Introduction to Moment Generating Function
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Characteristics of the Moment Generating Function
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[{"Name":"Introduction to Moment Generating Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"2m 45s","ChapterTopicVideoID":12688,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12688.jpeg","UploadDate":"2018-07-30T13:34:03.3600000","DurationForVideoObject":"PT2M45S","Description":null,"MetaTitle":"Tutorial: Video + Workbook | Proprep","MetaDescription":"Moment Generating Function - Introduction to Moment Generating Function. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/probability/moment-generating-function/introduction-to-moment-generating-function/vid13158","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this chapter, we\u0027ll be talking about Moment Generating Functions."},{"Start":"00:03.900 ","End":"00:09.990","Text":"Now, a moment generating function of a random variable is defined as the following."},{"Start":"00:09.990 ","End":"00:14.280","Text":"That\u0027s M_x of t. Well,"},{"Start":"00:14.280 ","End":"00:18.600","Text":"that\u0027s the moment generating function of a random variable x"},{"Start":"00:18.600 ","End":"00:24.465","Text":"that equals to the expectation of e^t times x."},{"Start":"00:24.465 ","End":"00:28.095","Text":"Now, for dealing with discrete random variables,"},{"Start":"00:28.095 ","End":"00:31.560","Text":"then the moment generating function equals the expectation,"},{"Start":"00:31.560 ","End":"00:33.920","Text":"as we said of e^tx,"},{"Start":"00:33.920 ","End":"00:37.570","Text":"and that equals to the sum over all of k,"},{"Start":"00:37.570 ","End":"00:47.165","Text":"where x equals k of e^tk times the probability of x equaling k. Now,"},{"Start":"00:47.165 ","End":"00:49.790","Text":"if we\u0027re dealing with a continuous random variable,"},{"Start":"00:49.790 ","End":"00:52.820","Text":"then the moment generating function equals again"},{"Start":"00:52.820 ","End":"00:57.110","Text":"the expectation of e^t times x and that equals to"},{"Start":"00:57.110 ","End":"01:05.245","Text":"the integral over all of x of e^t x times the density function of x dx."},{"Start":"01:05.245 ","End":"01:08.525","Text":"Now, this may sound a bit confusing,"},{"Start":"01:08.525 ","End":"01:11.840","Text":"but it\u0027ll all get cleared up once we go to an example."},{"Start":"01:11.840 ","End":"01:13.715","Text":"Let\u0027s just continue for now."},{"Start":"01:13.715 ","End":"01:20.275","Text":"The nth moment is defined as the expectation of x^n."},{"Start":"01:20.275 ","End":"01:24.860","Text":"The nth moment of a random variable x is obtained by taking"},{"Start":"01:24.860 ","End":"01:30.080","Text":"the nth derivative by t of the moment generating function,"},{"Start":"01:30.080 ","End":"01:33.815","Text":"write M_x of t at the point t equals 0."},{"Start":"01:33.815 ","End":"01:41.765","Text":"That means that M to the nth derivative of M_x of t, where t equals 0,"},{"Start":"01:41.765 ","End":"01:46.340","Text":"well that equals to the expectation of x to the power of n. For example,"},{"Start":"01:46.340 ","End":"01:48.695","Text":"if we\u0027re taking the first derivative,"},{"Start":"01:48.695 ","End":"01:53.345","Text":"then we\u0027ll get the expectation of x^1 or the expectation of x."},{"Start":"01:53.345 ","End":"01:56.105","Text":"Now for taking the second derivative,"},{"Start":"01:56.105 ","End":"02:00.735","Text":"then we\u0027ll get the expectation of x squared."},{"Start":"02:00.735 ","End":"02:03.170","Text":"That\u0027s all where t equals 0."},{"Start":"02:03.170 ","End":"02:06.440","Text":"Now there\u0027s a theorem that says that there\u0027s"},{"Start":"02:06.440 ","End":"02:12.724","Text":"a 1-to-1 correspondence between a random variable and its moment generating function."},{"Start":"02:12.724 ","End":"02:15.110","Text":"Now, at the end of this chapter,"},{"Start":"02:15.110 ","End":"02:19.640","Text":"there\u0027s a list of special distributions with their moment generating functions."},{"Start":"02:19.640 ","End":"02:24.560","Text":"Now also, because you need to know how to take a derivative of functions."},{"Start":"02:24.560 ","End":"02:27.080","Text":"Well, I have here"},{"Start":"02:27.080 ","End":"02:34.790","Text":"a list of various reminders of how to take a derivative."},{"Start":"02:34.790 ","End":"02:38.615","Text":"For example, the derivative of x^n,"},{"Start":"02:38.615 ","End":"02:42.675","Text":"well that equals to n times x^n minus 1,"},{"Start":"02:42.675 ","End":"02:45.160","Text":"and so on and so forth."}],"ID":13158},{"Watched":false,"Name":"Example","Duration":"12m ","ChapterTopicVideoID":12689,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this example, we\u0027re asked to show that the moment-generating function of"},{"Start":"00:04.230 ","End":"00:11.175","Text":"the exponential probability distribution is lambda divided by lambda minus t. Secondly,"},{"Start":"00:11.175 ","End":"00:16.300","Text":"we\u0027re asked to find the first and second moments of the probability distribution."},{"Start":"00:16.610 ","End":"00:19.545","Text":"Let\u0027s just remind ourselves."},{"Start":"00:19.545 ","End":"00:24.340","Text":"The moment-generating function of x at t,"},{"Start":"00:24.340 ","End":"00:35.030","Text":"that equals to the integral from all of x of e to the power of t times x,"},{"Start":"00:35.030 ","End":"00:39.045","Text":"times the density function of x dx."},{"Start":"00:39.045 ","End":"00:41.370","Text":"Now, let\u0027s just remind ourselves,"},{"Start":"00:41.370 ","End":"00:43.170","Text":"the density function of x,"},{"Start":"00:43.170 ","End":"00:46.130","Text":"where x is an exponential distribution,"},{"Start":"00:46.130 ","End":"00:52.085","Text":"well that equals lambda e to the power of minus lambda x,"},{"Start":"00:52.085 ","End":"00:55.655","Text":"where x is greater or equal to 0."},{"Start":"00:55.655 ","End":"00:59.125","Text":"Now, let\u0027s just plug in this."},{"Start":"00:59.125 ","End":"01:01.680","Text":"This density function here."},{"Start":"01:01.680 ","End":"01:06.200","Text":"Now that equals the integral over all of x."},{"Start":"01:06.200 ","End":"01:07.700","Text":"Well, what\u0027s all of x?"},{"Start":"01:07.700 ","End":"01:10.069","Text":"x is between 0 and infinity,"},{"Start":"01:10.069 ","End":"01:16.400","Text":"so that means that\u0027s 0 and that\u0027s an infinity of e to the power of t times x,"},{"Start":"01:16.400 ","End":"01:25.110","Text":"times lambda times e to the power of minus lambda x dx."},{"Start":"01:25.360 ","End":"01:29.360","Text":"Now, let\u0027s just simplify that."},{"Start":"01:29.360 ","End":"01:33.890","Text":"That\u0027s lambda. We\u0027re taking lambda out of the integral that\u0027s a constant."},{"Start":"01:33.890 ","End":"01:43.040","Text":"Now, we have the integral from 0 to plus infinity of e to the power of what?"},{"Start":"01:43.040 ","End":"01:46.940","Text":"Well, let\u0027s take minus x outright now,"},{"Start":"01:46.940 ","End":"01:55.720","Text":"minus x and we have here lambda minus t. That\u0027s dx right here."},{"Start":"01:55.720 ","End":"02:03.620","Text":"Now, here we have the integral of e to the power of some constant times x."},{"Start":"02:03.620 ","End":"02:08.535","Text":"Now again, let\u0027s remind ourselves of the integration rules here."},{"Start":"02:08.535 ","End":"02:14.165","Text":"We have the integral of e to the power of ax dx,"},{"Start":"02:14.165 ","End":"02:20.435","Text":"that equals to e to the power of ax divided by a."},{"Start":"02:20.435 ","End":"02:25.455","Text":"Let\u0027s just work with this and apply it here."},{"Start":"02:25.455 ","End":"02:28.410","Text":"Now, that equals lambda."},{"Start":"02:28.410 ","End":"02:33.830","Text":"Now e to the power of minus x times a constant."},{"Start":"02:33.830 ","End":"02:38.855","Text":"Well, that\u0027s a. That\u0027s minus lambda minus t, that\u0027s a."},{"Start":"02:38.855 ","End":"02:46.715","Text":"That\u0027ll be equal to e to the power of minus x lambda minus t,"},{"Start":"02:46.715 ","End":"02:53.280","Text":"divided by minus lambda minus t,"},{"Start":"02:53.410 ","End":"03:01.280","Text":"where x has to be between 0 and infinity."},{"Start":"03:01.280 ","End":"03:06.740","Text":"Now, when we plug in infinity here instead of x,"},{"Start":"03:06.740 ","End":"03:10.115","Text":"then in order for the numerator to be 0,"},{"Start":"03:10.115 ","End":"03:14.690","Text":"I need for lambda to be"},{"Start":"03:14.690 ","End":"03:20.510","Text":"greater or equal to t. When that happens,"},{"Start":"03:20.510 ","End":"03:24.085","Text":"then we have here lambda,"},{"Start":"03:24.085 ","End":"03:28.710","Text":"let\u0027s plugin infinity lambda times 0."},{"Start":"03:28.710 ","End":"03:34.000","Text":"It\u0027ll be 0 divided by minus lambda minus t,"},{"Start":"03:34.000 ","End":"03:38.420","Text":"minus, now what happens when we plug in 0 here?"},{"Start":"03:38.420 ","End":"03:43.535","Text":"That\u0027ll be lambda times e to the power of 0,"},{"Start":"03:43.535 ","End":"03:52.940","Text":"divided by minus lambda minus t. Now,"},{"Start":"03:52.940 ","End":"03:58.170","Text":"this minus and this minus cancels each other out."},{"Start":"03:58.960 ","End":"04:08.829","Text":"We have here lambda divided by lambda minus t. This"},{"Start":"04:08.829 ","End":"04:13.880","Text":"then is the moment generating function of x where x"},{"Start":"04:13.880 ","End":"04:19.445","Text":"is distributed with an exponential distribution with parameter lambda."},{"Start":"04:19.445 ","End":"04:23.370","Text":"This is what we\u0027re meant to show in this section."},{"Start":"04:23.740 ","End":"04:26.120","Text":"In this section, we\u0027re asked to find"},{"Start":"04:26.120 ","End":"04:29.935","Text":"the first and second moments of the probability distribution."},{"Start":"04:29.935 ","End":"04:33.945","Text":"Before that, let\u0027s just recall what we did."},{"Start":"04:33.945 ","End":"04:36.800","Text":"We\u0027ve calculated the moment generating function,"},{"Start":"04:36.800 ","End":"04:38.990","Text":"that\u0027s m sub x of t,"},{"Start":"04:38.990 ","End":"04:44.885","Text":"that equals to lambda divided by lambda minus t. Now,"},{"Start":"04:44.885 ","End":"04:48.590","Text":"how do we get from this function to"},{"Start":"04:48.590 ","End":"04:52.370","Text":"the first and second moments of the probability distribution?"},{"Start":"04:52.370 ","End":"04:58.020","Text":"Well, let\u0027s just recall what we read in the text, and here\u0027s the text."