 5 separate days and place 2 games each time,"},{"Start":"07:32.040 ","End":"07:33.995","Text":"2 games in each day."},{"Start":"07:33.995 ","End":"07:40.585","Text":"What\u0027s the probability that he will win both games in only 1 of the 5 times?"},{"Start":"07:40.585 ","End":"07:42.380","Text":"Let\u0027s get started here."},{"Start":"07:42.380 ","End":"07:48.050","Text":"We have here a binomial probability distribution, why is that?"},{"Start":"07:48.050 ","End":"07:52.685","Text":"Well, let\u0027s take a look at the conditions of a binomial distribution."},{"Start":"07:52.685 ","End":"07:55.755","Text":"Well, here are the conditions."},{"Start":"07:55.755 ","End":"07:58.425","Text":"We have the same Bernoulli trial."},{"Start":"07:58.425 ","End":"08:00.470","Text":"What\u0027s the Bernoulli trial in our case?"},{"Start":"08:00.470 ","End":"08:05.365","Text":"That\u0027s Joe entering the casino on each separate day."},{"Start":"08:05.365 ","End":"08:07.730","Text":"That\u0027s repeated independently."},{"Start":"08:07.730 ","End":"08:13.055","Text":"Well, no day is dependent on the previous day. They\u0027re independent."},{"Start":"08:13.055 ","End":"08:15.950","Text":"The trial is repeated n times where, yes,"},{"Start":"08:15.950 ","End":"08:20.120","Text":"here we have entering the casino 5 times."},{"Start":"08:20.120 ","End":"08:23.860","Text":"What else do we know in a Bernoulli trial?"},{"Start":"08:23.860 ","End":"08:25.695","Text":"We have success and failure."},{"Start":"08:25.695 ","End":"08:29.295","Text":"What\u0027s our definition of success?"},{"Start":"08:29.295 ","End":"08:34.940","Text":"Well, our definition of success is a win in both games,"},{"Start":"08:34.940 ","End":"08:37.130","Text":"in 2 games,"},{"Start":"08:37.150 ","End":"08:43.055","Text":"and the probability of winning 2 games, but that\u0027s 0.03."},{"Start":"08:43.055 ","End":"08:45.650","Text":"Now, how did I come up with this number?"},{"Start":"08:45.650 ","End":"08:50.795","Text":"Well, I calculated it in the previous section in the probability matrix."},{"Start":"08:50.795 ","End":"08:55.550","Text":"Let\u0027s define y as the number of"},{"Start":"08:55.550 ","End":"09:02.460","Text":"days where I have 2 wins."},{"Start":"09:02.690 ","End":"09:05.565","Text":"If that\u0027s the case,"},{"Start":"09:05.565 ","End":"09:11.130","Text":"then y is distributed with a binomial distribution,"},{"Start":"09:11.130 ","End":"09:13.190","Text":"or what else did I forget?"},{"Start":"09:13.190 ","End":"09:15.635","Text":"I forgot that n here equals 5."},{"Start":"09:15.635 ","End":"09:18.470","Text":"That\u0027s the 5 times that he entered the casino,"},{"Start":"09:18.470 ","End":"09:28.300","Text":"so y has a binomial distribution where n equals 5 and p equals 0.03."},{"Start":"09:28.700 ","End":"09:35.075","Text":"Let\u0027s just remember the calculations for a binomial distribution."},{"Start":"09:35.075 ","End":"09:38.660","Text":"The probability of x equaling k,"},{"Start":"09:38.660 ","End":"09:42.490","Text":"but that equals to n over k,"},{"Start":"09:42.490 ","End":"09:48.860","Text":"p^k times 1 minus p^n minus k. Well,"},{"Start":"09:48.860 ","End":"09:55.445","Text":"in our case, we\u0027re looking for the probability of y being equal to 1."},{"Start":"09:55.445 ","End":"10:00.200","Text":"Now, why is that? Because we\u0027re looking for the probability that he will win"},{"Start":"10:00.200 ","End":"10:04.820","Text":"both games and we\u0027ll have a success in only 1 of the 5 days."},{"Start":"10:04.820 ","End":"10:06.740","Text":"Well, that\u0027s this 1 right here,"},{"Start":"10:06.740 ","End":"10:07.925","Text":"this 1 right here."},{"Start":"10:07.925 ","End":"10:09.605","Text":"That\u0027s this guy right here."},{"Start":"10:09.605 ","End":"10:12.375","Text":"Now, let\u0027s plug in the numbers."},{"Start":"10:12.375 ","End":"10:13.710","Text":"That\u0027s n over k,"},{"Start":"10:13.710 ","End":"10:16.560","Text":"that\u0027s 5 over 1."},{"Start":"10:16.560 ","End":"10:22.160","Text":"P is 0.03 so that\u0027s 0.03^k."},{"Start":"10:22.160 ","End":"10:25.010","Text":"That\u0027s 1 times 1 minus p,"},{"Start":"10:25.010 ","End":"10:27.245","Text":"that\u0027s 1 minus 0.03,"},{"Start":"10:27.245 ","End":"10:34.440","Text":"that\u0027s 0.97^5 minus k. 5 minus 1,"},{"Start":"10:34.440 ","End":"10:40.160","Text":"that\u0027s 4, and that equals to 0.1328."}],"ID":13054},{"Watched":false,"Name":"Exercise 3","Duration":"10m 40s","ChapterTopicVideoID":12576,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"In this question, we\u0027ll be trying to open a door."},{"Start":"00:02.700 ","End":"00:05.775","Text":"Adam has a key ring with 5 keys,"},{"Start":"00:05.775 ","End":"00:09.135","Text":"where only 1 key fits the door to his home."},{"Start":"00:09.135 ","End":"00:13.500","Text":"He tries to open his door by choosing 1 key after another at random."},{"Start":"00:13.500 ","End":"00:15.090","Text":"After trying a given key,"},{"Start":"00:15.090 ","End":"00:18.465","Text":"he takes it off the ring so that he will not use it again."},{"Start":"00:18.465 ","End":"00:23.715","Text":"Now let X be the number of attempts it takes him until he opens the door."},{"Start":"00:23.715 ","End":"00:28.980","Text":"In section a we\u0027re asked to construct the probability function of X."},{"Start":"00:28.980 ","End":"00:31.005","Text":"Let\u0027s get started."},{"Start":"00:31.005 ","End":"00:33.135","Text":"In his first try,"},{"Start":"00:33.135 ","End":"00:36.390","Text":"that\u0027s X equal 1,"},{"Start":"00:36.390 ","End":"00:42.915","Text":"he either succeeds, that\u0027s a plus or he doesn\u0027t succeed, that\u0027s a minus."},{"Start":"00:42.915 ","End":"00:44.600","Text":"How many keys are there?"},{"Start":"00:44.600 ","End":"00:48.515","Text":"There are 5 keys and only 1 key fits the door."},{"Start":"00:48.515 ","End":"00:52.355","Text":"The probability of him succeeding,"},{"Start":"00:52.355 ","End":"00:56.510","Text":"of choosing the right key at random is 1 over 5."},{"Start":"00:56.510 ","End":"00:58.205","Text":"If he doesn\u0027t succeed,"},{"Start":"00:58.205 ","End":"01:00.785","Text":"that\u0027s 4 over 5."},{"Start":"01:00.785 ","End":"01:03.650","Text":"If he doesn\u0027t succeed, he\u0027ll try again."},{"Start":"01:03.650 ","End":"01:08.415","Text":"That means that X here will equal to 2."},{"Start":"01:08.415 ","End":"01:12.310","Text":"Again, if he succeeds, how many keys are left?"},{"Start":"01:12.310 ","End":"01:14.035","Text":"Well, 4 keys are left."},{"Start":"01:14.035 ","End":"01:18.115","Text":"If he succeeds, that\u0027ll be 1 over 4 and if he doesn\u0027t succeed,"},{"Start":"01:18.115 ","End":"01:20.730","Text":"that\u0027ll be 3 over 4,"},{"Start":"01:20.730 ","End":"01:24.740","Text":"and so on and so forth until he does open the door."},{"Start":"01:24.740 ","End":"01:28.260","Text":"Let\u0027s see how the whole tree looks."},{"Start":"01:29.560 ","End":"01:32.885","Text":"Here\u0027s the whole tree where again,"},{"Start":"01:32.885 ","End":"01:34.715","Text":"in his first try,"},{"Start":"01:34.715 ","End":"01:38.240","Text":"he can succeed with a probability of 1 over 5,"},{"Start":"01:38.240 ","End":"01:41.450","Text":"and he fails with a probability of 4 over 5."},{"Start":"01:41.450 ","End":"01:43.745","Text":"That\u0027s a plus and that\u0027s a minus right here."},{"Start":"01:43.745 ","End":"01:45.305","Text":"In his second try,"},{"Start":"01:45.305 ","End":"01:47.620","Text":"there are 4 keys."},{"Start":"01:47.620 ","End":"01:52.170","Text":"He can succeed with a probability of 1 over 4,"},{"Start":"01:52.170 ","End":"01:56.820","Text":"and he doesn\u0027t succeed with a probability of 3 over 4."},{"Start":"01:56.820 ","End":"01:59.190","Text":"In his third try,"},{"Start":"01:59.190 ","End":"02:01.740","Text":"there are 3 keys now."},{"Start":"02:01.740 ","End":"02:10.410","Text":"He can succeed with a probability of 1 over 3 and fail with a probability of 2 over 3."},{"Start":"02:10.410 ","End":"02:14.040","Text":"His fourth try X equals 4,"},{"Start":"02:14.040 ","End":"02:16.125","Text":"there are only 2 keys left."},{"Start":"02:16.125 ","End":"02:20.585","Text":"He can succeed or fail with a probability of a half,"},{"Start":"02:20.585 ","End":"02:22.445","Text":"and on his fifth trial,"},{"Start":"02:22.445 ","End":"02:23.810","Text":"well, he has to succeed."},{"Start":"02:23.810 ","End":"02:28.820","Text":"That\u0027s the only key left and that will be the key that will open the door."},{"Start":"02:28.820 ","End":"02:31.040","Text":"X, as we can see,"},{"Start":"02:31.040 ","End":"02:33.980","Text":"can have the values of 1,"},{"Start":"02:33.980 ","End":"02:36.095","Text":"2, 3, 4, and 5."},{"Start":"02:36.095 ","End":"02:38.225","Text":"Let\u0027s write that down."},{"Start":"02:38.225 ","End":"02:43.105","Text":"X can have the values of 1,"},{"Start":"02:43.105 ","End":"02:48.365","Text":"2, 3, 4, and 5."},{"Start":"02:48.365 ","End":"02:51.410","Text":"This is the probability of X."},{"Start":"02:51.410 ","End":"02:56.580","Text":"What\u0027s the probability now of X equaling 1?"},{"Start":"02:56.580 ","End":"02:58.600","Text":"When X equals 1,"},{"Start":"02:58.600 ","End":"03:01.576","Text":"that means that he succeeds on the first try,"},{"Start":"03:01.576 ","End":"03:03.995","Text":"so this is this branch right here."},{"Start":"03:03.995 ","End":"03:07.470","Text":"The probability is 1 over 5."},{"Start":"03:08.570 ","End":"03:12.760","Text":"What\u0027s the probability of X equaling 2?"},{"Start":"03:12.760 ","End":"03:16.165","Text":"But that means he succeeds on the second try."},{"Start":"03:16.165 ","End":"03:20.830","Text":"That means he has to fail in the first try and succeed in the second try."},{"Start":"03:20.830 ","End":"03:23.455","Text":"That\u0027ll be 4 over 5,"},{"Start":"03:23.455 ","End":"03:26.470","Text":"that\u0027s this probability right here, that\u0027s this branch,"},{"Start":"03:26.470 ","End":"03:28.540","Text":"and he succeeds on the second try,"},{"Start":"03:28.540 ","End":"03:30.370","Text":"so this is this branch right here,"},{"Start":"03:30.370 ","End":"03:35.950","Text":"times 1 over 4 and that will equal to 1 over 5."},{"Start":"03:35.950 ","End":"03:38.500","Text":"Let\u0027s put that here as well."},{"Start":"03:38.500 ","End":"03:42.635","Text":"What about the probability of X equaling 3?"},{"Start":"03:42.635 ","End":"03:45.315","Text":"That means he succeeds on his third try."},{"Start":"03:45.315 ","End":"03:47.635","Text":"That means he fails in his first try,"},{"Start":"03:47.635 ","End":"03:51.220","Text":"fails on his second try and succeeds on his third try."},{"Start":"03:51.220 ","End":"03:56.325","Text":"That means that 4 over 5, that\u0027s this branch."},{"Start":"03:56.325 ","End":"03:58.980","Text":"If he doesn\u0027t succeed on a second try,"},{"Start":"03:58.980 ","End":"04:03.390","Text":"that\u0027s times 3 over 4,"},{"Start":"04:03.390 ","End":"04:05.760","Text":"but he does succeed on his third try,"},{"Start":"04:05.760 ","End":"04:07.470","Text":"times 1 over 3."},{"Start":"04:07.470 ","End":"04:15.965","Text":"Again, these guys cancel out and we come up with the probability of 1 over 5."},{"Start":"04:15.965 ","End":"04:23.005","Text":"You know what, we\u0027ll get 1 over 5 for the other values of X as well."},{"Start":"04:23.005 ","End":"04:30.360","Text":"This is the probability function of X using the probability tree"},{"Start":"04:30.360 ","End":"04:35.010","Text":"to understand the logic"},{"Start":"04:35.010 ","End":"04:42.150","Text":"and we call this probability a uniform probability distribution."},{"Start":"04:42.650 ","End":"04:46.550","Text":"In section B, we\u0027re asked to calculate the expectation and"},{"Start":"04:46.550 ","End":"04:50.375","Text":"variance of X. Let\u0027s get to it."},{"Start":"04:50.375 ","End":"04:52.610","Text":"The expectation of X,"},{"Start":"04:52.610 ","End":"04:58.190","Text":"that\u0027s defined as the sum of X times its probability."},{"Start":"04:58.190 ","End":"05:02.150","Text":"That equals to 1 times 1 over 5,"},{"Start":"05:02.150 ","End":"05:05.500","Text":"plus 2 times 1 over 5,"},{"Start":"05:05.500 ","End":"05:08.735","Text":"plus 3 times 1 over 5,"},{"Start":"05:08.735 ","End":"05:10.685","Text":"and so on and so forth."},{"Start":"05:10.685 ","End":"05:13.914","Text":"That equals to 3."},{"Start":"05:13.914 ","End":"05:16.815","Text":"What about the variance of X?"},{"Start":"05:16.815 ","End":"05:22.240","Text":"That\u0027s defined as the sum of X squared times its probability,"},{"Start":"05:22.240 ","End":"05:26.410","Text":"minus the expectation squared of X."},{"Start":"05:26.410 ","End":"05:31.400","Text":"That equals to 1 squared times 1 over 5,"},{"Start":"05:31.400 ","End":"05:35.015","Text":"plus 2 squared times 1 over 5,"},{"Start":"05:35.015 ","End":"05:39.180","Text":"and so on and so forth minus 3 squared."},{"Start":"05:39.180 ","End":"05:42.495","Text":"That\u0027s the expectation square."},{"Start":"05:42.495 ","End":"05:48.360","Text":"That turns out to be equal to 2."},{"Start":"05:48.360 ","End":"05:54.665","Text":"This is the expectation and this is the variance of X."},{"Start":"05:54.665 ","End":"05:57.575","Text":"In section C we\u0027re given,"},{"Start":"05:57.575 ","End":"06:01.040","Text":"that each attempt to open the door takes half of a minute."},{"Start":"06:01.040 ","End":"06:03.350","Text":"We\u0027re asked what are the expectation and"},{"Start":"06:03.350 ","End":"06:06.840","Text":"variance of the total time needed to open the door?"},{"Start":"06:07.210 ","End":"06:14.165","Text":"Here we\u0027re dealing basically with the linear transformation. Why is that?"},{"Start":"06:14.165 ","End":"06:19.640","Text":"We\u0027re moving away from the random variable that talks about"},{"Start":"06:19.640 ","End":"06:24.850","Text":"the times that it takes Adam to open the door and we\u0027re moving from that,"},{"Start":"06:24.850 ","End":"06:27.945","Text":"to the total time needed."},{"Start":"06:27.945 ","End":"06:30.315","Text":"Time in minutes."},{"Start":"06:30.315 ","End":"06:32.780","Text":"We\u0027re talking about now,"},{"Start":"06:32.780 ","End":"06:38.105","Text":"a new random variable that talks about the total time in minutes"},{"Start":"06:38.105 ","End":"06:44.470","Text":"and we\u0027re talking about the expectation and variance of the time needed to open the door."},{"Start":"06:44.470 ","End":"06:48.520","Text":"Let\u0027s just write this out here."},{"Start":"06:49.370 ","End":"06:52.425","Text":"We\u0027re looking at X,"},{"Start":"06:52.425 ","End":"06:57.440","Text":"which is the number of times it takes to open"},{"Start":"06:57.440 ","End":"07:02.630","Text":"the door and we\u0027re moving from that to a new random variable,"},{"Start":"07:02.630 ","End":"07:03.935","Text":"let\u0027s call it Y,"},{"Start":"07:03.935 ","End":"07:14.050","Text":"which is the total number of minutes."},{"Start":"07:14.050 ","End":"07:17.060","Text":"When we\u0027re dealing with the linear transformation,"},{"Start":"07:17.060 ","End":"07:19.865","Text":"there are steps to go through in order to,"},{"Start":"07:19.865 ","End":"07:22.865","Text":"first of all, understand what the linear transformation is,"},{"Start":"07:22.865 ","End":"07:25.505","Text":"and then to calculate the expectation variance."},{"Start":"07:25.505 ","End":"07:28.420","Text":"Let\u0027s take a look at the steps."},{"Start":"07:28.420 ","End":"07:31.610","Text":"Here they are. In the first step,"},{"Start":"07:31.610 ","End":"07:34.790","Text":"we have to recognize that we\u0027re dealing with the linear transformation."},{"Start":"07:34.790 ","End":"07:36.215","Text":"I think we\u0027ve done that."},{"Start":"07:36.215 ","End":"07:40.610","Text":"We know that we move from the X random variable,"},{"Start":"07:40.610 ","End":"07:45.410","Text":"which deals with a number of times it takes until we open the door,"},{"Start":"07:45.410 ","End":"07:49.600","Text":"to the total number of minutes it takes to open the door."},{"Start":"07:49.600 ","End":"07:53.960","Text":"Let\u0027s write the transformation rule according to the data and the question."},{"Start":"07:53.960 ","End":"08:02.105","Text":"We say that the number of times for each time that Adam tries to open the door,"},{"Start":"08:02.105 ","End":"08:03.380","Text":"it takes half a minute,"},{"Start":"08:03.380 ","End":"08:06.680","Text":"so if we multiply 0.5,"},{"Start":"08:06.680 ","End":"08:08.420","Text":"that\u0027s half a minute,"},{"Start":"08:08.420 ","End":"08:09.770","Text":"the units here are in minutes,"},{"Start":"08:09.770 ","End":"08:13.885","Text":"so half a minute times the number of tries here,"},{"Start":"08:13.885 ","End":"08:16.695","Text":"well, that will equal to Y."},{"Start":"08:16.695 ","End":"08:19.325","Text":"According to the data,"},{"Start":"08:19.325 ","End":"08:21.470","Text":"this is a linear transformation."},{"Start":"08:21.470 ","End":"08:26.045","Text":"But, what\u0027s the general form of a linear transformation?"},{"Start":"08:26.045 ","End":"08:32.200","Text":"That\u0027s Y equals a, X plus b."},{"Start":"08:32.630 ","End":"08:34.760","Text":"What are we missing here?"},{"Start":"08:34.760 ","End":"08:36.680","Text":"We\u0027re missing here, the b,"},{"Start":"08:36.680 ","End":"08:41.415","Text":"so we can write this out by adding 0."},{"Start":"08:41.415 ","End":"08:45.585","Text":"When we do that, we\u0027ve brought this to the general form."},{"Start":"08:45.585 ","End":"08:53.160","Text":"Now we can identify that a equals 0.5 and b equals 0."},{"Start":"08:53.160 ","End":"08:55.610","Text":"Why do we need to do that?"},{"Start":"08:55.610 ","End":"09:02.760","Text":"Because we need to calculate the expectation of Y and the variance of Y."},{"Start":"09:02.760 ","End":"09:04.875","Text":"How do we do that?"},{"Start":"09:04.875 ","End":"09:06.405","Text":"Let\u0027s take again,"},{"Start":"09:06.405 ","End":"09:09.800","Text":"a look at the general form or"},{"Start":"09:09.800 ","End":"09:16.375","Text":"the general equations for the expectation and variance of a linear transformation."},{"Start":"09:16.375 ","End":"09:23.675","Text":"Here it is. The expectation of Y in a linear transformation is"},{"Start":"09:23.675 ","End":"09:31.520","Text":"equal to a times the expectation of X plus b and the variance of Y,"},{"Start":"09:31.520 ","End":"09:35.900","Text":"that equals to a squared times the variance of X."},{"Start":"09:35.900 ","End":"09:39.780","Text":"Let\u0027s get back into our question,"},{"Start":"09:39.780 ","End":"09:44.205","Text":"where y equals 0.5 times X plus 0."},{"Start":"09:44.205 ","End":"09:46.050","Text":"The expectation of y,"},{"Start":"09:46.050 ","End":"09:47.600","Text":"that\u0027ll be a,"},{"Start":"09:47.600 ","End":"09:51.530","Text":"a is 0.5 times the expectation of X."},{"Start":"09:51.530 ","End":"09:54.125","Text":"The expectation of X is 3,"},{"Start":"09:54.125 ","End":"09:55.925","Text":"we\u0027ve calculated that here."},{"Start":"09:55.925 ","End":"09:59.150","Text":"That\u0027s times 3 plus 0."},{"Start":"09:59.150 ","End":"10:01.940","Text":"0 is 0."},{"Start":"10:01.940 ","End":"10:06.965","Text":"This is the expectation of Y and that equals to 1.5."},{"Start":"10:06.965 ","End":"10:08.585","Text":"What are the units?"},{"Start":"10:08.585 ","End":"10:12.360","Text":"Minutes. What about the variance of Y?"},{"Start":"10:12.360 ","End":"10:14.410","Text":"That\u0027s a squared times the variance of X,"},{"Start":"10:14.410 ","End":"10:18.725","Text":"so that\u0027s 0.5 squared times the variance of X."},{"Start":"10:18.725 ","End":"10:20.980","Text":"The variance of X is 2."},{"Start":"10:20.980 ","End":"10:24.345","Text":"That equals to 0.5."},{"Start":"10:24.345 ","End":"10:26.575","Text":"What are the units here?"},{"Start":"10:26.575 ","End":"10:29.195","Text":"Minutes squared."},{"Start":"10:29.195 ","End":"10:32.690","Text":"Again, this is the expectation of Y."},{"Start":"10:32.690 ","End":"10:34.390","Text":"That\u0027s how long, on average,"},{"Start":"10:34.390 ","End":"10:39.980","Text":"it\u0027ll take him to open the door and this is the variance of Y."}],"ID":13055},{"Watched":false,"Name":"Exercise 4","Duration":"8m 8s","ChapterTopicVideoID":12577,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"In this question, we\u0027ll be talking about the number of malfunctions in a TV station."},{"Start":"00:05.100 ","End":"00:09.315","Text":"Now, the number of broadcasts malfunctions on Channel 1"},{"Start":"00:09.315 ","End":"00:13.820","Text":"has a Poisson distribution with the rate of 6 malfunctions per day."},{"Start":"00:13.820 ","End":"00:15.870","Text":"We\u0027re asked in section A,"},{"Start":"00:15.870 ","End":"00:22.169","Text":"what\u0027s the probability of at least 1 malfunction on a given day?"},{"Start":"00:22.169 ","End":"00:24.840","Text":"Let\u0027s write down what we\u0027re given."},{"Start":"00:24.840 ","End":"00:32.620","Text":"First of all, let\u0027s define x as the number of malfunctions."},{"Start":"00:33.800 ","End":"00:38.300","Text":"We\u0027re given that x has"},{"Start":"00:38.300 ","End":"00:44.610","Text":"a Poisson distribution where Lambda with parameter Lambda being equal to 6."},{"Start":"00:44.610 ","End":"00:46.290","Text":"Why 6?"},{"Start":"00:46.290 ","End":"00:48.540","Text":"Well, that\u0027s 6 malfunctions per day."},{"Start":"00:48.540 ","End":"00:51.995","Text":"That\u0027s the rate of the malfunctions."},{"Start":"00:51.995 ","End":"00:59.990","Text":"Now, let\u0027s just recall the probability of x equaling k in a Poisson distribution that"},{"Start":"00:59.990 ","End":"01:03.665","Text":"will equal to e to the power of minus Lambda times"},{"Start":"01:03.665 ","End":"01:07.970","Text":"Lambda to the power of k over k factorial,"},{"Start":"01:07.970 ","End":"01:10.490","Text":"where k equals 0,"},{"Start":"01:10.490 ","End":"01:14.585","Text":"1, 2, and so on and so forth till infinity."},{"Start":"01:14.585 ","End":"01:18.455","Text":"In our case, what are we asked?"},{"Start":"01:18.455 ","End":"01:23.300","Text":"Well, we\u0027re asked what\u0027s the probability of at least 1 malfunction in a given day."},{"Start":"01:23.300 ","End":"01:29.530","Text":"That means that we\u0027re looking for the probability of x being greater or equal to 1."},{"Start":"01:29.530 ","End":"01:32.655","Text":"Now, why greater or equal to 1?"},{"Start":"01:32.655 ","End":"01:35.095","Text":"That\u0027s at least 1 malfunction,"},{"Start":"01:35.095 ","End":"01:37.535","Text":"it could be 1, it could be 2, and so on and so forth."},{"Start":"01:37.535 ","End":"01:41.915","Text":"But because the value of x can go until infinity,"},{"Start":"01:41.915 ","End":"01:47.300","Text":"it\u0027s easier for us to calculate the complimentary probability here,"},{"Start":"01:47.300 ","End":"01:52.820","Text":"that will be 1 minus the probability of x being equal to 0."},{"Start":"01:52.820 ","End":"01:55.895","Text":"Now, in our case,"},{"Start":"01:55.895 ","End":"01:57.680","Text":"that\u0027s 1 minus,"},{"Start":"01:57.680 ","End":"02:00.550","Text":"now what\u0027s the probability of x being equal to 0?"},{"Start":"02:00.550 ","End":"02:04.160","Text":"That\u0027s e to the power of minus 6,"},{"Start":"02:04.160 ","End":"02:06.980","Text":"minus Lambda, times Lambda to the power of k,"},{"Start":"02:06.980 ","End":"02:13.490","Text":"that\u0027s 6 to the power of k here 0 divided by 0 factorial,"},{"Start":"02:13.490 ","End":"02:20.525","Text":"k factorial, and that comes out to 0.9975."},{"Start":"02:20.525 ","End":"02:27.750","Text":"That\u0027s the probability of getting at least 1 malfunction on a given day."},{"Start":"02:27.760 ","End":"02:33.005","Text":"In section B, we\u0027re asked what\u0027s the probability that during 1 week,"},{"Start":"02:33.005 ","End":"02:35.045","Text":"that\u0027s 7 broadcasting days,"},{"Start":"02:35.045 ","End":"02:40.880","Text":"at least 1 malfunction occurs on exactly 6 of the 7 days?"},{"Start":"02:40.880 ","End":"02:44.600","Text":"Here we have a binomial distribution, and why is that?"},{"Start":"02:44.600 ","End":"02:49.670","Text":"Well, let\u0027s look at the conditions of a binomial distribution and see if we meet them."},{"Start":"02:49.670 ","End":"02:53.990","Text":"Here are the conditions where the first"},{"Start":"02:53.990 ","End":"02:58.145","Text":"one says that the same Bernoulli trial is repeated independently,"},{"Start":"02:58.145 ","End":"03:01.115","Text":"the second is that the trial is repeated n times,"},{"Start":"03:01.115 ","End":"03:05.690","Text":"and we define x as that total number of successes obtained."},{"Start":"03:05.690 ","End":"03:07.415","Text":"Well, in our case,"},{"Start":"03:07.415 ","End":"03:10.820","Text":"a Bernoulli trial would be each day."},{"Start":"03:10.820 ","End":"03:14.015","Text":"Now, each day is independent of the last day."},{"Start":"03:14.015 ","End":"03:18.980","Text":"Now, what success in a Bernoulli trial?"},{"Start":"03:18.980 ","End":"03:27.150","Text":"Well, in our case success is having at least 1 malfunction."},{"Start":"03:27.650 ","End":"03:31.670","Text":"That\u0027s it, that\u0027s malfunction."},{"Start":"03:31.670 ","End":"03:34.310","Text":"Now what\u0027s the probability of the success?"},{"Start":"03:34.310 ","End":"03:40.865","Text":"Well, we\u0027ve calculated that in section a, that\u0027s 0.9975."},{"Start":"03:40.865 ","End":"03:46.085","Text":"That\u0027s the probability of having at least 1 malfunction per day."},{"Start":"03:46.085 ","End":"03:50.555","Text":"Now, how many times are we repeating this trial?"},{"Start":"03:50.555 ","End":"03:52.760","Text":"Well, we have 7 broadcasting days,"},{"Start":"03:52.760 ","End":"03:55.640","Text":"so n here equals to 7."},{"Start":"03:55.640 ","End":"04:00.080","Text":"Now, let\u0027s define the random variable. We\u0027ll use y."},{"Start":"04:00.080 ","End":"04:06.720","Text":"Now that would be the number of successful days."},{"Start":"04:10.420 ","End":"04:15.200","Text":"These are the conditions for the binomial distribution."},{"Start":"04:15.200 ","End":"04:20.900","Text":"That means that y here has a binomial distribution where n"},{"Start":"04:20.900 ","End":"04:28.460","Text":"equals 7 and the probability is 0.9975."},{"Start":"04:28.720 ","End":"04:39.050","Text":"Let\u0027s recall that the probability of y being equal to k in a binomial distribution, well,"},{"Start":"04:39.050 ","End":"04:41.350","Text":"that\u0027s n over k,"},{"Start":"04:41.350 ","End":"04:48.020","Text":"p to the power of k times 1 minus p to the power of n minus k. Again,"},{"Start":"04:48.020 ","End":"04:55.175","Text":"in our case, we\u0027re looking at the probability of y being equal to what?"},{"Start":"04:55.175 ","End":"05:00.560","Text":"Now here it says that we\u0027re looking for the probability that we have"},{"Start":"05:00.560 ","End":"05:06.320","Text":"at least 1 malfunction that occurs on exactly 6 of the 7 days."},{"Start":"05:06.320 ","End":"05:10.160","Text":"So y here has to equal to 6."},{"Start":"05:10.160 ","End":"05:12.560","Text":"It says exactly 6."