},{"Start":"04:58.020 ","End":"05:03.635","Text":"The nth moment is defined as the expectation of x to the power of n,"},{"Start":"05:03.635 ","End":"05:08.780","Text":"and the nth moment is obtained by taking the nth derivative by"},{"Start":"05:08.780 ","End":"05:15.100","Text":"t of the moment-generating function and then equating t to 0."},{"Start":"05:15.100 ","End":"05:17.685","Text":"Let\u0027s just do that here."},{"Start":"05:17.685 ","End":"05:21.030","Text":"We\u0027re looking, for now, the first moment."},{"Start":"05:21.030 ","End":"05:25.245","Text":"That\u0027s the expectation of x to the power of 1,"},{"Start":"05:25.245 ","End":"05:27.765","Text":"n now equals to 1."},{"Start":"05:27.765 ","End":"05:32.075","Text":"Well, that equals to the expectation of x."},{"Start":"05:32.075 ","End":"05:33.920","Text":"Now, how do we get that?"},{"Start":"05:33.920 ","End":"05:39.620","Text":"Well, that\u0027s the first derivative of the moment-generating function."},{"Start":"05:39.620 ","End":"05:44.390","Text":"We have the first derivative of this function."},{"Start":"05:44.390 ","End":"05:49.793","Text":"That means we\u0027re taking lambda divided by lambda minus t,"},{"Start":"05:49.793 ","End":"05:51.710","Text":"and we\u0027re taking the first derivative of that."},{"Start":"05:51.710 ","End":"05:53.375","Text":"Now let\u0027s just recall,"},{"Start":"05:53.375 ","End":"05:56.150","Text":"whenever we have a function a divided by b,"},{"Start":"05:56.150 ","End":"05:58.340","Text":"and we want to take the derivative of that."},{"Start":"05:58.340 ","End":"06:04.370","Text":"Well, that equals to the derivative of a times"},{"Start":"06:04.370 ","End":"06:11.245","Text":"b minus a times the derivative of b divided by b squared."},{"Start":"06:11.245 ","End":"06:16.820","Text":"This is just a quick reminder of how we take a derivative of this guy right here."},{"Start":"06:16.820 ","End":"06:19.220","Text":"Now, that equals to,"},{"Start":"06:19.220 ","End":"06:22.190","Text":"well, the derivative of lambda,"},{"Start":"06:22.190 ","End":"06:26.450","Text":"don\u0027t forget we\u0027re taking a derivative by t. That equals to 0,"},{"Start":"06:26.450 ","End":"06:29.590","Text":"so it\u0027s 0 times b."},{"Start":"06:29.590 ","End":"06:33.265","Text":"That\u0027ll be lambda minus t. That equals 0,"},{"Start":"06:33.265 ","End":"06:37.445","Text":"minus a, that\u0027s lambda in our case,"},{"Start":"06:37.445 ","End":"06:39.350","Text":"times the derivative of b."},{"Start":"06:39.350 ","End":"06:40.898","Text":"Well, the derivative of b,"},{"Start":"06:40.898 ","End":"06:43.088","Text":"this is a derivative of lambda minus t,"},{"Start":"06:43.088 ","End":"06:46.050","Text":"that equals minus 1."},{"Start":"06:46.050 ","End":"06:50.510","Text":"Lambda times minus 1 divided by b squared."},{"Start":"06:50.510 ","End":"06:54.305","Text":"Well in our case, that would be lambda minus t squared."},{"Start":"06:54.305 ","End":"06:56.600","Text":"Now let\u0027s just simplify that."},{"Start":"06:56.600 ","End":"06:58.130","Text":"That equals 0."},{"Start":"06:58.130 ","End":"07:01.610","Text":"Minus minus cancels each other out."},{"Start":"07:01.610 ","End":"07:09.080","Text":"We have lambda here divided by lambda minus t squared."},{"Start":"07:09.080 ","End":"07:15.510","Text":"Now, we want to equate t to 0. That\u0027s right here."},{"Start":"07:15.510 ","End":"07:21.140","Text":"That means that we have lambda divided by lambda squared,"},{"Start":"07:21.140 ","End":"07:24.115","Text":"or 1 divided by lambda."},{"Start":"07:24.115 ","End":"07:29.885","Text":"This then is the first derivative of the moment-generating function,"},{"Start":"07:29.885 ","End":"07:33.575","Text":"which is the expectation of x."},{"Start":"07:33.575 ","End":"07:38.450","Text":"If we recall, whenever we have a random variable"},{"Start":"07:38.450 ","End":"07:43.655","Text":"that\u0027s distributed with an exponential distribution with parameter lambda,"},{"Start":"07:43.655 ","End":"07:49.350","Text":"then the expectation of that random variable is 1 divided by lambda."},{"Start":"07:50.000 ","End":"07:52.715","Text":"What about the second moment?"},{"Start":"07:52.715 ","End":"07:55.895","Text":"Well, that means that n now has to be equal to 2."},{"Start":"07:55.895 ","End":"08:00.075","Text":"We\u0027re looking for the expectation of x squared."},{"Start":"08:00.075 ","End":"08:02.235","Text":"That\u0027s this guy right here."},{"Start":"08:02.235 ","End":"08:04.505","Text":"Now, how do we get that?"},{"Start":"08:04.505 ","End":"08:09.605","Text":"Well, we have to take the second derivative of the moment-generating function."},{"Start":"08:09.605 ","End":"08:12.259","Text":"Taking the moment-generating function,"},{"Start":"08:12.259 ","End":"08:14.500","Text":"taking the second derivative."},{"Start":"08:14.500 ","End":"08:17.450","Text":"Now, the first derivative is this guy right here."},{"Start":"08:17.450 ","End":"08:19.625","Text":"All we need to do is to take"},{"Start":"08:19.625 ","End":"08:24.860","Text":"another derivative of this expression right here in order to get the second derivative."},{"Start":"08:24.860 ","End":"08:26.315","Text":"Let\u0027s do that."},{"Start":"08:26.315 ","End":"08:31.900","Text":"That\u0027ll be equal to lambda divided by lambda minus t squared."},{"Start":"08:31.900 ","End":"08:37.310","Text":"We want to take the derivative of that to get the second derivative."},{"Start":"08:37.310 ","End":"08:44.120","Text":"Now, Let\u0027s use this technique again to calculate this ratio right here."},{"Start":"08:44.120 ","End":"08:46.639","Text":"a would be lambda,"},{"Start":"08:46.639 ","End":"08:51.350","Text":"and now b would be lambda minus t squared."},{"Start":"08:51.350 ","End":"08:57.545","Text":"That\u0027ll be 0 times lambda minus t squared."},{"Start":"08:57.545 ","End":"09:07.054","Text":"That\u0027ll be the derivative of a of lambda times b of lambda minus t squared,"},{"Start":"09:07.054 ","End":"09:13.160","Text":"minus now a that\u0027ll be our lambda times the derivative of the denominator,"},{"Start":"09:13.160 ","End":"09:15.500","Text":"that\u0027s lambda minus t squared without b"},{"Start":"09:15.500 ","End":"09:21.905","Text":"2 times lambda minus t times the internal derivative times minus 1."},{"Start":"09:21.905 ","End":"09:26.965","Text":"All this then has to go over lambda minus t squared,"},{"Start":"09:26.965 ","End":"09:29.020","Text":"squared, that\u0027s b squared."},{"Start":"09:29.020 ","End":"09:32.750","Text":"That\u0027ll be lambda minus t squared."},{"Start":"09:32.750 ","End":"09:35.914","Text":"That\u0027s our b and we want b squared,"},{"Start":"09:35.914 ","End":"09:38.425","Text":"so that\u0027s it right here."},{"Start":"09:38.425 ","End":"09:41.075","Text":"Now, let\u0027s simplify that."},{"Start":"09:41.075 ","End":"09:45.175","Text":"That will be equal to 2 times lambda,"},{"Start":"09:45.175 ","End":"09:50.825","Text":"times lambda minus t divided by,"},{"Start":"09:50.825 ","End":"09:54.590","Text":"now lambda minus t to the power of 4,"},{"Start":"09:54.590 ","End":"09:58.640","Text":"and we want t to be equal to 0."},{"Start":"09:58.640 ","End":"10:02.455","Text":"Let\u0027s just plug in 0 where t is."},{"Start":"10:02.455 ","End":"10:06.000","Text":"Here we go. It\u0027ll be 2."},{"Start":"10:06.000 ","End":"10:15.355","Text":"That be equal to 2 times lambda squared divided by lambda to the 4th,"},{"Start":"10:15.355 ","End":"10:17.070","Text":"or in essence,"},{"Start":"10:17.070 ","End":"10:19.755","Text":"that\u0027ll be 2 divided by lambda squared."},{"Start":"10:19.755 ","End":"10:23.930","Text":"Again, that\u0027ll be equal to the expectation of x squared,"},{"Start":"10:23.930 ","End":"10:26.030","Text":"which is the second moment."},{"Start":"10:26.030 ","End":"10:32.140","Text":"Now, we have the second moment and we\u0027ve calculated the first moment."},{"Start":"10:32.140 ","End":"10:34.255","Text":"That\u0027s the expectation of x."},{"Start":"10:34.255 ","End":"10:35.930","Text":"Now I haven\u0027t asked this of you,"},{"Start":"10:35.930 ","End":"10:41.570","Text":"but let\u0027s use those results in order to calculate the variance of x."},{"Start":"10:41.570 ","End":"10:47.585","Text":"The variance of x, we know that that equals to the expectation of x squared,"},{"Start":"10:47.585 ","End":"10:51.860","Text":"minus the expectation squared of x."},{"Start":"10:51.860 ","End":"10:54.200","Text":"Now, the expectation of x squared,"},{"Start":"10:54.200 ","End":"10:55.340","Text":"that\u0027s this guy right here."},{"Start":"10:55.340 ","End":"10:59.565","Text":"That\u0027s 2 divided by lambda squared minus,"},{"Start":"10:59.565 ","End":"11:01.260","Text":"well, what\u0027s the expectation of x?"},{"Start":"11:01.260 ","End":"11:02.790","Text":"Well, that\u0027s right here."},{"Start":"11:02.790 ","End":"11:06.450","Text":"That will be 1 divided by lambda squared."},{"Start":"11:06.450 ","End":"11:12.380","Text":"That means that we\u0027re looking at 2 minus 1 divided by lambda squared."},{"Start":"11:12.380 ","End":"11:15.620","Text":"That means that we\u0027re looking at 1 divided by lambda squared."},{"Start":"11:15.620 ","End":"11:22.205","Text":"This then is the variance of x where x is distributed with an exponential distribution."},{"Start":"11:22.205 ","End":"11:26.210","Text":"Now, again, how did we arrive at this?"},{"Start":"11:26.210 ","End":"11:31.760","Text":"By calculating the expectation of x squared and the expectation of x."},{"Start":"11:31.760 ","End":"11:37.505","Text":"That\u0027s the second moment and the first moment of the distribution."},{"Start":"11:37.505 ","End":"11:44.390","Text":"Now, this may be a long way around to calculate the variance of the known distribution,"},{"Start":"11:44.390 ","End":"11:47.390","Text":"but for many complicated distributions,"},{"Start":"11:47.390 ","End":"11:51.065","Text":"the moment-generating function,"},{"Start":"11:51.065 ","End":"11:57.365","Text":"the derivation of the moments using the moments then generating function,"},{"Start":"11:57.365 ","End":"12:00.510","Text":"that\u0027s the easiest way to go."}],"ID":13159},{"Watched":false,"Name":"Exercise 1","Duration":"3m 52s","ChapterTopicVideoID":12690,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this question, the following probability function"},{"Start":"00:02.640 ","End":"00:04.570","Text":"is given for discrete random variable."},{"Start":"00:04.570 ","End":"00:06.045","Text":"We have our Xs,"},{"Start":"00:06.045 ","End":"00:08.460","Text":"x equals 1 or 2 or 3,"},{"Start":"00:08.460 ","End":"00:12.167","Text":"and we have the probability for x 1/2,"},{"Start":"00:12.167 ","End":"00:13.380","Text":"1/3, and 1/6 respectively."