},{"Start":"05:12.560 ","End":"05:14.300","Text":"Now, if that\u0027s the case,"},{"Start":"05:14.300 ","End":"05:15.995","Text":"let\u0027s just plug in the numbers,"},{"Start":"05:15.995 ","End":"05:18.690","Text":"n here equals 7,"},{"Start":"05:18.690 ","End":"05:22.179","Text":"so that\u0027s 7 over k, k is 6,"},{"Start":"05:22.179 ","End":"05:31.250","Text":"times the probability, that\u0027s times 0.9975 to the power of k,"},{"Start":"05:31.250 ","End":"05:40.760","Text":"to the power of 6 times 1 minus 0.9975 to the power of n minus k,"},{"Start":"05:40.760 ","End":"05:42.140","Text":"well, that\u0027s 7 minus 6,"},{"Start":"05:42.140 ","End":"05:49.890","Text":"that\u0027s 1 and that equals to 0.0172."},{"Start":"05:51.080 ","End":"05:56.240","Text":"In section C, we\u0027re asked what\u0027s the expectation of the number of days from this day"},{"Start":"05:56.240 ","End":"06:00.755","Text":"until the first day on which at least 1 malfunction occurs?"},{"Start":"06:00.755 ","End":"06:03.950","Text":"Well, here we have a geometric distribution."},{"Start":"06:03.950 ","End":"06:05.360","Text":"Now, let\u0027s see why."},{"Start":"06:05.360 ","End":"06:10.250","Text":"Let\u0027s look at the conditions of the geometric distribution see if we meet them."},{"Start":"06:10.250 ","End":"06:12.560","Text":"Here are the conditions."},{"Start":"06:12.560 ","End":"06:14.090","Text":"Now, the first condition"},{"Start":"06:14.090 ","End":"06:18.260","Text":"says that we\u0027re looking at the same Bernoulli trial that\u0027s repeated"},{"Start":"06:18.260 ","End":"06:21.980","Text":"independently and we\u0027re looking at x as being"},{"Start":"06:21.980 ","End":"06:26.980","Text":"defined as the number of trials carried out until the first success."},{"Start":"06:26.980 ","End":"06:29.975","Text":"When we have a Bernoulli trial,"},{"Start":"06:29.975 ","End":"06:31.610","Text":"we have success and failures."},{"Start":"06:31.610 ","End":"06:35.450","Text":"Success in our case, well,"},{"Start":"06:35.450 ","End":"06:45.810","Text":"that would be a day with at least 1 malfunction."},{"Start":"06:46.860 ","End":"06:51.940","Text":"Now, what\u0027s the probability of this day happening?"},{"Start":"06:51.940 ","End":"06:55.480","Text":"Well, that\u0027s 0.9975."},{"Start":"06:55.480 ","End":"06:58.945","Text":"We calculated that in the previous sections."},{"Start":"06:58.945 ","End":"07:06.460","Text":"Let\u0027s define w, a new random variable as the number"},{"Start":"07:06.460 ","End":"07:13.930","Text":"of days until the first success."},{"Start":"07:13.930 ","End":"07:19.150","Text":"That\u0027s the number of days until we get at least 1 malfunction."},{"Start":"07:19.150 ","End":"07:21.010","Text":"If that\u0027s the case,"},{"Start":"07:21.010 ","End":"07:30.120","Text":"we say that w has a geometric distribution with the probability being 0.9975."},{"Start":"07:30.120 ","End":"07:36.560","Text":"If we recall the expectation of w,"},{"Start":"07:36.560 ","End":"07:41.315","Text":"the expectation of a random variable with a geometric distribution"},{"Start":"07:41.315 ","End":"07:46.520","Text":"that equals to 1 over p. In our case,"},{"Start":"07:46.520 ","End":"07:48.695","Text":"let\u0027s just see what we have here."},{"Start":"07:48.695 ","End":"07:53.730","Text":"Well, the expectation of w would be equal to 1 over p,"},{"Start":"07:53.730 ","End":"07:59.100","Text":"that equals to 1 over 0.9975,"},{"Start":"07:59.100 ","End":"08:04.870","Text":"and that equals to 1.0025 days."}],"ID":13056},{"Watched":false,"Name":"Exercise 5 Parts a-b","Duration":"7m 34s","ChapterTopicVideoID":12578,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.315","Text":"In this question, we\u0027ll be dealing with a retail store."},{"Start":"00:03.315 ","End":"00:07.110","Text":"Now, the owner of a large store in a shopping mall notices that"},{"Start":"00:07.110 ","End":"00:11.475","Text":"40 percent of the products in his store were purchased for children,"},{"Start":"00:11.475 ","End":"00:13.845","Text":"35 percent were purchased for women,"},{"Start":"00:13.845 ","End":"00:16.590","Text":"and 25 percent were purchased for men."},{"Start":"00:16.590 ","End":"00:19.830","Text":"Now, 10 percent of the products purchased for children,"},{"Start":"00:19.830 ","End":"00:23.040","Text":"and 60 percent of the products that were purchased for women,"},{"Start":"00:23.040 ","End":"00:27.900","Text":"and 50 percent of the products purchased from men were made outside of the USA,"},{"Start":"00:27.900 ","End":"00:29.400","Text":"they were foreign made."},{"Start":"00:29.400 ","End":"00:30.870","Text":"They were made abroad."},{"Start":"00:30.870 ","End":"00:33.000","Text":"Now, in section a we\u0027re asked,"},{"Start":"00:33.000 ","End":"00:36.210","Text":"what\u0027s the probability that a product sold in the store is"},{"Start":"00:36.210 ","End":"00:39.850","Text":"manufactured outside of the United States?"},{"Start":"00:39.850 ","End":"00:43.975","Text":"Well, in order to answer this question we need to describe our data."},{"Start":"00:43.975 ","End":"00:49.385","Text":"The best way to do that is through a probability tree. Why is that?"},{"Start":"00:49.385 ","End":"00:53.203","Text":"Well, here the data is presented to us in 2 levels,"},{"Start":"00:53.203 ","End":"00:56.562","Text":"the 1st level being the level of people,"},{"Start":"00:56.562 ","End":"00:59.420","Text":"women, men, and children."},{"Start":"00:59.420 ","End":"01:02.885","Text":"The 2nd level is the level of the products,"},{"Start":"01:02.885 ","End":"01:07.250","Text":"whether they were made in the USA or whether they were made abroad."},{"Start":"01:07.250 ","End":"01:08.780","Text":"If that\u0027s the case,"},{"Start":"01:08.780 ","End":"01:12.925","Text":"let\u0027s start constructing our probability tree."},{"Start":"01:12.925 ","End":"01:15.285","Text":"Here\u0027s the first level,"},{"Start":"01:15.285 ","End":"01:16.640","Text":"and again, as we can see,"},{"Start":"01:16.640 ","End":"01:18.395","Text":"we have a branch for children,"},{"Start":"01:18.395 ","End":"01:19.760","Text":"a branch for women,"},{"Start":"01:19.760 ","End":"01:21.530","Text":"and a branch for men."},{"Start":"01:21.530 ","End":"01:26.360","Text":"Now, we also have the probabilities,"},{"Start":"01:26.360 ","End":"01:33.125","Text":"40 percent of the products in the store were purchased for children,"},{"Start":"01:33.125 ","End":"01:35.390","Text":"35 for women, and 25 for men,"},{"Start":"01:35.390 ","End":"01:40.220","Text":"that\u0027s the 35 percent here and 25 percent here."},{"Start":"01:40.220 ","End":"01:42.970","Text":"Now, what about the 2nd level?"},{"Start":"01:42.970 ","End":"01:45.180","Text":"Here\u0027s the 2nd level."},{"Start":"01:45.180 ","End":"01:48.095","Text":"Again, let\u0027s just take a look, for example,"},{"Start":"01:48.095 ","End":"01:50.585","Text":"the children\u0027s branch right here, well,"},{"Start":"01:50.585 ","End":"01:54.620","Text":"10 percent of the products for children were made abroad."},{"Start":"01:54.620 ","End":"01:56.940","Text":"That\u0027s the 10 percent right here,"},{"Start":"01:56.940 ","End":"01:58.905","Text":"and there it is here,"},{"Start":"01:58.905 ","End":"02:02.645","Text":"and 60 percent for women that\u0027s here,"},{"Start":"02:02.645 ","End":"02:06.015","Text":"and 50 percent that\u0027s here."},{"Start":"02:06.015 ","End":"02:12.755","Text":"I\u0027ve marked all the probabilities of all the products that were made abroad."},{"Start":"02:12.755 ","End":"02:17.859","Text":"Now, let\u0027s just complete the probabilities for the products made in the USA."},{"Start":"02:17.859 ","End":"02:21.980","Text":"Well, for children, if 10 percent of the products were made abroad,"},{"Start":"02:21.980 ","End":"02:25.880","Text":"that means that 90 percent were made in the USA."},{"Start":"02:25.880 ","End":"02:29.705","Text":"For women, if 60 percent of the products were made abroad,"},{"Start":"02:29.705 ","End":"02:33.550","Text":"then that means that 40 percent were made in the USA,"},{"Start":"02:33.550 ","End":"02:40.739","Text":"and for men, if 50 percent were made abroad then 50 percent were made in the USA."},{"Start":"02:40.740 ","End":"02:46.620","Text":"If that\u0027s the case, we can go ahead and start solving section a."},{"Start":"02:46.760 ","End":"02:49.055","Text":"In section a, again,"},{"Start":"02:49.055 ","End":"02:51.980","Text":"we\u0027re asked what\u0027s the probability that a product sold in"},{"Start":"02:51.980 ","End":"02:55.325","Text":"the store is manufactured outside of the United States?"},{"Start":"02:55.325 ","End":"03:02.445","Text":"We\u0027re basically asking what\u0027s the probability that a product to sold abroad?"},{"Start":"03:02.445 ","End":"03:05.520","Text":"Now, what\u0027s that equal to?"},{"Start":"03:05.520 ","End":"03:09.050","Text":"If we look at our probability tree,"},{"Start":"03:09.050 ","End":"03:13.055","Text":"we see that we have this branch right here,"},{"Start":"03:13.055 ","End":"03:15.305","Text":"and this branch right here,"},{"Start":"03:15.305 ","End":"03:16.820","Text":"and this branch right here."},{"Start":"03:16.820 ","End":"03:20.905","Text":"Now, these are the branches that we\u0027re going to be interested in."},{"Start":"03:20.905 ","End":"03:23.685","Text":"How do we calculate this?"},{"Start":"03:23.685 ","End":"03:28.249","Text":"Well, we multiply the probabilities along the branches"},{"Start":"03:28.249 ","End":"03:33.445","Text":"and we sum the probabilities between the branches."},{"Start":"03:33.445 ","End":"03:39.328","Text":"This would be the branch that we\u0027ll be interested in for children,"},{"Start":"03:39.328 ","End":"03:41.300","Text":"and this will be the branch for women,"},{"Start":"03:41.300 ","End":"03:44.280","Text":"and this would be the branch for men."},{"Start":"03:45.650 ","End":"03:53.300","Text":"The probability of a product being made abroad, that would be,"},{"Start":"03:53.300 ","End":"03:56.270","Text":"for children now let\u0027s take a look at this branch,"},{"Start":"03:56.270 ","End":"03:58.070","Text":"that\u0027ll be 0.4,"},{"Start":"03:58.070 ","End":"04:02.665","Text":"40 percent times 0.1,10 percent of the products."},{"Start":"04:02.665 ","End":"04:04.740","Text":"Plus now for women,"},{"Start":"04:04.740 ","End":"04:10.770","Text":"that\u0027s 0.35, that\u0027s the 35 percent here times 0.6,"},{"Start":"04:10.770 ","End":"04:15.110","Text":"60 percent of the products being made abroad plus"},{"Start":"04:15.110 ","End":"04:21.510","Text":"25 percent of the men so that\u0027s 0.25 times 0.5."},{"Start":"04:21.510 ","End":"04:24.900","Text":"That\u0027s 25 times 50 percent."},{"Start":"04:24.900 ","End":"04:31.515","Text":"Now, that equals to 0.375."},{"Start":"04:31.515 ","End":"04:38.085","Text":"The products that are being sold in the store,"},{"Start":"04:38.085 ","End":"04:43.220","Text":"that\u0027s the probability of these products being made abroad."},{"Start":"04:43.220 ","End":"04:46.460","Text":"In section b, we\u0027re given that X is"},{"Start":"04:46.460 ","End":"04:48.830","Text":"the number of products sold in the store from when it"},{"Start":"04:48.830 ","End":"04:54.040","Text":"opens on Monday inclusive until the first local product to sold."},{"Start":"04:54.040 ","End":"04:57.895","Text":"We\u0027re asked what\u0027s the probability function of X?"},{"Start":"04:57.895 ","End":"05:01.925","Text":"This looks like a geometric distribution. Why is that?"},{"Start":"05:01.925 ","End":"05:04.265","Text":"Let\u0027s take a look at the conditions"},{"Start":"05:04.265 ","End":"05:07.700","Text":"of the geometric distribution and see if we meet them."},{"Start":"05:07.700 ","End":"05:10.208","Text":"Here are the conditions,"},{"Start":"05:10.208 ","End":"05:14.540","Text":"the 1st 1 says that we have the same Bernoulli trial that\u0027s repeated independently,"},{"Start":"05:14.540 ","End":"05:16.910","Text":"2nd, saying that X is defined as"},{"Start":"05:16.910 ","End":"05:20.380","Text":"the number of trials carried out until the first success."},{"Start":"05:20.380 ","End":"05:25.545","Text":"The Bernoulli trial that we have here is a sale."},{"Start":"05:25.545 ","End":"05:32.730","Text":"Success would be defined as"},{"Start":"05:32.730 ","End":"05:42.820","Text":"selling a local product because that\u0027s what we\u0027re interested in."},{"Start":"05:47.420 ","End":"05:50.450","Text":"In the previous section,"},{"Start":"05:50.450 ","End":"05:55.550","Text":"we\u0027ve calculated the probability of a foreign product."},{"Start":"05:55.550 ","End":"05:59.342","Text":"A local product would be 1 minus a foreign product,"},{"Start":"05:59.342 ","End":"06:02.940","Text":"so that\u0027s 1 minus 0.375,"},{"Start":"06:03.250 ","End":"06:08.060","Text":"and that would equal to 0.625."},{"Start":"06:08.060 ","End":"06:12.290","Text":"Now again, this would be the probability of"},{"Start":"06:12.290 ","End":"06:18.305","Text":"having a local product and this would be the probability of having a foreign product."},{"Start":"06:18.305 ","End":"06:21.020","Text":"Now, what about X?"},{"Start":"06:21.020 ","End":"06:28.860","Text":"Well, X is defined as the number of products sold,"},{"Start":"06:30.610 ","End":"06:38.190","Text":"until the first local product."},{"Start":"06:38.810 ","End":"06:40.995","Text":"If that\u0027s the case,"},{"Start":"06:40.995 ","End":"06:51.115","Text":"then X has a geometric probability where the parameter p is equal to 0.625."},{"Start":"06:51.115 ","End":"06:54.790","Text":"Now, let\u0027s just remind ourselves of"},{"Start":"06:54.790 ","End":"06:57.970","Text":"the probability distribution function that equals"},{"Start":"06:57.970 ","End":"07:03.565","Text":"to p times 1 minus p to the power of k minus 1."},{"Start":"07:03.565 ","End":"07:09.129","Text":"In our case, the probability of X being equal to k,"},{"Start":"07:09.129 ","End":"07:10.810","Text":"let\u0027s plug in the numbers."},{"Start":"07:10.810 ","End":"07:17.350","Text":"P is 0.625, so that\u0027s 0.625 times now"},{"Start":"07:17.350 ","End":"07:25.350","Text":"1 minus p is 0.375 to the power of k minus 1 so that\u0027s k minus 1."},{"Start":"07:25.350 ","End":"07:28.410","Text":"We want to leave it as a function,"},{"Start":"07:28.410 ","End":"07:34.440","Text":"so this would be the probability function of X."}],"ID":13057},{"Watched":false,"Name":"Exercise 5 Parts c-d","Duration":"6m 38s","ChapterTopicVideoID":12579,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.800","Text":"In Section C we\u0027re asked,"},{"Start":"00:01.800 ","End":"00:05.010","Text":"what\u0027s the expectation of the number of foreign-made products"},{"Start":"00:05.010 ","End":"00:09.360","Text":"sold until the first US manufactured product is sold?"},{"Start":"00:09.360 ","End":"00:11.550","Text":"Well, before answering this question,"},{"Start":"00:11.550 ","End":"00:13.710","Text":"let\u0027s just look at this thing graphically."},{"Start":"00:13.710 ","End":"00:17.160","Text":"Let\u0027s try to understand what we\u0027re expected to do here."},{"Start":"00:17.160 ","End":"00:18.855","Text":"In order to do that,"},{"Start":"00:18.855 ","End":"00:23.805","Text":"let\u0027s take a look at the stream of sales along the timeline."},{"Start":"00:23.805 ","End":"00:29.180","Text":"Well, we\u0027re told here that we"},{"Start":"00:29.180 ","End":"00:34.250","Text":"have to stop at the first US manufactured product that is sold."},{"Start":"00:34.250 ","End":"00:38.625","Text":"If this was the last product sold,"},{"Start":"00:38.625 ","End":"00:44.905","Text":"so that\u0027ll be the first product that\u0027s local."},{"Start":"00:44.905 ","End":"00:48.430","Text":"Now, we\u0027ll define that as x."},{"Start":"00:48.430 ","End":"00:51.350","Text":"But we\u0027ve already not only defined it here as x,"},{"Start":"00:51.350 ","End":"00:52.910","Text":"we defined it,"},{"Start":"00:52.910 ","End":"00:56.045","Text":"previously in the previous section."},{"Start":"00:56.045 ","End":"01:02.585","Text":"Now, if that\u0027s the first local product that\u0027s being sold,"},{"Start":"01:02.585 ","End":"01:09.630","Text":"that means that all the previous sales had been a foreign products."},{"Start":"01:11.330 ","End":"01:16.195","Text":"Now, how many sales of foreign products that we have?"},{"Start":"01:16.195 ","End":"01:19.220","Text":"We had x minus 1."},{"Start":"01:19.400 ","End":"01:21.640","Text":"What are we looking for?"},{"Start":"01:21.640 ","End":"01:25.570","Text":"We\u0027re looking for the expectation of exactly this number,"},{"Start":"01:25.570 ","End":"01:29.890","Text":"the expectation of the number of foreign made products sold until"},{"Start":"01:29.890 ","End":"01:35.160","Text":"the first US manufactured product that\u0027s been sold."},{"Start":"01:35.160 ","End":"01:39.250","Text":"This is the number that we\u0027re looking at and we\u0027re"},{"Start":"01:39.250 ","End":"01:44.890","Text":"looking to calculate the expectation of this number. Let\u0027s define it."},{"Start":"01:44.890 ","End":"01:51.765","Text":"Let\u0027s define y as a new random variable that equals to x minus 1."},{"Start":"01:51.765 ","End":"01:56.390","Text":"Now, this is a linear transformation of x."},{"Start":"01:56.390 ","End":"02:00.680","Text":"Let\u0027s take a look at the general form of a linear transformation."},{"Start":"02:00.680 ","End":"02:06.370","Text":"That\u0027s y equals to a times x plus b."},{"Start":"02:06.370 ","End":"02:11.420","Text":"We have here the same form as the general form here."},{"Start":"02:11.420 ","End":"02:19.040","Text":"We can easily identify a as being 1 and b as being minus 1."},{"Start":"02:19.040 ","End":"02:22.535","Text":"Now, what\u0027s the expectation of y?"},{"Start":"02:22.535 ","End":"02:29.000","Text":"Well, that equals to a times the expectation of x plus b."},{"Start":"02:29.000 ","End":"02:32.750","Text":"But what is the expectation of x?"},{"Start":"02:32.750 ","End":"02:35.300","Text":"As we said, we\u0027ve defined x previously."},{"Start":"02:35.300 ","End":"02:45.620","Text":"x was defined as having a geometrical distribution where the probability equals 0.625."},{"Start":"02:45.620 ","End":"02:49.940","Text":"What\u0027s the expectation of x?"},{"Start":"02:49.940 ","End":"02:53.105","Text":"That\u0027s equal to 1 over p,"},{"Start":"02:53.105 ","End":"02:57.920","Text":"that equals to 1 over 0.625."},{"Start":"02:57.920 ","End":"03:03.895","Text":"Now, let\u0027s get back to our case here, our linear transformation."},{"Start":"03:03.895 ","End":"03:06.330","Text":"The expectation of y,"},{"Start":"03:06.330 ","End":"03:08.570","Text":"that\u0027s what we\u0027re asked to find out."},{"Start":"03:08.570 ","End":"03:13.220","Text":"What\u0027s the expectation of the number forming products sold and so on and so forth."},{"Start":"03:13.220 ","End":"03:16.670","Text":"That\u0027s y, that equals to a,"},{"Start":"03:16.670 ","End":"03:20.150","Text":"a is 1 times the expectation of x."},{"Start":"03:20.150 ","End":"03:26.490","Text":"Well, that\u0027s 1 divided by 0.625 plus b,"},{"Start":"03:26.490 ","End":"03:28.330","Text":"where b is minus 1,"},{"Start":"03:28.330 ","End":"03:31.550","Text":"and that equals to 0.6."},{"Start":"03:31.550 ","End":"03:33.005","Text":"What\u0027s the units?"},{"Start":"03:33.005 ","End":"03:39.410","Text":"Products. The expectation of the number of"},{"Start":"03:39.410 ","End":"03:47.645","Text":"foreign-made products sold is 0.6 until we sell the first local product."},{"Start":"03:47.645 ","End":"03:53.240","Text":"In Section D, we\u0027re given that 7 products were sold in the store in Tuesday,"},{"Start":"03:53.240 ","End":"03:59.830","Text":"and we\u0027re asked what\u0027s the probability that exactly 3 of them are foreign made?"},{"Start":"03:59.830 ","End":"04:03.245","Text":"Here we have a binomial distribution."},{"Start":"04:03.245 ","End":"04:05.390","Text":"But let\u0027s take a look at the conditions for"},{"Start":"04:05.390 ","End":"04:08.730","Text":"the binomial distributions and see if we meet them."},{"Start":"04:09.710 ","End":"04:12.005","Text":"Let\u0027s take a look at them."},{"Start":"04:12.005 ","End":"04:16.625","Text":"This, we have the same Bernoulli trial that\u0027s repeated independently."},{"Start":"04:16.625 ","End":"04:22.160","Text":"The trials repeated n times and x is defined as the total number of successes obtained."},{"Start":"04:22.160 ","End":"04:24.875","Text":"What\u0027s a Bernoulli trial?"},{"Start":"04:24.875 ","End":"04:27.030","Text":"Well, that\u0027s a sale."},{"Start":"04:27.130 ","End":"04:29.990","Text":"What\u0027s our success?"},{"Start":"04:29.990 ","End":"04:34.670","Text":"We define success as"},{"Start":"04:34.670 ","End":"04:41.935","Text":"a foreign product that\u0027s been sold."},{"Start":"04:41.935 ","End":"04:46.720","Text":"What\u0027s the probability of selling a foreign product where"},{"Start":"04:46.720 ","End":"04:51.780","Text":"we\u0027ve calculated that in Section 8, that\u0027s 0.375."},{"Start":"04:51.780 ","End":"04:54.850","Text":"Now, this trials repeated n times."},{"Start":"04:54.850 ","End":"04:56.721","Text":"7 products were sold,"},{"Start":"04:56.721 ","End":"04:59.420","Text":"that means that n equals 7."},{"Start":"04:59.420 ","End":"05:05.260","Text":"Let\u0027s define a random variable w. That\u0027s the number"},{"Start":"05:05.260 ","End":"05:07.120","Text":"of foreign"},{"Start":"05:07.120 ","End":"05:16.680","Text":"products sold."},{"Start":"05:16.680 ","End":"05:17.800","Text":"If that\u0027s the case,"},{"Start":"05:17.800 ","End":"05:22.700","Text":"then w has a binomial distribution where n"},{"Start":"05:22.700 ","End":"05:29.320","Text":"equals 7 and p equals 0.375."},{"Start":"05:29.900 ","End":"05:36.950","Text":"Let\u0027s just remind ourselves of the distribution function for a binomial distribution."},{"Start":"05:36.950 ","End":"05:40.610","Text":"The probability of w being equal to k,"},{"Start":"05:40.610 ","End":"05:43.415","Text":"that equals to n over k,"},{"Start":"05:43.415 ","End":"05:48.845","Text":"p to the power of k times q to the power of n minus k. Now,"},{"Start":"05:48.845 ","End":"05:53.555","Text":"q is 1 minus p. In our case,"},{"Start":"05:53.555 ","End":"05:59.690","Text":"we\u0027re looking for the probability of w being equal to exactly 3,"},{"Start":"05:59.690 ","End":"06:02.095","Text":"that\u0027s equaling to 3."},{"Start":"06:02.095 ","End":"06:06.555","Text":"Let\u0027s just plug the numbers in, n equals 7."},{"Start":"06:06.555 ","End":"06:16.065","Text":"That\u0027s 7 over 3. k equals 3. What\u0027s p?"},{"Start":"06:16.065 ","End":"06:22.850","Text":"0.375 to the power of k, 3,"},{"Start":"06:22.850 ","End":"06:28.760","Text":"times 0.625 to the power of n minus k,"},{"Start":"06:28.760 ","End":"06:31.430","Text":"it was 7 minus 3, that\u0027s 4."},{"Start":"06:31.430 ","End":"06:35.970","Text":"Now, that will equal to 0.282."}],"ID":13058},{"Watched":false,"Name":"Exercise 6 Part a","Duration":"8m 18s","ChapterTopicVideoID":12580,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.734","Text":"In this question, we will be trying to sell movies."},{"Start":"00:03.734 ","End":"00:07.305","Text":"Now, a production company makes 3 movies, The Deer,"},{"Start":"00:07.305 ","End":"00:11.415","Text":"Never, and Sudden Death for a local television station."},{"Start":"00:11.415 ","End":"00:15.255","Text":"The production company is trying to sell these movies overseas."},{"Start":"00:15.255 ","End":"00:19.245","Text":"The probability of selling the movies overseas are as follows."},{"Start":"00:19.245 ","End":"00:23.670","Text":"The chances of selling the movie The Deer overseas are 0.6,"},{"Start":"00:23.670 ","End":"00:28.050","Text":"the chances of selling the movie Never overseas are 0.7,"},{"Start":"00:28.050 ","End":"00:32.685","Text":"and the chances of selling the movie Sudden Death overseas are 0.2."},{"Start":"00:32.685 ","End":"00:37.740","Text":"Each movie costs $500,000 to produce and generated"},{"Start":"00:37.740 ","End":"00:43.325","Text":"$200,000 of revenue per movie through a local television station."},{"Start":"00:43.325 ","End":"00:46.310","Text":"Now, if the movies are sold overseas,"},{"Start":"00:46.310 ","End":"00:51.335","Text":"each movie will bring revenue of $600,000 per movie."},{"Start":"00:51.335 ","End":"00:53.940","Text":"We\u0027re asked in Section A to construct"},{"Start":"00:53.940 ","End":"00:58.920","Text":"the probability function of the number of movies sold overseas."},{"Start":"00:59.150 ","End":"01:03.815","Text":"The first thing that we have to do is define a random variable."},{"Start":"01:03.815 ","End":"01:05.420","Text":"Now, what are we asked?"},{"Start":"01:05.420 ","End":"01:12.220","Text":"We\u0027re asked to construct the probability function of the number of movies sold overseas."},{"Start":"01:12.220 ","End":"01:15.885","Text":"Again, let\u0027s write that down."},{"Start":"01:15.885 ","End":"01:27.190","Text":"x, a random variable will be the number of movies sold overseas."},{"Start":"01:27.230 ","End":"01:29.885","Text":"Now, if that\u0027s the case,"},{"Start":"01:29.885 ","End":"01:33.080","Text":"how many movies can we sell overseas?"},{"Start":"01:33.080 ","End":"01:36.349","Text":"Well, we can sell up to 3 movies,"},{"Start":"01:36.349 ","End":"01:38.780","Text":"and we can also sell no movies at all."},{"Start":"01:38.780 ","End":"01:44.150","Text":"So x can have the values of 0,"},{"Start":"01:44.150 ","End":"01:46.230","Text":"1, 2,"},{"Start":"01:46.230 ","End":"01:55.040","Text":"and 3, and what we want to do right now is to find or calculate the probabilities of x,"},{"Start":"01:55.040 ","End":"01:57.245","Text":"where x equals these values."},{"Start":"01:57.245 ","End":"01:59.600","Text":"Now, how can we do this?"},{"Start":"01:59.600 ","End":"02:03.005","Text":"Well, this looks like a binomial distribution."},{"Start":"02:03.005 ","End":"02:06.740","Text":"But although it looks like a binomial distribution, it\u0027s not."},{"Start":"02:06.740 ","End":"02:14.510","Text":"Why is that? Because if we define our success as a movie being sold overseas,"},{"Start":"02:14.510 ","End":"02:21.500","Text":"then one of the conditions is that the probability for success would be the same."},{"Start":"02:21.500 ","End":"02:23.930","Text":"It\u0027ll be a constant,"},{"Start":"02:23.930 ","End":"02:25.790","Text":"it will be one number,"},{"Start":"02:25.790 ","End":"02:27.650","Text":"that\u0027ll be the probability."},{"Start":"02:27.650 ","End":"02:29.630","Text":"But here, we don\u0027t have that condition."},{"Start":"02:29.630 ","End":"02:33.665","Text":"We\u0027re not meeting that condition because for each movie,"},{"Start":"02:33.665 ","End":"02:38.960","Text":"we have different probabilities of it being sold overseas."},{"Start":"02:38.960 ","End":"02:42.140","Text":"We\u0027re going to have to do this the long way."},{"Start":"02:42.140 ","End":"02:46.700","Text":"Now, another thing that we need to do is we need to"},{"Start":"02:46.700 ","End":"02:52.790","Text":"assume that there\u0027s independence between the movies when they\u0027re sold overseas."