},{"Start":"00:13.380 ","End":"00:17.920","Text":"We\u0027re asked to find the moment generating function of x."},{"Start":"00:17.930 ","End":"00:23.670","Text":"We know that the moment generating function of x,"},{"Start":"00:23.670 ","End":"00:30.715","Text":"that equals to the expectation of e to the power of t times x."},{"Start":"00:30.715 ","End":"00:33.000","Text":"Now, that equals,"},{"Start":"00:33.000 ","End":"00:35.990","Text":"because we\u0027re dealing in a discrete random variable,"},{"Start":"00:35.990 ","End":"00:41.900","Text":"that\u0027s the sum then of e to the power of t times k times"},{"Start":"00:41.900 ","End":"00:48.770","Text":"the probability of x being equal to that k over all k. Now,"},{"Start":"00:48.770 ","End":"00:53.845","Text":"let\u0027s calculate this moment generating function."},{"Start":"00:53.845 ","End":"01:04.175","Text":"That\u0027ll be equal to M_x of t. That equals to e to the power of t. Now,"},{"Start":"01:04.175 ","End":"01:10.810","Text":"k here equals 1 times 1 times the probability of x being equal to 1."},{"Start":"01:10.810 ","End":"01:15.220","Text":"Well, that\u0027s 1/2, times 1/2 plus,"},{"Start":"01:15.220 ","End":"01:16.990","Text":"now let\u0027s go to the second expression,"},{"Start":"01:16.990 ","End":"01:24.370","Text":"that\u0027s e to the power of t times 2 times the probability of x being equal to 2."},{"Start":"01:24.370 ","End":"01:25.895","Text":"Well, that\u0027s the 1/3,"},{"Start":"01:25.895 ","End":"01:31.155","Text":"plus e to the power of t times 3."},{"Start":"01:31.155 ","End":"01:36.040","Text":"Now, k equals 3 times the probability of x being equal to 3,"},{"Start":"01:36.040 ","End":"01:38.755","Text":"that equals to 1/6."},{"Start":"01:38.755 ","End":"01:44.770","Text":"This then is the moment generating function that we\u0027re looking for."},{"Start":"01:44.770 ","End":"01:46.630","Text":"In this section, we\u0027re asked to derive"},{"Start":"01:46.630 ","End":"01:49.450","Text":"the expectation from the moment generating function."},{"Start":"01:49.450 ","End":"01:55.215","Text":"If we recall, this was a function that we\u0027ve calculated in section A."},{"Start":"01:55.215 ","End":"02:00.470","Text":"We\u0027re looking to derive the expectation of x."},{"Start":"02:00.470 ","End":"02:05.525","Text":"Now, we weren\u0027t asked to calculate the expectation of x using the regular technique,"},{"Start":"02:05.525 ","End":"02:10.430","Text":"summing the values of x times the probabilities."},{"Start":"02:10.430 ","End":"02:15.220","Text":"But we want to base our answer on this moment generating function."},{"Start":"02:15.220 ","End":"02:16.880","Text":"Let\u0027s do that. Now,"},{"Start":"02:16.880 ","End":"02:18.380","Text":"the expectation of x,"},{"Start":"02:18.380 ","End":"02:20.045","Text":"that\u0027s the first moment."},{"Start":"02:20.045 ","End":"02:22.750","Text":"N here has to be equal to 1."},{"Start":"02:22.750 ","End":"02:29.030","Text":"The implication of that is that we have to take the first derivative of this function,"},{"Start":"02:29.030 ","End":"02:31.715","Text":"set t to be equal to 0,"},{"Start":"02:31.715 ","End":"02:36.275","Text":"and that\u0027ll give us the first moment or the expectation of x."},{"Start":"02:36.275 ","End":"02:42.140","Text":"We\u0027re looking for the first derivative of the moment generating function."},{"Start":"02:42.140 ","End":"02:44.390","Text":"Let\u0027s just derive that."},{"Start":"02:44.390 ","End":"02:53.780","Text":"That\u0027ll be 1/2 times e to the power of t plus 1/3 e to the power of"},{"Start":"02:53.780 ","End":"03:04.325","Text":"2t times 2 plus 1/6 e to the power of 3t times 3."},{"Start":"03:04.325 ","End":"03:08.915","Text":"Now, here we have to set t to be equal to 0."},{"Start":"03:08.915 ","End":"03:12.730","Text":"Now, that means that this will be 1/2."},{"Start":"03:12.730 ","End":"03:18.840","Text":"E to the power of 0 is 1 so that\u0027ll be 1/2 times 1."},{"Start":"03:18.840 ","End":"03:23.225","Text":"1/2 plus 1/3 times 1."},{"Start":"03:23.225 ","End":"03:26.840","Text":"E to the power of 2 times 0 is 1 times 2."},{"Start":"03:26.840 ","End":"03:34.025","Text":"That\u0027ll be 2/3 plus 1/6 times e to the power of 0 times 3."},{"Start":"03:34.025 ","End":"03:35.930","Text":"Well, that\u0027s 3 over 6,"},{"Start":"03:35.930 ","End":"03:38.065","Text":"and that\u0027s again 1/2."},{"Start":"03:38.065 ","End":"03:41.630","Text":"That means that the expectation of x,"},{"Start":"03:41.630 ","End":"03:44.905","Text":"that equals to 1 and 2/3."},{"Start":"03:44.905 ","End":"03:48.035","Text":"Here we go. That\u0027s the expectation of x"},{"Start":"03:48.035 ","End":"03:52.890","Text":"when we derived it from the moment generating function."}],"ID":13160},{"Watched":false,"Name":"Exercise 2","Duration":"9m 41s","ChapterTopicVideoID":12691,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.074","Text":"This question we\u0027re asked to find the moment"},{"Start":"00:02.074 ","End":"00:04.755","Text":"generating function of the binomial distribution,"},{"Start":"00:04.755 ","End":"00:08.310","Text":"where x has a binomial distribution with parameters n"},{"Start":"00:08.310 ","End":"00:13.470","Text":"and p and to find the first and second moments of the function."},{"Start":"00:13.470 ","End":"00:19.770","Text":"We\u0027re looking then for the moment generating function m_x of"},{"Start":"00:19.770 ","End":"00:26.320","Text":"t and that\u0027s defined as the expectation of e^tx."},{"Start":"00:26.320 ","End":"00:29.180","Text":"Now, because we\u0027re dealing in"},{"Start":"00:29.180 ","End":"00:33.230","Text":"a discrete random variable then the moment generating function,"},{"Start":"00:33.230 ","End":"00:37.535","Text":"that\u0027s the sum over all values of k,"},{"Start":"00:37.535 ","End":"00:45.575","Text":"e^tk times the probability of x being equal to k. Now,"},{"Start":"00:45.575 ","End":"00:48.960","Text":"let\u0027s just recall that the probability of x being"},{"Start":"00:48.960 ","End":"00:53.750","Text":"equal to k in a binomial distribution that equals to n/k,"},{"Start":"00:53.750 ","End":"00:58.580","Text":"p^k, and 1 minus p^n minus"},{"Start":"00:58.580 ","End":"01:04.100","Text":"k. Let\u0027s just plug in this expression instead of this guy right here."},{"Start":"01:04.100 ","End":"01:06.595","Text":"That will be equal to the sum."},{"Start":"01:06.595 ","End":"01:09.735","Text":"Now, k here goes from 0-n,"},{"Start":"01:09.735 ","End":"01:16.700","Text":"so that means that\u0027s the sum from k equaling 0-n of what?"},{"Start":"01:16.700 ","End":"01:20.285","Text":"Of e^tk."},{"Start":"01:20.285 ","End":"01:23.690","Text":"Now, this guy is this expression right here,"},{"Start":"01:23.690 ","End":"01:27.780","Text":"so that means that that\u0027s times n/k,"},{"Start":"01:27.880 ","End":"01:34.010","Text":"p^k times 1 minus p^n minus k. Now,"},{"Start":"01:34.010 ","End":"01:37.640","Text":"let\u0027s do a little bit of algebraic cosmetics here."},{"Start":"01:37.640 ","End":"01:45.180","Text":"That equals to the sum k going from 0-n. Now,"},{"Start":"01:45.180 ","End":"01:48.405","Text":"n/k we\u0027ll put that first."},{"Start":"01:48.405 ","End":"01:53.950","Text":"Now here we have p^k and also e^tk."},{"Start":"01:54.380 ","End":"01:58.365","Text":"Let\u0027s just write this out like this."},{"Start":"01:58.365 ","End":"02:07.150","Text":"That will be equal to e^t times p^k"},{"Start":"02:07.340 ","End":"02:16.440","Text":"times 1 minus p^n minus k. Here we\u0027ve"},{"Start":"02:16.440 ","End":"02:20.360","Text":"gotten about as far as we can go without bringing in"},{"Start":"02:20.360 ","End":"02:25.340","Text":"some new knowledge and this new knowledge is Newton\u0027s binomial."},{"Start":"02:25.340 ","End":"02:29.550","Text":"Now, if we recall from a math class,"},{"Start":"02:29.930 ","End":"02:35.720","Text":"Newton\u0027s binomial looks like this."},{"Start":"02:35.720 ","End":"02:43.340","Text":"We have a plus b^n and that equals to the sum of what?"},{"Start":"02:43.340 ","End":"02:49.145","Text":"Of n/k times a^k"},{"Start":"02:49.145 ","End":"02:53.540","Text":"times b^n minus k. Now,"},{"Start":"02:53.540 ","End":"02:56.940","Text":"this is exactly what we have here."},{"Start":"02:56.940 ","End":"03:00.070","Text":"This is our a and this is our b."},{"Start":"03:00.070 ","End":"03:07.310","Text":"A then equals to e^t times p, and b,"},{"Start":"03:07.310 ","End":"03:11.080","Text":"that equals to 1 minus p. Now,"},{"Start":"03:11.080 ","End":"03:12.925","Text":"the minute we recognize that,"},{"Start":"03:12.925 ","End":"03:17.290","Text":"then we can write this very long expression."},{"Start":"03:17.290 ","End":"03:18.730","Text":"We can write it out like this."},{"Start":"03:18.730 ","End":"03:27.835","Text":"That equals to e^t times p plus, and a plus b, well,"},{"Start":"03:27.835 ","End":"03:35.095","Text":"that\u0027s 1 minus p. All that has to be to the power of n. Now,"},{"Start":"03:35.095 ","End":"03:39.110","Text":"if you didn\u0027t learn the Newton\u0027s binomial,"},{"Start":"03:39.110 ","End":"03:41.345","Text":"you don\u0027t know how to work with it,"},{"Start":"03:41.345 ","End":"03:44.280","Text":"then you can\u0027t solve this problem."},{"Start":"03:44.330 ","End":"03:51.595","Text":"There we go this then is the moment generating function of x,"},{"Start":"03:51.595 ","End":"03:55.710","Text":"that\u0027s distributed with a binomial distribution."},{"Start":"03:56.800 ","End":"04:01.145","Text":"Now that we have our moment generating function right here,"},{"Start":"04:01.145 ","End":"04:05.115","Text":"let\u0027s calculate the first and second moments of the function."},{"Start":"04:05.115 ","End":"04:07.785","Text":"First of all, let\u0027s take a look the first moment,"},{"Start":"04:07.785 ","End":"04:12.380","Text":"that means that we\u0027re looking for the expectation of x."},{"Start":"04:12.380 ","End":"04:13.865","Text":"Now how did we get that?"},{"Start":"04:13.865 ","End":"04:18.860","Text":"Well, we have to take the first derivative of the moment generating function."},{"Start":"04:18.860 ","End":"04:26.240","Text":"That\u0027s the derivative of m_x t. Having taken the derivative,"},{"Start":"04:26.240 ","End":"04:31.870","Text":"we have to set t to 0 in order to get our expectation."},{"Start":"04:31.870 ","End":"04:34.780","Text":"Let\u0027s do that."