},{"Start":"02:52.790 ","End":"02:54.290","Text":"That means that if I sell,"},{"Start":"02:54.290 ","End":"02:56.765","Text":"let\u0027s say, for example, The Deer,"},{"Start":"02:56.765 ","End":"02:59.210","Text":"then that will have no impact"},{"Start":"02:59.210 ","End":"03:05.285","Text":"whatsoever on the movie Never or Sudden Death when they\u0027re sold overseas."},{"Start":"03:05.285 ","End":"03:09.410","Text":"Otherwise, I won\u0027t be able to calculate the probabilities."},{"Start":"03:09.410 ","End":"03:11.720","Text":"Now, having said that,"},{"Start":"03:11.720 ","End":"03:16.790","Text":"let\u0027s see how we can actually calculate these probabilities."},{"Start":"03:16.790 ","End":"03:22.510","Text":"Well, the probability, let\u0027s say of x being equal to 3,"},{"Start":"03:22.510 ","End":"03:26.485","Text":"that means that all movies would be sold."},{"Start":"03:26.485 ","End":"03:29.910","Text":"That means that The Deer will be sold overseas and"},{"Start":"03:29.910 ","End":"03:33.135","Text":"Never will be sold overseas and Sudden Death will be sold overseas."},{"Start":"03:33.135 ","End":"03:38.915","Text":"Well, because we assumed independence,"},{"Start":"03:38.915 ","End":"03:43.280","Text":"then all we have to do is multiply these probabilities."},{"Start":"03:43.280 ","End":"03:47.255","Text":"That\u0027s 0.6 for The Deer times"},{"Start":"03:47.255 ","End":"03:53.620","Text":"0.74 for Never times 0.2 for Sudden Death,"},{"Start":"03:53.620 ","End":"04:00.395","Text":"and that will equal to 0.084."},{"Start":"04:00.395 ","End":"04:04.880","Text":"Now, this is this probability right here."},{"Start":"04:04.880 ","End":"04:12.770","Text":"Let\u0027s take a look at the other end of the spectrum where x equals 0."},{"Start":"04:12.770 ","End":"04:19.970","Text":"Well, that means that I don\u0027t sell any movie abroad or overseas."},{"Start":"04:19.970 ","End":"04:25.325","Text":"All I have to do is take the complimentary set of these probabilities."},{"Start":"04:25.325 ","End":"04:31.609","Text":"If I sold The Deer at a probability of 0.6 overseas,"},{"Start":"04:31.609 ","End":"04:34.880","Text":"then I do not sell it with a probability of"},{"Start":"04:34.880 ","End":"04:39.590","Text":"0.4 and so on and so forth for Never and Sudden Death."},{"Start":"04:39.590 ","End":"04:41.870","Text":"That\u0027ll be 0.4,"},{"Start":"04:41.870 ","End":"04:44.165","Text":"that\u0027s 1 minus 0.6,"},{"Start":"04:44.165 ","End":"04:48.795","Text":"times 0.3, that\u0027s 1 minus 0.7,"},{"Start":"04:48.795 ","End":"04:55.485","Text":"times 0.8, that\u0027s 1 minus 0.2,"},{"Start":"04:55.485 ","End":"05:03.010","Text":"and that equals to 0.096."},{"Start":"05:03.980 ","End":"05:10.715","Text":"Now let\u0027s take a look at the probability where x equals 1. Now what does that mean?"},{"Start":"05:10.715 ","End":"05:14.860","Text":"This means that I am selling only 1 movie abroad,"},{"Start":"05:14.860 ","End":"05:19.190","Text":"and it could be The Deer with a probability of 0.6 or"},{"Start":"05:19.190 ","End":"05:24.365","Text":"Never with a probability of 0.7 or Sudden Death with a probability of 0.2."},{"Start":"05:24.365 ","End":"05:26.755","Text":"Only one of them."},{"Start":"05:26.755 ","End":"05:29.150","Text":"What\u0027s this probability?"},{"Start":"05:29.150 ","End":"05:34.160","Text":"Well, if I sell The Deer with a probability of 0.6,"},{"Start":"05:34.160 ","End":"05:39.090","Text":"then I do not sell the movie Never and Sudden Death,"},{"Start":"05:39.090 ","End":"05:47.875","Text":"so that\u0027s times 0.3 times 0.8."},{"Start":"05:47.875 ","End":"05:52.925","Text":"Now, that\u0027s the probability of selling The Deer."},{"Start":"05:52.925 ","End":"05:56.705","Text":"Now what happens if I sell the movie Never?"},{"Start":"05:56.705 ","End":"06:06.740","Text":"Well, that will be 0.4 times 0.7 times 0.8."},{"Start":"06:06.740 ","End":"06:10.920","Text":"Now why is that? I do not sell the movie The Deer,"},{"Start":"06:10.920 ","End":"06:14.040","Text":"that\u0027s 1 minus 0.6, that\u0027s 0.4."},{"Start":"06:14.040 ","End":"06:18.230","Text":"I do sell the movie Never with a probability of"},{"Start":"06:18.230 ","End":"06:25.155","Text":"0.7 and I do not sell the movie Sudden Death with a probability of 0.8."},{"Start":"06:25.155 ","End":"06:28.620","Text":"Now, what happens if I do sell Sudden Death?"},{"Start":"06:28.620 ","End":"06:33.495","Text":"Well, I don\u0027t sell The Deer, that\u0027s 0.4,"},{"Start":"06:33.495 ","End":"06:37.245","Text":"times not selling Never, that\u0027s 0.3,"},{"Start":"06:37.245 ","End":"06:42.525","Text":"times selling Sudden Death, that\u0027s 0.2."},{"Start":"06:42.525 ","End":"06:50.050","Text":"Now, all this comes out to 0.392."},{"Start":"06:52.280 ","End":"06:55.670","Text":"Let\u0027s just input these numbers,"},{"Start":"06:55.670 ","End":"06:58.550","Text":"these probabilities into this table."},{"Start":"06:58.550 ","End":"07:01.790","Text":"Well, for x equals 0, that\u0027s this one,"},{"Start":"07:01.790 ","End":"07:05.260","Text":"so that\u0027s 0.096,"},{"Start":"07:05.260 ","End":"07:10.770","Text":"and for x equals 3, that\u0027s 0.084."},{"Start":"07:10.770 ","End":"07:17.335","Text":"For x equals 1, that\u0027s 0.392."},{"Start":"07:17.335 ","End":"07:23.210","Text":"Now, in order to calculate the probability of x equals 2,"},{"Start":"07:24.090 ","End":"07:26.740","Text":"well, I can do this the long way,"},{"Start":"07:26.740 ","End":"07:29.290","Text":"or I can do this,"},{"Start":"07:29.290 ","End":"07:31.765","Text":"I can do 1 minus"},{"Start":"07:31.765 ","End":"07:37.480","Text":"the sum of the probabilities because the sum of the probabilities have to add up to 1,"},{"Start":"07:37.480 ","End":"07:41.700","Text":"so it\u0027s 1 minus 0.084"},{"Start":"07:41.700 ","End":"07:49.160","Text":"plus 0.096 plus 0.392."},{"Start":"07:49.160 ","End":"07:53.960","Text":"That comes out to 0.428."},{"Start":"07:54.020 ","End":"07:58.540","Text":"We\u0027ll plug that in right here. That\u0027s 0.428."},{"Start":"08:00.350 ","End":"08:05.960","Text":"Here we have the probability function."},{"Start":"08:05.960 ","End":"08:11.990","Text":"We\u0027ve constructed the probability function of the number of movies sold overseas."},{"Start":"08:11.990 ","End":"08:17.280","Text":"It\u0027s right here. These are the probabilities."}],"ID":13059},{"Watched":false,"Name":"Exercise 6 Parts b-c","Duration":"6m 37s","ChapterTopicVideoID":12581,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.590","Text":"In Section B, we\u0027re asked,"},{"Start":"00:01.590 ","End":"00:05.685","Text":"what is the expectation and variance of the number of movies sold?"},{"Start":"00:05.685 ","End":"00:09.900","Text":"X, as we recall is the number of movies sold overseas,"},{"Start":"00:09.900 ","End":"00:16.215","Text":"and this was the probability distribution of x that we calculated in Section A."},{"Start":"00:16.215 ","End":"00:21.915","Text":"Now we\u0027re asked, what\u0027s the expectation of x and the variance?"},{"Start":"00:21.915 ","End":"00:25.550","Text":"That will equal, what\u0027s the definition of the expectation?"},{"Start":"00:25.550 ","End":"00:30.310","Text":"That\u0027s the sum of x times its probability."},{"Start":"00:30.310 ","End":"00:33.720","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"00:33.720 ","End":"00:35.340","Text":"0 times this, so that\u0027s 0."},{"Start":"00:35.340 ","End":"00:40.020","Text":"We won\u0027t write this down. We\u0027ll start with 1."},{"Start":"00:40.020 ","End":"00:45.750","Text":"That\u0027s 1 times 0.392 plus 2 times"},{"Start":"00:45.750 ","End":"00:53.520","Text":"0.428 plus 3 times 0.084,"},{"Start":"00:53.520 ","End":"00:57.330","Text":"and that comes out to 1.5,"},{"Start":"00:57.330 ","End":"01:00.645","Text":"and the units are movies."},{"Start":"01:00.645 ","End":"01:03.920","Text":"What about the variance of x?"},{"Start":"01:03.920 ","End":"01:07.510","Text":"Well, that equals to the sum of x squared times"},{"Start":"01:07.510 ","End":"01:12.395","Text":"the probability minus the expectation squared of x."},{"Start":"01:12.395 ","End":"01:15.220","Text":"Again, we won\u0027t relate to 0 here,"},{"Start":"01:15.220 ","End":"01:16.380","Text":"but we\u0027ll start with 1."},{"Start":"01:16.380 ","End":"01:23.925","Text":"That\u0027s 1 squared times 0.392 plus 2 squared times"},{"Start":"01:23.925 ","End":"01:28.380","Text":"0.428 plus 3 squared times"},{"Start":"01:28.380 ","End":"01:34.965","Text":"0.084 minus 1.5 squared."},{"Start":"01:34.965 ","End":"01:36.750","Text":"Let\u0027s not forget that."},{"Start":"01:36.750 ","End":"01:40.080","Text":"That comes out to 0.61,"},{"Start":"01:40.080 ","End":"01:45.480","Text":"and the units here are movies squared."},{"Start":"01:46.310 ","End":"01:48.840","Text":"In Section C, we\u0027re asked,"},{"Start":"01:48.840 ","End":"01:51.050","Text":"what is the expectation and standard deviation of"},{"Start":"01:51.050 ","End":"01:54.230","Text":"the profits made by the production company?"},{"Start":"01:54.230 ","End":"01:57.620","Text":"We want that in units of hundreds of thousands of dollars."},{"Start":"01:57.620 ","End":"02:01.415","Text":"First, just let\u0027s mark what we\u0027re asked to do."},{"Start":"02:01.415 ","End":"02:06.560","Text":"We\u0027re asked to find the expectation and standard deviation of the profits."},{"Start":"02:06.560 ","End":"02:08.135","Text":"Now, up till now,"},{"Start":"02:08.135 ","End":"02:11.855","Text":"we\u0027ve been talking about the number of movies being sold abroad,"},{"Start":"02:11.855 ","End":"02:13.685","Text":"that was the definition of x."},{"Start":"02:13.685 ","End":"02:16.490","Text":"Here, we\u0027re looking for profits."},{"Start":"02:16.490 ","End":"02:19.400","Text":"This is basically a new random variable."},{"Start":"02:19.400 ","End":"02:20.930","Text":"Let\u0027s call that y,"},{"Start":"02:20.930 ","End":"02:25.925","Text":"and that will be defined as the profits."},{"Start":"02:25.925 ","End":"02:34.260","Text":"Now, the profits are defined as income minus expenses."},{"Start":"02:35.200 ","End":"02:38.270","Text":"Now, if that\u0027s the case,"},{"Start":"02:38.270 ","End":"02:40.130","Text":"then, first of all,"},{"Start":"02:40.130 ","End":"02:45.815","Text":"let\u0027s take a look at the data that was provided for us in the question."},{"Start":"02:45.815 ","End":"02:52.050","Text":"We\u0027re given that each movie costs $500,000 to produce,"},{"Start":"02:52.050 ","End":"02:59.645","Text":"and it generated $200,000 of revenue per movie through a local television station."},{"Start":"02:59.645 ","End":"03:01.850","Text":"Now if the movies are sold overseas,"},{"Start":"03:01.850 ","End":"03:08.450","Text":"each movie would bring a revenue of $600,000 per movie."},{"Start":"03:08.450 ","End":"03:12.200","Text":"Let\u0027s get back. In essence,"},{"Start":"03:12.200 ","End":"03:16.400","Text":"what we have here is the linear transformation from x,"},{"Start":"03:16.400 ","End":"03:21.440","Text":"that\u0027s the number of movies sold"},{"Start":"03:21.440 ","End":"03:27.855","Text":"abroad to y,"},{"Start":"03:27.855 ","End":"03:31.080","Text":"which is the profits."},{"Start":"03:31.080 ","End":"03:33.645","Text":"Now, how do we do that?"},{"Start":"03:33.645 ","End":"03:41.160","Text":"Well, y was defined as income minus expenses."},{"Start":"03:41.210 ","End":"03:44.415","Text":"Well, we have 2 types of income,"},{"Start":"03:44.415 ","End":"03:50.135","Text":"income from the local market and income from the market abroad."},{"Start":"03:50.135 ","End":"03:52.100","Text":"From the local market,"},{"Start":"03:52.100 ","End":"03:55.740","Text":"we have $200,000 per movie,"},{"Start":"03:55.740 ","End":"03:58.990","Text":"and we have 3 movies."},{"Start":"03:59.090 ","End":"04:05.195","Text":"That\u0027s the income in hundreds of thousands of dollars from the local market,"},{"Start":"04:05.195 ","End":"04:09.770","Text":"plus $600,000 times x,"},{"Start":"04:09.770 ","End":"04:13.230","Text":"that\u0027s the number of movies sold abroad."},{"Start":"04:13.520 ","End":"04:19.080","Text":"That\u0027s the income parameters minus now the expenses."},{"Start":"04:19.080 ","End":"04:23.785","Text":"Now, the expenses are $500,000."},{"Start":"04:23.785 ","End":"04:30.175","Text":"That\u0027s the cost of making the movie or producing a movie times the 3 movies."},{"Start":"04:30.175 ","End":"04:36.840","Text":"That equals to 6 plus 6x minus 15,"},{"Start":"04:36.840 ","End":"04:40.975","Text":"that equals to 6x minus 9."},{"Start":"04:40.975 ","End":"04:49.250","Text":"Now, let\u0027s take a look at the general form of a linear transformation."},{"Start":"04:49.250 ","End":"04:53.210","Text":"Well, that\u0027s y equals ax plus b."},{"Start":"04:53.210 ","End":"05:00.915","Text":"In our case, a equals 6 and b equals minus 9."},{"Start":"05:00.915 ","End":"05:02.760","Text":"We can see that right here."},{"Start":"05:02.760 ","End":"05:05.830","Text":"What\u0027s the expectation of y?"},{"Start":"05:05.830 ","End":"05:12.335","Text":"Well, that equals to a times the expectation of x plus b,"},{"Start":"05:12.335 ","End":"05:14.975","Text":"and the variance of y, well,"},{"Start":"05:14.975 ","End":"05:19.130","Text":"that equals to a squared times the variance of x."},{"Start":"05:19.130 ","End":"05:23.480","Text":"Let\u0027s get back to this side of the fence."},{"Start":"05:23.480 ","End":"05:26.765","Text":"We\u0027re looking for the expectation of y."},{"Start":"05:26.765 ","End":"05:30.180","Text":"Now, that\u0027s a is 6,"},{"Start":"05:30.180 ","End":"05:32.805","Text":"6 times the expectation of x."},{"Start":"05:32.805 ","End":"05:34.655","Text":"Now, the expectation of x,"},{"Start":"05:34.655 ","End":"05:37.085","Text":"if you recall that\u0027s 1.5."},{"Start":"05:37.085 ","End":"05:40.715","Text":"We\u0027ve calculated that in Section B, minus b,"},{"Start":"05:40.715 ","End":"05:43.230","Text":"b is minus 9,"},{"Start":"05:43.230 ","End":"05:44.400","Text":"so it\u0027s minus 9,"},{"Start":"05:44.400 ","End":"05:46.845","Text":"and that equals to 0."},{"Start":"05:46.845 ","End":"05:54.080","Text":"The expected profits of the production company is $0."},{"Start":"05:54.080 ","End":"05:58.030","Text":"Now, let\u0027s take a look at the variance."},{"Start":"05:58.030 ","End":"06:00.850","Text":"The variance of y."},{"Start":"06:00.850 ","End":"06:04.310","Text":"Well, that equals to a squared times the variance of x."},{"Start":"06:04.310 ","End":"06:05.510","Text":"Now a is 6,"},{"Start":"06:05.510 ","End":"06:13.675","Text":"so that\u0027s 6 squared times the variance of x was 0.61,"},{"Start":"06:13.675 ","End":"06:19.580","Text":"and that comes out to 21.96."},{"Start":"06:19.580 ","End":"06:21.380","Text":"Now, we weren\u0027t asked for the variance,"},{"Start":"06:21.380 ","End":"06:24.260","Text":"we\u0027re asked for the standard deviation of y."},{"Start":"06:24.260 ","End":"06:29.420","Text":"Well, that\u0027s fine because that\u0027s just the square root of the variance,"},{"Start":"06:29.420 ","End":"06:35.490","Text":"and that comes out to 4.68."}],"ID":13060},{"Watched":false,"Name":"Exercise 7","Duration":"10m 9s","ChapterTopicVideoID":12582,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.625","Text":"In this question, we\u0027ll be dealing with candy."},{"Start":"00:02.625 ","End":"00:06.960","Text":"Now, 20 percent of the candy in a candy factory are strawberry flavored."},{"Start":"00:06.960 ","End":"00:09.750","Text":"The factory uses mass production."},{"Start":"00:09.750 ","End":"00:12.810","Text":"The rest of the candies have various flavors."},{"Start":"00:12.810 ","End":"00:15.420","Text":"There are exactly 5 candies and a bag,"},{"Start":"00:15.420 ","End":"00:17.115","Text":"and we\u0027re asked in section a,"},{"Start":"00:17.115 ","End":"00:22.680","Text":"if a bag is selected and it\u0027s known that there are fewer than 3 strawberry candies in it."},{"Start":"00:22.680 ","End":"00:27.970","Text":"What\u0027s the probability that there\u0027s 1 strawberry candy in the bag?"},{"Start":"00:28.490 ","End":"00:32.310","Text":"By now, I hope that you can identify"},{"Start":"00:32.310 ","End":"00:35.800","Text":"that we\u0027re dealing here with a binomial distribution."},{"Start":"00:35.800 ","End":"00:37.100","Text":"Now, why is that?"},{"Start":"00:37.100 ","End":"00:42.515","Text":"Well, let\u0027s take a look at the conditions for a binomial distribution, and here it is."},{"Start":"00:42.515 ","End":"00:45.710","Text":"We can see that we\u0027re dealing"},{"Start":"00:45.710 ","End":"00:49.715","Text":"with the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:49.715 ","End":"00:52.288","Text":"The trial is repeated n times,"},{"Start":"00:52.288 ","End":"00:55.760","Text":"and X is defined as the total number of successes obtained."},{"Start":"00:55.760 ","End":"00:57.755","Text":"Well, let\u0027s take a look."},{"Start":"00:57.755 ","End":"00:59.030","Text":"We have a Bernoulli trial."},{"Start":"00:59.030 ","End":"01:00.635","Text":"What\u0027s a Bernoulli trial?"},{"Start":"01:00.635 ","End":"01:03.440","Text":"That\u0027s a trial that has success or failure,"},{"Start":"01:03.440 ","End":"01:04.760","Text":"so what\u0027s our success?"},{"Start":"01:04.760 ","End":"01:08.360","Text":"How do we define success?"},{"Start":"01:08.360 ","End":"01:18.360","Text":"That\u0027ll be the strawberry-flavored candy."},{"Start":"01:19.300 ","End":"01:22.780","Text":"Now, what\u0027s the probability of success?"},{"Start":"01:22.780 ","End":"01:27.620","Text":"We\u0027re given that 20 percent of the candies in the candy factory are strawberry flavored,"},{"Start":"01:27.620 ","End":"01:30.700","Text":"so the probability is 0.2."},{"Start":"01:30.700 ","End":"01:33.940","Text":"The trial is repeated n times,"},{"Start":"01:33.940 ","End":"01:41.580","Text":"we\u0027re talking about we have exactly 5 candies in a bag,"},{"Start":"01:41.580 ","End":"01:44.355","Text":"so n here equals 5."},{"Start":"01:44.355 ","End":"01:48.360","Text":"Now, let\u0027s define X as"},{"Start":"01:48.360 ","End":"01:56.470","Text":"the number of successes in a bag."},{"Start":"01:57.170 ","End":"02:05.750","Text":"That means that we\u0027re dealing with X that has a binomial distribution where n equals 5,"},{"Start":"02:05.750 ","End":"02:08.900","Text":"and p equals 0.2."},{"Start":"02:08.900 ","End":"02:13.670","Text":"Now let\u0027s just recall the probability of X being equal to k,"},{"Start":"02:13.670 ","End":"02:16.445","Text":"well that equals to n over k,"},{"Start":"02:16.445 ","End":"02:22.790","Text":"p^k times 1 minus p^n minus"},{"Start":"02:22.790 ","End":"02:32.040","Text":"k. Let\u0027s continue on from here and see how we can solve section a."},{"Start":"02:32.510 ","End":"02:37.940","Text":"In this section, we have a conditional probability, and why is that?"},{"Start":"02:37.940 ","End":"02:41.600","Text":"Because we\u0027re given, it\u0027s known that there are"},{"Start":"02:41.600 ","End":"02:45.220","Text":"fewer than 3 strawberry candies in the bag,"},{"Start":"02:45.220 ","End":"02:50.210","Text":"and we\u0027re asked for the probability that there\u0027s 1 strawberry candy in the bag."},{"Start":"02:50.210 ","End":"02:52.970","Text":"Well, let\u0027s construct the conditional probability."},{"Start":"02:52.970 ","End":"02:56.460","Text":"That\u0027s the probability, now what\u0027s given,"},{"Start":"02:56.460 ","End":"03:01.410","Text":"that X is less than 3 candies in the bag,"},{"Start":"03:01.410 ","End":"03:05.930","Text":"and we want to know what\u0027s the probability of X being equal to 1."},{"Start":"03:05.930 ","End":"03:10.415","Text":"Well, we know how to deal with this."},{"Start":"03:10.415 ","End":"03:14.060","Text":"In the denominator, we put the probability of what\u0027s given,"},{"Start":"03:14.060 ","End":"03:19.095","Text":"that\u0027s the probability of X being less than 3."},{"Start":"03:19.095 ","End":"03:21.990","Text":"In the numerator, that\u0027s the intersect of these guys,"},{"Start":"03:21.990 ","End":"03:25.325","Text":"so that\u0027s the probability of X equaling 1,"},{"Start":"03:25.325 ","End":"03:28.875","Text":"and X less than 3."},{"Start":"03:28.875 ","End":"03:31.350","Text":"Now, if X is less than 3,"},{"Start":"03:31.350 ","End":"03:34.035","Text":"then X is equal to 0,"},{"Start":"03:34.035 ","End":"03:36.525","Text":"or 1 or 2."},{"Start":"03:36.525 ","End":"03:41.485","Text":"We have to intersect that with X equaling 1."},{"Start":"03:41.485 ","End":"03:44.540","Text":"In essence, what we have here,"},{"Start":"03:44.540 ","End":"03:48.200","Text":"that\u0027s the probability of X equaling 1,"},{"Start":"03:48.200 ","End":"03:50.405","Text":"because that\u0027s the intersect,"},{"Start":"03:50.405 ","End":"03:56.550","Text":"divided by the probability of X being less than 3."},{"Start":"03:57.080 ","End":"04:02.750","Text":"We have these probabilities so all we have to do is plug in the numbers to"},{"Start":"04:02.750 ","End":"04:09.150","Text":"this formula so we can calculate the conditional probability."},{"Start":"04:10.330 ","End":"04:15.550","Text":"Let\u0027s do that. What\u0027s the probability of X equaling 1?"},{"Start":"04:15.550 ","End":"04:18.900","Text":"That\u0027s n over k,"},{"Start":"04:18.900 ","End":"04:22.515","Text":"n is 5 and k is 1."},{"Start":"04:22.515 ","End":"04:27.060","Text":"It\u0027s 5 over 1 times p,"},{"Start":"04:27.060 ","End":"04:30.320","Text":"p is 0.2^k,"},{"Start":"04:30.320 ","End":"04:31.880","Text":"k is 1,"},{"Start":"04:31.880 ","End":"04:34.895","Text":"times 1 minus p, that\u0027s 0.8,"},{"Start":"04:34.895 ","End":"04:39.470","Text":"1 minus 0.8^n minus k,"},{"Start":"04:39.470 ","End":"04:42.565","Text":"5 minus 1, that\u0027s 4."},{"Start":"04:42.565 ","End":"04:47.390","Text":"That is the probability of X equal 1."},{"Start":"04:47.390 ","End":"04:54.595","Text":"Now, this has to go over the probability of X being less than 3."},{"Start":"04:54.595 ","End":"04:59.540","Text":"That means that we have to calculate the probability of X equaling 0,"},{"Start":"04:59.540 ","End":"05:02.855","Text":"1, and 2. Let\u0027s get to it."},{"Start":"05:02.855 ","End":"05:05.840","Text":"That\u0027s 5/0, well,"},{"Start":"05:05.840 ","End":"05:13.070","Text":"k equals 0. 0.2^0 times 0.8^5."},{"Start":"05:13.070 ","End":"05:18.890","Text":"That\u0027s the probability of X being equal to 0,"},{"Start":"05:18.890 ","End":"05:22.670","Text":"plus now the probability of X equaling 1,"},{"Start":"05:22.670 ","End":"05:24.865","Text":"that\u0027s 5 over 1,"},{"Start":"05:24.865 ","End":"05:32.150","Text":"0.2^1 times 0.8^5 minus 1, that\u0027s 4,"},{"Start":"05:32.150 ","End":"05:36.920","Text":"plus now the probability of X equaling 2,"},{"Start":"05:36.920 ","End":"05:47.675","Text":"that\u0027s 5/2, 0.2 squared times 0.8^5 minus 2, that\u0027s 3."},{"Start":"05:47.675 ","End":"05:53.640","Text":"All that equals to 0.4348."},{"Start":"05:58.220 ","End":"06:02.780","Text":"In section b, we\u0027re given that 1 bag after another is randomly"},{"Start":"06:02.780 ","End":"06:07.175","Text":"selected in order to find a bag without any strawberry candies."},{"Start":"06:07.175 ","End":"06:12.545","Text":"We\u0027re asked, what\u0027s the probability that they will have to sample more than 6 bags?"},{"Start":"06:12.545 ","End":"06:16.910","Text":"Well, this is obviously a geometric distribution. Why is that?"},{"Start":"06:16.910 ","End":"06:19.765","Text":"Well, let\u0027s take a look at the conditions."},{"Start":"06:19.765 ","End":"06:23.660","Text":"The conditions for a geometric distribution is that we"},{"Start":"06:23.660 ","End":"06:27.275","Text":"have the same Bernoulli trial that\u0027s repeated independently,"},{"Start":"06:27.275 ","End":"06:30.290","Text":"and we have a random variable that\u0027s defined as the number"},{"Start":"06:30.290 ","End":"06:33.208","Text":"of trials carried out until the first success,"},{"Start":"06:33.208 ","End":"06:34.610","Text":"and this is what we have here,"},{"Start":"06:34.610 ","End":"06:39.575","Text":"we have to sample bag after bag until the success,"},{"Start":"06:39.575 ","End":"06:42.965","Text":"is finding a bag without any strawberry candies."},{"Start":"06:42.965 ","End":"06:44.525","Text":"Again, let\u0027s write this down."},{"Start":"06:44.525 ","End":"06:49.470","Text":"Success is a bag"},{"Start":"06:49.510 ","End":"07:00.810","Text":"with no strawberry-flavored candy."},{"Start":"07:02.360 ","End":"07:07.375","Text":"What\u0027s the probability of success here?"},{"Start":"07:07.375 ","End":"07:10.810","Text":"Well, in order to find the probability of"},{"Start":"07:10.810 ","End":"07:15.100","Text":"a bag that doesn\u0027t have a strawberry flavored candy,"},{"Start":"07:15.100 ","End":"07:19.630","Text":"we have to go back to section a, and why is that?"},{"Start":"07:19.630 ","End":"07:21.445","Text":"Because in section a,"},{"Start":"07:21.445 ","End":"07:32.410","Text":"we defined X is the number of strawberry flavored candy in a bag."},{"Start":"07:33.440 ","End":"07:37.030","Text":"We know that X was defined with"},{"Start":"07:37.030 ","End":"07:43.255","Text":"a binomial distribution where n equals 5 and p equals 0.2."},{"Start":"07:43.255 ","End":"07:48.910","Text":"Now, what\u0027s the probability then of X equaling 0,"},{"Start":"07:48.910 ","End":"07:52.570","Text":"the number of strawberry flavored candy in the bag equaling 0?"},{"Start":"07:52.570 ","End":"07:54.940","Text":"Because it\u0027s a binomial distribution,"},{"Start":"07:54.940 ","End":"07:59.650","Text":"let\u0027s write this down."},{"Start":"07:59.650 ","End":"08:05.775","Text":"That\u0027s 5 over 0, 0.2,"},{"Start":"08:05.775 ","End":"08:10.825","Text":"p^0 times 0.8^5,"},{"Start":"08:10.825 ","End":"08:15.680","Text":"that equals to 0.328."},{"Start":"08:16.490 ","End":"08:20.360","Text":"Here, in this section,"},{"Start":"08:20.360 ","End":"08:23.900","Text":"where success is the bag with no strawberry flavored candy with"},{"Start":"08:23.900 ","End":"08:28.320","Text":"a probability of that is 0.328."},{"Start":"08:28.390 ","End":"08:32.210","Text":"Now, we have here a random variable"},{"Start":"08:32.210 ","End":"08:35.390","Text":"that\u0027s defined as the number of trials carried out until the first success."},{"Start":"08:35.390 ","End":"08:36.680","Text":"What\u0027s a trial here?"},{"Start":"08:36.680 ","End":"08:45.620","Text":"A trial is sampling a bag until we find a bag with no strawberry candies."},{"Start":"08:45.620 ","End":"08:47.405","Text":"Let\u0027s define Y,"},{"Start":"08:47.405 ","End":"08:49.625","Text":"a new random variable,"},{"Start":"08:49.625 ","End":"08:59.830","Text":"as the number of bags with no strawberry flavored candy."},{"Start":"09:01.010 ","End":"09:06.905","Text":"That means according to the conditions that Y has"},{"Start":"09:06.905 ","End":"09:13.920","Text":"a geometric distribution where the probability is 0.328."},{"Start":"09:15.460 ","End":"09:18.380","Text":"What are we looking for?"},{"Start":"09:18.380 ","End":"09:24.410","Text":"We\u0027re looking for the probability of Y being greater than 6."},{"Start":"09:24.410 ","End":"09:26.800","Text":"This is what we\u0027re looking for."