},{"Start":"04:35.420 ","End":"04:39.860","Text":"We have to take the derivative of this expression right here with"},{"Start":"04:39.860 ","End":"04:44.180","Text":"respect to t. That will be equal to"},{"Start":"04:44.180 ","End":"04:48.865","Text":"n times this expression right here"},{"Start":"04:48.865 ","End":"04:55.250","Text":"e^t times p plus 1 minus p^n"},{"Start":"04:55.250 ","End":"05:01.085","Text":"minus 1 times the internal derivative right here,"},{"Start":"05:01.085 ","End":"05:08.430","Text":"and that\u0027s times e^t times p. Now,"},{"Start":"05:08.430 ","End":"05:12.040","Text":"let\u0027s set t to be equal to 0."},{"Start":"05:12.040 ","End":"05:22.470","Text":"That means that we have here n times p plus 1 minus p. That will be equal"},{"Start":"05:22.470 ","End":"05:28.710","Text":"to p plus 1 minus p^n"},{"Start":"05:28.710 ","End":"05:35.525","Text":"minus 1 times e^0 times p. Now,"},{"Start":"05:35.525 ","End":"05:38.485","Text":"that equals to the following,"},{"Start":"05:38.485 ","End":"05:41.895","Text":"p minus p that cancels each other out."},{"Start":"05:41.895 ","End":"05:44.790","Text":"1^n minus 1, that\u0027s 1,"},{"Start":"05:44.790 ","End":"05:48.225","Text":"so that will be n times,"},{"Start":"05:48.225 ","End":"05:52.845","Text":"e^0 is 1 times p, so np."},{"Start":"05:52.845 ","End":"05:57.440","Text":"Here we go. Now, if we recall from"},{"Start":"05:57.440 ","End":"06:01.805","Text":"our lessons on binomial distribution the expectation of x,"},{"Start":"06:01.805 ","End":"06:05.370","Text":"that\u0027s n times p. There it is."},{"Start":"06:05.930 ","End":"06:09.650","Text":"Let\u0027s take a look at the second moment here."},{"Start":"06:09.650 ","End":"06:13.570","Text":"We\u0027re looking at the expectation of x squared,"},{"Start":"06:13.570 ","End":"06:16.580","Text":"and that\u0027s our second moment. How do we get that?"},{"Start":"06:16.580 ","End":"06:20.675","Text":"Well, we take the second derivative of the moment generating function."},{"Start":"06:20.675 ","End":"06:23.495","Text":"Now, if we look here,"},{"Start":"06:23.495 ","End":"06:26.600","Text":"this is a first derivative of the moment generating functions."},{"Start":"06:26.600 ","End":"06:30.950","Text":"All we need to do is just take another derivative of this expression right here."},{"Start":"06:30.950 ","End":"06:32.990","Text":"But before we can do that,"},{"Start":"06:32.990 ","End":"06:36.380","Text":"let\u0027s just remind ourselves if we have 2 functions,"},{"Start":"06:36.380 ","End":"06:43.920","Text":"let\u0027s say a and b and we want to calculate the derivative of a times b,"},{"Start":"06:43.920 ","End":"06:51.335","Text":"that equals to the derivative of a times b plus a times the derivative of b."},{"Start":"06:51.335 ","End":"06:54.515","Text":"Now, this is what we have here."},{"Start":"06:54.515 ","End":"06:56.450","Text":"This is our first function right here,"},{"Start":"06:56.450 ","End":"06:58.025","Text":"and this is our second function."},{"Start":"06:58.025 ","End":"07:01.835","Text":"That means that a equals to n times"},{"Start":"07:01.835 ","End":"07:09.360","Text":"e^t times p plus 1 minus p^n minus 1 and b here,"},{"Start":"07:09.360 ","End":"07:15.845","Text":"that equals to e^t times p. Having understood that,"},{"Start":"07:15.845 ","End":"07:20.135","Text":"let\u0027s go ahead and take the derivative of this guy right here."},{"Start":"07:20.135 ","End":"07:23.180","Text":"That equals to the derivative of a,"},{"Start":"07:23.180 ","End":"07:29.669","Text":"so that\u0027s n times n minus 1 times"},{"Start":"07:29.669 ","End":"07:36.570","Text":"e^t times p plus 1 minus p^n minus 2"},{"Start":"07:36.570 ","End":"07:43.010","Text":"times e^t times p. That\u0027s an internal derivative right"},{"Start":"07:43.010 ","End":"07:50.285","Text":"here times B times e^t times p plus,"},{"Start":"07:50.285 ","End":"07:52.990","Text":"we\u0027ve done this now we have to do this."},{"Start":"07:52.990 ","End":"07:55.595","Text":"Now, plus a,"},{"Start":"07:55.595 ","End":"07:57.380","Text":"where a is this guy right here,"},{"Start":"07:57.380 ","End":"08:05.990","Text":"so that\u0027s n times e^t times p plus 1 minus p^n minus 1,"},{"Start":"08:05.990 ","End":"08:11.960","Text":"that\u0027s a, times the derivative of b."},{"Start":"08:11.960 ","End":"08:15.840","Text":"That will be e^tp."},{"Start":"08:15.840 ","End":"08:22.500","Text":"Now, all this has to be when we set t equaling 0."},{"Start":"08:22.500 ","End":"08:27.070","Text":"Let\u0027s just plug in 0 for t and see what we get."},{"Start":"08:28.010 ","End":"08:31.045","Text":"This then equals to the following."},{"Start":"08:31.045 ","End":"08:34.180","Text":"That will be n times n minus 1."},{"Start":"08:34.180 ","End":"08:38.880","Text":"Now e^t, where t equals 0 every time I see that, that will be 1."},{"Start":"08:38.880 ","End":"08:43.120","Text":"That will be 1 times p plus 1 minus p,"},{"Start":"08:43.120 ","End":"08:46.850","Text":"that equals to 1^n minus 2."},{"Start":"08:46.850 ","End":"08:54.850","Text":"That will be 1 times p times 1 minus p,"},{"Start":"08:54.850 ","End":"09:00.510","Text":"that means that we have p squared plus."},{"Start":"09:00.510 ","End":"09:02.820","Text":"Now this expression right here,"},{"Start":"09:02.820 ","End":"09:07.960","Text":"that will be n. Now e^0,"},{"Start":"09:07.960 ","End":"09:13.080","Text":"that\u0027s 1, so that\u0027s 1 times p plus 1 minus p^n minus 1."},{"Start":"09:13.080 ","End":"09:14.520","Text":"That\u0027s 1."},{"Start":"09:14.520 ","End":"09:19.385","Text":"That will be n times 1 times p. Well that\u0027s np."},{"Start":"09:19.385 ","End":"09:21.865","Text":"Now, let\u0027s just clean that up."},{"Start":"09:21.865 ","End":"09:23.740","Text":"That will be equal to"},{"Start":"09:23.740 ","End":"09:34.120","Text":"np times n minus 1 times p plus 1."},{"Start":"09:34.120 ","End":"09:41.050","Text":"This then is our second moment for x."}],"ID":13161},{"Watched":false,"Name":"Exercise 3","Duration":"10m 12s","ChapterTopicVideoID":12692,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this question, we\u0027re asked to find the moment generating function of"},{"Start":"00:03.030 ","End":"00:06.660","Text":"the geometric distribution where we have a random variable, X,"},{"Start":"00:06.660 ","End":"00:11.280","Text":"and it\u0027s distributed with a geometric distribution with parameter p. We\u0027re also asked to"},{"Start":"00:11.280 ","End":"00:12.810","Text":"calculate the expectation of"},{"Start":"00:12.810 ","End":"00:16.845","Text":"the probability distribution from the moment generating function."},{"Start":"00:16.845 ","End":"00:22.140","Text":"Let\u0027s get started. We have our moment generating function,"},{"Start":"00:22.140 ","End":"00:30.920","Text":"that\u0027s m sub x of t. That\u0027s defined as the expectation of e to the power of t times x."},{"Start":"00:30.920 ","End":"00:35.445","Text":"Now, that equals to what?"},{"Start":"00:35.445 ","End":"00:39.960","Text":"Well, let\u0027s first recall that the probability of x"},{"Start":"00:39.960 ","End":"00:44.510","Text":"being equal to some k in a geometric distribution, well,"},{"Start":"00:44.510 ","End":"00:51.520","Text":"that equals to 1 minus p to the power of k minus 1 times p,"},{"Start":"00:51.520 ","End":"00:56.795","Text":"for all of k from 1 until infinity."},{"Start":"00:56.795 ","End":"01:01.925","Text":"Well, because we\u0027re dealing with a discrete random variable,"},{"Start":"01:01.925 ","End":"01:07.610","Text":"then this step becomes the sum of k being equaling"},{"Start":"01:07.610 ","End":"01:14.470","Text":"1 to infinity of e to the power of t times k,"},{"Start":"01:14.470 ","End":"01:16.360","Text":"that\u0027s the value that\u0027s running here,"},{"Start":"01:16.360 ","End":"01:20.995","Text":"times the probability of x being equal to k. Now,"},{"Start":"01:20.995 ","End":"01:24.265","Text":"let\u0027s just substitute this expression"},{"Start":"01:24.265 ","End":"01:27.925","Text":"instead of the probability of x being equal to k, and now,"},{"Start":"01:27.925 ","End":"01:36.310","Text":"that is equal to the sum of k being equal to 1 till infinity of e to the power of t times"},{"Start":"01:36.310 ","End":"01:46.300","Text":"k times 1 minus p to the power of k minus 1 times p. What do we have here?"},{"Start":"01:46.300 ","End":"01:48.190","Text":"Let\u0027s take a look at this expression right here."},{"Start":"01:48.190 ","End":"01:50.745","Text":"Well, first of all, we can take p out of the sum."},{"Start":"01:50.745 ","End":"01:55.970","Text":"Secondly, we have here e to the power of tk,"},{"Start":"01:55.970 ","End":"01:59.990","Text":"and here we have an expression to the power of k minus 1."},{"Start":"01:59.990 ","End":"02:06.020","Text":"What I want to do, is I want to make this an expression to the power of k minus 1."},{"Start":"02:06.020 ","End":"02:07.490","Text":"Now, how do I do that?"},{"Start":"02:07.490 ","End":"02:10.460","Text":"Well, I\u0027ll use 1 of"},{"Start":"02:10.460 ","End":"02:16.655","Text":"the algebraic manipulation where I multiply and I divide by the same value."},{"Start":"02:16.655 ","End":"02:20.660","Text":"Basically, I\u0027m multiplying this expression by 1."},{"Start":"02:20.660 ","End":"02:22.100","Text":"Let\u0027s see how this work. Well, first of all,"},{"Start":"02:22.100 ","End":"02:23.930","Text":"we said we\u0027ll take p out."},{"Start":"02:23.930 ","End":"02:30.290","Text":"The sum of k being equal to 1 till infinity,"},{"Start":"02:30.290 ","End":"02:34.745","Text":"e to the power of t times k, but now,"},{"Start":"02:34.745 ","End":"02:41.750","Text":"I\u0027ll multiply by e to the power t and divide by e to the power of t."},{"Start":"02:41.750 ","End":"02:50.550","Text":"Then I have 1 minus p to the power of k minus 1."},{"Start":"02:51.640 ","End":"02:54.595","Text":"Let\u0027s see what that equals."},{"Start":"02:54.595 ","End":"02:58.360","Text":"Well, that equals to p. Now,"},{"Start":"02:58.360 ","End":"03:03.540","Text":"let\u0027s take e to the power of t outside of the sum."},{"Start":"03:03.540 ","End":"03:06.450","Text":"That\u0027s a constant, e to the power of t,"},{"Start":"03:06.450 ","End":"03:09.835","Text":"sum, now what do I have here?"},{"Start":"03:09.835 ","End":"03:15.865","Text":"Well, I have e to the power of t times"},{"Start":"03:15.865 ","End":"03:22.715","Text":"k minus 1 times 1 minus p to the power of k minus 1."},{"Start":"03:22.715 ","End":"03:25.755","Text":"We follow that. Now,"},{"Start":"03:25.755 ","End":"03:30.315","Text":"that then equals to p times e to the power of t,"},{"Start":"03:30.315 ","End":"03:38.370","Text":"the sum of e to the power of t,"},{"Start":"03:38.370 ","End":"03:39.885","Text":"let\u0027s just fix that up,"},{"Start":"03:39.885 ","End":"03:45.800","Text":"times 1 minus p and all this to the power of k minus 1."