},{"Start":"09:26.800 ","End":"09:29.450","Text":"Sampling more than 6 bags,"},{"Start":"09:29.450 ","End":"09:31.790","Text":"that\u0027s the probability that we\u0027re looking for."},{"Start":"09:31.790 ","End":"09:36.485","Text":"There\u0027s a special characteristic in the geometric distribution"},{"Start":"09:36.485 ","End":"09:43.115","Text":"which says that if we have a random variable which is greater than k,"},{"Start":"09:43.115 ","End":"09:46.865","Text":"well that equals to q^k."},{"Start":"09:46.865 ","End":"09:52.755","Text":"Where q equals 1 minus p. Let\u0027s get back here."},{"Start":"09:52.755 ","End":"09:54.930","Text":"P is 0.328,"},{"Start":"09:54.930 ","End":"10:00.990","Text":"so that\u0027ll be 1 minus 0.328^k,"},{"Start":"10:00.990 ","End":"10:02.813","Text":"k is 6,"},{"Start":"10:02.813 ","End":"10:05.950","Text":"and that equals to 0.0923."}],"ID":13061},{"Watched":false,"Name":"Exercise 8","Duration":"10m 16s","ChapterTopicVideoID":12583,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"In this question, we\u0027ll be talking about results in an exam."},{"Start":"00:03.870 ","End":"00:07.380","Text":"Now, an exam is composed of 2 parts, A and B."},{"Start":"00:07.380 ","End":"00:11.475","Text":"Both part A and part B have 10 questions each."},{"Start":"00:11.475 ","End":"00:15.990","Text":"A student prepares for only part A and his chances of answering"},{"Start":"00:15.990 ","End":"00:20.850","Text":"each question correctly in this part are 80 percent or 0.8."},{"Start":"00:20.850 ","End":"00:25.425","Text":"In part B, there are 4 answers to each question,"},{"Start":"00:25.425 ","End":"00:27.630","Text":"only 1 of which is correct."},{"Start":"00:27.630 ","End":"00:30.120","Text":"The student guesses all the answers to"},{"Start":"00:30.120 ","End":"00:32.955","Text":"this part of the exam and we\u0027re asked in section A,"},{"Start":"00:32.955 ","End":"00:40.300","Text":"what\u0027s the probability that he answers exactly 7 questions correctly on the first part."},{"Start":"00:40.300 ","End":"00:43.685","Text":"Here we have a binomial distribution,"},{"Start":"00:43.685 ","End":"00:46.810","Text":"and let\u0027s just take a look why."},{"Start":"00:46.810 ","End":"00:51.935","Text":"These are the conditions that we have to meet for the binomial distribution."},{"Start":"00:51.935 ","End":"00:55.865","Text":"We have the same Bernoulli trial that\u0027s repeated independently,"},{"Start":"00:55.865 ","End":"00:58.295","Text":"the trial is repeated n times,"},{"Start":"00:58.295 ","End":"01:02.360","Text":"and x would be defined as the total number of successes obtained."},{"Start":"01:02.360 ","End":"01:05.840","Text":"Well, our Bernoulli trial\u0027s answering questions."},{"Start":"01:05.840 ","End":"01:13.340","Text":"In a success would be defined as answering the questions correctly,"},{"Start":"01:13.340 ","End":"01:18.510","Text":"so that\u0027s the correct answer."},{"Start":"01:18.670 ","End":"01:24.470","Text":"Now, what\u0027s the probability of answering the question correctly?"},{"Start":"01:24.470 ","End":"01:28.530","Text":"Now this is correct answer in part A."},{"Start":"01:28.530 ","End":"01:30.315","Text":"This is what we\u0027re talking about."},{"Start":"01:30.315 ","End":"01:33.920","Text":"The probability of getting a correct answer in part A,"},{"Start":"01:33.920 ","End":"01:37.060","Text":"well, that\u0027s 0.8, that\u0027s right here."},{"Start":"01:37.060 ","End":"01:40.100","Text":"Now, how many questions do we have in part A?"},{"Start":"01:40.100 ","End":"01:43.220","Text":"Well, we have 10 questions."},{"Start":"01:43.220 ","End":"01:52.950","Text":"If we define x as the number of successes in part A,"},{"Start":"01:52.950 ","End":"01:55.850","Text":"then x would be distributed with"},{"Start":"01:55.850 ","End":"02:01.415","Text":"a binomial distribution where n equals 10 and p equals 0.8."},{"Start":"02:01.415 ","End":"02:07.675","Text":"Now let\u0027s recall the probability of x equaling k in a binomial distribution, well,"},{"Start":"02:07.675 ","End":"02:09.860","Text":"that\u0027s n over k,"},{"Start":"02:09.860 ","End":"02:16.894","Text":"p^k times 1 minus p^n minus k. In our case,"},{"Start":"02:16.894 ","End":"02:18.110","Text":"what are we looking for?"},{"Start":"02:18.110 ","End":"02:23.695","Text":"We\u0027re looking for the probability of x equaling 7."},{"Start":"02:23.695 ","End":"02:26.610","Text":"Now why is that? Well, right here."},{"Start":"02:26.610 ","End":"02:31.790","Text":"What\u0027s the probability that he answers exactly 7 questions? That\u0027s right here."},{"Start":"02:31.790 ","End":"02:33.350","Text":"Now, if that\u0027s the case,"},{"Start":"02:33.350 ","End":"02:34.910","Text":"let\u0027s plug in the numbers,"},{"Start":"02:34.910 ","End":"02:38.045","Text":"n is 10 over k,"},{"Start":"02:38.045 ","End":"02:41.825","Text":"where k is 7, times p,"},{"Start":"02:41.825 ","End":"02:47.090","Text":"p is 0.8^7 times 1 minus p,"},{"Start":"02:47.090 ","End":"02:50.330","Text":"that\u0027s 0.2^n minus k. Well,"},{"Start":"02:50.330 ","End":"02:52.310","Text":"that\u0027s 10 minus 7, that\u0027s 3,"},{"Start":"02:52.310 ","End":"02:57.390","Text":"so this equals to 0.2013."},{"Start":"02:58.970 ","End":"03:01.230","Text":"In section B, we\u0027re asked,"},{"Start":"03:01.230 ","End":"03:03.060","Text":"what\u0027s the probability that he answers"},{"Start":"03:03.060 ","End":"03:06.735","Text":"less than 3 questions correctly on the second part."},{"Start":"03:06.735 ","End":"03:13.880","Text":"Well, let\u0027s define y as the number of"},{"Start":"03:13.880 ","End":"03:17.610","Text":"correct answers"},{"Start":"03:18.130 ","End":"03:24.420","Text":"in part B."},{"Start":"03:24.420 ","End":"03:25.770","Text":"If that\u0027s the case,"},{"Start":"03:25.770 ","End":"03:28.990","Text":"then just like in section A,"},{"Start":"03:28.990 ","End":"03:31.340","Text":"just like in the random variable x,"},{"Start":"03:31.340 ","End":"03:34.730","Text":"y is also distributed with"},{"Start":"03:34.730 ","End":"03:39.830","Text":"a binomial distribution where n equals 10 because we have 10 questions in section B,"},{"Start":"03:39.830 ","End":"03:47.950","Text":"but now the probability of getting a correct answer is 0.25."},{"Start":"03:47.950 ","End":"03:50.900","Text":"Now why is that? Because for each question,"},{"Start":"03:50.900 ","End":"03:53.135","Text":"we had 4 possible answers,"},{"Start":"03:53.135 ","End":"03:54.680","Text":"only 1 of which was right,"},{"Start":"03:54.680 ","End":"03:56.510","Text":"so that\u0027s 1 out of 4,"},{"Start":"03:56.510 ","End":"03:59.305","Text":"1 out of 4 that\u0027s 25 percent."},{"Start":"03:59.305 ","End":"04:04.652","Text":"Let\u0027s just recall probability of y being equal to k. Well,"},{"Start":"04:04.652 ","End":"04:08.140","Text":"that equals to n over k,"},{"Start":"04:08.170 ","End":"04:15.300","Text":"p^k times 1 minus p^n minus k. That\u0027s a general formula here."},{"Start":"04:15.300 ","End":"04:16.905","Text":"What are we asked?"},{"Start":"04:16.905 ","End":"04:22.445","Text":"We\u0027re asked for the probability of y being less than 3."},{"Start":"04:22.445 ","End":"04:27.380","Text":"Less than 3 questions are correctly answered."},{"Start":"04:27.380 ","End":"04:34.190","Text":"Now, that equals to the probability of y being equal to 0,"},{"Start":"04:34.190 ","End":"04:38.330","Text":"plus the probability of y being equal to 1,"},{"Start":"04:38.330 ","End":"04:43.105","Text":"plus the probability of y being equal to 2."},{"Start":"04:43.105 ","End":"04:48.090","Text":"Now, for each 1 of these expressions,"},{"Start":"04:48.090 ","End":"04:53.765","Text":"all we need to do is to plug in the numbers into this equation. Let\u0027s get to work."},{"Start":"04:53.765 ","End":"04:58.715","Text":"n equals 10. Here in this expression,"},{"Start":"04:58.715 ","End":"05:01.475","Text":"k equals 0, so it\u0027s 10 over 0."},{"Start":"05:01.475 ","End":"05:02.825","Text":"What\u0027s our p here?"},{"Start":"05:02.825 ","End":"05:09.790","Text":"p is 0.25, so that\u0027s 0.25^0 times 1 minus p. Well,"},{"Start":"05:09.790 ","End":"05:14.447","Text":"that\u0027s 0.75^10 plus,"},{"Start":"05:14.447 ","End":"05:16.550","Text":"let\u0027s go over to this expression right here."},{"Start":"05:16.550 ","End":"05:18.835","Text":"That\u0027s 10 over 1 now."},{"Start":"05:18.835 ","End":"05:25.380","Text":"0.25^1 times 0.75^10 minus 1,"},{"Start":"05:25.380 ","End":"05:27.795","Text":"that\u0027s 9, plus,"},{"Start":"05:27.795 ","End":"05:31.385","Text":"now here we go to the next expression where y equals 2."},{"Start":"05:31.385 ","End":"05:33.430","Text":"That\u0027s 10 over 2,"},{"Start":"05:33.430 ","End":"05:41.210","Text":"0.25 squared times 0.75^8,"},{"Start":"05:41.210 ","End":"05:49.740","Text":"and that comes out to 0.5256."},{"Start":"05:49.740 ","End":"05:51.970","Text":"In section C, we\u0027re asked,"},{"Start":"05:51.970 ","End":"05:56.690","Text":"what are the expectation and variance of the number of correct answers in the first part?"},{"Start":"05:56.690 ","End":"05:59.725","Text":"Well, if we recall in section A,"},{"Start":"05:59.725 ","End":"06:09.120","Text":"x was defined as the number of correct answers in part A."},{"Start":"06:10.100 ","End":"06:19.445","Text":"Now, x had a binomial distribution where n equals 10 and p equals 0.8."},{"Start":"06:19.445 ","End":"06:22.065","Text":"Now, the expectation of x,"},{"Start":"06:22.065 ","End":"06:25.085","Text":"where x has a binomial distribution that equals to"},{"Start":"06:25.085 ","End":"06:29.495","Text":"n times p. Let\u0027s plug in the numbers and it\u0027s 10,"},{"Start":"06:29.495 ","End":"06:32.357","Text":"and p is 0.8."},{"Start":"06:32.357 ","End":"06:35.340","Text":"10 times 0.8 equals 8."},{"Start":"06:35.340 ","End":"06:38.110","Text":"Well, the units here are answers."},{"Start":"06:38.110 ","End":"06:41.240","Text":"Now, what about the variance of x?"},{"Start":"06:41.240 ","End":"06:46.295","Text":"Well, the variance of x is defined as n times p times q,"},{"Start":"06:46.295 ","End":"06:49.400","Text":"where q equals 1 minus p. Well,"},{"Start":"06:49.400 ","End":"06:52.490","Text":"again, let\u0027s plug in the numbers here."},{"Start":"06:52.490 ","End":"06:57.615","Text":"n is 10 times p is 0.8 times q,"},{"Start":"06:57.615 ","End":"07:00.060","Text":"that\u0027s 1 minus 0.8,"},{"Start":"07:00.060 ","End":"07:04.395","Text":"that\u0027s 0.2, and that equals to 1.6."},{"Start":"07:04.395 ","End":"07:08.290","Text":"The units are answers squared."},{"Start":"07:08.570 ","End":"07:10.830","Text":"In section D, we\u0027re asked,"},{"Start":"07:10.830 ","End":"07:13.000","Text":"what are the expectation and variance of the number of"},{"Start":"07:13.000 ","End":"07:16.045","Text":"correct answers on the entire exam?"},{"Start":"07:16.045 ","End":"07:19.060","Text":"Well, in order to do that first,"},{"Start":"07:19.060 ","End":"07:23.510","Text":"let\u0027s calculate the expectation of y,"},{"Start":"07:23.510 ","End":"07:26.710","Text":"the number of correct answers in the second part."},{"Start":"07:26.710 ","End":"07:30.745","Text":"Now that equals to n times p. If we recall,"},{"Start":"07:30.745 ","End":"07:39.380","Text":"y is distributed with a binomial distribution where n equals 10 and p equals 0.25."},{"Start":"07:39.680 ","End":"07:48.285","Text":"Here that\u0027ll be 10 times 0.25 and that equals to 2.5."},{"Start":"07:48.285 ","End":"07:51.730","Text":"What about the variance of y?"},{"Start":"07:51.730 ","End":"07:55.850","Text":"Well, that equals to n times p times q."},{"Start":"07:55.850 ","End":"08:01.745","Text":"That equals to 10 times 0.25 times 0.75,"},{"Start":"08:01.745 ","End":"08:06.570","Text":"and that equals to 1.875."},{"Start":"08:07.340 ","End":"08:09.360","Text":"What are we asked here?"},{"Start":"08:09.360 ","End":"08:12.560","Text":"We\u0027re asked about the expectation and variance for"},{"Start":"08:12.560 ","End":"08:17.270","Text":"the total number of correct answers in the entire exam."},{"Start":"08:17.270 ","End":"08:23.340","Text":"We\u0027ll define T, the total as being x plus y."},{"Start":"08:23.340 ","End":"08:25.460","Text":"That\u0027s the number of correct answers in part"},{"Start":"08:25.460 ","End":"08:29.045","Text":"1 and the number of correct answers in part 2."},{"Start":"08:29.045 ","End":"08:35.125","Text":"We\u0027re asked for the expectation of T. Well,"},{"Start":"08:35.125 ","End":"08:39.535","Text":"that equals to the expectation of x plus y."},{"Start":"08:39.535 ","End":"08:45.410","Text":"If we recall, the expectation of the sum equals the sum of the expectations,"},{"Start":"08:45.410 ","End":"08:49.990","Text":"that\u0027s the expectation of x plus the expectation of y."},{"Start":"08:49.990 ","End":"08:53.535","Text":"Now, the expectation of x was 8."},{"Start":"08:53.535 ","End":"08:59.280","Text":"We\u0027ve calculated that in section C plus the expectation of y."},{"Start":"08:59.280 ","End":"09:02.500","Text":"Well, that\u0027s right here, that\u0027s 2.5."},{"Start":"09:02.570 ","End":"09:13.040","Text":"We\u0027re expecting to get 10.5 correct answers if we\u0027re in the whole of the exam."},{"Start":"09:13.040 ","End":"09:16.100","Text":"Now what about the variance of T?"},{"Start":"09:16.100 ","End":"09:18.700","Text":"Well, the variance of T,"},{"Start":"09:18.700 ","End":"09:23.239","Text":"that equals to the variance of the sum x plus y."},{"Start":"09:23.239 ","End":"09:28.655","Text":"Now, when does this equal this?"},{"Start":"09:28.655 ","End":"09:34.805","Text":"When does the variance of the sum equals the sum of the variance?"},{"Start":"09:34.805 ","End":"09:38.240","Text":"Only when x and y are independent."},{"Start":"09:38.240 ","End":"09:41.480","Text":"Now, are the number of correct answers in"},{"Start":"09:41.480 ","End":"09:43.490","Text":"the first section dependent on the number of"},{"Start":"09:43.490 ","End":"09:46.265","Text":"correct answers in the second question? No, they\u0027re not."},{"Start":"09:46.265 ","End":"09:48.950","Text":"That means that x and y are independent."},{"Start":"09:48.950 ","End":"09:54.620","Text":"Now, the variance for x was 1.6."},{"Start":"09:54.620 ","End":"09:59.345","Text":"We\u0027ve calculated that in section C. The variance for y,"},{"Start":"09:59.345 ","End":"10:03.130","Text":"well, that\u0027s 1.875."},{"Start":"10:03.130 ","End":"10:05.720","Text":"Now, if that\u0027s the case,"},{"Start":"10:05.720 ","End":"10:11.765","Text":"then that equals to 3.475."},{"Start":"10:11.765 ","End":"10:16.020","Text":"The units here are answers squared."}],"ID":13062},{"Watched":false,"Name":"Exercise 9","Duration":"3m 49s","ChapterTopicVideoID":12584,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"In this question, we\u0027re given that x is"},{"Start":"00:02.400 ","End":"00:06.975","Text":"a random variable where the expectation of x is 2 and the variance of x is 1,"},{"Start":"00:06.975 ","End":"00:11.655","Text":"and we\u0027re asked to calculate the expectation of x minus 5 squared."},{"Start":"00:11.655 ","End":"00:14.535","Text":"Well, this is a play with algebra."},{"Start":"00:14.535 ","End":"00:16.875","Text":"Let\u0027s just get to it."},{"Start":"00:16.875 ","End":"00:21.345","Text":"The expectation of x minus 5 squared."},{"Start":"00:21.345 ","End":"00:26.010","Text":"Well, we know from algebra that a minus b squared,"},{"Start":"00:26.010 ","End":"00:33.120","Text":"well that equals to a squared minus 2ab plus b squared."},{"Start":"00:33.120 ","End":"00:35.460","Text":"Let\u0027s just expand that."},{"Start":"00:35.460 ","End":"00:43.630","Text":"That\u0027s the expectation of x squared minus 10x plus 25."},{"Start":"00:43.910 ","End":"00:51.740","Text":"Here now we have the expectation of a sum of variables."},{"Start":"00:51.740 ","End":"00:56.840","Text":"Now again, the expectation of a sum x_1 plus x_2,"},{"Start":"00:56.840 ","End":"01:00.950","Text":"for example, that equals to the sum of the expectations."},{"Start":"01:00.950 ","End":"01:07.190","Text":"That\u0027s the expectation of x_1 plus the expectation of x_2."},{"Start":"01:07.190 ","End":"01:09.790","Text":"Let\u0027s just expand that."},{"Start":"01:09.790 ","End":"01:14.630","Text":"That\u0027ll be the expectation of x squared"},{"Start":"01:14.630 ","End":"01:21.360","Text":"plus the expectation of minus 10x plus 25."},{"Start":"01:22.000 ","End":"01:28.090","Text":"Let\u0027s take a look at each 1 of these expressions independently."},{"Start":"01:28.090 ","End":"01:34.825","Text":"Now, let\u0027s try to find out what the value of x with the expectation of x squared is."},{"Start":"01:34.825 ","End":"01:36.650","Text":"In order to do that,"},{"Start":"01:36.650 ","End":"01:41.570","Text":"let\u0027s remind ourselves of the equation for the variance."},{"Start":"01:41.570 ","End":"01:43.960","Text":"The equation for the variance,"},{"Start":"01:43.960 ","End":"01:47.750","Text":"that equals to the sum of x squared times"},{"Start":"01:47.750 ","End":"01:53.610","Text":"its probability minus the expectation squared of x."},{"Start":"01:53.610 ","End":"01:57.370","Text":"Now, there\u0027s another expression."},{"Start":"01:57.920 ","End":"02:05.820","Text":"The other expectation is the expectation of x squared minus the expectation squared of x."},{"Start":"02:05.820 ","End":"02:11.030","Text":"Now, what\u0027s the value of the variance where we\u0027re given that?"},{"Start":"02:11.030 ","End":"02:12.200","Text":"That\u0027s 1 right here."},{"Start":"02:12.200 ","End":"02:14.030","Text":"That\u0027ll equal 1."},{"Start":"02:14.030 ","End":"02:16.790","Text":"Now, the expectation of x squared,"},{"Start":"02:16.790 ","End":"02:19.440","Text":"this is what we\u0027re trying to find out."},{"Start":"02:19.570 ","End":"02:23.210","Text":"What are we looking for here?"},{"Start":"02:23.210 ","End":"02:24.920","Text":"Well, here we have,"},{"Start":"02:24.920 ","End":"02:26.930","Text":"what\u0027s the expectation squared of x?"},{"Start":"02:26.930 ","End":"02:31.910","Text":"Well, the expectation of x equals 2,"},{"Start":"02:31.910 ","End":"02:34.605","Text":"so we\u0027re looking at 2 squared."},{"Start":"02:34.605 ","End":"02:40.965","Text":"That means that the expectation of x squared that would equal to 5."},{"Start":"02:40.965 ","End":"02:44.020","Text":"Let\u0027s put 5 down here."},{"Start":"02:44.020 ","End":"02:48.760","Text":"Now, what about the expectation of this expression right here?"},{"Start":"02:48.760 ","End":"02:51.745","Text":"Well, this looks like a linear transformation of x."},{"Start":"02:51.745 ","End":"02:54.775","Text":"Now, what do we know about linear transformations?"},{"Start":"02:54.775 ","End":"02:58.770","Text":"The expectation of a linear transformation,"},{"Start":"02:58.770 ","End":"03:06.645","Text":"ax plus b well that equals to a times the expectation of x plus b."},{"Start":"03:06.645 ","End":"03:12.615","Text":"In our case, a is minus 10 and b is 25,"},{"Start":"03:12.615 ","End":"03:18.955","Text":"so it\u0027ll be minus 10."},{"Start":"03:18.955 ","End":"03:20.935","Text":"Now, what\u0027s the expectation of x?"},{"Start":"03:20.935 ","End":"03:22.660","Text":"The expectation of x is 2,"},{"Start":"03:22.660 ","End":"03:25.000","Text":"that\u0027ll be 2 plus b."},{"Start":"03:25.000 ","End":"03:27.020","Text":"Well, that\u0027s 25."},{"Start":"03:27.020 ","End":"03:32.960","Text":"That equals to 5 minus 20 plus 25."},{"Start":"03:32.960 ","End":"03:36.605","Text":"That means that this equals to 10."},{"Start":"03:36.605 ","End":"03:41.295","Text":"The expectation of x minus 5 squared,"},{"Start":"03:41.295 ","End":"03:46.065","Text":"where the expectation of x is 2 and the variance of x is 1,"},{"Start":"03:46.065 ","End":"03:49.540","Text":"well that expression equals 10."}],"ID":13063},{"Watched":false,"Name":"Exercise 10 Part a","Duration":"7m 22s","ChapterTopicVideoID":12585,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"In this question, we\u0027ll be talking about passing a driving test."},{"Start":"00:03.705 ","End":"00:07.260","Text":"Now the chances of passing a driving test are p."},{"Start":"00:07.260 ","End":"00:12.360","Text":"4 drivers are randomly selected and the probability of 2 of them passing"},{"Start":"00:12.360 ","End":"00:16.890","Text":"the test is 8/3 higher than the chances that all"},{"Start":"00:16.890 ","End":"00:22.320","Text":"4 will pass the test and we\u0027re asked to calculate the value of p. Well,"},{"Start":"00:22.320 ","End":"00:25.215","Text":"again, here we have a binomial distribution,"},{"Start":"00:25.215 ","End":"00:29.430","Text":"but let\u0027s take a look at the conditions to make sure."},{"Start":"00:29.430 ","End":"00:32.910","Text":"Okay, so here we have the conditions where we have"},{"Start":"00:32.910 ","End":"00:36.540","Text":"the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:36.540 ","End":"00:42.605","Text":"The trial is repeated n times and x is defined as the total number of successes obtained."},{"Start":"00:42.605 ","End":"00:44.510","Text":"Okay, so what do we have here?"},{"Start":"00:44.510 ","End":"00:50.315","Text":"Well, a Bernoulli trial is a driving test and success is passing the driving test."},{"Start":"00:50.315 ","End":"00:57.180","Text":"Now, the number of Bernoulli trials that we have are 4."},{"Start":"00:57.180 ","End":"00:58.830","Text":"We have 4 people,"},{"Start":"00:58.830 ","End":"01:01.588","Text":"4 drivers randomly selected,"},{"Start":"01:01.588 ","End":"01:06.460","Text":"and we\u0027re looking for p. That\u0027s what we\u0027re looking for and"},{"Start":"01:06.460 ","End":"01:13.655","Text":"X would be the number of drivers that have passed."},{"Start":"01:13.655 ","End":"01:15.770","Text":"If that\u0027s the case,"},{"Start":"01:15.770 ","End":"01:23.105","Text":"then X has a binomial distribution where n equals 4 and p,"},{"Start":"01:23.105 ","End":"01:24.590","Text":"we don\u0027t know what that is."},{"Start":"01:24.590 ","End":"01:27.270","Text":"That\u0027s what we\u0027re looking for."},{"Start":"01:27.490 ","End":"01:36.920","Text":"Let\u0027s just recall that the probability of x equaling k. That equals to n/k,"},{"Start":"01:36.920 ","End":"01:45.665","Text":"p^k times 1 minus p^n minus k. But what do we know?"},{"Start":"01:45.665 ","End":"01:50.330","Text":"We know that the probability of 2 of them passing the test"},{"Start":"01:50.330 ","End":"01:55.385","Text":"is 8/3 higher than the chances that all 4 will pass the test."},{"Start":"01:55.385 ","End":"01:57.455","Text":"Now, what does that mean?"},{"Start":"01:57.455 ","End":"02:02.000","Text":"That means that the probability of X equaling 2,"},{"Start":"02:02.000 ","End":"02:05.135","Text":"well that equals to 8/3."},{"Start":"02:05.135 ","End":"02:14.710","Text":"8 divided by 3 or 8/3 times the probability that x will equal 4."},{"Start":"02:15.350 ","End":"02:20.540","Text":"All we have to do right now is just to calculate the probabilities where X equals"},{"Start":"02:20.540 ","End":"02:25.385","Text":"2 and X equals 4 and solve for p. Okay,"},{"Start":"02:25.385 ","End":"02:28.910","Text":"so what\u0027s the probability of X equaling 2?"},{"Start":"02:28.910 ","End":"02:31.280","Text":"Well, let\u0027s just plug in the numbers right here."},{"Start":"02:31.280 ","End":"02:32.420","Text":"n equals 4, right?"},{"Start":"02:32.420 ","End":"02:41.540","Text":"That\u0027ll be 4 over 2p^2 times 1 minus p^n minus k,"},{"Start":"02:41.540 ","End":"02:42.770","Text":"4 minus 2, that\u0027s 2."},{"Start":"02:42.770 ","End":"02:45.700","Text":"That equals to 8/3,"},{"Start":"02:45.700 ","End":"02:49.025","Text":"times the probability of x equaling 4."},{"Start":"02:49.025 ","End":"02:58.850","Text":"So that\u0027ll be 4/4 p^4 times 1 minus p^4 minus 4, that\u0027s 0."},{"Start":"02:58.850 ","End":"03:01.220","Text":"Let\u0027s simplify this a bit."},{"Start":"03:01.220 ","End":"03:09.980","Text":"4/2 that\u0027s 6 times p squared times 1 minus p squared and that equals to 8/3."},{"Start":"03:09.980 ","End":"03:15.170","Text":"Now 4/4 is 1 and 1 minus p^0, that\u0027s 1."},{"Start":"03:15.170 ","End":"03:18.870","Text":"So we\u0027re left with p^4."},{"Start":"03:19.340 ","End":"03:22.755","Text":"Let\u0027s just move things around here."},{"Start":"03:22.755 ","End":"03:27.560","Text":"That\u0027ll be 6p squared times"},{"Start":"03:27.560 ","End":"03:33.980","Text":"1 minus p squared minus 8/3 p^4,"},{"Start":"03:33.980 ","End":"03:35.959","Text":"that will equal to 0."},{"Start":"03:35.959 ","End":"03:44.330","Text":"Now, let\u0027s take p squared out as a common factor and so that\u0027ll be p squared times"},{"Start":"03:44.330 ","End":"03:48.574","Text":"6 times 1 minus p squared"},{"Start":"03:48.574 ","End":"03:55.280","Text":"minus 8/3 p squared and that has to equal to 0."},{"Start":"03:55.280 ","End":"03:58.610","Text":"Now in order for this expression to be equal to 0,"},{"Start":"03:58.610 ","End":"04:03.380","Text":"then either p squared has to be 0 or p equal to 0,"},{"Start":"04:03.380 ","End":"04:06.080","Text":"or this expression within the square brackets,"},{"Start":"04:06.080 ","End":"04:08.230","Text":"that has to equal 0."},{"Start":"04:08.230 ","End":"04:14.940","Text":"Let\u0027s just solve for this expression within the square brackets."},{"Start":"04:15.470 ","End":"04:17.730","Text":"Let\u0027s write this out again."},{"Start":"04:17.730 ","End":"04:24.514","Text":"That\u0027s 6 times 1 minus p squared minus 8/3 p squared."},{"Start":"04:24.514 ","End":"04:27.800","Text":"We want that to be equal to 0 and we\u0027ll solve for"},{"Start":"04:27.800 ","End":"04:35.355","Text":"p. Let\u0027s first multiply both sides by 3 just to get rid of the fraction here."},{"Start":"04:35.355 ","End":"04:41.180","Text":"That\u0027s 18 times 1 minus p squared minus 8p squared,"},{"Start":"04:41.180 ","End":"04:43.010","Text":"and that will equal to 0."},{"Start":"04:43.010 ","End":"04:46.160","Text":"Now, if we remember our algebra,"},{"Start":"04:46.160 ","End":"04:49.040","Text":"a minus b squared,"},{"Start":"04:49.040 ","End":"04:53.990","Text":"that equals to a squared minus 2ab plus b squared, right?"},{"Start":"04:53.990 ","End":"04:56.600","Text":"Let\u0027s just expand this."},{"Start":"04:56.600 ","End":"05:04.070","Text":"That\u0027ll be 18 times 1 minus 2p plus p"},{"Start":"05:04.070 ","End":"05:08.445","Text":"squared minus 8p squared."},{"Start":"05:08.445 ","End":"05:10.440","Text":"That equals to 0."},{"Start":"05:10.440 ","End":"05:12.180","Text":"Let\u0027s just do this."},{"Start":"05:12.180 ","End":"05:19.230","Text":"That\u0027s 18 minus 36p plus"},{"Start":"05:19.230 ","End":"05:26.589","Text":"18p squared minus 8p squared and that equals to 0."},{"Start":"05:26.589 ","End":"05:29.575","Text":"Let\u0027s just collect everything here."},{"Start":"05:29.575 ","End":"05:32.980","Text":"We have the following,"},{"Start":"05:32.980 ","End":"05:39.249","Text":"that\u0027ll be 10p squared"},{"Start":"05:39.249 ","End":"05:45.955","Text":"minus 36p plus 18,"},{"Start":"05:45.955 ","End":"05:48.475","Text":"and that will equal 0."},{"Start":"05:48.475 ","End":"05:56.440","Text":"Now, we have here a quadratic equation and we can solve for p. Now if we remember,"},{"Start":"05:56.440 ","End":"05:59.605","Text":"the solution for quadratic equation,"},{"Start":"05:59.605 ","End":"06:04.940","Text":"X_1,2 that equals to minus b plus minus the square root of"},{"Start":"06:04.940 ","End":"06:09.905","Text":"b squared minus 4ac divided by 2a."},{"Start":"06:09.905 ","End":"06:14.040","Text":"a in our case will be 10,"},{"Start":"06:14.040 ","End":"06:17.130","Text":"b will be 36 or minus 36,"},{"Start":"06:17.130 ","End":"06:19.590","Text":"and c would be 18."},{"Start":"06:19.590 ","End":"06:26.225","Text":"Now let\u0027s just plug in the numbers and we see what we get."},{"Start":"06:26.225 ","End":"06:28.550","Text":"Well, we get minus b,"},{"Start":"06:28.550 ","End":"06:34.460","Text":"so that\u0027ll be 36 plus minus the square root of b squared."},{"Start":"06:34.460 ","End":"06:42.420","Text":"Well, that\u0027s minus 36 squared minus 4 times a times"},{"Start":"06:42.420 ","End":"06:51.765","Text":"c. That\u0027s 4 times 10 times 18 divided by 2a,"},{"Start":"06:51.765 ","End":"06:55.740","Text":"well that\u0027ll be 2 times 10."},{"Start":"06:55.740 ","End":"07:03.610","Text":"That will equal to p1 being equal to 3,"},{"Start":"07:03.610 ","End":"07:07.340","Text":"and p2 being equal to 0.6."},{"Start":"07:07.340 ","End":"07:10.565","Text":"Now, since p is the probability,"},{"Start":"07:10.565 ","End":"07:15.533","Text":"then this is not an acceptable answer,"},{"Start":"07:15.533 ","End":"07:22.890","Text":"so p2 is the correct 1 where the probability is 0.6."}],"ID":13064},{"Watched":false,"Name":"Exercise 10 Parts b-e","Duration":"9m 16s","ChapterTopicVideoID":12586,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"Joan, 1 of the students taking the driving test"},{"Start":"00:02.880 ","End":"00:05.865","Text":"continues taking the test until she passes."},{"Start":"00:05.865 ","End":"00:07.860","Text":"We\u0027re asked in section b, what\u0027s"},{"Start":"00:07.860 ","End":"00:11.595","Text":"the probability of passing the driving test on the fourth trial?"},{"Start":"00:11.595 ","End":"00:15.225","Text":"Well, here, we have a geometric distribution."},{"Start":"00:15.225 ","End":"00:18.960","Text":"Let\u0027s take a look at the conditions to make sure."},{"Start":"00:18.960 ","End":"00:21.480","Text":"Here are the conditions."},{"Start":"00:21.480 ","End":"00:23.310","Text":"Let\u0027s take a look at the first 1."},{"Start":"00:23.310 ","End":"00:26.640","Text":"We have the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:26.640 ","End":"00:30.490","Text":"Our Bernoulli trial is taking the test"},{"Start":"00:30.490 ","End":"00:35.475","Text":"and we have an underlying assumption that the tests are independent."},{"Start":"00:35.475 ","End":"00:40.730","Text":"X is defined as the number of trials carried out until the first success."},{"Start":"00:40.730 ","End":"00:45.380","Text":"Instead of X, we\u0027ll be using Y. Y"},{"Start":"00:45.380 ","End":"00:51.350","Text":"then has a geometric distribution with a probability of 0.6."},{"Start":"00:51.350 ","End":"00:54.899","Text":"That\u0027s a probability that we\u0027ve calculated in section a."},{"Start":"00:54.899 ","End":"01:01.400","Text":"Let\u0027s just recall the probability of Y equaling k. In a geometric distribution,"},{"Start":"01:01.400 ","End":"01:02.600","Text":"that\u0027s the probability,"},{"Start":"01:02.600 ","End":"01:09.275","Text":"that\u0027s p times 1 minus p to the power of k minus 1."},{"Start":"01:09.275 ","End":"01:16.895","Text":"We want to find out the probability of Y being equal to 4."},{"Start":"01:16.895 ","End":"01:18.665","Text":"Why is that?"},{"Start":"01:18.665 ","End":"01:23.330","Text":"Because we\u0027re asked for the probability of passing the driving test and the fourth trial."},{"Start":"01:23.330 ","End":"01:24.755","Text":"Well, that equals to,"},{"Start":"01:24.755 ","End":"01:25.820","Text":"let\u0027s plug in the numbers."},{"Start":"01:25.820 ","End":"01:29.480","Text":"That\u0027s 0.6 times 1 minus p,"},{"Start":"01:29.480 ","End":"01:32.600","Text":"that\u0027s 0.4 to the power of k minus 1,"},{"Start":"01:32.600 ","End":"01:35.255","Text":"that\u0027s 4 minus 1, that\u0027s 3,"},{"Start":"01:35.255 ","End":"01:42.000","Text":"and that equals to 0.0384."},{"Start":"01:42.580 ","End":"01:46.400","Text":"In section c, we\u0027re asked what\u0027s the probability that she"},{"Start":"01:46.400 ","End":"01:49.512","Text":"will have to take the test at least 5 times."},{"Start":"01:49.512 ","End":"01:53.690","Text":"Let\u0027s just recall from previous sections that x has"},{"Start":"01:53.690 ","End":"01:59.825","Text":"a geometric distribution where the probability P equals 0.6."},{"Start":"01:59.825 ","End":"02:01.640","Text":"As such, what do we ask?"},{"Start":"02:01.640 ","End":"02:07.460","Text":"We\u0027re asked, what\u0027s the probability of x being greater or equal to 5?"},{"Start":"02:07.460 ","End":"02:09.740","Text":"1 of the special characteristics of"},{"Start":"02:09.740 ","End":"02:15.430","Text":"the geometric distribution is that the probability of x being greater than k,"},{"Start":"02:15.430 ","End":"02:21.230","Text":"that equals to 1 minus p to the power of k. But here,"},{"Start":"02:21.230 ","End":"02:25.100","Text":"we can\u0027t use this special characteristic because here,"},{"Start":"02:25.100 ","End":"02:27.710","Text":"we have greater than or equal to and here,"},{"Start":"02:27.710 ","End":"02:29.900","Text":"we have just greater than."},{"Start":"02:29.900 ","End":"02:33.725","Text":"But because x is a discrete random variable,"},{"Start":"02:33.725 ","End":"02:40.535","Text":"this equals the probability of x being greater than 4."},{"Start":"02:40.535 ","End":"02:46.061","Text":"Because again, x is a discrete random variable."},{"Start":"02:46.061 ","End":"02:50.360","Text":"Now, we can use this special characteristic and that equals"},{"Start":"02:50.360 ","End":"02:54.990","Text":"to 1 minus 0.6 to the power of k,"},{"Start":"02:54.990 ","End":"02:56.730","Text":"now, k is 4."},{"Start":"02:56.730 ","End":"03:03.790","Text":"That equals to 0.0256."},{"Start":"03:04.090 ","End":"03:06.860","Text":"In this section, we\u0027re asked what are"},{"Start":"03:06.860 ","End":"03:10.384","Text":"the expectation variance of the number of failed attempts?"},{"Start":"03:10.384 ","End":"03:14.110","Text":"Let\u0027s just mark this out."},{"Start":"03:14.110 ","End":"03:20.383","Text":"We want to find the expectation and the variance of the number of failed attempts."},{"Start":"03:20.383 ","End":"03:21.560","Text":"Why is that important?"},{"Start":"03:21.560 ","End":"03:23.015","Text":"Because up till now,"},{"Start":"03:23.015 ","End":"03:26.330","Text":"we\u0027ve been dealing with the random variable X,"},{"Start":"03:26.330 ","End":"03:36.120","Text":"which is the number of attempts until I pass."},{"Start":"03:36.120 ","End":"03:40.849","Text":"If X is a number of attempts until it passed,"},{"Start":"03:40.849 ","End":"03:47.240","Text":"that means that x minus 1 is the number of failed attempts."},{"Start":"03:47.240 ","End":"03:54.750","Text":"So Y would be the number of failed attempts."},{"Start":"03:55.460 ","End":"03:59.100","Text":"Here, we have a linear transformation."},{"Start":"03:59.100 ","End":"04:01.640","Text":"What\u0027s the general form of a linear transformation?"},{"Start":"04:01.640 ","End":"04:05.690","Text":"Y equals ax plus b."},{"Start":"04:05.690 ","End":"04:12.110","Text":"Well, here, we can identify that a equals 1 and b equals minus 1."},{"Start":"04:12.110 ","End":"04:17.663","Text":"Here\u0027s our 1, that\u0027s 1 times X and minus 1, that\u0027s b."},{"Start":"04:17.663 ","End":"04:20.715","Text":"What\u0027s the expectation of y?"},{"Start":"04:20.715 ","End":"04:28.400","Text":"Well, that\u0027ll be a times the expectation of x plus b and the variance of Y,"},{"Start":"04:28.400 ","End":"04:34.620","Text":"well, that equals to a squared times the variance of x."},{"Start":"04:35.070 ","End":"04:45.350","Text":"We know that x has a geometric distribution with the parameter p equals 0.6."},{"Start":"04:45.350 ","End":"04:46.610","Text":"If that\u0027s the case,"},{"Start":"04:46.610 ","End":"04:48.740","Text":"the expectation of x,"},{"Start":"04:48.740 ","End":"04:53.570","Text":"that equals to 1 over p or 1 divided by"},{"Start":"04:53.570 ","End":"05:00.920","Text":"0.6 and that equals to 1.67 and the variance of x,"},{"Start":"05:00.920 ","End":"05:07.340","Text":"that equals to q over p squared."},{"Start":"05:07.340 ","End":"05:12.260","Text":"Now, q is minus p. That\u0027s 0.4 divided"},{"Start":"05:12.260 ","End":"05:17.786","Text":"by 0.6 squared and that equals to 1.11."},{"Start":"05:17.786 ","End":"05:24.940","Text":"Having said that, we want to find the expectation of Y,"},{"Start":"05:24.940 ","End":"05:27.880","Text":"the linear transformation right here."},{"Start":"05:27.880 ","End":"05:29.785","Text":"That equals to,"},{"Start":"05:29.785 ","End":"05:31.090","Text":"let\u0027s plug in the numbers,"},{"Start":"05:31.090 ","End":"05:34.480","Text":"a here\u0027s 1 times the expectation of x,"},{"Start":"05:34.480 ","End":"05:41.130","Text":"that\u0027s 1.67 plus b."},{"Start":"05:41.130 ","End":"05:43.555","Text":"B is minus 1, so that\u0027s minus 1,"},{"Start":"05:43.555 ","End":"05:46.157","Text":"that equals to 0.67."},{"Start":"05:46.157 ","End":"05:49.615","Text":"What about the variance of Y?"},{"Start":"05:49.615 ","End":"05:53.275","Text":"Well, that equals to a squared times the variance of x."},{"Start":"05:53.275 ","End":"05:55.390","Text":"A squared here is 1,"},{"Start":"05:55.390 ","End":"06:00.759","Text":"so that\u0027s 1 squared times the variance of x, that\u0027s 1.11."},{"Start":"06:00.759 ","End":"06:05.290","Text":"That equals again, 1.11."},{"Start":"06:05.570 ","End":"06:13.018","Text":"The expectation of Y is the expected number of failed attempts."},{"Start":"06:13.018 ","End":"06:21.320","Text":"The units here is failed attempts and the units here for the variance,"},{"Start":"06:21.320 ","End":"06:30.930","Text":"that\u0027ll be 1.11 failed attempts squared."},{"Start":"06:32.020 ","End":"06:35.570","Text":"In Section e, we\u0027re given that we know that"},{"Start":"06:35.570 ","End":"06:38.465","Text":"Joan took the test 3 times and still didn\u0027t pass."},{"Start":"06:38.465 ","End":"06:42.470","Text":"We\u0027re asked what are chances of passing the test under fifth try?"},{"Start":"06:42.470 ","End":"06:45.695","Text":"Well, here, we have a conditional probability."},{"Start":"06:45.695 ","End":"06:47.210","Text":"Let\u0027s just write this out."},{"Start":"06:47.210 ","End":"06:49.355","Text":"That\u0027s the probability,"},{"Start":"06:49.355 ","End":"06:55.190","Text":"what\u0027s given that Joan took the test 3 times and still didn\u0027t pass,"},{"Start":"06:55.190 ","End":"06:58.385","Text":"that means that x has to be greater than 3."},{"Start":"06:58.385 ","End":"07:03.005","Text":"Again, let\u0027s remind ourselves what\u0027s x. X is the number"},{"Start":"07:03.005 ","End":"07:10.395","Text":"of attempts until we pass."},{"Start":"07:10.395 ","End":"07:17.900","Text":"We also know that X has a geometric distribution where p equals 0.6."},{"Start":"07:17.900 ","End":"07:21.740","Text":"The probability of x equaling k,"},{"Start":"07:21.740 ","End":"07:27.060","Text":"that\u0027s p times 1 minus p to the power of k"},{"Start":"07:27.060 ","End":"07:32.975","Text":"minus 1 and we also know that the probability of x being greater than k,"},{"Start":"07:32.975 ","End":"07:39.445","Text":"that\u0027s 1 minus p to the power of k. Having recalled all that,"},{"Start":"07:39.445 ","End":"07:41.360","Text":"what do we have here?"},{"Start":"07:41.360 ","End":"07:44.645","Text":"Well, we have the probability of what\u0027s given,"},{"Start":"07:44.645 ","End":"07:46.940","Text":"we\u0027re given that x is greater than 3,"},{"Start":"07:46.940 ","End":"07:52.355","Text":"we\u0027re trying to find out what are the chances of passing the test on a fifth trial."},{"Start":"07:52.355 ","End":"07:56.530","Text":"That means that x has to equal to 5."},{"Start":"07:56.570 ","End":"07:58.775","Text":"If that\u0027s the case,"},{"Start":"07:58.775 ","End":"08:00.275","Text":"we know how to solve this."},{"Start":"08:00.275 ","End":"08:03.650","Text":"In the numerator, there\u0027s a probability of x being equal to"},{"Start":"08:03.650 ","End":"08:08.505","Text":"5 intersect x greater than 3."},{"Start":"08:08.505 ","End":"08:13.693","Text":"In the denominator, there\u0027s the probability of x being greater than 3."},{"Start":"08:13.693 ","End":"08:20.960","Text":"What\u0027s the probability of x equal 5 intersect x greater than 3?"},{"Start":"08:20.960 ","End":"08:24.970","Text":"Well, that\u0027s the probability of x equal to 5."},{"Start":"08:24.970 ","End":"08:27.440","Text":"Then in the denominator,"},{"Start":"08:27.440 ","End":"08:32.280","Text":"we still have the probability of x being greater than 3."},{"Start":"08:32.710 ","End":"08:39.055","Text":"Let\u0027s now use this equation right here when x equals 5."},{"Start":"08:39.055 ","End":"08:41.720","Text":"Now, p is 0.6,"},{"Start":"08:41.720 ","End":"08:45.045","Text":"so the probability of x being equal to 5,"},{"Start":"08:45.045 ","End":"08:52.085","Text":"that\u0027s 0.6 times 0.4 to the power of k minus 1, 5 minus 1,"},{"Start":"08:52.085 ","End":"08:58.490","Text":"that\u0027s 4 divided by the probability of x being greater than 3,"},{"Start":"08:58.490 ","End":"08:59.900","Text":"that\u0027s 1 minus p,"},{"Start":"08:59.900 ","End":"09:04.185","Text":"that\u0027s 0.4 to the power of k. K is 3."},{"Start":"09:04.185 ","End":"09:07.010","Text":"Now, these guys cancel each other out,"},{"Start":"09:07.010 ","End":"09:15.450","Text":"so we have 0.6 times 0.4 and that equals to 0.24."}],"ID":13065},{"Watched":false,"Name":"Exercise 11","Duration":"6m 32s","ChapterTopicVideoID":12587,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this question, we\u0027re given that a robot is at the 0 point on the axis,"},{"Start":"00:04.290 ","End":"00:06.255","Text":"that means he\u0027s on the origins."},{"Start":"00:06.255 ","End":"00:07.965","Text":"Now it takes n steps."},{"Start":"00:07.965 ","End":"00:11.745","Text":"In every step, the probability of going 1 unit to the right is p,"},{"Start":"00:11.745 ","End":"00:16.905","Text":"and the probability of going 1 unit to the left is 1 minus p. Now,"},{"Start":"00:16.905 ","End":"00:22.430","Text":"we define X as the number on which the robot lands after taking n steps,"},{"Start":"00:22.430 ","End":"00:30.155","Text":"and we\u0027re asked to write the probability function of X as a function of p and n. Now,"},{"Start":"00:30.155 ","End":"00:32.636","Text":"this is by no means a simple question,"},{"Start":"00:32.636 ","End":"00:35.945","Text":"then it does take a little bit of creative thinking."},{"Start":"00:35.945 ","End":"00:39.620","Text":"Let\u0027s get to it. First of all,"},{"Start":"00:39.620 ","End":"00:49.290","Text":"let\u0027s define y as the number of steps taken to the right."},{"Start":"00:49.290 ","End":"00:54.260","Text":"Now, we can look at the decision to turn"},{"Start":"00:54.260 ","End":"00:59.255","Text":"right or left as a Bernoulli trial with success and failure."},{"Start":"00:59.255 ","End":"01:05.705","Text":"So let\u0027s define our success here as turning right."},{"Start":"01:05.705 ","End":"01:11.690","Text":"We could have easily have decided that success was to turn left but we\u0027ll go with right."},{"Start":"01:11.690 ","End":"01:15.975","Text":"Now, the probability of turning right is p,"},{"Start":"01:15.975 ","End":"01:17.700","Text":"that\u0027s given right here."},{"Start":"01:17.700 ","End":"01:19.520","Text":"How many trials do we have?"},{"Start":"01:19.520 ","End":"01:25.550","Text":"Well, we have n. We have n trials which are independent of each other,"},{"Start":"01:25.550 ","End":"01:30.810","Text":"each decision to turn right or left is independent of the other."},{"Start":"01:30.910 ","End":"01:34.160","Text":"If we have success,"},{"Start":"01:34.160 ","End":"01:37.865","Text":"we defined success with the same probability for steps for"},{"Start":"01:37.865 ","End":"01:41.900","Text":"all the trials and we have n trials that are independent of each other."},{"Start":"01:41.900 ","End":"01:48.440","Text":"Well, that means that our random variable is distributed with"},{"Start":"01:48.440 ","End":"01:52.115","Text":"a binomial distribution with n and"},{"Start":"01:52.115 ","End":"01:59.150","Text":"p. Now let\u0027s take a look at X."},{"Start":"01:59.150 ","End":"02:01.765","Text":"Don\u0027t forget we wanted X."},{"Start":"02:01.765 ","End":"02:03.360","Text":"We\u0027re looking at X."},{"Start":"02:03.360 ","End":"02:07.435","Text":"Let X be the number which the robot lands after taking n steps."},{"Start":"02:07.435 ","End":"02:15.140","Text":"So what we want to do is we want to find the relationship between X and Y."},{"Start":"02:15.140 ","End":"02:18.585","Text":"Now, X, as we said,"},{"Start":"02:18.585 ","End":"02:27.420","Text":"is the number that we land on after n steps."},{"Start":"02:27.420 ","End":"02:34.075","Text":"Well, if we take Y steps to the right,"},{"Start":"02:34.075 ","End":"02:37.160","Text":"and then we subtract from it,"},{"Start":"02:37.160 ","End":"02:40.160","Text":"the number of steps taken to the left,"},{"Start":"02:40.160 ","End":"02:45.740","Text":"then that would be the number that we will land on after n steps."},{"Start":"02:45.740 ","End":"02:47.705","Text":"So Y, as we said,"},{"Start":"02:47.705 ","End":"02:50.990","Text":"is the number of steps to the right."},{"Start":"02:50.990 ","End":"02:55.640","Text":"Now we subtract from that the number of steps taken to the left."},{"Start":"02:55.640 ","End":"02:57.680","Text":"Now, what\u0027s that? Well,"},{"Start":"02:57.680 ","End":"03:03.830","Text":"that\u0027s the total of n steps minus the number of steps taken to the right."},{"Start":"03:03.830 ","End":"03:06.200","Text":"So that\u0027s n minus Y,"},{"Start":"03:06.200 ","End":"03:11.140","Text":"and that would be the number of steps taken to the left."},{"Start":"03:11.140 ","End":"03:14.595","Text":"Now, having understood that,"},{"Start":"03:14.595 ","End":"03:16.400","Text":"we can simplify this."},{"Start":"03:16.400 ","End":"03:18.560","Text":"But first of all,"},{"Start":"03:18.560 ","End":"03:20.390","Text":"in order to understand that,"},{"Start":"03:20.390 ","End":"03:22.310","Text":"let\u0027s take an example."},{"Start":"03:22.310 ","End":"03:24.800","Text":"Assume that n equals 10."},{"Start":"03:24.800 ","End":"03:29.005","Text":"We have a total of 10 steps that the robot has taken."},{"Start":"03:29.005 ","End":"03:32.430","Text":"We also assume that Y equals 3,"},{"Start":"03:32.430 ","End":"03:35.915","Text":"the robot took 3 steps to the right."},{"Start":"03:35.915 ","End":"03:38.690","Text":"Then how many steps did he take to the left?"},{"Start":"03:38.690 ","End":"03:41.060","Text":"That\u0027s n minus Y."},{"Start":"03:41.060 ","End":"03:43.120","Text":"That would be 7."},{"Start":"03:43.120 ","End":"03:45.780","Text":"So what would be X?"},{"Start":"03:45.780 ","End":"03:52.250","Text":"X would be 3 steps to the right minus 7 steps to the left."},{"Start":"03:52.250 ","End":"03:56.495","Text":"So that\u0027s 3 minus 7 and that would be minus 4."},{"Start":"03:56.495 ","End":"04:00.485","Text":"So after 10 steps under these conditions,"},{"Start":"04:00.485 ","End":"04:04.990","Text":"the robot would be on number minus 4."},{"Start":"04:04.990 ","End":"04:07.580","Text":"We\u0027ve understood this."},{"Start":"04:07.580 ","End":"04:09.800","Text":"Now let\u0027s just simplify that."},{"Start":"04:09.800 ","End":"04:14.210","Text":"X then would be equal to 2Y minus"},{"Start":"04:14.210 ","End":"04:21.855","Text":"n. Let\u0027s get back to our random variable Y."},{"Start":"04:21.855 ","End":"04:26.120","Text":"Now if Y had a binomial distribution with n and p,"},{"Start":"04:26.120 ","End":"04:34.820","Text":"then the probability of Y equaling k or we can say that the probability of y,"},{"Start":"04:34.820 ","End":"04:37.325","Text":"that would be equal to n/y"},{"Start":"04:37.325 ","End":"04:47.360","Text":"p^y times 1 minus p to the power of n minus y."},{"Start":"04:47.360 ","End":"04:53.725","Text":"But we don\u0027t want this probability function as a function of y."},{"Start":"04:53.725 ","End":"04:55.870","Text":"We want to see it as a function of x,"},{"Start":"04:55.870 ","End":"04:57.790","Text":"that\u0027s what we were asked for."},{"Start":"04:57.790 ","End":"04:59.770","Text":"So if that\u0027s the case,"},{"Start":"04:59.770 ","End":"05:03.730","Text":"let\u0027s take a look at the relationship between x and y."},{"Start":"05:03.730 ","End":"05:08.800","Text":"In here, we have x as a function of y but let\u0027s now turn this"},{"Start":"05:08.800 ","End":"05:14.365","Text":"around and see if we can get y as a function of x."},{"Start":"05:14.365 ","End":"05:22.160","Text":"Well, if we do that and y would be equal to x plus n over 2."},{"Start":"05:23.690 ","End":"05:27.910","Text":"In this function, every time we see a y,"},{"Start":"05:27.910 ","End":"05:30.350","Text":"we\u0027ll replace it with this expression."},{"Start":"05:30.350 ","End":"05:32.330","Text":"So let\u0027s do that."},{"Start":"05:32.330 ","End":"05:34.905","Text":"That\u0027ll be n over,"},{"Start":"05:34.905 ","End":"05:36.830","Text":"now that\u0027s instead of y,"},{"Start":"05:36.830 ","End":"05:38.480","Text":"that\u0027ll be x plus n over 2."},{"Start":"05:38.480 ","End":"05:45.140","Text":"So that\u0027ll be x plus n divided by 2 times p^y."},{"Start":"05:45.140 ","End":"05:48.770","Text":"That\u0027s to the power of x plus n divided by"},{"Start":"05:48.770 ","End":"05:56.182","Text":"2 times 1 minus p to the power of n minus,"},{"Start":"05:56.182 ","End":"05:57.475","Text":"now, n minus y."},{"Start":"05:57.475 ","End":"06:03.310","Text":"So that\u0027ll be n minus x plus n divided by 2."},{"Start":"06:03.310 ","End":"06:06.180","Text":"Let\u0027s just simplify that."},{"Start":"06:06.180 ","End":"06:10.770","Text":"That\u0027ll be n over x plus n divided by"},{"Start":"06:10.770 ","End":"06:20.090","Text":"2 times p to the power of x plus n divided by 2 times 1 minus p to the power."},{"Start":"06:20.090 ","End":"06:27.440","Text":"Now, this expression up here would be equal to n minus x divided by 2."},{"Start":"06:27.440 ","End":"06:31.890","Text":"So this is the answer."}],"ID":13066},{"Watched":false,"Name":"Exercise 12","Duration":"11m 10s","ChapterTopicVideoID":12588,"CourseChapterTopicPlaylistID":64491,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.633","Text":"In this question, we\u0027ll be tossing a coin."},{"Start":"00:02.633 ","End":"00:08.070","Text":"The chances of getting heads when tossing a coin is P. If the first toss is heads,"},{"Start":"00:08.070 ","End":"00:10.155","Text":"you lose a dollar and the game is over."},{"Start":"00:10.155 ","End":"00:13.560","Text":"Otherwise, the coin is tossed over and over again."},{"Start":"00:13.560 ","End":"00:16.200","Text":"You win 1 dollar for each tales that you"},{"Start":"00:16.200 ","End":"00:19.265","Text":"get from the beginning of the game until you get your first heads,"},{"Start":"00:19.265 ","End":"00:24.375","Text":"and we\u0027re asked to construct a probability function of the profits."},{"Start":"00:24.375 ","End":"00:35.350","Text":"Let\u0027s define first n as the number of tosses and x would be our profits."},{"Start":"00:35.480 ","End":"00:40.110","Text":"Now, let\u0027s just create a table to see how this game works."},{"Start":"00:40.110 ","End":"00:45.119","Text":"That\u0027s n, that\u0027ll be x,"},{"Start":"00:45.119 ","End":"00:48.280","Text":"and that\u0027ll be the probabilities."},{"Start":"00:48.280 ","End":"00:50.260","Text":"How many tosses do we have?"},{"Start":"00:50.260 ","End":"00:51.775","Text":"Well, we have the first toss,"},{"Start":"00:51.775 ","End":"00:52.975","Text":"the second toss,"},{"Start":"00:52.975 ","End":"00:55.345","Text":"the third toss, and so on and so forth."},{"Start":"00:55.345 ","End":"00:59.050","Text":"What happens to our profits?"},{"Start":"00:59.050 ","End":"01:02.365","Text":"If we toss the coin only once and stop,"},{"Start":"01:02.365 ","End":"01:04.270","Text":"that means that we got heads."},{"Start":"01:04.270 ","End":"01:06.370","Text":"If we got heads on the first toss,"},{"Start":"01:06.370 ","End":"01:10.565","Text":"we lose a dollar so that\u0027s minus 1 and profit is minus 1 dollar."},{"Start":"01:10.565 ","End":"01:12.580","Text":"What\u0027s the probability of getting a head?"},{"Start":"01:12.580 ","End":"01:16.650","Text":"Well that\u0027s p. What about 2 tosses?"},{"Start":"01:16.650 ","End":"01:19.230","Text":"Well, if we have 2 tosses and then we stop,"},{"Start":"01:19.230 ","End":"01:24.900","Text":"then we must have tails on the first toss and heads the second toss."},{"Start":"01:24.900 ","End":"01:29.715","Text":"If we have 1 tail in the 2 tosses,"},{"Start":"01:29.715 ","End":"01:32.280","Text":"that means that I win 1 dollar."},{"Start":"01:32.280 ","End":"01:34.140","Text":"Because I win 1 dollar for each tail,"},{"Start":"01:34.140 ","End":"01:36.705","Text":"so my profit here is 1."},{"Start":"01:36.705 ","End":"01:38.540","Text":"What\u0027s the probability here?"},{"Start":"01:38.540 ","End":"01:40.150","Text":"Well, that would be 1 minus p,"},{"Start":"01:40.150 ","End":"01:45.005","Text":"that\u0027s the probability of getting a tail times the probability of getting a head."},{"Start":"01:45.005 ","End":"01:47.720","Text":"What about 3 tosses?"},{"Start":"01:47.720 ","End":"01:51.020","Text":"Well, when I have 3 tosses,"},{"Start":"01:51.020 ","End":"01:52.700","Text":"I have at first,"},{"Start":"01:52.700 ","End":"01:54.200","Text":"the first tossing, the second toss,"},{"Start":"01:54.200 ","End":"01:55.940","Text":"I have both of them tails,"},{"Start":"01:55.940 ","End":"01:58.325","Text":"and the third toss being heads."},{"Start":"01:58.325 ","End":"01:59.975","Text":"How many tails do I have?"},{"Start":"01:59.975 ","End":"02:02.720","Text":"2. What\u0027s my profit?"},{"Start":"02:02.720 ","End":"02:04.110","Text":"Well,1 dollar for each tail,"},{"Start":"02:04.110 ","End":"02:06.450","Text":"so I have 2 dollars of profits,"},{"Start":"02:06.450 ","End":"02:08.310","Text":"my profit is 2 dollars."},{"Start":"02:08.310 ","End":"02:10.250","Text":"What\u0027s the probability here?"},{"Start":"02:10.250 ","End":"02:15.770","Text":"Well, getting 2 tails is 1 minus p squared times p getting a head."},{"Start":"02:15.770 ","End":"02:18.665","Text":"Here, we can see a pattern emerging,"},{"Start":"02:18.665 ","End":"02:24.660","Text":"that\u0027s the probability of x being equal to k. Here,"},{"Start":"02:24.660 ","End":"02:27.180","Text":"look at the values of k. That\u0027s minus 1,"},{"Start":"02:27.180 ","End":"02:29.100","Text":"and then it jumps to 1,"},{"Start":"02:29.100 ","End":"02:30.900","Text":"2, and 3 and so forth."