},{"Start":"03:45.800 ","End":"03:50.820","Text":"Again, where k goes from 1 till infinity."},{"Start":"03:51.310 ","End":"03:58.320","Text":"We\u0027ve gotten this far and let\u0027s see how we can continue from this point on."},{"Start":"03:59.780 ","End":"04:05.885","Text":"Let\u0027s just look at this part of the expression from the sum sign onwards."},{"Start":"04:05.885 ","End":"04:09.920","Text":"Now, here we have a geometric series."},{"Start":"04:09.920 ","End":"04:13.740","Text":"Now, what\u0027s the sum of a geometric series?"},{"Start":"04:14.150 ","End":"04:21.690","Text":"That\u0027s the sum that equals to a_1 divided by 1 minus q."},{"Start":"04:21.690 ","End":"04:26.025","Text":"Now, what\u0027s a_1?"},{"Start":"04:26.025 ","End":"04:28.340","Text":"a_1 is the first member of the series."},{"Start":"04:28.340 ","End":"04:30.335","Text":"That\u0027s where k equals to 1."},{"Start":"04:30.335 ","End":"04:34.100","Text":"We have an expression to the power of k equaling 1,"},{"Start":"04:34.100 ","End":"04:35.360","Text":"so it\u0027s 1 minus 1,"},{"Start":"04:35.360 ","End":"04:37.000","Text":"to the power of 0."},{"Start":"04:37.000 ","End":"04:42.605","Text":"That means that the first member of the series equals to 1."},{"Start":"04:42.605 ","End":"04:43.970","Text":"Then what about q? Well,"},{"Start":"04:43.970 ","End":"04:45.545","Text":"q is our multiplier."},{"Start":"04:45.545 ","End":"04:50.240","Text":"That means that it\u0027s all this expression right here within the square brackets."},{"Start":"04:50.240 ","End":"04:52.310","Text":"q here, well,"},{"Start":"04:52.310 ","End":"05:00.195","Text":"that equals to e to the power of t times 1 minus p. Now,"},{"Start":"05:00.195 ","End":"05:07.965","Text":"having said that, I need for q to be less than 1 in order for the series to converge."},{"Start":"05:07.965 ","End":"05:15.980","Text":"That means that the condition here is that e to the power of t times 1 minus p,"},{"Start":"05:15.980 ","End":"05:18.920","Text":"well that has to be less than 1."},{"Start":"05:18.920 ","End":"05:23.195","Text":"Let\u0027s just do a little bit of manipulation here."},{"Start":"05:23.195 ","End":"05:25.370","Text":"That\u0027ll be e to the power of t,"},{"Start":"05:25.370 ","End":"05:28.685","Text":"that\u0027s less than 1 divided by 1 minus p,"},{"Start":"05:28.685 ","End":"05:33.230","Text":"divided both sides by 1 minus p. Take the lan of both sides,"},{"Start":"05:33.230 ","End":"05:37.065","Text":"so the lan of e to the power of t, well that\u0027s t,"},{"Start":"05:37.065 ","End":"05:40.965","Text":"and that has to be less than the lan of 1 divided by"},{"Start":"05:40.965 ","End":"05:50.210","Text":"1 minus p. This is the condition that we need to have in order to continue this thing,"},{"Start":"05:50.210 ","End":"05:54.730","Text":"in order to make this a geometric series."},{"Start":"05:54.730 ","End":"05:59.420","Text":"Having understood that, then this expression right"},{"Start":"05:59.420 ","End":"06:04.010","Text":"here becomes p times e to the power of t,"},{"Start":"06:04.010 ","End":"06:05.300","Text":"that this part of it,"},{"Start":"06:05.300 ","End":"06:08.090","Text":"and instead of this geometric series,"},{"Start":"06:08.090 ","End":"06:12.120","Text":"so I can put the sum of the geometric series here,"},{"Start":"06:12.120 ","End":"06:15.455","Text":"that will be times a_1 is 1 as we said,"},{"Start":"06:15.455 ","End":"06:18.110","Text":"and that\u0027ll be 1 minus q,"},{"Start":"06:18.110 ","End":"06:24.285","Text":"where q is e to the power of t times 1 minus p. Then"},{"Start":"06:24.285 ","End":"06:32.590","Text":"this becomes the moment generating function of a geometric distribution."},{"Start":"06:32.590 ","End":"06:37.850","Text":"Now, we have to find the first moment of the geometric distribution."},{"Start":"06:37.850 ","End":"06:40.460","Text":"Well, this is a moment generating function right here."},{"Start":"06:40.460 ","End":"06:42.410","Text":"Now, how do we find the first moment?"},{"Start":"06:42.410 ","End":"06:44.980","Text":"Well, first moment is E,"},{"Start":"06:44.980 ","End":"06:46.815","Text":"the expectation of x,"},{"Start":"06:46.815 ","End":"06:48.470","Text":"and how do we get that?"},{"Start":"06:48.470 ","End":"06:51.560","Text":"Well, we take the moment generating function,"},{"Start":"06:51.560 ","End":"06:54.530","Text":"and we take the derivative,"},{"Start":"06:54.530 ","End":"06:57.650","Text":"the first derivative of the moment generating function,"},{"Start":"06:57.650 ","End":"07:01.220","Text":"and we have to set t to equal to 0."},{"Start":"07:01.220 ","End":"07:06.320","Text":"Now, let\u0027s just recall that if we have 2 functions, a and b,"},{"Start":"07:06.320 ","End":"07:09.335","Text":"and we want to take the derivative of a divided by b,"},{"Start":"07:09.335 ","End":"07:13.100","Text":"well that equals to the derivative of a times b"},{"Start":"07:13.100 ","End":"07:16.790","Text":"minus a times the derivative of b divided by b squared."},{"Start":"07:16.790 ","End":"07:18.440","Text":"In our case,"},{"Start":"07:18.440 ","End":"07:23.690","Text":"a here is p times e to the power of t, and b, well,"},{"Start":"07:23.690 ","End":"07:27.470","Text":"that equals to 1 minus e to the power of t times"},{"Start":"07:27.470 ","End":"07:31.730","Text":"1 minus p. This is just a quick reminder."},{"Start":"07:31.730 ","End":"07:34.790","Text":"Now, let\u0027s get back to our question here."},{"Start":"07:34.790 ","End":"07:39.480","Text":"We\u0027re taking the first derivative of this function right here."},{"Start":"07:39.590 ","End":"07:44.660","Text":"That means that we\u0027re looking at the derivative of a times b."},{"Start":"07:44.660 ","End":"07:51.830","Text":"Well, the derivative of a is p times e to the power of t times b."},{"Start":"07:51.830 ","End":"08:00.065","Text":"Well, that\u0027s 1 minus e to the power of t times 1 minus p. Minus a,"},{"Start":"08:00.065 ","End":"08:05.195","Text":"that\u0027s p times e to the power of t times the derivative of b."},{"Start":"08:05.195 ","End":"08:10.505","Text":"Well, derivative of 1 is 0 and the derivative of this guy right here,"},{"Start":"08:10.505 ","End":"08:12.500","Text":"well, the minus cancels this out,"},{"Start":"08:12.500 ","End":"08:18.200","Text":"that becomes a plus and that means that we have here"},{"Start":"08:18.200 ","End":"08:26.685","Text":"1 minus p times e to the power of t. All this has to be over,"},{"Start":"08:26.685 ","End":"08:33.830","Text":"1 minus e to the power of t times 1 minus p squared."},{"Start":"08:33.830 ","End":"08:36.240","Text":"I hope you\u0027re still following me."},{"Start":"08:36.240 ","End":"08:44.125","Text":"Now, this all has to be set to 0. t here has to be set to 0."},{"Start":"08:44.125 ","End":"08:46.415","Text":"To make life easier,"},{"Start":"08:46.415 ","End":"08:51.620","Text":"whenever we see e to the power of t or e to the power of 0, that becomes 1."},{"Start":"08:51.620 ","End":"08:54.395","Text":"Let\u0027s just simplify this."},{"Start":"08:54.395 ","End":"09:00.110","Text":"That means that we have p times 1 minus,"},{"Start":"09:00.110 ","End":"09:09.590","Text":"now 1 minus p plus p times 1 minus p,"},{"Start":"09:09.590 ","End":"09:14.715","Text":"divided by 1 minus,"},{"Start":"09:14.715 ","End":"09:17.925","Text":"now 1 minus p squared,"},{"Start":"09:17.925 ","End":"09:20.090","Text":"all this becomes squared."},{"Start":"09:20.090 ","End":"09:23.250","Text":"Now, let\u0027s just simplify that."},{"Start":"09:23.440 ","End":"09:29.420","Text":"This now becomes p squared plus p"},{"Start":"09:29.420 ","End":"09:35.240","Text":"minus p squared divided by p squared,"},{"Start":"09:35.240 ","End":"09:37.610","Text":"so these cancels each other out."},{"Start":"09:37.610 ","End":"09:40.715","Text":"We have p divided by p squared."},{"Start":"09:40.715 ","End":"09:44.135","Text":"That means that we have 1 over p. Now,"},{"Start":"09:44.135 ","End":"09:50.100","Text":"this then is the first moment of the geometric distribution."},{"Start":"09:50.100 ","End":"09:51.885","Text":"Now, if we recall,"},{"Start":"09:51.885 ","End":"09:55.940","Text":"whenever we have a random variable x with parameter p"},{"Start":"09:55.940 ","End":"10:00.190","Text":"and x is distributed with the geometric distribution,"},{"Start":"10:00.190 ","End":"10:05.409","Text":"then the expectation of x is 1 divided by p. Here,"},{"Start":"10:05.409 ","End":"10:08.150","Text":"we\u0027ve calculated the first moment or the expectation of"},{"Start":"10:08.150 ","End":"10:12.750","Text":"x using the moment generating function."}],"ID":13162},{"Watched":false,"Name":"Exercise 4","Duration":"7m 53s","ChapterTopicVideoID":12693,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this question, we\u0027re asked to find the moment"},{"Start":"00:01.980 ","End":"00:04.635","Text":"generating function of the Poisson distribution,"},{"Start":"00:04.635 ","End":"00:08.160","Text":"where X has a Poisson distribution with parameter Lambda."},{"Start":"00:08.160 ","End":"00:10.539","Text":"Next, we\u0027re asked to find the first,"},{"Start":"00:10.539 ","End":"00:14.260","Text":"and second moments of the probability distribution."},{"Start":"00:14.280 ","End":"00:20.670","Text":"We\u0027re asked for the moment generating function of x."},{"Start":"00:20.670 ","End":"00:26.250","Text":"That\u0027s defined as the expectation of e to the power of t times x."},{"Start":"00:26.250 ","End":"00:31.950","Text":"That equals to the sum of all values of k. Why sum?"},{"Start":"00:31.950 ","End":"00:34.870","Text":"Because we\u0027re dealing with a discrete random variable."},{"Start":"00:34.870 ","End":"00:39.560","Text":"The sum of e to the power of t times k times"},{"Start":"00:39.560 ","End":"00:46.340","Text":"the probability of x being equal to k. For a Poisson distribution,"},{"Start":"00:46.340 ","End":"00:50.120","Text":"the probability of x being equal to k,"},{"Start":"00:50.120 ","End":"00:53.330","Text":"that equals to e to the power of"},{"Start":"00:53.330 ","End":"01:01.175","Text":"minus Lambda times Lambda to the power of k divided by k factorial."},{"Start":"01:01.175 ","End":"01:08.790","Text":"That\u0027s for k equaling 0 until infinity."},{"Start":"01:09.070 ","End":"01:14.465","Text":"Let\u0027s just plug in this expression instead of this."