},{"Start":"02:30.900 ","End":"02:34.425","Text":"Well, that equals to p,"},{"Start":"02:34.425 ","End":"02:36.195","Text":"the probability p,"},{"Start":"02:36.195 ","End":"02:41.640","Text":"if k is equal to minus 1,"},{"Start":"02:41.640 ","End":"02:48.680","Text":"and the probability of 1 minus p^k minus"},{"Start":"02:48.680 ","End":"02:56.195","Text":"1 times p for k equaling 2,"},{"Start":"02:56.195 ","End":"02:59.395","Text":"3, and so on and so forth."},{"Start":"02:59.395 ","End":"03:01.940","Text":"This is a probability function."},{"Start":"03:01.940 ","End":"03:05.450","Text":"This looks a lot like the geometric distribution,"},{"Start":"03:05.450 ","End":"03:12.215","Text":"but here this first task ruins it all and that\u0027s the way it is."},{"Start":"03:12.215 ","End":"03:14.765","Text":"We\u0027re going to have to deal with this,"},{"Start":"03:14.765 ","End":"03:19.670","Text":"and we\u0027ll do that in the next sections but this right here,"},{"Start":"03:19.670 ","End":"03:23.990","Text":"that\u0027s the probability function of our profits."},{"Start":"03:23.990 ","End":"03:28.730","Text":"In section b, we\u0027re asked to express the expected profits in terms of"},{"Start":"03:28.730 ","End":"03:34.090","Text":"p. We\u0027re looking here for the expectation of the profits,"},{"Start":"03:34.090 ","End":"03:35.915","Text":"the expectation of x."},{"Start":"03:35.915 ","End":"03:41.390","Text":"Let\u0027s first recall what the probability function of x was."},{"Start":"03:41.390 ","End":"03:44.795","Text":"That was the probability of x being equal to k,"},{"Start":"03:44.795 ","End":"03:51.300","Text":"but that equal to p where k was equal to minus 1,"},{"Start":"03:51.300 ","End":"03:58.250","Text":"and 1 minus p^k minus 1 times p where k equal 2,"},{"Start":"03:58.250 ","End":"04:02.400","Text":"3, 4, and so on and so forth."},{"Start":"04:03.710 ","End":"04:10.055","Text":"How can we use that to calculate the expectation of x?"},{"Start":"04:10.055 ","End":"04:14.960","Text":"Well, what\u0027s the formal formula for the expectation?"},{"Start":"04:14.960 ","End":"04:20.300","Text":"Well, that\u0027s the sum of x times the probability of x."},{"Start":"04:20.300 ","End":"04:22.610","Text":"Let\u0027s just expand that."},{"Start":"04:22.610 ","End":"04:29.320","Text":"That will equal to minus 1 times p plus,"},{"Start":"04:29.320 ","End":"04:32.215","Text":"we\u0027re summing things up, now k equals 2."},{"Start":"04:32.215 ","End":"04:39.320","Text":"That\u0027ll be 2 times 1 minus p to the power of 2 minus 1,"},{"Start":"04:39.320 ","End":"04:41.630","Text":"that\u0027s 1, times p,"},{"Start":"04:41.630 ","End":"04:46.505","Text":"plus 3 times 1 minus p squared times"},{"Start":"04:46.505 ","End":"04:52.580","Text":"p plus 4 times 1 minus p cubed times p,"},{"Start":"04:52.580 ","End":"04:56.534","Text":"and so on and so forth."},{"Start":"04:56.534 ","End":"04:59.075","Text":"Here, as we said,"},{"Start":"04:59.075 ","End":"05:04.940","Text":"we have a thing that looks very similar to a geometric distribution."},{"Start":"05:04.940 ","End":"05:06.530","Text":"What ruins it for us,"},{"Start":"05:06.530 ","End":"05:08.480","Text":"is this guy right here."},{"Start":"05:08.480 ","End":"05:11.930","Text":"But, again, how can we utilize this similarity?"},{"Start":"05:11.930 ","End":"05:13.340","Text":"Well, let\u0027s take a look."},{"Start":"05:13.340 ","End":"05:16.210","Text":"Let\u0027s put this on hold for a bit."},{"Start":"05:16.210 ","End":"05:21.575","Text":"Take a look at a general random variable, let\u0027s call it w,"},{"Start":"05:21.575 ","End":"05:26.180","Text":"that has a geometric distribution with parameter p. We know that"},{"Start":"05:26.180 ","End":"05:31.790","Text":"the expectation of w on 1 hand equals 1 over p but on the other hand,"},{"Start":"05:31.790 ","End":"05:33.980","Text":"let\u0027s just expand this out here."},{"Start":"05:33.980 ","End":"05:41.525","Text":"Well, that would be 1 times p plus 2 times 1 minus p times p,"},{"Start":"05:41.525 ","End":"05:46.505","Text":"plus 3 times 1 minus p squared times p,"},{"Start":"05:46.505 ","End":"05:50.075","Text":"plus, and so on and so forth till infinity."},{"Start":"05:50.075 ","End":"05:56.000","Text":"What\u0027s similar between this expression,"},{"Start":"05:56.000 ","End":"05:59.750","Text":"this infinite series of the general random variable w,"},{"Start":"05:59.750 ","End":"06:04.100","Text":"and this series for our random variable x?"},{"Start":"06:04.100 ","End":"06:13.580","Text":"Well, we see that this series from this expression onwards till infinity,"},{"Start":"06:13.580 ","End":"06:17.135","Text":"that equals to this series, it\u0027s the same."},{"Start":"06:17.135 ","End":"06:19.550","Text":"What\u0027s missing here?"},{"Start":"06:19.550 ","End":"06:24.110","Text":"We\u0027re missing this expression 1 times p. If"},{"Start":"06:24.110 ","End":"06:28.610","Text":"we add this expression here and then subtracted again,"},{"Start":"06:28.610 ","End":"06:31.655","Text":"basically, we\u0027re leaving it alone on one hand,"},{"Start":"06:31.655 ","End":"06:37.790","Text":"but it helps us in defining what this infinite series is."},{"Start":"06:37.790 ","End":"06:41.705","Text":"Let\u0027s do this. Let\u0027s just take this guy,"},{"Start":"06:41.705 ","End":"06:46.610","Text":"and that would equal 2 now minus 1 times p,"},{"Start":"06:46.610 ","End":"06:49.115","Text":"now let\u0027s add this expression,"},{"Start":"06:49.115 ","End":"06:53.180","Text":"plus 1 times p plus,"},{"Start":"06:53.180 ","End":"06:54.500","Text":"let\u0027s continue on here,"},{"Start":"06:54.500 ","End":"07:03.890","Text":"that\u0027s 2 times 1 minus p times p plus 3 times 1 minus p squared times p,"},{"Start":"07:03.890 ","End":"07:05.555","Text":"and so on and so forth,"},{"Start":"07:05.555 ","End":"07:10.520","Text":"but let\u0027s not forget to subtract the same expression that we added,"},{"Start":"07:10.520 ","End":"07:16.085","Text":"that\u0027s minus 1 times p. What do we have here?"},{"Start":"07:16.085 ","End":"07:21.470","Text":"We see this expression right here, this infinite series,"},{"Start":"07:21.470 ","End":"07:26.771","Text":"it equals to this infinite series right here."},{"Start":"07:26.771 ","End":"07:34.300","Text":"We know that this is the infinite series that defines the expectation of x,"},{"Start":"07:34.300 ","End":"07:37.880","Text":"or defines the expectation of a random variable with"},{"Start":"07:37.880 ","End":"07:41.780","Text":"the geometric distribution but that equals to"},{"Start":"07:41.780 ","End":"07:45.830","Text":"1 divided by p. We can take this right here,"},{"Start":"07:45.830 ","End":"07:50.745","Text":"this infinite series and replace it with this expression right here."},{"Start":"07:50.745 ","End":"07:55.540","Text":"That will equal to minus 1 times p plus now,"},{"Start":"07:55.540 ","End":"08:01.130","Text":"let\u0027s replace this whole series with this expression because they\u0027re the same,"},{"Start":"08:01.130 ","End":"08:04.085","Text":"they\u0027re equivalent, that\u0027s 1 divided by p,"},{"Start":"08:04.085 ","End":"08:06.425","Text":"minus 1 times p,"},{"Start":"08:06.425 ","End":"08:07.730","Text":"that\u0027s this guy right here,"},{"Start":"08:07.730 ","End":"08:09.125","Text":"we didn\u0027t forget that."},{"Start":"08:09.125 ","End":"08:11.120","Text":"Let\u0027s just simplify this."},{"Start":"08:11.120 ","End":"08:17.130","Text":"That\u0027ll be minus 2p plus 1 divided by p,"},{"Start":"08:17.130 ","End":"08:19.850","Text":"and when we simplify that even more,"},{"Start":"08:19.850 ","End":"08:26.395","Text":"that becomes 1 minus 2 times p squared over p."},{"Start":"08:26.395 ","End":"08:35.360","Text":"That\u0027s the expectation of the profits of x in terms of p. In section c,"},{"Start":"08:35.360 ","End":"08:39.425","Text":"we\u0027re asked for which values of p is it worthwhile to play the game?"},{"Start":"08:39.425 ","End":"08:49.090","Text":"Well, the meaning of worthwhile is that the expectation of x where x is our profits,"},{"Start":"08:52.340 ","End":"08:58.095","Text":"the expected profits has to be greater than 0 otherwise we\u0027d be losing."},{"Start":"08:58.095 ","End":"09:00.890","Text":"In section b above,"},{"Start":"09:00.890 ","End":"09:06.710","Text":"we\u0027ve calculated the expectation of x to be 1 minus 2 times"},{"Start":"09:06.710 ","End":"09:12.440","Text":"p squared divided by p and that has to be greater than 0, that\u0027s what we want."},{"Start":"09:12.440 ","End":"09:19.205","Text":"We want this expectation to be greater than 0."},{"Start":"09:19.205 ","End":"09:22.100","Text":"First of all, let\u0027s clean things up."},{"Start":"09:22.100 ","End":"09:30.620","Text":"Let\u0027s multiply both sides by p. We have 1 minus 2 times p squared that\u0027s greater than 0."},{"Start":"09:30.620 ","End":"09:33.005","Text":"Let\u0027s move things around,"},{"Start":"09:33.005 ","End":"09:38.570","Text":"that\u0027s 1 is greater than 2 times p squared."},{"Start":"09:38.570 ","End":"09:42.845","Text":"Let\u0027s divide both sides by 2,"},{"Start":"09:42.845 ","End":"09:48.695","Text":"so that\u0027ll be 1/2 that\u0027ll be greater than p squared."},{"Start":"09:48.695 ","End":"09:56.000","Text":"Let\u0027s just remember algebra,"},{"Start":"09:56.000 ","End":"09:57.590","Text":"and if we clean this up a bit,"},{"Start":"09:57.590 ","End":"09:59.090","Text":"let\u0027s just switch sides,"},{"Start":"09:59.090 ","End":"10:03.580","Text":"that\u0027s p squared less than a half; that\u0027s the same thing."},{"Start":"10:03.580 ","End":"10:07.100","Text":"Whenever we have this type of expression,"},{"Start":"10:07.100 ","End":"10:14.930","Text":"then we know that p has to be between these guys right here."},{"Start":"10:14.930 ","End":"10:22.280","Text":"That\u0027ll be the square root of 1/2 and minus the square root of 1/2."},{"Start":"10:22.280 ","End":"10:24.725","Text":"What else do we know?"},{"Start":"10:24.725 ","End":"10:33.050","Text":"We know that p has to be greater than 0 because p is the probability."},{"Start":"10:33.050 ","End":"10:34.895","Text":"On our scale here,"},{"Start":"10:34.895 ","End":"10:42.515","Text":"where we have minus a half and plus a half, this is 0."},{"Start":"10:42.515 ","End":"10:46.819","Text":"This range talks about this guy right here,"},{"Start":"10:46.819 ","End":"10:50.960","Text":"and this range talks about this guy right here."},{"Start":"10:50.960 ","End":"10:55.800","Text":"We\u0027re looking at the probability"},{"Start":"10:55.940 ","End":"11:03.260","Text":"between 0 and the square root of 1/2."},{"Start":"11:03.260 ","End":"11:09.840","Text":"That\u0027s the range that will make this game worthwhile for us to play."}],"ID":13067}],"Thumbnail":null,"ID":64491},{"Name":"Special Discrete Probability Distributions - Poisson Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"3m 30s","ChapterTopicVideoID":12589,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this chapter, we\u0027ll be talking about the special discrete probabilities,"},{"Start":"00:03.600 ","End":"00:06.525","Text":"specifically the Poisson distribution."},{"Start":"00:06.525 ","End":"00:10.035","Text":"Now, why are they special discrete probabilities?"},{"Start":"00:10.035 ","End":"00:15.450","Text":"Because we\u0027re given the equation for the probability function"},{"Start":"00:15.450 ","End":"00:21.810","Text":"and the equations in order to calculate the variance and the expectation."},{"Start":"00:21.810 ","End":"00:25.230","Text":"Okay, so what\u0027s the Poisson distribution?"},{"Start":"00:25.230 ","End":"00:30.075","Text":"Well, the Poisson distribution characterizes events occurring over time."},{"Start":"00:30.075 ","End":"00:31.710","Text":"Now, it could be any event."},{"Start":"00:31.710 ","End":"00:33.180","Text":"It could be, for example,"},{"Start":"00:33.180 ","End":"00:37.095","Text":"the number of earthquakes in a year or"},{"Start":"00:37.095 ","End":"00:41.979","Text":"the number of clients entering a store in a 24-hour period."},{"Start":"00:41.979 ","End":"00:45.915","Text":"Now, Lambda, which is a Greek letter,"},{"Start":"00:45.915 ","End":"00:49.880","Text":"is a parameter of the probability distribution."},{"Start":"00:49.880 ","End":"00:55.130","Text":"Now this parameter represents the rate that the event occurs over time,"},{"Start":"00:55.130 ","End":"01:00.050","Text":"meaning how many events on average occurred during the specified timeframe."},{"Start":"01:00.050 ","End":"01:01.730","Text":"Now in our example of the earthquakes,"},{"Start":"01:01.730 ","End":"01:06.380","Text":"it could be 5 earthquakes in a year or it could"},{"Start":"01:06.380 ","End":"01:14.210","Text":"be 500 clients entering the store in a 24-hour period."},{"Start":"01:14.210 ","End":"01:16.820","Text":"Now, as we said,"},{"Start":"01:16.820 ","End":"01:21.109","Text":"when X has a Poisson distribution,"},{"Start":"01:21.109 ","End":"01:22.460","Text":"then we can write it like this."},{"Start":"01:22.460 ","End":"01:27.470","Text":"X is distributed with a Poisson distribution with parameter Lambda."},{"Start":"01:27.470 ","End":"01:30.425","Text":"Now, I prefer writing it like this."},{"Start":"01:30.425 ","End":"01:35.555","Text":"X is distributed with a Poisson distribution right here,"},{"Start":"01:35.555 ","End":"01:37.505","Text":"with a parameter Lambda."},{"Start":"01:37.505 ","End":"01:40.325","Text":"We can write p instead of pois."},{"Start":"01:40.325 ","End":"01:42.440","Text":"It\u0027s much cleaner."},{"Start":"01:42.440 ","End":"01:46.220","Text":"Now, the Poisson distribution has to appear"},{"Start":"01:46.220 ","End":"01:49.830","Text":"as an assumption in the question so you don\u0027t have to identify by"},{"Start":"01:49.830 ","End":"01:58.370","Text":"itself like you did in the geometric or in the binomial or in the uniform distributions."},{"Start":"01:58.370 ","End":"02:06.930","Text":"Here it has to be explicitly stated that X is distributed with a Poisson distribution."},{"Start":"02:07.120 ","End":"02:09.845","Text":"Now, having said that,"},{"Start":"02:09.845 ","End":"02:13.460","Text":"what\u0027s the probability function of this distribution?"},{"Start":"02:13.460 ","End":"02:17.790","Text":"Well, the probability function is p."},{"Start":"02:17.790 ","End":"02:23.940","Text":"The probability that X equals K because it\u0027s a discrete distribution."},{"Start":"02:23.940 ","End":"02:27.270","Text":"That equals to e to the power of minus Lambda,"},{"Start":"02:27.270 ","End":"02:34.880","Text":"our parameter, times Lambda to the power of k divided by K factorial."},{"Start":"02:34.880 ","End":"02:37.374","Text":"Now, for those of you who don\u0027t know,"},{"Start":"02:37.374 ","End":"02:44.015","Text":"e is an irrational number equaling 2.718 and so on and so forth till infinity."},{"Start":"02:44.015 ","End":"02:47.855","Text":"If you have to calculate e to the power of something,"},{"Start":"02:47.855 ","End":"02:51.940","Text":"then all you have to do is go to your calculator, so it\u0027s there."},{"Start":"02:51.940 ","End":"02:58.505","Text":"Now, again, this is the probability function of X,"},{"Start":"02:58.505 ","End":"03:05.285","Text":"where X is distributed with a Poisson distribution and K goes from 0,"},{"Start":"03:05.285 ","End":"03:08.585","Text":"1, 2, and so on and so forth till infinity."},{"Start":"03:08.585 ","End":"03:12.920","Text":"Now, another characteristic is that the expectation"},{"Start":"03:12.920 ","End":"03:17.780","Text":"and the variance equal one another and both of them equal Lambda."},{"Start":"03:17.780 ","End":"03:19.955","Text":"Now, apart from that,"},{"Start":"03:19.955 ","End":"03:24.020","Text":"there is some special characteristics of the probability function"},{"Start":"03:24.020 ","End":"03:29.550","Text":"that will encounter these when we get into the example."}],"ID":13068},{"Watched":false,"Name":"Example","Duration":"9m 44s","ChapterTopicVideoID":12590,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.680 ","End":"00:03.239","Text":"This is our example."},{"Start":"00:03.239 ","End":"00:04.530","Text":"At a call center,"},{"Start":"00:04.530 ","End":"00:08.010","Text":"calls arrive at the rate of 5 calls per minute."},{"Start":"00:08.010 ","End":"00:11.640","Text":"The number of calls has a Poisson distribution."},{"Start":"00:11.640 ","End":"00:14.850","Text":"We\u0027re asked, what\u0027s the probability that 1 call was"},{"Start":"00:14.850 ","End":"00:18.930","Text":"received in a given minute? Well, what do we know?"},{"Start":"00:18.930 ","End":"00:27.190","Text":"Well, first of all, let\u0027s define x as the number of calls per minute."},{"Start":"00:27.190 ","End":"00:35.640","Text":"Now, we\u0027re given that x is distributed with a Poisson distribution where Lambda equals 5."},{"Start":"00:35.640 ","End":"00:37.805","Text":"Now again, how do I know this?"},{"Start":"00:37.805 ","End":"00:41.134","Text":"Well, again, we\u0027re given that it\u0027s a Poisson distribution,"},{"Start":"00:41.134 ","End":"00:43.205","Text":"and Lambda being equal to 5,"},{"Start":"00:43.205 ","End":"00:44.330","Text":"well, we\u0027re given that right here."},{"Start":"00:44.330 ","End":"00:47.790","Text":"That\u0027s a rate of 5 calls per minute."},{"Start":"00:48.830 ","End":"00:50.855","Text":"Now, having said that,"},{"Start":"00:50.855 ","End":"00:57.735","Text":"what\u0027s the probability of x being equal to k?"},{"Start":"00:57.735 ","End":"01:02.360","Text":"Well, that equals to e to the power of minus Lambda,"},{"Start":"01:02.360 ","End":"01:05.270","Text":"times Lambda to the power of k,"},{"Start":"01:05.270 ","End":"01:08.580","Text":"divided by k factorial."},{"Start":"01:09.040 ","End":"01:13.180","Text":"Once we know all this, let\u0027s answer a."},{"Start":"01:13.180 ","End":"01:18.650","Text":"A ask us, what\u0027s the probability that 1 call was received in a given minute?"},{"Start":"01:18.650 ","End":"01:24.495","Text":"Well, that\u0027s the probability of x being equal to 1."},{"Start":"01:24.495 ","End":"01:26.705","Text":"Well, let\u0027s just plug in the numbers."},{"Start":"01:26.705 ","End":"01:29.120","Text":"That\u0027s e to the power of minus Lambda."},{"Start":"01:29.120 ","End":"01:30.950","Text":"Now, Lambda equals 5,"},{"Start":"01:30.950 ","End":"01:32.575","Text":"so that\u0027s minus 5,"},{"Start":"01:32.575 ","End":"01:35.670","Text":"times Lambda to the power of k. Well,"},{"Start":"01:35.670 ","End":"01:37.440","Text":"that\u0027s 5 to the power of k,"},{"Start":"01:37.440 ","End":"01:42.420","Text":"k equals 1, divided by k factorial."},{"Start":"01:42.420 ","End":"01:44.565","Text":"Now, that\u0027s 1 factorial,"},{"Start":"01:44.565 ","End":"01:49.270","Text":"and that equals to 0.0337."},{"Start":"01:52.510 ","End":"01:54.755","Text":"In section b we\u0027re asked,"},{"Start":"01:54.755 ","End":"01:58.835","Text":"what\u0027s the probability that 12 calls arrive in 2 minutes?"},{"Start":"01:58.835 ","End":"02:02.750","Text":"Now, here we have to pay attention because originally we"},{"Start":"02:02.750 ","End":"02:06.935","Text":"were talking about the number of calls per 1 minute,"},{"Start":"02:06.935 ","End":"02:12.095","Text":"and here, we\u0027re asked about the number of calls in 2 minutes."},{"Start":"02:12.095 ","End":"02:13.564","Text":"Now, this is something special."},{"Start":"02:13.564 ","End":"02:18.399","Text":"This is something new because the units of time have changed."},{"Start":"02:18.399 ","End":"02:21.620","Text":"How does the Poisson distribution deal with that?"},{"Start":"02:21.620 ","End":"02:26.930","Text":"Well, there\u0027s a special characteristics that actually talks about how"},{"Start":"02:26.930 ","End":"02:32.800","Text":"Lambda is affected by the new measure of time."},{"Start":"02:32.800 ","End":"02:35.480","Text":"In these special characteristics,"},{"Start":"02:35.480 ","End":"02:41.659","Text":"we see that the parameter Lambda is proportional to the interval of time being discussed."},{"Start":"02:41.659 ","End":"02:46.270","Text":"That means that if originally Lambda was equal to 5,"},{"Start":"02:46.270 ","End":"02:50.135","Text":"when we were talking about the number of calls per 1 minute,"},{"Start":"02:50.135 ","End":"02:53.019","Text":"then lambda would be equal to 10"},{"Start":"02:53.019 ","End":"02:57.215","Text":"when we\u0027re talking about the number of calls for 2 minutes,"},{"Start":"02:57.215 ","End":"03:00.320","Text":"or 15 for the number of calls per 3 minutes,"},{"Start":"03:00.320 ","End":"03:01.640","Text":"and so on and so forth."},{"Start":"03:01.640 ","End":"03:07.820","Text":"This is how the Poisson distribution deals with this."},{"Start":"03:07.820 ","End":"03:13.310","Text":"Lambda will be proportional to the interval of time being discussed."},{"Start":"03:13.310 ","End":"03:17.240","Text":"Let\u0027s get back to our question. Here we go."},{"Start":"03:17.240 ","End":"03:22.385","Text":"We\u0027re asked again, what\u0027s the probability that 12 calls arrive in 2 minutes?"},{"Start":"03:22.385 ","End":"03:28.500","Text":"Let\u0027s define a new variable, call it y,"},{"Start":"03:28.500 ","End":"03:36.075","Text":"and that would equal the number of calls per 2 minutes."},{"Start":"03:36.075 ","End":"03:43.250","Text":"Again, we said that if Lambda would equal to 5 for x,"},{"Start":"03:43.250 ","End":"03:46.450","Text":"where x is the number of calls per 1 minute,"},{"Start":"03:46.450 ","End":"03:52.500","Text":"then Lambda would be equal to 10 when we\u0027re talking about 2 minutes."},{"Start":"03:52.570 ","End":"03:57.385","Text":"Lambda would be equal to 15 for 3 minutes,"},{"Start":"03:57.385 ","End":"03:59.265","Text":"and so on and so forth."},{"Start":"03:59.265 ","End":"04:03.214","Text":"Here we\u0027re talking about the number of calls in 2 minutes,"},{"Start":"04:03.214 ","End":"04:08.275","Text":"so that means that Lambda here would be equal to 10."},{"Start":"04:08.275 ","End":"04:14.700","Text":"If that\u0027s the case, let\u0027s just write out the probability that we have to figure out."},{"Start":"04:14.750 ","End":"04:22.190","Text":"We\u0027re asked, what\u0027s the probability that y would be equal to 12?"},{"Start":"04:22.530 ","End":"04:25.190","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"04:25.190 ","End":"04:30.555","Text":"That\u0027s e to the power of minus Lambda minus 10,"},{"Start":"04:30.555 ","End":"04:32.865","Text":"times Lambda, that\u0027s 10,"},{"Start":"04:32.865 ","End":"04:35.625","Text":"to the power of k, now k is 12,"},{"Start":"04:35.625 ","End":"04:40.745","Text":"divided by k factorial."},{"Start":"04:40.745 ","End":"04:44.270","Text":"That means that we\u0027re dealing with 12 factorial here,"},{"Start":"04:44.270 ","End":"04:53.450","Text":"and that would equal to 0.0948."},{"Start":"04:53.450 ","End":"04:55.025","Text":"In section c, we\u0027re asked,"},{"Start":"04:55.025 ","End":"04:58.655","Text":"what\u0027s the probability that 1 call arrives in 1 minute,"},{"Start":"04:58.655 ","End":"05:03.410","Text":"and 12 calls will arrive during the following 2 minutes?"},{"Start":"05:03.720 ","End":"05:07.585","Text":"Let\u0027s first take a look at this graphically."},{"Start":"05:07.585 ","End":"05:11.195","Text":"Let\u0027s assume that this is a timeline."},{"Start":"05:11.195 ","End":"05:14.710","Text":"Starting from 0, we have 1 minute,"},{"Start":"05:14.710 ","End":"05:16.499","Text":"we have 2 minutes,"},{"Start":"05:16.499 ","End":"05:18.560","Text":"3 minutes, 4 minutes,"},{"Start":"05:18.560 ","End":"05:20.045","Text":"and so on and so forth."},{"Start":"05:20.045 ","End":"05:21.635","Text":"Now, what do we want?"},{"Start":"05:21.635 ","End":"05:27.090","Text":"We want the event that 1 call will arrive in 1 minute."},{"Start":"05:27.090 ","End":"05:28.790","Text":"Again, assume that we start at 0."},{"Start":"05:28.790 ","End":"05:32.480","Text":"In this interval, we want to have 1 call,"},{"Start":"05:32.480 ","End":"05:36.210","Text":"and in the following 2 minutes,"},{"Start":"05:36.210 ","End":"05:38.325","Text":"we should have 12 calls."},{"Start":"05:38.325 ","End":"05:41.480","Text":"Again, what would be the time intervals here?"},{"Start":"05:41.480 ","End":"05:45.020","Text":"Well, the 1st interval would be this 1 right here, from 0-1,"},{"Start":"05:45.020 ","End":"05:46.990","Text":"assuming that we start from 0,"},{"Start":"05:46.990 ","End":"05:49.290","Text":"and in the following 2 minutes,"},{"Start":"05:49.290 ","End":"05:53.490","Text":"that means that this is the 2nd time period."},{"Start":"05:53.490 ","End":"05:55.005","Text":"The following 2 minutes,"},{"Start":"05:55.005 ","End":"05:58.390","Text":"we should have 12 calls."},{"Start":"05:58.430 ","End":"06:02.885","Text":"Well, if we recall from the previous sections,"},{"Start":"06:02.885 ","End":"06:05.600","Text":"the number of calls per 1 minute,"},{"Start":"06:05.600 ","End":"06:08.330","Text":"that\u0027s defined by the random variable x,"},{"Start":"06:08.330 ","End":"06:13.560","Text":"and the number of calls per 2 minutes,"},{"Start":"06:13.560 ","End":"06:15.260","Text":"that was in section b,"},{"Start":"06:15.260 ","End":"06:17.750","Text":"that was defined as y."},{"Start":"06:17.750 ","End":"06:21.965","Text":"Now, having understood this,"},{"Start":"06:21.965 ","End":"06:27.665","Text":"what I want to do is I want to write out the probability that we have to calculate."},{"Start":"06:27.665 ","End":"06:32.090","Text":"Well, that\u0027s the probability of x being equal to 1."},{"Start":"06:32.090 ","End":"06:36.885","Text":"That means that in the 1st interval we have 1 call,"},{"Start":"06:36.885 ","End":"06:40.170","Text":"and y equals 12."},{"Start":"06:40.170 ","End":"06:42.120","Text":"That means in the 2nd interval,"},{"Start":"06:42.120 ","End":"06:43.770","Text":"that\u0027s a 2-minute interval,"},{"Start":"06:43.770 ","End":"06:46.530","Text":"we should have 12 calls."},{"Start":"06:46.530 ","End":"06:48.905","Text":"Now, in order to calculate this,"},{"Start":"06:48.905 ","End":"06:50.210","Text":"I want to use 1 of"},{"Start":"06:50.210 ","End":"06:55.860","Text":"the special characteristics of the Poisson distribution, and here it is."},{"Start":"06:56.140 ","End":"06:59.780","Text":"1 of the special characteristics is that"},{"Start":"06:59.780 ","End":"07:05.150","Text":"non-overlapping time intervals are independent of each other."},{"Start":"07:05.150 ","End":"07:08.000","Text":"If they\u0027re overlapping, they\u0027re dependent."},{"Start":"07:08.000 ","End":"07:10.190","Text":"If they\u0027re not overlapping,"},{"Start":"07:10.190 ","End":"07:13.355","Text":"then they\u0027re independent of each other."},{"Start":"07:13.355 ","End":"07:17.245","Text":"Let\u0027s go back now to our question."},{"Start":"07:17.245 ","End":"07:21.770","Text":"Here, we have 2 time intervals,"},{"Start":"07:21.770 ","End":"07:23.600","Text":"the yellow 1 and the green 1."},{"Start":"07:23.600 ","End":"07:25.805","Text":"They\u0027re not overlapping."},{"Start":"07:25.805 ","End":"07:34.420","Text":"We can say then that x and y are independent."