},{"Start":"01:14.465 ","End":"01:18.560","Text":"That makes this sum over all k,"},{"Start":"01:18.560 ","End":"01:26.120","Text":"from 0 to plus infinity of e to the power of t times k. Let\u0027s substitute."},{"Start":"01:26.120 ","End":"01:30.380","Text":"Instead of this, we\u0027ll substitute this expression times e to the power of"},{"Start":"01:30.380 ","End":"01:36.260","Text":"minus Lambda times Lambda to the power of k divided by k factorial."},{"Start":"01:36.260 ","End":"01:41.520","Text":"Let\u0027s just make this a little bit easier on the eye."},{"Start":"01:41.520 ","End":"01:46.955","Text":"We\u0027ll take e to the power of minus Lambda l because it\u0027s not dependent on k,"},{"Start":"01:46.955 ","End":"01:48.800","Text":"e to the power of minus Lambda,"},{"Start":"01:48.800 ","End":"01:52.260","Text":"sum k equaling 0."},{"Start":"01:52.260 ","End":"01:57.650","Text":"From 0 to infinity of e"},{"Start":"01:57.650 ","End":"02:03.755","Text":"to the power of t times Lambda all of that,"},{"Start":"02:03.755 ","End":"02:09.870","Text":"to the power of k divided by k factorial."},{"Start":"02:11.060 ","End":"02:18.410","Text":"If we take a look at the sum of this expression right here,"},{"Start":"02:18.410 ","End":"02:22.865","Text":"not including e to the power of minus Lambda just from the sum onwards,"},{"Start":"02:22.865 ","End":"02:26.165","Text":"this looks like the Taylor series."},{"Start":"02:26.165 ","End":"02:29.134","Text":"Let\u0027s just remind ourselves."},{"Start":"02:29.134 ","End":"02:34.235","Text":"Taylor series that equals to the following."},{"Start":"02:34.235 ","End":"02:38.480","Text":"That\u0027s the sum of k being equal to 0 to"},{"Start":"02:38.480 ","End":"02:45.005","Text":"infinity of a to the power of k divided by k factorial,"},{"Start":"02:45.005 ","End":"02:48.635","Text":"that equals to e to the power of a."},{"Start":"02:48.635 ","End":"02:56.090","Text":"In our case, a equals to e to the power of t times Lambda."},{"Start":"02:56.090 ","End":"02:58.805","Text":"This is just a reminder here."},{"Start":"02:58.805 ","End":"03:01.460","Text":"Instead of this guy right here,"},{"Start":"03:01.460 ","End":"03:03.815","Text":"we can just substitute that for this."},{"Start":"03:03.815 ","End":"03:06.535","Text":"That means that this,"},{"Start":"03:06.535 ","End":"03:11.010","Text":"equals to e to the power of"},{"Start":"03:11.010 ","End":"03:17.550","Text":"minus Lambda times e to the power of a."},{"Start":"03:17.550 ","End":"03:23.400","Text":"A is this. That\u0027s e to the power of t times Lambda."},{"Start":"03:23.400 ","End":"03:29.945","Text":"Let\u0027s just make this a little bit better."},{"Start":"03:29.945 ","End":"03:34.400","Text":"That equals to e to the power of Lambda,"},{"Start":"03:34.400 ","End":"03:39.265","Text":"times e to the power of t minus 1."},{"Start":"03:39.265 ","End":"03:47.070","Text":"This then is the moment generating function of the Poisson distribution."},{"Start":"03:47.380 ","End":"03:52.220","Text":"We\u0027re asked to find the first and second moments."},{"Start":"03:52.220 ","End":"03:54.540","Text":"Let\u0027s do that."},{"Start":"03:54.770 ","End":"03:59.300","Text":"This is a moment generating function of the Poisson distribution."},{"Start":"03:59.300 ","End":"04:01.580","Text":"We\u0027re asked to find the first moment,"},{"Start":"04:01.580 ","End":"04:03.890","Text":"where the first moment is the expectation of x."},{"Start":"04:03.890 ","End":"04:05.510","Text":"How do we do that?"},{"Start":"04:05.510 ","End":"04:10.865","Text":"We take our moment generating function and we take the first derivative of that,"},{"Start":"04:10.865 ","End":"04:13.775","Text":"then set t to equal 0."},{"Start":"04:13.775 ","End":"04:16.235","Text":"That\u0027s how we get the first moment."},{"Start":"04:16.235 ","End":"04:18.350","Text":"Let\u0027s take this function,"},{"Start":"04:18.350 ","End":"04:19.940","Text":"take the first derivative,"},{"Start":"04:19.940 ","End":"04:26.030","Text":"and that equals to e to the power of Lambda times e to the power of t"},{"Start":"04:26.030 ","End":"04:36.940","Text":"minus 1 times Lambda e to the power of t. We have to set t to equal to 0."},{"Start":"04:36.940 ","End":"04:38.735","Text":"That equals to what?"},{"Start":"04:38.735 ","End":"04:43.310","Text":"E to the power of 0 minus 1,"},{"Start":"04:43.310 ","End":"04:45.230","Text":"that\u0027s 1 minus 1, that\u0027s 0."},{"Start":"04:45.230 ","End":"04:47.210","Text":"E to the power of Lambda times 0,"},{"Start":"04:47.210 ","End":"04:49.670","Text":"that\u0027s e to the power of 0,"},{"Start":"04:49.670 ","End":"04:54.800","Text":"times Lambda e to the power of 0,"},{"Start":"04:54.800 ","End":"04:59.080","Text":"and that equals to Lambda."},{"Start":"04:59.080 ","End":"05:05.540","Text":"Here, that\u0027s the first moment of the Poisson distribution."},{"Start":"05:05.540 ","End":"05:09.815","Text":"If we recall, if we have a Poisson distribution with parameter Lambda,"},{"Start":"05:09.815 ","End":"05:15.185","Text":"then the expectation of that distribution is Lambda."},{"Start":"05:15.185 ","End":"05:17.900","Text":"What about the second moment?"},{"Start":"05:17.900 ","End":"05:20.990","Text":"That\u0027s the expectation of x squared."},{"Start":"05:20.990 ","End":"05:25.370","Text":"Again, we have to take our moment generating function,"},{"Start":"05:25.370 ","End":"05:28.505","Text":"and we have to take the second derivative."},{"Start":"05:28.505 ","End":"05:31.095","Text":"Here\u0027s the first derivative."},{"Start":"05:31.095 ","End":"05:34.010","Text":"All we need to do is to derive that."},{"Start":"05:34.010 ","End":"05:35.900","Text":"Let\u0027s do that."},{"Start":"05:35.900 ","End":"05:42.180","Text":"That will be equal to Lambda times e to the power of"},{"Start":"05:42.180 ","End":"05:50.510","Text":"Lambda times e to the power of t minus 1 times Lambda e to the power of t,"},{"Start":"05:50.510 ","End":"05:55.385","Text":"plus e to the power of t times"},{"Start":"05:55.385 ","End":"06:01.320","Text":"e to the power of Lambda times e to the power of t minus 1."},{"Start":"06:01.670 ","End":"06:07.325","Text":"Here, again, we have to set t to equal to 0."},{"Start":"06:07.325 ","End":"06:09.140","Text":"There is no way around it."},{"Start":"06:09.140 ","End":"06:12.485","Text":"We need to know our derivatives."},{"Start":"06:12.485 ","End":"06:18.465","Text":"That equals to Lambda, t equals 0."},{"Start":"06:18.465 ","End":"06:21.650","Text":"That\u0027ll be e to the power of 0 minus 1,"},{"Start":"06:21.650 ","End":"06:23.315","Text":"that\u0027s 1 minus 1, that\u0027s 0."},{"Start":"06:23.315 ","End":"06:25.340","Text":"That\u0027ll be e to the power of 0,"},{"Start":"06:25.340 ","End":"06:29.810","Text":"times Lambda e to the power of 0,"},{"Start":"06:29.810 ","End":"06:39.120","Text":"plus e to the power 0 times e to the power Lambda times 0."},{"Start":"06:39.380 ","End":"06:47.895","Text":"That equals to Lambda times Lambda plus 1."},{"Start":"06:47.895 ","End":"06:57.100","Text":"This is Lambda and this is 1. Here we go."},{"Start":"06:57.100 ","End":"06:59.455","Text":"That\u0027s our second moment."},{"Start":"06:59.455 ","End":"07:04.120","Text":"Let\u0027s calculate the variance."},{"Start":"07:04.120 ","End":"07:09.830","Text":"The variance of x,"},{"Start":"07:09.830 ","End":"07:19.060","Text":"that equals to the expectation of x squared minus the expectation squared of x."},{"Start":"07:19.060 ","End":"07:21.100","Text":"This is our second moment,"},{"Start":"07:21.100 ","End":"07:22.510","Text":"and this is a first moment."},{"Start":"07:22.510 ","End":"07:29.855","Text":"That equals to Lambda times Lambda plus 1 minus Lambda squared."},{"Start":"07:29.855 ","End":"07:38.465","Text":"That equals to Lambda squared plus Lambda minus Lambda squared and that equals to Lambda."},{"Start":"07:38.465 ","End":"07:41.810","Text":"Again, this then is our variance."},{"Start":"07:41.810 ","End":"07:49.775","Text":"If we recall, in a Poisson distribution the expectation and the variance are the same,"},{"Start":"07:49.775 ","End":"07:53.100","Text":"and they both equal Lambda."}],"ID":13163},{"Watched":false,"Name":"Exercise 5","Duration":"6m 35s","ChapterTopicVideoID":12694,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"Let X be a random variable with the following density function:"},{"Start":"00:03.510 ","End":"00:07.200","Text":"f of x equals A times e to the power of minus x,"},{"Start":"00:07.200 ","End":"00:10.635","Text":"where x is between 0 and 7, and 0 otherwise."},{"Start":"00:10.635 ","End":"00:15.480","Text":"We\u0027re asked to find the value of a. f of x,"},{"Start":"00:15.480 ","End":"00:20.535","Text":"well that equals to A times e to the power of minus x,"},{"Start":"00:20.535 ","End":"00:28.155","Text":"where x is between 0 and 7, and 0 otherwise."},{"Start":"00:28.155 ","End":"00:35.100","Text":"Now we know that the integral between minus infinity and infinity of a function."},{"Start":"00:35.100 ","End":"00:39.450","Text":"If we want this function to be a density function,"},{"Start":"00:39.450 ","End":"00:42.670","Text":"that integral has to be equal to 1."},{"Start":"00:42.740 ","End":"00:46.940","Text":"What we\u0027re going to do is we\u0027re going to take the integral of"},{"Start":"00:46.940 ","End":"00:52.370","Text":"this expression equated to 1 in order to calculate our a."},{"Start":"00:52.370 ","End":"00:53.920","Text":"Let\u0027s do that."},{"Start":"00:53.920 ","End":"00:57.290","Text":"That\u0027s the integral from 0 to 7."},{"Start":"00:57.290 ","End":"01:03.845","Text":"That\u0027s where a function is defined of A times e to the power of minus x_dx."},{"Start":"01:03.845 ","End":"01:05.765","Text":"That equals to,"},{"Start":"01:05.765 ","End":"01:09.425","Text":"we\u0027ll take A out of the integral because it\u0027s a constant,"},{"Start":"01:09.425 ","End":"01:11.465","Text":"0 to 7,"},{"Start":"01:11.465 ","End":"01:14.995","Text":"of e to the power of minus x_dx."},{"Start":"01:14.995 ","End":"01:18.195","Text":"That equals to a times,"},{"Start":"01:18.195 ","End":"01:20.510","Text":"what\u0027s the integral of e to the power of minus x?"},{"Start":"01:20.510 ","End":"01:25.910","Text":"That\u0027s e to the power of minus x divided by minus 1."},{"Start":"01:25.910 ","End":"01:28.850","Text":"That equals to what?"},{"Start":"01:28.850 ","End":"01:31.780","Text":"We\u0027re going to have to do that from 0 to 7,"},{"Start":"01:31.780 ","End":"01:36.290","Text":"we\u0027re going to have to calculate this thing in this range right here."},{"Start":"01:36.290 ","End":"01:39.130","Text":"That equals to minus a."},{"Start":"01:39.130 ","End":"01:43.250","Text":"Now, e to the power of the substitutes 7,"},{"Start":"01:43.250 ","End":"01:45.500","Text":"for x minus 7,"},{"Start":"01:45.500 ","End":"01:50.065","Text":"minus e to the power of minus 0."},{"Start":"01:50.065 ","End":"01:53.105","Text":"That equals to minus a,"},{"Start":"01:53.105 ","End":"01:56.990","Text":"e to the power of minus 7, minus 1."},{"Start":"01:56.990 ","End":"02:01.080","Text":"Let\u0027s just put the minus back in here,"},{"Start":"02:01.080 ","End":"02:08.055","Text":"and that equals to a times 1 minus e to the power of minus 7."},{"Start":"02:08.055 ","End":"02:15.170","Text":"Now we know that we want to equate that to 1 in order to calculate our a."},{"Start":"02:15.170 ","End":"02:24.190","Text":"That means that a now is equal to 1 divided by 1 minus e to the power of minus 7,"},{"Start":"02:24.190 ","End":"02:26.930","Text":"Or we can do this,"},{"Start":"02:26.930 ","End":"02:32.800","Text":"we can multiply the numerator and the denominator by e to the power of 7,"},{"Start":"02:32.800 ","End":"02:39.490","Text":"so we\u0027ll get e to the power of 7 divided by e to the power of 7 minus 1."},{"Start":"02:39.490 ","End":"02:43.610","Text":"Here then is our value for a."},{"Start":"02:43.610 ","End":"02:47.960","Text":"Now we can use this expression or this expression really doesn\u0027t matter, it\u0027s the same."},{"Start":"02:47.960 ","End":"02:51.720","Text":"Again, we found the value of a."},{"Start":"02:51.720 ","End":"02:55.900","Text":"In this section, we\u0027re asked to find the moment generating function of X."},{"Start":"02:55.900 ","End":"02:59.440","Text":"Well, we found the value of a in the previous sections."},{"Start":"02:59.440 ","End":"03:02.920","Text":"So the density function equals e to the power"},{"Start":"03:02.920 ","End":"03:06.610","Text":"of 7 divided by e to the power of 7 minus 1."},{"Start":"03:06.610 ","End":"03:10.210","Text":"That was our a times e to the power of minus x,"},{"Start":"03:10.210 ","End":"03:17.325","Text":"where x is between 0 and 7, and 0 otherwise."},{"Start":"03:17.325 ","End":"03:19.090","Text":"That\u0027s a density function."},{"Start":"03:19.090 ","End":"03:21.895","Text":"We\u0027re asked to find the moment-generating function."},{"Start":"03:21.895 ","End":"03:25.670","Text":"The moment generating function of x that equals to"},{"Start":"03:25.670 ","End":"03:30.370","Text":"the expectation of E to the power of t times x."},{"Start":"03:30.370 ","End":"03:34.025","Text":"That means that we\u0027re dealing with the integral, okay?"},{"Start":"03:34.025 ","End":"03:38.105","Text":"Now, integral because we\u0027re it have a continuous random variable."},{"Start":"03:38.105 ","End":"03:44.450","Text":"The range is from 0 to 7 of e to the power"},{"Start":"03:44.450 ","End":"03:50.675","Text":"of t times x times the density function dx."},{"Start":"03:50.675 ","End":"03:53.150","Text":"Now, instead of this,"},{"Start":"03:53.150 ","End":"03:55.805","Text":"we\u0027ll put in this expression right here."},{"Start":"03:55.805 ","End":"04:01.850","Text":"That means that we\u0027re dealing with the integral from 0 to 7 of e to the power of"},{"Start":"04:01.850 ","End":"04:08.480","Text":"tx times e to the power of 7 divided by e to the power 7 minus 1,"},{"Start":"04:08.480 ","End":"04:12.230","Text":"times e to the power of minus x_dx."},{"Start":"04:12.230 ","End":"04:14.165","Text":"Let\u0027s simplify that."},{"Start":"04:14.165 ","End":"04:16.775","Text":"We\u0027ll take this constant out of the integral."},{"Start":"04:16.775 ","End":"04:22.190","Text":"That\u0027s e to the power of 7 divided by e to the power of 7 minus 1."},{"Start":"04:22.190 ","End":"04:26.630","Text":"The integral from 0 to 7 of e to the power of"},{"Start":"04:26.630 ","End":"04:33.290","Text":"tx times e to the power of minus x_dx."},{"Start":"04:33.290 ","End":"04:35.530","Text":"Let\u0027s simplify that."},{"Start":"04:35.530 ","End":"04:42.470","Text":"That equals e to the power of 7 divided by e to the power of 7 minus 1."},{"Start":"04:42.470 ","End":"04:48.230","Text":"The integral from 0 to 7 of e to the power of x,"},{"Start":"04:48.230 ","End":"04:53.210","Text":"we took x out, and that\u0027ll be t minus 1 dx."},{"Start":"04:53.210 ","End":"04:57.320","Text":"Now, all we have to do is to calculate this"},{"Start":"04:57.320 ","End":"05:01.580","Text":"integral in the range between 0 and 7. Let\u0027s do that."},{"Start":"05:01.580 ","End":"05:06.560","Text":"We have e to the power of 7 divided by e to the power of 7 minus 1."},{"Start":"05:06.560 ","End":"05:10.550","Text":"Now the integral of this guy is e to the power of"},{"Start":"05:10.550 ","End":"05:20.205","Text":"x times t minus 1 divided now by t minus 1,"},{"Start":"05:20.205 ","End":"05:24.320","Text":"from 0 to 7."},{"Start":"05:24.320 ","End":"05:28.495","Text":"Let\u0027s continue on here."},{"Start":"05:28.495 ","End":"05:31.240","Text":"Let\u0027s just plug in the numbers for x."},{"Start":"05:31.240 ","End":"05:39.160","Text":"We have here e to the power of 7 divided by e to the power of 7 minus 1,"},{"Start":"05:39.160 ","End":"05:45.815","Text":"that will be e to the power of 7,"},{"Start":"05:45.815 ","End":"05:48.090","Text":"t minus 1,"},{"Start":"05:48.090 ","End":"05:50.775","Text":"divided by t minus 1,"},{"Start":"05:50.775 ","End":"05:59.420","Text":"minus e to the power of 0 times t minus 1 divided by t minus 1."},{"Start":"05:59.960 ","End":"06:02.745","Text":"Let\u0027s simplify that."},{"Start":"06:02.745 ","End":"06:09.200","Text":"That equals e to the power of 7 divided by e to the power of 7 minus 1."},{"Start":"06:09.200 ","End":"06:13.175","Text":"We\u0027ll take a common denominator,"},{"Start":"06:13.175 ","End":"06:17.120","Text":"that would be times t minus 1."},{"Start":"06:17.120 ","End":"06:18.754","Text":"That\u0027s a common denominator."},{"Start":"06:18.754 ","End":"06:26.450","Text":"That\u0027ll be e to the power of 7 times t minus 1, minus 1."},{"Start":"06:26.450 ","End":"06:36.120","Text":"This then is our moment generating function for the density function."}],"ID":13164},{"Watched":false,"Name":"Exercise 6","Duration":"6m 37s","ChapterTopicVideoID":12695,"CourseChapterTopicPlaylistID":245053,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"Let X be a random variable with an expectation of 5 and a variance of 16,"},{"Start":"00:04.890 ","End":"00:09.180","Text":"and let M sub x of t be the moment generating function of X."},{"Start":"00:09.180 ","End":"00:14.000","Text":"Now, Y is a random variable with a moment generating function M sub y of t,"},{"Start":"00:14.000 ","End":"00:18.240","Text":"and we assume that M sub y of t equals to t times"},{"Start":"00:18.240 ","End":"00:23.760","Text":"M sub x of t. We\u0027re asked to calculate the expectation and variance of Y."},{"Start":"00:23.760 ","End":"00:26.505","Text":"First of all, let\u0027s write down what we know."},{"Start":"00:26.505 ","End":"00:31.785","Text":"The expectation of x that equals to 5, that\u0027s right here."},{"Start":"00:31.785 ","End":"00:36.045","Text":"The variance of x that equals to 16, that\u0027s right here,"},{"Start":"00:36.045 ","End":"00:41.600","Text":"and the connection between the moment generating functions and sub y of t,"},{"Start":"00:41.600 ","End":"00:48.575","Text":"that equals to t times M sub x of t. That\u0027s given to us right here."},{"Start":"00:48.575 ","End":"00:53.465","Text":"Now, we\u0027re asked to calculate the expectation and variance of Y."},{"Start":"00:53.465 ","End":"00:55.640","Text":"Let\u0027s start with the expectation of Y."},{"Start":"00:55.640 ","End":"00:58.834","Text":"When the expectation is the first moment,"},{"Start":"00:58.834 ","End":"01:04.865","Text":"we need to take the first derivative of this thing right here,"},{"Start":"01:04.865 ","End":"01:07.280","Text":"the moment generating function of Y,"},{"Start":"01:07.280 ","End":"01:09.050","Text":"and set t to 0."},{"Start":"01:09.050 ","End":"01:10.700","Text":"That means that we have to take"},{"Start":"01:10.700 ","End":"01:16.310","Text":"the first derivative of the moment generating function of Y."},{"Start":"01:16.310 ","End":"01:18.350","Text":"Now, before we can do that,"},{"Start":"01:18.350 ","End":"01:22.460","Text":"let\u0027s just remind ourselves of some of the derivation rules."},{"Start":"01:22.460 ","End":"01:25.370","Text":"If we have 2 functions, A and B,"},{"Start":"01:25.370 ","End":"01:29.960","Text":"and we want to take the derivative of a times b, well,"},{"Start":"01:29.960 ","End":"01:35.855","Text":"that equals to the derivative a times b plus a times the derivative of b."},{"Start":"01:35.855 ","End":"01:37.190","Text":"Now in our case,"},{"Start":"01:37.190 ","End":"01:39.140","Text":"a would be equal to t,"},{"Start":"01:39.140 ","End":"01:44.615","Text":"and b will be equal to M sub x of t. Now,"},{"Start":"01:44.615 ","End":"01:49.250","Text":"having said that, let\u0027s derive this expression right here."},{"Start":"01:49.250 ","End":"01:54.545","Text":"Well, that equals to the derivative of a, that\u0027s 1,"},{"Start":"01:54.545 ","End":"01:57.860","Text":"times M x at t"},{"Start":"01:57.860 ","End":"02:06.500","Text":"plus t times the derivative of M sub x of t. Now,"},{"Start":"02:06.500 ","End":"02:09.410","Text":"we want to set t to be equal to 0,"},{"Start":"02:09.410 ","End":"02:17.880","Text":"so that means that we have M sub x of 0 plus 0 times M,"},{"Start":"02:17.880 ","End":"02:21.756","Text":"the derivative of M sub x of t,"},{"Start":"02:21.756 ","End":"02:26.315","Text":"and that equals to M sub x of 0."},{"Start":"02:26.315 ","End":"02:28.610","Text":"Now, what is that equal to?"},{"Start":"02:28.610 ","End":"02:32.340","Text":"Well, let\u0027s calculate this guy right here."},{"Start":"02:37.250 ","End":"02:40.155","Text":"M sub x of t,"},{"Start":"02:40.155 ","End":"02:44.340","Text":"that\u0027s defined as the integral over all of x"},{"Start":"02:44.340 ","End":"02:49.839","Text":"of e to the power of t times x times the density function dx."},{"Start":"02:49.839 ","End":"02:53.530","Text":"Now, that\u0027s assuming we have a continuous random"},{"Start":"02:53.530 ","End":"02:57.790","Text":"variable and if we have a discrete random variable,"},{"Start":"02:57.790 ","End":"03:03.125","Text":"then instead of the integral, we\u0027ll use the sigma sign."},{"Start":"03:03.125 ","End":"03:06.345","Text":"But for now, we\u0027ll assume that x is continuous."