},{"Start":"07:35.000 ","End":"07:37.280","Text":"Now, once we said this,"},{"Start":"07:37.280 ","End":"07:40.850","Text":"then this should be easy to calculate because the probability of"},{"Start":"07:40.850 ","End":"07:42.860","Text":"the intersection of 2 events is"},{"Start":"07:42.860 ","End":"07:46.640","Text":"the multiplication of the probabilities of each 1 of the events."},{"Start":"07:46.640 ","End":"07:52.130","Text":"That equals to the probability of x being equal to 1,"},{"Start":"07:52.130 ","End":"07:57.515","Text":"times the probability of y being equal to 12."},{"Start":"07:57.515 ","End":"08:01.970","Text":"Now, as I said, the probability of x being equal to 1,"},{"Start":"08:01.970 ","End":"08:04.885","Text":"we\u0027ve calculated that in section 1."},{"Start":"08:04.885 ","End":"08:10.355","Text":"That was 0.0337,"},{"Start":"08:10.355 ","End":"08:13.910","Text":"and the probability of y being equal to 12, well,"},{"Start":"08:13.910 ","End":"08:15.920","Text":"we calculated that in section b,"},{"Start":"08:15.920 ","End":"08:21.390","Text":"that was equal to 0.0948."},{"Start":"08:24.050 ","End":"08:26.810","Text":"Now, when we figure these things out,"},{"Start":"08:26.810 ","End":"08:32.130","Text":"this comes out to 0.0032."},{"Start":"08:33.370 ","End":"08:35.675","Text":"In section d we\u0027re asked,"},{"Start":"08:35.675 ","End":"08:36.793","Text":"what are the expectation,"},{"Start":"08:36.793 ","End":"08:39.625","Text":"and variance of the number of calls in a minute?"},{"Start":"08:39.625 ","End":"08:44.840","Text":"Well, the number of calls in a minute was defined by the random variable x,"},{"Start":"08:44.840 ","End":"08:52.520","Text":"and we know that that was distributed with a Poisson distribution where Lambda equals 5."},{"Start":"08:52.520 ","End":"09:02.240","Text":"Now, we know that the expectation of x equals the variance of x,"},{"Start":"09:02.240 ","End":"09:04.040","Text":"and that equals to Lambda,"},{"Start":"09:04.040 ","End":"09:07.250","Text":"whenever x is distributed with a Poisson distribution,"},{"Start":"09:07.250 ","End":"09:08.450","Text":"and that\u0027s what we have here."},{"Start":"09:08.450 ","End":"09:12.190","Text":"The expectation of x would be equal to 5,"},{"Start":"09:12.190 ","End":"09:17.125","Text":"the variance of x would also be equal to 5,"},{"Start":"09:17.125 ","End":"09:20.147","Text":"and the standard deviation of x, well,"},{"Start":"09:20.147 ","End":"09:23.480","Text":"that\u0027s the square root of the variance,"},{"Start":"09:23.480 ","End":"09:28.795","Text":"and that would equal to 2.24."},{"Start":"09:28.795 ","End":"09:33.165","Text":"Now, the units here are calls per minute."},{"Start":"09:33.165 ","End":"09:35.255","Text":"That\u0027s for the expectation,"},{"Start":"09:35.255 ","End":"09:38.570","Text":"and also for the standard deviation,"},{"Start":"09:38.570 ","End":"09:43.800","Text":"that\u0027ll be 2.24 calls per minute."}],"ID":13069},{"Watched":false,"Name":"Exercise 1","Duration":"4m 55s","ChapterTopicVideoID":12591,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this question, we\u0027ll be talking about calls at a call center."},{"Start":"00:03.390 ","End":"00:08.270","Text":"Now, a call center receives calls at a rate of 5 calls per minute."},{"Start":"00:08.270 ","End":"00:15.045","Text":"The number of calls in a minute has a Poisson probability distribution."},{"Start":"00:15.045 ","End":"00:16.995","Text":"First of all, let\u0027s write down what we know."},{"Start":"00:16.995 ","End":"00:23.730","Text":"Let\u0027s define x as the number of calls per minute,"},{"Start":"00:23.730 ","End":"00:32.735","Text":"and we\u0027re given that x is distributed with a Poisson distribution with Lambda equals 5."},{"Start":"00:32.735 ","End":"00:34.565","Text":"Now, how do I know that?"},{"Start":"00:34.565 ","End":"00:36.245","Text":"Well, let\u0027s take a look."},{"Start":"00:36.245 ","End":"00:39.155","Text":"We\u0027re looking at the number of calls in a minute,"},{"Start":"00:39.155 ","End":"00:40.885","Text":"that\u0027s our x right here,"},{"Start":"00:40.885 ","End":"00:46.845","Text":"and we\u0027re given that it has a Poisson probability that\u0027s right here,"},{"Start":"00:46.845 ","End":"00:55.315","Text":"and the parameter Lambda is 5 calls per minute, that\u0027s right here."},{"Start":"00:55.315 ","End":"01:02.960","Text":"When we know that the probability of x being equal to k equals to E to the power of"},{"Start":"01:02.960 ","End":"01:10.665","Text":"minus Lambda times Lambda^k over k factorial."},{"Start":"01:10.665 ","End":"01:13.490","Text":"Having written all this down,"},{"Start":"01:13.490 ","End":"01:15.950","Text":"let\u0027s go ahead and answer the questions."},{"Start":"01:15.950 ","End":"01:22.280","Text":"Section a, what\u0027s the probability that 1 call will be received in a minute?"},{"Start":"01:22.280 ","End":"01:26.395","Text":"Well, that\u0027s the probability of x being equal to 1."},{"Start":"01:26.395 ","End":"01:29.955","Text":"Now let\u0027s plug in the numbers."},{"Start":"01:29.955 ","End":"01:33.920","Text":"That equals to e to the power of minus Lambda."},{"Start":"01:33.920 ","End":"01:35.540","Text":"Now Lambda is 5,"},{"Start":"01:35.540 ","End":"01:39.830","Text":"so that\u0027s minus 5 times 5 Lambda^k."},{"Start":"01:39.830 ","End":"01:41.693","Text":"Now k is 1."},{"Start":"01:41.693 ","End":"01:44.405","Text":"Divided by 1 factorial,"},{"Start":"01:44.405 ","End":"01:50.245","Text":"and that comes out to 0.0337."},{"Start":"01:50.245 ","End":"01:54.230","Text":"In section b, we\u0027re asked what\u0027s the probability that"},{"Start":"01:54.230 ","End":"01:58.940","Text":"at least 1 call will be received in a minute."},{"Start":"01:58.940 ","End":"02:08.160","Text":"We\u0027re looking at the probability of x being greater or equal to 1, at least 1."},{"Start":"02:08.160 ","End":"02:13.620","Text":"Now we know that the values of x can be"},{"Start":"02:13.620 ","End":"02:19.355","Text":"0 or 1 or 2 up to infinity."},{"Start":"02:19.355 ","End":"02:23.750","Text":"If that\u0027s the case, let\u0027s just take the complimentary set here. It\u0027ll be much easier."},{"Start":"02:23.750 ","End":"02:31.235","Text":"That\u0027ll be 1 minus the probability of x being equal to 0."},{"Start":"02:31.235 ","End":"02:32.891","Text":"If that\u0027s the case, that\u0027s 1 minus,"},{"Start":"02:32.891 ","End":"02:35.755","Text":"now let\u0027s plug in the numbers."},{"Start":"02:35.755 ","End":"02:39.400","Text":"That\u0027s e to the power of minus Lambda,"},{"Start":"02:39.400 ","End":"02:44.097","Text":"that\u0027s minus 5 times 5 to the power of k,"},{"Start":"02:44.097 ","End":"02:48.870","Text":"k is 0 divided by 0 factorial."},{"Start":"02:48.870 ","End":"02:55.540","Text":"Now, that\u0027ll equal to 0.9933."},{"Start":"02:56.150 ","End":"03:00.770","Text":"In section c, we\u0027re asked what\u0027s the probability that at"},{"Start":"03:00.770 ","End":"03:06.170","Text":"most 2 calls will be received in a minute."},{"Start":"03:06.170 ","End":"03:14.420","Text":"We\u0027re looking at the probability where x is less than or equal to 2, at most 2."},{"Start":"03:14.420 ","End":"03:19.670","Text":"Well, we can break this down into 3 components."},{"Start":"03:19.670 ","End":"03:24.680","Text":"The probability where x equals 2 plus"},{"Start":"03:24.680 ","End":"03:32.135","Text":"the probability of x being equal to 1 plus the probability of x being equal to 0."},{"Start":"03:32.135 ","End":"03:35.660","Text":"Now, the probability of x being equal to 2,"},{"Start":"03:35.660 ","End":"03:37.580","Text":"well, let\u0027s plug in the numbers."},{"Start":"03:37.580 ","End":"03:41.875","Text":"That equals to e to the power of minus 5,"},{"Start":"03:41.875 ","End":"03:45.080","Text":"Lambda is 5, times 5 to the power of 2,"},{"Start":"03:45.080 ","End":"03:50.480","Text":"5 squared divided by 2 factorial plus,"},{"Start":"03:50.480 ","End":"03:53.150","Text":"well, what\u0027s the probability of x being equal to 1?"},{"Start":"03:53.150 ","End":"03:55.130","Text":"Well, we figured that out right here."},{"Start":"03:55.130 ","End":"04:00.520","Text":"That\u0027s 0.0337. That\u0027s right here."},{"Start":"04:00.520 ","End":"04:04.460","Text":"Plus the probability of x being equal to 0."},{"Start":"04:04.460 ","End":"04:07.145","Text":"Well, we\u0027ve calculated that right here,"},{"Start":"04:07.145 ","End":"04:15.475","Text":"and that equals to 0.0067."},{"Start":"04:15.475 ","End":"04:17.810","Text":"Now, when we calculate everything out,"},{"Start":"04:17.810 ","End":"04:24.365","Text":"that turns out to be 0.1246."},{"Start":"04:24.365 ","End":"04:31.170","Text":"In section d, we\u0027re asked what\u0027s the variance of the number of calls in a minute."},{"Start":"04:31.170 ","End":"04:37.475","Text":"Well, when x is distributed with a Poisson distribution,"},{"Start":"04:37.475 ","End":"04:44.700","Text":"we know that the expectation of x equals the variance of x and that equals to Lambda."},{"Start":"04:44.700 ","End":"04:48.285","Text":"In our case, Lambda equals 5."},{"Start":"04:48.285 ","End":"04:54.040","Text":"So the variance of x would be equal to 5."}],"ID":13070},{"Watched":false,"Name":"Exercise 2 Part a","Duration":"3m 46s","ChapterTopicVideoID":12592,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this question, we\u0027ll be talking about the number of errors in a newspaper."},{"Start":"00:04.005 ","End":"00:07.500","Text":"Now, the number of errors per newspaper page has"},{"Start":"00:07.500 ","End":"00:12.390","Text":"a Poisson probability distribution with an average of 4 errors per page."},{"Start":"00:12.390 ","End":"00:16.935","Text":"There are 5 pages in a given section of the newspaper and we\u0027re asked,"},{"Start":"00:16.935 ","End":"00:21.825","Text":"what\u0027s the probability that there will be exactly 18 errors in this section?"},{"Start":"00:21.825 ","End":"00:24.465","Text":"Well first, let\u0027s write down the data that we know."},{"Start":"00:24.465 ","End":"00:32.830","Text":"We know that x is the number of errors per page."},{"Start":"00:33.170 ","End":"00:42.835","Text":"We also know that x has a Poisson probability where Lambda equals 4."},{"Start":"00:42.835 ","End":"00:45.110","Text":"Now, why do I know this?"},{"Start":"00:45.110 ","End":"00:48.950","Text":"Well, here, this is the random variable,"},{"Start":"00:48.950 ","End":"00:51.575","Text":"the number of errors per newspaper page."},{"Start":"00:51.575 ","End":"00:55.345","Text":"We\u0027re given that it has a Poisson probability."},{"Start":"00:55.345 ","End":"00:57.500","Text":"We\u0027re also given the perimeter,"},{"Start":"00:57.500 ","End":"01:01.530","Text":"an average of 4 errors per page."},{"Start":"01:01.660 ","End":"01:04.040","Text":"Now, since we know that,"},{"Start":"01:04.040 ","End":"01:08.430","Text":"we know that the probability of x being equal to k,"},{"Start":"01:08.430 ","End":"01:12.275","Text":"well that would be equal to e to the power of minus Lambda"},{"Start":"01:12.275 ","End":"01:18.410","Text":"times Lambda to the power of k/k factorial."},{"Start":"01:18.410 ","End":"01:22.145","Text":"Now, what do we asked in section a?"},{"Start":"01:22.145 ","End":"01:25.264","Text":"We\u0027re not asked about the number of errors per page,"},{"Start":"01:25.264 ","End":"01:28.840","Text":"but the number of errors per section."},{"Start":"01:28.840 ","End":"01:32.840","Text":"First of all, let\u0027s define a new variable."},{"Start":"01:32.840 ","End":"01:41.090","Text":"We\u0027ll call this R, and that would be the number of errors per section."},{"Start":"01:41.090 ","End":"01:44.495","Text":"Now, this is a new random variable."},{"Start":"01:44.495 ","End":"01:49.640","Text":"It will have a new value for Lambda."},{"Start":"01:49.640 ","End":"01:51.860","Text":"Now, what would that value be?"},{"Start":"01:51.860 ","End":"01:53.630","Text":"Well, in order to find that,"},{"Start":"01:53.630 ","End":"01:56.600","Text":"let\u0027s just rely on 1 of the special characteristics"},{"Start":"01:56.600 ","End":"02:01.025","Text":"of the Poisson distribution, and here it is."},{"Start":"02:01.025 ","End":"02:04.760","Text":"We\u0027re told that the parameter Lambda is"},{"Start":"02:04.760 ","End":"02:08.330","Text":"proportional to the interval of time being discussed."},{"Start":"02:08.330 ","End":"02:15.714","Text":"In our case, Lambda would be proportional to the number of errors per page."},{"Start":"02:15.714 ","End":"02:22.565","Text":"If we have an average of 4 errors per page and we have in a section 5 pages,"},{"Start":"02:22.565 ","End":"02:25.295","Text":"then the number of errors per section would be"},{"Start":"02:25.295 ","End":"02:29.570","Text":"4 errors per page times 5 pages in a section,"},{"Start":"02:29.570 ","End":"02:32.315","Text":"there\u0027ll be 20 years per section."},{"Start":"02:32.315 ","End":"02:36.720","Text":"Having understood that, let\u0027s get back to our question."},{"Start":"02:36.880 ","End":"02:42.545","Text":"Here we are, if r is the number of errors per section,"},{"Start":"02:42.545 ","End":"02:47.185","Text":"we know that r would have"},{"Start":"02:47.185 ","End":"02:56.080","Text":"a Poisson distribution where Lambda would be 4 errors per page times 5 pages per session,"},{"Start":"02:56.080 ","End":"02:58.930","Text":"there\u0027ll be equal to 20."},{"Start":"02:58.930 ","End":"03:01.845","Text":"Now, if that\u0027s the case,"},{"Start":"03:01.845 ","End":"03:03.955","Text":"let\u0027s solve section a."},{"Start":"03:03.955 ","End":"03:09.805","Text":"We\u0027re looking at the probability where r is equal to 18."},{"Start":"03:09.805 ","End":"03:14.890","Text":"The probability that there will be exactly 18 errors in a section."},{"Start":"03:14.890 ","End":"03:17.035","Text":"Well, let\u0027s just plug in the numbers."},{"Start":"03:17.035 ","End":"03:21.970","Text":"That\u0027s e to the power of minus Lambda, now what\u0027s Lambda?"},{"Start":"03:21.970 ","End":"03:26.980","Text":"Lambda here is 20, so that\u0027s minus 20 times Lambda to the power of k. Well,"},{"Start":"03:26.980 ","End":"03:28.940","Text":"that\u0027s 20. And what\u0027s k?"},{"Start":"03:28.940 ","End":"03:35.875","Text":"K here is 18 divided by 18 factorial."},{"Start":"03:35.875 ","End":"03:45.690","Text":"That comes out to 0.084 errors per section."}],"ID":13071},{"Watched":false,"Name":"Exercise 2 Part b","Duration":"7m 54s","ChapterTopicVideoID":12593,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In section B, we\u0027re asked: If there are no errors on the first page,"},{"Start":"00:05.175 ","End":"00:10.555","Text":"what\u0027s the probability that the section will have a total of 15 errors?"},{"Start":"00:10.555 ","End":"00:14.190","Text":"Before we solve this,"},{"Start":"00:14.190 ","End":"00:16.980","Text":"let\u0027s just recall some of our definitions."},{"Start":"00:16.980 ","End":"00:21.840","Text":"We know that X is the number of errors per"},{"Start":"00:21.840 ","End":"00:31.420","Text":"page and R is the number of errors per section."},{"Start":"00:33.980 ","End":"00:36.290","Text":"Again, while reading this,"},{"Start":"00:36.290 ","End":"00:39.860","Text":"we note that this is a conditional probability."},{"Start":"00:39.860 ","End":"00:42.665","Text":"We\u0027re looking for the probability."},{"Start":"00:42.665 ","End":"00:44.225","Text":"Now, what\u0027s given to us?"},{"Start":"00:44.225 ","End":"00:46.070","Text":"There are no errors on the first page,"},{"Start":"00:46.070 ","End":"00:49.890","Text":"that means that X equals 0 and what else?"},{"Start":"00:49.890 ","End":"00:54.290","Text":"We\u0027re looking for the probability that the section will have a total of 15 errors."},{"Start":"00:54.290 ","End":"00:57.520","Text":"That means R equals 15."},{"Start":"00:57.520 ","End":"01:01.000","Text":"Well, we know how to solve this."},{"Start":"01:01.790 ","End":"01:05.770","Text":"On the denominator we have the probability of what\u0027s given,"},{"Start":"01:05.770 ","End":"01:09.130","Text":"that\u0027s the probability of X being equal to 0."},{"Start":"01:09.130 ","End":"01:12.920","Text":"In the numerator, we have the intersect of these 2 guys."},{"Start":"01:12.920 ","End":"01:22.300","Text":"That\u0027s the probability of R equals 15 and X being equal to 0."},{"Start":"01:22.730 ","End":"01:26.570","Text":"Now, we know what this means."},{"Start":"01:26.570 ","End":"01:27.830","Text":"Well, what does this mean?"},{"Start":"01:27.830 ","End":"01:32.285","Text":"What does R equals 15 and X equals 0,"},{"Start":"01:32.285 ","End":"01:36.350","Text":"what does that mean with respect to pages and errors?"},{"Start":"01:36.350 ","End":"01:40.170","Text":"Well, let\u0027s take a look at this graphically."},{"Start":"01:40.220 ","End":"01:43.805","Text":"We have here 5 pages."},{"Start":"01:43.805 ","End":"01:44.926","Text":"This is the first page,"},{"Start":"01:44.926 ","End":"01:47.270","Text":"this is the second page,"},{"Start":"01:47.270 ","End":"01:49.685","Text":"the third page, the fourth page,"},{"Start":"01:49.685 ","End":"01:51.050","Text":"and the fifth page."},{"Start":"01:51.050 ","End":"01:58.155","Text":"We\u0027re saying that on 1 hand here, X equals 0."},{"Start":"01:58.155 ","End":"02:02.870","Text":"There are no errors on the first page,"},{"Start":"02:02.870 ","End":"02:08.650","Text":"and we\u0027re also saying that here in all the section,"},{"Start":"02:08.650 ","End":"02:12.255","Text":"we have R that equals to 15."},{"Start":"02:12.255 ","End":"02:17.285","Text":"But, we can see here that we have an overlap."},{"Start":"02:17.285 ","End":"02:22.670","Text":"It is page number 1 that\u0027s overlapping between the definition of X"},{"Start":"02:22.670 ","End":"02:28.010","Text":"and the definition of R. That means that they\u0027re dependent on each other."},{"Start":"02:28.010 ","End":"02:36.330","Text":"That means that the number of errors on the first page will influence R, and vice versa."},{"Start":"02:36.350 ","End":"02:39.530","Text":"We don\u0027t really know how to handle this,"},{"Start":"02:39.530 ","End":"02:46.230","Text":"but we do know that when we have non-overlapping units,"},{"Start":"02:46.230 ","End":"02:49.520","Text":"or pages in our question,"},{"Start":"02:49.520 ","End":"02:54.210","Text":"then the variables would be independent."},{"Start":"02:54.530 ","End":"02:59.110","Text":"Let\u0027s see if we can make this a little bit simpler."},{"Start":"02:59.110 ","End":"03:02.355","Text":"Again, we have 1,"},{"Start":"03:02.355 ","End":"03:03.705","Text":"2, 3,"},{"Start":"03:03.705 ","End":"03:06.160","Text":"4, and 5."},{"Start":"03:06.880 ","End":"03:12.590","Text":"Now, if we say that in the first page we have"},{"Start":"03:12.590 ","End":"03:18.160","Text":"no errors, it\u0027s right here."},{"Start":"03:18.160 ","End":"03:21.400","Text":"We\u0027re looking for an event where we have"},{"Start":"03:21.400 ","End":"03:26.110","Text":"no errors on the first page and we have 15 errors in the whole section."},{"Start":"03:26.110 ","End":"03:30.205","Text":"Basically, what we\u0027re saying is we have no errors on the first page,"},{"Start":"03:30.205 ","End":"03:35.645","Text":"but in pages 2-5,"},{"Start":"03:35.645 ","End":"03:38.790","Text":"we have 15 errors."},{"Start":"03:38.790 ","End":"03:43.465","Text":"Let\u0027s now define a new variable called Y."},{"Start":"03:43.465 ","End":"03:46.075","Text":"That\u0027ll be the number of"},{"Start":"03:46.075 ","End":"03:55.800","Text":"errors on pages 2-5."},{"Start":"03:55.800 ","End":"03:58.395","Text":"Now, the minute we did this,"},{"Start":"03:58.395 ","End":"04:06.680","Text":"we can rewrite this in such a way that it will help us to solve this much easier."},{"Start":"04:06.680 ","End":"04:08.480","Text":"We can do this."},{"Start":"04:08.480 ","End":"04:14.745","Text":"We can say that the probability of R equals 15,"},{"Start":"04:14.745 ","End":"04:20.420","Text":"and X equals 0 over the probability of X being equal to 0,"},{"Start":"04:20.420 ","End":"04:28.050","Text":"that\u0027s equivalent of saying that the probability of X equals 0, that\u0027s this guy."},{"Start":"04:28.050 ","End":"04:31.330","Text":"That instead of R equals 15,"},{"Start":"04:31.330 ","End":"04:40.205","Text":"we say Y equals 15 over the probability of X being equal to 0."},{"Start":"04:40.205 ","End":"04:42.185","Text":"Now, again, what does this mean?"},{"Start":"04:42.185 ","End":"04:50.855","Text":"This means that we have no errors on the first page and 15 errors on pages 2-5."},{"Start":"04:50.855 ","End":"04:54.680","Text":"Here we\u0027re saying we have no errors on the first page,"},{"Start":"04:54.680 ","End":"04:57.559","Text":"and we have 15 errors in the whole section."},{"Start":"04:57.559 ","End":"05:01.055","Text":"But, if we have no errors on the first page,"},{"Start":"05:01.055 ","End":"05:03.590","Text":"then we have 15 errors on pages 2-5,"},{"Start":"05:03.590 ","End":"05:07.655","Text":"which is exactly the definition of Y."},{"Start":"05:07.655 ","End":"05:12.020","Text":"Now this equals to what?"},{"Start":"05:12.020 ","End":"05:19.175","Text":"Now we know again because the section page number 1,"},{"Start":"05:19.175 ","End":"05:23.980","Text":"and the other sections pages 2-5,"},{"Start":"05:23.980 ","End":"05:26.840","Text":"they\u0027re independent of each other."},{"Start":"05:26.840 ","End":"05:28.675","Text":"They\u0027re not overlapping,"},{"Start":"05:28.675 ","End":"05:31.490","Text":"and because they\u0027re not overlapping, they\u0027re independent."},{"Start":"05:31.490 ","End":"05:33.550","Text":"We can write this out like this,"},{"Start":"05:33.550 ","End":"05:39.350","Text":"as the probability of X being equal to 0 times the probability of"},{"Start":"05:39.350 ","End":"05:46.250","Text":"Y equals 15 over the probability of X being equal to 0."},{"Start":"05:46.250 ","End":"05:50.720","Text":"Now, this is simple because we can cancel these guys"},{"Start":"05:50.720 ","End":"05:56.945","Text":"out and they\u0027ll be equal to the probability of Y being equal to 15."},{"Start":"05:56.945 ","End":"06:01.960","Text":"We\u0027ve solved that conditional probability."},{"Start":"06:01.960 ","End":"06:03.860","Text":"Now, we didn\u0027t really solve it,"},{"Start":"06:03.860 ","End":"06:05.780","Text":"we just simplified it."},{"Start":"06:05.780 ","End":"06:09.275","Text":"Now, what is this equal to?"},{"Start":"06:09.275 ","End":"06:16.835","Text":"Well, we know that Y has a Poisson distribution,"},{"Start":"06:16.835 ","End":"06:19.130","Text":"but we don\u0027t know what the lambda is."},{"Start":"06:19.130 ","End":"06:25.054","Text":"We know that we have 4 errors per page on average,"},{"Start":"06:25.054 ","End":"06:31.855","Text":"but that\u0027s the parameter for X,"},{"Start":"06:31.855 ","End":"06:33.475","Text":"but not for Y."},{"Start":"06:33.475 ","End":"06:39.650","Text":"Now, remember what we said about lambda being proportional to the units."},{"Start":"06:39.650 ","End":"06:43.385","Text":"If we have 4 errors per page,"},{"Start":"06:43.385 ","End":"06:45.350","Text":"now how many pages do we have for Y?"},{"Start":"06:45.350 ","End":"06:48.350","Text":"Well, Y is defined for pages 2-5."},{"Start":"06:48.350 ","End":"06:50.330","Text":"We have 2, 3, 4, 5,"},{"Start":"06:50.330 ","End":"06:52.935","Text":"we have 4 pages."},{"Start":"06:52.935 ","End":"06:57.640","Text":"Now we have 4 errors per page times 4 pages."},{"Start":"06:58.310 ","End":"07:02.560","Text":"Lambda would equals 16."},{"Start":"07:02.720 ","End":"07:08.906","Text":"We\u0027re saying that the probability of Y being equal to k, well,"},{"Start":"07:08.906 ","End":"07:13.240","Text":"that would equal to e, first of all,"},{"Start":"07:13.240 ","End":"07:14.545","Text":"to the minus lambda,"},{"Start":"07:14.545 ","End":"07:18.085","Text":"lambda to the power of k over k factorial."},{"Start":"07:18.085 ","End":"07:22.735","Text":"In our case, what are we looking for?"},{"Start":"07:22.735 ","End":"07:29.090","Text":"We\u0027re looking for the probability of Y being equal to 15."},{"Start":"07:29.090 ","End":"07:31.450","Text":"Let\u0027s just plug in the numbers."},{"Start":"07:31.450 ","End":"07:33.700","Text":"That\u0027s e to the power of minus lambda."},{"Start":"07:33.700 ","End":"07:36.130","Text":"Now lambda here in our case that\u0027s 16,"},{"Start":"07:36.130 ","End":"07:46.090","Text":"so that\u0027s minus 16 times 16 to the power of k.15 divided by k factorial,"},{"Start":"07:46.090 ","End":"07:53.990","Text":"that\u0027s 15 factorial, and that turns out to be 0.099."}],"ID":13072},{"Watched":false,"Name":"Exercise 2 Part c","Duration":"7m 28s","ChapterTopicVideoID":12594,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.133","Text":"In section C, we\u0027re asked if there are a total of 18 errors in this section,"},{"Start":"00:04.133 ","End":"00:08.235","Text":"what\u0027s the probability that there are 5 errors on the page?"},{"Start":"00:08.235 ","End":"00:13.755","Text":"Well, first let\u0027s recall some of the definitions from the previous sections."},{"Start":"00:13.755 ","End":"00:20.655","Text":"X was the number of errors per page."},{"Start":"00:20.655 ","End":"00:28.900","Text":"R was the number of errors per section."},{"Start":"00:29.630 ","End":"00:32.955","Text":"Y was the number of"},{"Start":"00:32.955 ","End":"00:36.870","Text":"errors per"},{"Start":"00:36.870 ","End":"00:43.170","Text":"pages 2-5."},{"Start":"00:43.170 ","End":"00:50.345","Text":"We also know that X has a Poisson distribution where lambda equals 4."},{"Start":"00:50.345 ","End":"00:57.815","Text":"R has a Poisson distribution where lambda equals 20,"},{"Start":"00:57.815 ","End":"01:05.510","Text":"and y has a Poisson distribution where lambda equals 16."},{"Start":"01:05.510 ","End":"01:10.160","Text":"Now all this was done in the previous sections."},{"Start":"01:10.160 ","End":"01:15.170","Text":"We also know that the probability of x being equal to k,"},{"Start":"01:15.170 ","End":"01:19.010","Text":"where x has a Poisson distribution, well,"},{"Start":"01:19.010 ","End":"01:21.875","Text":"the probability function is this,"},{"Start":"01:21.875 ","End":"01:30.900","Text":"e to the power of minus lambda times lambda to the power of k divided by k factorial."},{"Start":"01:31.430 ","End":"01:34.610","Text":"Having done all this,"},{"Start":"01:34.610 ","End":"01:37.760","Text":"let\u0027s see if we can solve section C. Now,"},{"Start":"01:37.760 ","End":"01:39.295","Text":"in section C,"},{"Start":"01:39.295 ","End":"01:41.895","Text":"we have a conditional probability."},{"Start":"01:41.895 ","End":"01:45.080","Text":"Let\u0027s set this up. We have the probability."},{"Start":"01:45.080 ","End":"01:46.715","Text":"Now, what\u0027s known?"},{"Start":"01:46.715 ","End":"01:50.915","Text":"If there are a total of 18 errors in this section,"},{"Start":"01:50.915 ","End":"01:54.200","Text":"that means that if r equals 18,"},{"Start":"01:54.200 ","End":"01:55.895","Text":"that\u0027s given to us,"},{"Start":"01:55.895 ","End":"01:57.170","Text":"what are we looking for?"},{"Start":"01:57.170 ","End":"02:00.965","Text":"The probability that there were 5 errors on the first page,"},{"Start":"02:00.965 ","End":"02:03.695","Text":"that means that x equals 5."},{"Start":"02:03.695 ","End":"02:07.355","Text":"Now, we know how to solve this thing."},{"Start":"02:07.355 ","End":"02:14.135","Text":"In the denominator, we have the probability of what\u0027s given, r equals 18."},{"Start":"02:14.135 ","End":"02:18.425","Text":"In the numerator, we have the probability of the intersect."},{"Start":"02:18.425 ","End":"02:22.924","Text":"That\u0027s the probability of x being equal to 5,"},{"Start":"02:22.924 ","End":"02:28.355","Text":"and r equals to 18."},{"Start":"02:28.355 ","End":"02:32.720","Text":"Now, we know the meaning of r equals 18,"},{"Start":"02:32.720 ","End":"02:36.080","Text":"r equals 18 means that in the whole section,"},{"Start":"02:36.080 ","End":"02:38.495","Text":"there are 18 errors."},{"Start":"02:38.495 ","End":"02:40.340","Text":"Now, what does this mean?"},{"Start":"02:40.340 ","End":"02:44.390","Text":"The intersect where x equals 5 and r equals 18, well,"},{"Start":"02:44.390 ","End":"02:53.780","Text":"that\u0027s the intercept where on the first page we have 5 errors and in the whole section,"},{"Start":"02:53.780 ","End":"02:56.015","Text":"we have 18 errors."},{"Start":"02:56.015 ","End":"03:02.480","Text":"Now, we know that x and r are dependent variables. What do we want to do?"},{"Start":"03:02.480 ","End":"03:10.210","Text":"We want to split this up in such a way that we will have 2 independent variables here."},{"Start":"03:10.210 ","End":"03:12.580","Text":"How do we do that?"},{"Start":"03:12.580 ","End":"03:16.775","Text":"Well, again, let\u0027s look at this graphically."},{"Start":"03:16.775 ","End":"03:22.110","Text":"We have here our 5 pages,"},{"Start":"03:23.740 ","End":"03:27.765","Text":"1,2,3,4, and 5."},{"Start":"03:27.765 ","End":"03:34.010","Text":"We\u0027re saying that in the first page we have 5 errors."},{"Start":"03:34.010 ","End":"03:37.440","Text":"In the whole section,"},{"Start":"03:37.870 ","End":"03:45.500","Text":"we have 18 errors."},{"Start":"03:45.500 ","End":"03:48.710","Text":"Now, the question is,"},{"Start":"03:48.710 ","End":"03:51.680","Text":"do we have an overlap or not?"},{"Start":"03:51.680 ","End":"03:57.410","Text":"Well, obviously we do, x and r are dependent because of this overlap right here."},{"Start":"03:57.410 ","End":"03:59.690","Text":"But we can split this up."},{"Start":"03:59.690 ","End":"04:01.640","Text":"We can define the variable y,"},{"Start":"04:01.640 ","End":"04:04.160","Text":"which we already did in the last section."},{"Start":"04:04.160 ","End":"04:08.170","Text":"Now y are the number of errors on pages 2-5?"},{"Start":"04:08.170 ","End":"04:11.220","Text":"Now, if we have 18 errors on the whole section,"},{"Start":"04:11.220 ","End":"04:13.835","Text":"we have 5 errors on the first section,"},{"Start":"04:13.835 ","End":"04:21.460","Text":"then that means that we have 13 errors on pages 2-5."},{"Start":"04:21.460 ","End":"04:25.320","Text":"Now, x and y we know to be independent."},{"Start":"04:25.320 ","End":"04:29.090","Text":"Why is that? Because they have non-overlapping units."},{"Start":"04:29.090 ","End":"04:32.300","Text":"Page 1 does not overlap on pages 2,"},{"Start":"04:32.300 ","End":"04:36.500","Text":"3, 4, and 5, so x and y are independent."},{"Start":"04:36.500 ","End":"04:42.820","Text":"As such, we can now solve the conditional probability."},{"Start":"04:42.820 ","End":"04:46.245","Text":"This equals this."},{"Start":"04:46.245 ","End":"04:52.190","Text":"That would equal to the probability of x being equal to"},{"Start":"04:52.190 ","End":"05:00.965","Text":"5 times the probability of y being equal to what?"},{"Start":"05:00.965 ","End":"05:05.790","Text":"Well, again, 18 minus 5, that\u0027s 13."},{"Start":"05:06.020 ","End":"05:13.325","Text":"Now, over the probability of r being equal to 18."},{"Start":"05:13.325 ","End":"05:15.410","Text":"Again, why is that?"},{"Start":"05:15.410 ","End":"05:20.315","Text":"We could multiply the probability"},{"Start":"05:20.315 ","End":"05:24.740","Text":"of this intersect right here because x and r were dependent,"},{"Start":"05:24.740 ","End":"05:28.295","Text":"we can only do that when the 2 variables are independent."},{"Start":"05:28.295 ","End":"05:31.940","Text":"We had to switch r into something else."},{"Start":"05:31.940 ","End":"05:34.485","Text":"Now, What was that something else?"},{"Start":"05:34.485 ","End":"05:42.910","Text":"That was the new random variable y where it counted the number of errors on pages 2-5,"},{"Start":"05:42.910 ","End":"05:46.700","Text":"making x and y independent."},{"Start":"05:47.600 ","End":"05:52.990","Text":"The transformation from here to here was just to"},{"Start":"05:52.990 ","End":"05:58.580","Text":"split up the whole section into the first page and then the rest of the pages."},{"Start":"05:58.580 ","End":"06:04.840","Text":"There it is. This is a conditional probability that we have to calculate."},{"Start":"06:04.840 ","End":"06:08.260","Text":"Now again, we know that x was"},{"Start":"06:08.260 ","End":"06:14.805","Text":"distributed with a Poisson distribution with lambda equaling 4y,"},{"Start":"06:14.805 ","End":"06:19.055","Text":"a Poisson distribution with lambda equaling 16"},{"Start":"06:19.055 ","End":"06:23.570","Text":"and r a Poisson distribution with lambda equaling 20."},{"Start":"06:23.570 ","End":"06:24.950","Text":"If that\u0027s the case,"},{"Start":"06:24.950 ","End":"06:30.145","Text":"let\u0027s just plug in the numbers and calculate this probability."},{"Start":"06:30.145 ","End":"06:38.960","Text":"That would equal e to the power of what\u0027s lambda for x, that\u0027s 4,"},{"Start":"06:38.960 ","End":"06:41.315","Text":"so that\u0027s minus 4,"},{"Start":"06:41.315 ","End":"06:48.870","Text":"times 4 to the power of 5 over 5 factorial, k equals 5,"},{"Start":"06:48.870 ","End":"06:51.630","Text":"times e to the power of,"},{"Start":"06:51.630 ","End":"06:53.250","Text":"now lambda here with 16,"},{"Start":"06:53.250 ","End":"06:55.890","Text":"so it\u0027s minus 16,"},{"Start":"06:55.890 ","End":"06:58.450","Text":"times 16 to the power of k,"},{"Start":"06:58.450 ","End":"07:00.395","Text":"k here is 13,"},{"Start":"07:00.395 ","End":"07:05.905","Text":"over 13 factorial divided by,"},{"Start":"07:05.905 ","End":"07:08.775","Text":"the lambda here was 20."},{"Start":"07:08.775 ","End":"07:14.975","Text":"That was e to the power of minus 20 times 20 to the power of k,"},{"Start":"07:14.975 ","End":"07:20.390","Text":"18 over 18 factorial."},{"Start":"07:20.390 ","End":"07:24.970","Text":"All of this equals 0.151."}],"ID":13073},{"Watched":false,"Name":"Exercise 3","Duration":"7m 2s","ChapterTopicVideoID":12595,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this question will be talking about fatal accidents."},{"Start":"00:03.180 ","End":"00:06.330","Text":"Now, the number of fatal accidents in Florida has"},{"Start":"00:06.330 ","End":"00:11.940","Text":"a Poisson probability distribution with a standard deviation of 2 accidents per week."},{"Start":"00:11.940 ","End":"00:16.890","Text":"We\u0027re asked, what\u0027s the expectation of the number of accidents in a week?"},{"Start":"00:16.890 ","End":"00:19.575","Text":"Well, what are we given?"},{"Start":"00:19.575 ","End":"00:29.170","Text":"Well, first of all, X is the number of accidents per week."},{"Start":"00:29.750 ","End":"00:33.680","Text":"We\u0027re also given that X has"},{"Start":"00:33.680 ","End":"00:39.230","Text":"a Poisson distribution where the parameter Lambda isn\u0027t known."},{"Start":"00:39.230 ","End":"00:43.190","Text":"But we\u0027re asked what\u0027s the expectation of the number of accidents."},{"Start":"00:43.190 ","End":"00:49.129","Text":"Now, we know that if X is distributed with a Poisson distribution,"},{"Start":"00:49.129 ","End":"00:55.460","Text":"then the expectation of X equals the variance of X and that equals to Lambda."},{"Start":"00:55.460 ","End":"00:57.935","Text":"Now, what are we given here?"},{"Start":"00:57.935 ","End":"01:04.820","Text":"We\u0027re given that the standard deviation of X is 2 accidents per week."},{"Start":"01:04.820 ","End":"01:09.215","Text":"That means that the standard deviation of X equals 2."},{"Start":"01:09.215 ","End":"01:11.390","Text":"Now, if that\u0027s the case,"},{"Start":"01:11.390 ","End":"01:14.885","Text":"then the variance of X, well,"},{"Start":"01:14.885 ","End":"01:18.829","Text":"that equals to the standard deviation of X squared,"},{"Start":"01:18.829 ","End":"01:26.375","Text":"and that equals, in our case to 2 squared, that equals to 4."},{"Start":"01:26.375 ","End":"01:31.610","Text":"But since the variance equals the expectation and that equals to Lambda and then that"},{"Start":"01:31.610 ","End":"01:37.460","Text":"means that the expectation of X equals 4 and that equals to Lambda."},{"Start":"01:37.460 ","End":"01:46.050","Text":"Which means that X has a Poisson distribution where Lambda equals 4."},{"Start":"01:46.570 ","End":"01:51.770","Text":"In section B, we\u0027re asked what\u0027s the probability that during a month,"},{"Start":"01:51.770 ","End":"01:54.095","Text":"assuming that there are 4 weeks in a month,"},{"Start":"01:54.095 ","End":"01:59.434","Text":"there will be exactly 1 week with 3 fatal traffic accidents?"},{"Start":"01:59.434 ","End":"02:02.930","Text":"Let\u0027s first write down what we know."},{"Start":"02:02.930 ","End":"02:11.080","Text":"We know that X is the number of accidents per week."},{"Start":"02:11.180 ","End":"02:18.050","Text":"We know that X is distributed with a Poisson distribution where Lambda equals 4."},{"Start":"02:18.050 ","End":"02:20.785","Text":"We\u0027ve calculated that in section a."},{"Start":"02:20.785 ","End":"02:25.610","Text":"We know that the probability of X being equal to k,"},{"Start":"02:25.610 ","End":"02:29.840","Text":"that\u0027s the generic formula for the Poisson distribution."},{"Start":"02:29.840 ","End":"02:32.660","Text":"That\u0027s e to the power of minus Lambda,"},{"Start":"02:32.660 ","End":"02:37.340","Text":"Lambda to the power of k/k factorial."},{"Start":"02:37.340 ","End":"02:40.790","Text":"Now here what we want to know is that what\u0027s"},{"Start":"02:40.790 ","End":"02:46.355","Text":"the probability of having 3 fatal accidents in a week?"},{"Start":"02:46.355 ","End":"02:51.395","Text":"That\u0027s the probability of X being equal to 3."},{"Start":"02:51.395 ","End":"02:53.943","Text":"Now, let\u0027s just plug in the numbers,"},{"Start":"02:53.943 ","End":"02:57.050","Text":"that\u0027s e to the power of minus Lambda."},{"Start":"02:57.050 ","End":"02:58.760","Text":"Now, that\u0027s Lambda equals 4,"},{"Start":"02:58.760 ","End":"03:06.815","Text":"so that\u0027s minus 4 times 4 to the power of k. K equals 3/3 factorial,"},{"Start":"03:06.815 ","End":"03:12.065","Text":"and that comes out to 0.1954."},{"Start":"03:12.065 ","End":"03:20.415","Text":"Now, this is a probability of getting 3 fatal accidents in a week. What are we asked?"},{"Start":"03:20.415 ","End":"03:24.350","Text":"We\u0027re asked what\u0027s the probability that during a month,"},{"Start":"03:24.350 ","End":"03:31.030","Text":"4 weeks, there will be exactly 1 week with 3 fatal traffic accidents."},{"Start":"03:31.030 ","End":"03:34.520","Text":"Let\u0027s take a week as a trial,"},{"Start":"03:34.520 ","End":"03:38.555","Text":"a Bernoulli trial, having a success and failure."},{"Start":"03:38.555 ","End":"03:41.110","Text":"Now, what success?"},{"Start":"03:41.110 ","End":"03:45.930","Text":"How do we define success?"},{"Start":"03:45.930 ","End":"03:55.090","Text":"That would equal a week having 3 accidents."},{"Start":"03:56.660 ","End":"04:07.100","Text":"We\u0027ll define y as the number of successful weeks."},{"Start":"04:09.380 ","End":"04:12.415","Text":"Now, the probability of success,"},{"Start":"04:12.415 ","End":"04:14.350","Text":"well, that was calculated right here."},{"Start":"04:14.350 ","End":"04:18.340","Text":"The probability equals 0.1954."},{"Start":"04:18.340 ","End":"04:24.150","Text":"That was a probability of having 3 accidents in a week."},{"Start":"04:24.150 ","End":"04:27.925","Text":"Now, how many trials do we have?"},{"Start":"04:27.925 ","End":"04:32.410","Text":"Well, we\u0027re asked to look at the probability during a month,"},{"Start":"04:32.410 ","End":"04:35.810","Text":"and a month has 4 weeks right here."},{"Start":"04:35.810 ","End":"04:39.375","Text":"That means that n equals 4."},{"Start":"04:39.375 ","End":"04:44.120","Text":"That means that we have 4 trials and we\u0027re trying to find out what\u0027s"},{"Start":"04:44.120 ","End":"04:49.460","Text":"the probability that there will be 3 fatal accidents in exactly 1 week,"},{"Start":"04:49.460 ","End":"04:51.980","Text":"in the 4 weeks in the month."},{"Start":"04:51.980 ","End":"04:56.960","Text":"Now, this looks like a binomial distribution."},{"Start":"04:56.960 ","End":"04:59.990","Text":"We have y, the number of successful weeks,"},{"Start":"04:59.990 ","End":"05:03.515","Text":"and we have, this is what we\u0027re trying to count."},{"Start":"05:03.515 ","End":"05:05.765","Text":"We have 4 trials,"},{"Start":"05:05.765 ","End":"05:10.795","Text":"4 weeks with the probability of success."},{"Start":"05:10.795 ","End":"05:15.120","Text":"Having 3 accidents in a week, that\u0027s 0.1954."},{"Start":"05:15.120 ","End":"05:17.970","Text":"Now that\u0027s a binomial distribution."},{"Start":"05:17.970 ","End":"05:22.640","Text":"That means that y is distributed with a binomial distribution"},{"Start":"05:22.640 ","End":"05:29.915","Text":"where n equals 4 and p equals 0.1954."},{"Start":"05:29.915 ","End":"05:31.475","Text":"Now if that\u0027s the case,"},{"Start":"05:31.475 ","End":"05:35.210","Text":"the probability of y equaling k. Now,"},{"Start":"05:35.210 ","End":"05:38.225","Text":"this is the equation for the binomial distribution,"},{"Start":"05:38.225 ","End":"05:41.690","Text":"that\u0027s n/k, p^k,"},{"Start":"05:41.690 ","End":"05:45.800","Text":"q^n minus k,"},{"Start":"05:45.800 ","End":"05:49.615","Text":"now q is 1 minus p,"},{"Start":"05:49.615 ","End":"05:51.720","Text":"let\u0027s not forget that."},{"Start":"05:52.640 ","End":"06:00.710","Text":"Let\u0027s just plug in the numbers and see what\u0027s the probability of y being equal to 1?"},{"Start":"06:00.710 ","End":"06:01.910","Text":"This is what we\u0027re asked."},{"Start":"06:01.910 ","End":"06:07.145","Text":"We\u0027re asked what is the probability of having exactly 1 week,"},{"Start":"06:07.145 ","End":"06:09.095","Text":"with 3 fatal accidents?"},{"Start":"06:09.095 ","End":"06:11.630","Text":"Well, that equals to n/k,"},{"Start":"06:11.630 ","End":"06:14.730","Text":"where n here is 4/k,"},{"Start":"06:14.730 ","End":"06:18.545","Text":"that\u0027s 1, and this is 4."},{"Start":"06:18.545 ","End":"06:28.310","Text":"P, that\u0027s 0.1954^1, to the power of k,"},{"Start":"06:28.310 ","End":"06:32.870","Text":"times 1 minus p,"},{"Start":"06:32.870 ","End":"06:39.350","Text":"which is 1 minus 0.1954^n minus k,"},{"Start":"06:39.350 ","End":"06:42.485","Text":"that\u0027s 4 minus 1, that\u0027s 3."},{"Start":"06:42.485 ","End":"06:47.420","Text":"That would equal to 0.407,"},{"Start":"06:47.420 ","End":"06:54.050","Text":"so this is a probability of having exactly 1 week where we"},{"Start":"06:54.050 ","End":"07:02.340","Text":"have 3 fatal accidents in 1 week out of 4 weeks, out of a month."}],"ID":13074},{"Watched":false,"Name":"Exercise 4","Duration":"6m 37s","ChapterTopicVideoID":12596,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this question, we\u0027ll be talking about customers entering the neighborhood 7-11 store."},{"Start":"00:05.580 ","End":"00:10.590","Text":"Now, the number of customers entering the neighborhood 7-11 store has"},{"Start":"00:10.590 ","End":"00:15.840","Text":"a Poisson probability distribution with an average of 2 customers per minute,"},{"Start":"00:15.840 ","End":"00:18.780","Text":"and we\u0027re asked, what\u0027s the probability that there will"},{"Start":"00:18.780 ","End":"00:21.660","Text":"be exactly 3 customers in a given minute?"},{"Start":"00:21.660 ","End":"00:23.580","Text":"Well, first of all,"},{"Start":"00:23.580 ","End":"00:24.735","Text":"let\u0027s write down what we know."},{"Start":"00:24.735 ","End":"00:34.725","Text":"We know that x is the number of clients per minute,"},{"Start":"00:34.725 ","End":"00:44.290","Text":"and we also know that x has a Poisson probability distribution where Lambda equals 2."},{"Start":"00:44.290 ","End":"00:47.085","Text":"Again, how do we know this?"},{"Start":"00:47.085 ","End":"00:50.720","Text":"We know that this is a Poisson distribution,"},{"Start":"00:50.720 ","End":"00:53.720","Text":"we\u0027re looking at the number of customers,"},{"Start":"00:53.720 ","End":"00:58.140","Text":"and we have an average of 2 customers per minute."},{"Start":"00:58.910 ","End":"01:01.125","Text":"If that\u0027s the case,"},{"Start":"01:01.125 ","End":"01:05.686","Text":"in section a we\u0027re asked,"},{"Start":"01:05.686 ","End":"01:10.940","Text":"what\u0027s the probability that there will be exactly 3 customers in a given minute?"},{"Start":"01:10.940 ","End":"01:20.320","Text":"A asks us what\u0027s the probability that x equals 3?"},{"Start":"01:20.450 ","End":"01:30.350","Text":"Now, we know that the equation for a Poisson distribution where x equals k. Well,"},{"Start":"01:30.350 ","End":"01:35.025","Text":"that would equal to e to the power of minus Lambda,"},{"Start":"01:35.025 ","End":"01:40.265","Text":"times lambda to the power of k over k factorial."},{"Start":"01:40.265 ","End":"01:43.134","Text":"Let\u0027s just plug in the numbers here."},{"Start":"01:43.134 ","End":"01:46.910","Text":"We\u0027ll have e to the power of minus Lambda."},{"Start":"01:46.910 ","End":"01:49.265","Text":"Now Lambda is 2, so that\u0027s minus 2,"},{"Start":"01:49.265 ","End":"01:51.530","Text":"times 2 to the power of k,"},{"Start":"01:51.530 ","End":"01:56.060","Text":"k is 3, over 3 factorial,"},{"Start":"01:56.060 ","End":"02:02.340","Text":"and that would equal to 0.1804."},{"Start":"02:02.510 ","End":"02:05.015","Text":"Section b asks us,"},{"Start":"02:05.015 ","End":"02:10.490","Text":"what is the probability that at least 1 customer will arrive in a given minute?"},{"Start":"02:10.490 ","End":"02:18.370","Text":"We\u0027re looking for the probability that x would be greater or equal to 1,"},{"Start":"02:18.370 ","End":"02:20.910","Text":"that\u0027s at least, at least 1."},{"Start":"02:20.910 ","End":"02:25.760","Text":"Now, we know that x can have the values of 0,"},{"Start":"02:25.760 ","End":"02:30.350","Text":"1, 2, and so on and so forth until infinity."},{"Start":"02:30.350 ","End":"02:37.335","Text":"We\u0027re asked, what\u0027s the probability of x being greater or equal to 1?"},{"Start":"02:37.335 ","End":"02:40.295","Text":"Well, instead of calculating this,"},{"Start":"02:40.295 ","End":"02:42.080","Text":"let\u0027s take the complimentary set."},{"Start":"02:42.080 ","End":"02:43.415","Text":"I think that\u0027ll be easier."},{"Start":"02:43.415 ","End":"02:45.715","Text":"That\u0027ll be 1,"},{"Start":"02:45.715 ","End":"02:50.900","Text":"minus the probability of x equaling 0."},{"Start":"02:50.900 ","End":"02:54.029","Text":"We\u0027ll take this guy right here."},{"Start":"02:54.050 ","End":"02:56.120","Text":"If that\u0027s the case,"},{"Start":"02:56.120 ","End":"02:58.010","Text":"let\u0027s just plug in the numbers."},{"Start":"02:58.010 ","End":"03:02.990","Text":"That\u0027s 1, minus e to the power of minus Lambda."},{"Start":"03:02.990 ","End":"03:05.630","Text":"Now Lambda is 2, so that\u0027s minus 2,"},{"Start":"03:05.630 ","End":"03:10.020","Text":"times 2 to the power of k. Now k is 0."},{"Start":"03:10.020 ","End":"03:13.070","Text":"Over 0 factorial."},{"Start":"03:13.070 ","End":"03:17.580","Text":"That would equal to 0.8647."},{"Start":"03:21.560 ","End":"03:23.889","Text":"Section c asks,"},{"Start":"03:23.889 ","End":"03:29.660","Text":"what\u0027s the probability that there will be at most 2 customers in a given minute?"},{"Start":"03:29.660 ","End":"03:37.100","Text":"What we\u0027re asking for is the probability that there will be at most 2 customers."},{"Start":"03:37.100 ","End":"03:42.390","Text":"That means that x is less than or equal to 2."},{"Start":"03:43.160 ","End":"03:46.810","Text":"Now, what\u0027s that equal to?"},{"Start":"03:46.810 ","End":"03:54.405","Text":"Well, that equals to the probability of x being equal to 0,"},{"Start":"03:54.405 ","End":"03:59.415","Text":"plus the probability of x being equal to 1,"},{"Start":"03:59.415 ","End":"04:03.765","Text":"plus the probability of x being equal to 2."},{"Start":"04:03.765 ","End":"04:05.880","Text":"Now, why is that? Well,"},{"Start":"04:05.880 ","End":"04:07.605","Text":"as we said right here,"},{"Start":"04:07.605 ","End":"04:09.740","Text":"x is less than or equal to 2,"},{"Start":"04:09.740 ","End":"04:14.665","Text":"so x can be either 0 or 1 or 2."},{"Start":"04:14.665 ","End":"04:18.230","Text":"All we have to do is add up these probabilities."},{"Start":"04:18.230 ","End":"04:22.115","Text":"We\u0027ll have to calculate them and add them up, and we\u0027ll get the answer."},{"Start":"04:22.115 ","End":"04:26.243","Text":"The probability of x being equal to 0, well,"},{"Start":"04:26.243 ","End":"04:31.660","Text":"we\u0027ve calculated that right here."},{"Start":"04:31.660 ","End":"04:33.425","Text":"Let\u0027s just do it again."},{"Start":"04:33.425 ","End":"04:38.795","Text":"That\u0027s e to the power of minus 2,"},{"Start":"04:38.795 ","End":"04:45.860","Text":"times 2 to the power of 0, over 0 factorial."},{"Start":"04:45.860 ","End":"04:47.645","Text":"That\u0027s this guy right here."},{"Start":"04:47.645 ","End":"04:50.675","Text":"Plus what\u0027s the probability of x being equal to 1?"},{"Start":"04:50.675 ","End":"04:53.575","Text":"Again, that\u0027s e to the power of minus 2."},{"Start":"04:53.575 ","End":"04:56.785","Text":"Let\u0027s not forget that Lambda equals 2,"},{"Start":"04:56.785 ","End":"05:00.740","Text":"so it\u0027s e to the power of minus Lambda times 2."},{"Start":"05:00.740 ","End":"05:04.145","Text":"That\u0027s Lambda to the power of k, now k is 1."},{"Start":"05:04.145 ","End":"05:07.548","Text":"Over 1 factorial,"},{"Start":"05:07.548 ","End":"05:12.090","Text":"plus the probability of x being equal to 2."},{"Start":"05:12.090 ","End":"05:15.515","Text":"That\u0027ll be equal to e to the power of minus Lambda."},{"Start":"05:15.515 ","End":"05:17.645","Text":"Lambda is 2, so that\u0027s minus 2,"},{"Start":"05:17.645 ","End":"05:20.550","Text":"times 2 to the power of 2."},{"Start":"05:20.550 ","End":"05:24.600","Text":"That\u0027s 2 squared over 2 factorial."},{"Start":"05:24.600 ","End":"05:30.840","Text":"That comes out to 0.6767."},{"Start":"05:30.840 ","End":"05:33.140","Text":"In section d we\u0027re asked,"},{"Start":"05:33.140 ","End":"05:35.270","Text":"what are the expectation and standard deviation of"},{"Start":"05:35.270 ","End":"05:38.245","Text":"the number of customers entering the store in a minute?"},{"Start":"05:38.245 ","End":"05:41.465","Text":"Well, we defined x as the number of clients per minute,"},{"Start":"05:41.465 ","End":"05:46.720","Text":"and we know that x has a Poisson distribution where Lambda equals 2."},{"Start":"05:46.720 ","End":"05:51.560","Text":"We also know that if x is distributed with a Poisson distribution,"},{"Start":"05:51.560 ","End":"05:55.903","Text":"then the expectation of x equals the variance of x,"},{"Start":"05:55.903 ","End":"05:57.670","Text":"and that equals to Lambda."},{"Start":"05:57.670 ","End":"06:06.500","Text":"In our case, Lambda equals 2 so we can say that the expectation of x equals 2. 2 what?"},{"Start":"06:06.500 ","End":"06:10.280","Text":"Clients per minute."},{"Start":"06:10.280 ","End":"06:12.680","Text":"What\u0027s the variance of x?"},{"Start":"06:12.680 ","End":"06:15.065","Text":"Well, again, that equals to 2,"},{"Start":"06:15.065 ","End":"06:17.810","Text":"but we\u0027re not asked for the variance."},{"Start":"06:17.810 ","End":"06:19.610","Text":"We\u0027re asked for the standard deviation."},{"Start":"06:19.610 ","End":"06:25.865","Text":"The standard deviation of x is the square root of the variance, equaling 2."},{"Start":"06:25.865 ","End":"06:29.130","Text":"That equals to 1.414,"},{"Start":"06:29.630 ","End":"06:36.520","Text":"and the units here are clients per minute."}],"ID":13075},{"Watched":false,"Name":"Exercise 5 Parts a-b","Duration":"6m 42s","ChapterTopicVideoID":12598,"CourseChapterTopicPlaylistID":64492,"HasSubtitles":true,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"This question will be talking about the number of babies being born in a hospital."},{"Start":"00:04.170 ","End":"00:07.710","Text":"Now, the number of births in a certain hospital has"},{"Start":"00:07.710 ","End":"00:12.800","Text":"a Poisson probability distribution with an expectation of 8 births per day."},{"Start":"00:12.800 ","End":"00:17.145","Text":"We\u0027re asked, what\u0027s the probability that 10 babies were born on Sunday,"},{"Start":"00:17.145 ","End":"00:19.750","Text":"and 7 on Monday."},{"Start":"00:19.820 ","End":"00:30.640","Text":"Well, let\u0027s first define x_1 as the number of births on Sunday,"},{"Start":"00:31.070 ","End":"00:40.139","Text":"and x_2 will be the number of births on Monday."},{"Start":"00:40.139 ","End":"00:49.180","Text":"Now, x_1 has a Poisson distribution,"},{"Start":"00:49.180 ","End":"00:52.160","Text":"with lambda being equal to 8."},{"Start":"00:52.160 ","End":"00:54.489","Text":"Same thing for x_2."},{"Start":"00:54.489 ","End":"01:02.645","Text":"X_2 has a Poisson distribution where lambda also equals to 8, and we\u0027re asked,"},{"Start":"01:02.645 ","End":"01:08.610","Text":"what\u0027s the probability that x_1 is equal to 10,"},{"Start":"01:08.610 ","End":"01:12.670","Text":"and x_2 is equal to 7."},{"Start":"01:12.670 ","End":"01:14.585","Text":"Now, why is that?"},{"Start":"01:14.585 ","End":"01:18.320","Text":"Again, what\u0027s the probability that 10 babies were born on Sunday?"},{"Start":"01:18.320 ","End":"01:20.765","Text":"Well, Sunday that\u0027s the random variable x_1."},{"Start":"01:20.765 ","End":"01:27.130","Text":"We\u0027re looking for the probability that x_1 is equal to 10,"},{"Start":"01:27.130 ","End":"01:28.940","Text":"and that x_2,"},{"Start":"01:28.940 ","End":"01:32.245","Text":"the number of births on Monday equals 7."},{"Start":"01:32.245 ","End":"01:36.260","Text":"Now, there\u0027s a special characteristic in"},{"Start":"01:36.260 ","End":"01:41.360","Text":"the Poisson distribution where we\u0027re talking about time intervals."},{"Start":"01:41.360 ","End":"01:45.455","Text":"If the time intervals are not overlapping,"},{"Start":"01:45.455 ","End":"01:49.570","Text":"then the random variables would be independent,"},{"Start":"01:49.570 ","End":"01:52.000","Text":"and here, obviously Sunday,"},{"Start":"01:52.000 ","End":"01:54.965","Text":"the time interval is Sunday here, and Monday here,"},{"Start":"01:54.965 ","End":"01:58.580","Text":"the 2-time intervals are not overla