},{"Start":"03:06.345 ","End":"03:09.870","Text":"Now, M sub x at 0,"},{"Start":"03:09.870 ","End":"03:12.435","Text":"when t equals 0, that\u0027s the integral,"},{"Start":"03:12.435 ","End":"03:19.740","Text":"over all of x of e to the power of 0 times x times f of x dx."},{"Start":"03:19.740 ","End":"03:27.020","Text":"Now, that equals to the integral over all of x of 1 times the density function,"},{"Start":"03:27.020 ","End":"03:29.510","Text":"and that equals to 1."},{"Start":"03:29.510 ","End":"03:32.975","Text":"No matter what the density function is,"},{"Start":"03:32.975 ","End":"03:37.445","Text":"M sub x of 0 would be always 1."},{"Start":"03:37.445 ","End":"03:41.325","Text":"Here, that means that this is 1."},{"Start":"03:41.325 ","End":"03:46.580","Text":"That means that the expectation of Y now is 1."},{"Start":"03:46.580 ","End":"03:50.165","Text":"Now, let\u0027s calculate the variance of Y."},{"Start":"03:50.165 ","End":"03:52.805","Text":"The variance of Y,"},{"Start":"03:52.805 ","End":"04:00.990","Text":"that equals to the expectation of Y squared minus the expectation squared of Y."},{"Start":"04:00.990 ","End":"04:03.885","Text":"Now, we know what the expectation of Y, that\u0027s 1."},{"Start":"04:03.885 ","End":"04:05.915","Text":"What\u0027s the expectation of Y squared?"},{"Start":"04:05.915 ","End":"04:12.800","Text":"Well, that\u0027s the second derivative of the moment generating function of Y,"},{"Start":"04:12.800 ","End":"04:14.525","Text":"so let\u0027s do that."},{"Start":"04:14.525 ","End":"04:21.110","Text":"The second derivative of the moment generating function of Y,"},{"Start":"04:21.110 ","End":"04:22.640","Text":"well, that equals to this."},{"Start":"04:22.640 ","End":"04:24.695","Text":"This is a first derivative."},{"Start":"04:24.695 ","End":"04:28.820","Text":"All we have to do is take another derivative of"},{"Start":"04:28.820 ","End":"04:33.445","Text":"this expression right here to get the second derivative, so let\u0027s do that."},{"Start":"04:33.445 ","End":"04:36.816","Text":"That equals to M sub x of t,"},{"Start":"04:36.816 ","End":"04:40.275","Text":"the first derivative, that\u0027s the derivative of this guy,"},{"Start":"04:40.275 ","End":"04:42.030","Text":"plus, well,"},{"Start":"04:42.030 ","End":"04:48.050","Text":"that\u0027ll be t times the derivative of this guy,"},{"Start":"04:48.050 ","End":"04:55.760","Text":"which is this, plus 1 times this guy,"},{"Start":"04:55.760 ","End":"04:59.670","Text":"the derivative of t times this guy."},{"Start":"04:59.780 ","End":"05:06.090","Text":"Now, that\u0027s where t has to be equal to 0."},{"Start":"05:06.090 ","End":"05:12.390","Text":"Now, that equals to this."},{"Start":"05:12.390 ","End":"05:16.680","Text":"The first derivative of the moment generating function of x at"},{"Start":"05:16.680 ","End":"05:22.210","Text":"point 0 plus 0 times this,"},{"Start":"05:24.920 ","End":"05:31.805","Text":"plus the first derivative of the moment generating function of x at 0."},{"Start":"05:31.805 ","End":"05:34.955","Text":"Now, what is that equal to?"},{"Start":"05:34.955 ","End":"05:37.730","Text":"Well, this is the expectation of X."},{"Start":"05:37.730 ","End":"05:45.860","Text":"That\u0027s the first derivative of the moment generating function of x at t equal 0."},{"Start":"05:45.860 ","End":"05:48.620","Text":"Well, that equals to 5."},{"Start":"05:48.620 ","End":"05:51.140","Text":"Now this equals to 0,"},{"Start":"05:51.140 ","End":"05:53.450","Text":"and this is the same as this,"},{"Start":"05:53.450 ","End":"05:56.290","Text":"so that equals to 5 as well."},{"Start":"05:56.290 ","End":"06:02.840","Text":"We have here the expectation of Y squared,"},{"Start":"06:02.840 ","End":"06:05.090","Text":"well that equals to 10."},{"Start":"06:05.090 ","End":"06:07.495","Text":"We\u0027re not finished yet."},{"Start":"06:07.495 ","End":"06:11.315","Text":"We know then, that the variance of Y,"},{"Start":"06:11.315 ","End":"06:18.140","Text":"well that equals to the expectation of Y squared minus the expectation squared of Y."},{"Start":"06:18.140 ","End":"06:20.390","Text":"Well, the expectation of Y squared,"},{"Start":"06:20.390 ","End":"06:22.195","Text":"that\u0027s 10, that\u0027s right here."},{"Start":"06:22.195 ","End":"06:25.560","Text":"What about the expectation squared of Y?"},{"Start":"06:25.560 ","End":"06:27.540","Text":"Well, the expectation of Y, that\u0027s 1,"},{"Start":"06:27.540 ","End":"06:32.045","Text":"so there\u0027ll be minus 1 squared and so we have 10 minus 1,"},{"Start":"06:32.045 ","End":"06:33.695","Text":"that equals to 9."},{"Start":"06:33.695 ","End":"06:38.310","Text":"This then is the variance of Y."}],"ID":13165}],"Thumbnail":null,"ID":245053},{"Name":"Characteristics of the Moment Generating Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"4m 38s","ChapterTopicVideoID":12696,"CourseChapterTopicPlaylistID":245054,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.860","Text":"In this chapter, we\u0027ll be talking about the"},{"Start":"00:01.860 ","End":"00:04.725","Text":"characteristics of the moment generating functions."},{"Start":"00:04.725 ","End":"00:10.095","Text":"Now, there are 3 major characteristics that you should be familiar with."},{"Start":"00:10.095 ","End":"00:14.460","Text":"The first 1, is that there\u0027s a 1-1 correspondence between"},{"Start":"00:14.460 ","End":"00:18.825","Text":"a random variable and the moment generating function. What does that mean?"},{"Start":"00:18.825 ","End":"00:21.855","Text":"Well, that means that if I recognize"},{"Start":"00:21.855 ","End":"00:26.130","Text":"that I have a moment generating function of a certain type,"},{"Start":"00:26.130 ","End":"00:29.220","Text":"then I know for a fact that it would belong to"},{"Start":"00:29.220 ","End":"00:34.005","Text":"a specific random variable or a specific distribution,"},{"Start":"00:34.005 ","End":"00:35.455","Text":"and the other way around."},{"Start":"00:35.455 ","End":"00:39.920","Text":"If I have a specific distribution or random variable,"},{"Start":"00:39.920 ","End":"00:45.185","Text":"then it would have only 1 moment generating function."},{"Start":"00:45.185 ","End":"00:50.105","Text":"Now below, I\u0027ve given you a list of the major distributions,"},{"Start":"00:50.105 ","End":"00:51.845","Text":"the major random variables,"},{"Start":"00:51.845 ","End":"00:53.540","Text":"and their moment generating functions."},{"Start":"00:53.540 ","End":"00:56.280","Text":"Let\u0027s just take a look at some of them."},{"Start":"00:56.540 ","End":"01:01.610","Text":"Here for example, you can see the random variables,"},{"Start":"01:01.610 ","End":"01:03.350","Text":"from a binomial distribution,"},{"Start":"01:03.350 ","End":"01:04.700","Text":"a geometric distribution,"},{"Start":"01:04.700 ","End":"01:06.680","Text":"or a Poisson distribution."},{"Start":"01:06.680 ","End":"01:10.648","Text":"This then is the form of the moment"},{"Start":"01:10.648 ","End":"01:16.890","Text":"generating functions for each one of these distributions."},{"Start":"01:16.960 ","End":"01:19.955","Text":"Let\u0027s take a look at the second characteristic."},{"Start":"01:19.955 ","End":"01:24.980","Text":"Well, the effect of a linear transformation on a moment generating function is this;"},{"Start":"01:24.980 ","End":"01:27.290","Text":"we have the moment generating function of"},{"Start":"01:27.290 ","End":"01:30.080","Text":"a linear transformation of x at point t, well,"},{"Start":"01:30.080 ","End":"01:39.060","Text":"that equals to e^bt times the moment generating function at at."},{"Start":"01:39.060 ","End":"01:42.320","Text":"Now, since this is so easy to prove,"},{"Start":"01:42.320 ","End":"01:44.360","Text":"let\u0027s just do that right here."},{"Start":"01:44.360 ","End":"01:49.725","Text":"We\u0027re looking at the moment generating function of ax"},{"Start":"01:49.725 ","End":"01:55.290","Text":"plus b at point t. That\u0027s a linear transformation here."},{"Start":"01:55.290 ","End":"01:59.340","Text":"Where we have ax plus b,"},{"Start":"01:59.340 ","End":"02:02.335","Text":"that would be, for example, y."},{"Start":"02:02.335 ","End":"02:05.735","Text":"Now, that\u0027s equal to what?"},{"Start":"02:05.735 ","End":"02:08.510","Text":"Well, that\u0027s the expectation by definition,"},{"Start":"02:08.510 ","End":"02:16.365","Text":"that\u0027s expectation of e^t times ax plus b."},{"Start":"02:16.365 ","End":"02:19.535","Text":"Now, what does that equal to?"},{"Start":"02:19.535 ","End":"02:23.600","Text":"Well, that equals to the expectation of"},{"Start":"02:23.600 ","End":"02:32.185","Text":"e^t times ax times e^t times b."},{"Start":"02:32.185 ","End":"02:35.550","Text":"Now, e^t times b,"},{"Start":"02:35.550 ","End":"02:37.175","Text":"well, that\u0027s a constant."},{"Start":"02:37.175 ","End":"02:41.285","Text":"We can take that out of the expectation."},{"Start":"02:41.285 ","End":"02:46.400","Text":"So they\u0027ll be e^t times b times"},{"Start":"02:46.400 ","End":"02:53.405","Text":"the expectation of e^t times ax."},{"Start":"02:53.405 ","End":"02:57.440","Text":"Now that, this thing right here,"},{"Start":"02:57.440 ","End":"03:03.955","Text":"this is the moment generating function of the random variable x at point at."},{"Start":"03:03.955 ","End":"03:06.410","Text":"This whole expression will be equal to"},{"Start":"03:06.410 ","End":"03:15.280","Text":"e^tb times the moment generating function of x at point at."},{"Start":"03:15.280 ","End":"03:18.150","Text":"Third characteristics is this."},{"Start":"03:18.150 ","End":"03:21.615","Text":"We have x and y that are independent random variables,"},{"Start":"03:21.615 ","End":"03:26.030","Text":"then the moment generating function of the sum of these variables,"},{"Start":"03:26.030 ","End":"03:29.180","Text":"that would be equal to the multiplication of"},{"Start":"03:29.180 ","End":"03:32.915","Text":"the moment generating function of each one of the brand of variables."},{"Start":"03:32.915 ","End":"03:35.345","Text":"Now again, this is very easy to prove,"},{"Start":"03:35.345 ","End":"03:37.460","Text":"so let\u0027s do that."},{"Start":"03:37.460 ","End":"03:43.125","Text":"Now the moment generating function of x plus y at point t,"},{"Start":"03:43.125 ","End":"03:50.965","Text":"well that\u0027s defined as the expectation of e^t times x plus y."},{"Start":"03:50.965 ","End":"04:02.300","Text":"Now, that equals to the expectation of e^t times x times e^t times y."},{"Start":"04:02.300 ","End":"04:07.550","Text":"Now, because x and y are independent variables,"},{"Start":"04:07.550 ","End":"04:11.390","Text":"well that equals to the expectation of"},{"Start":"04:11.390 ","End":"04:19.115","Text":"e^tx times the expectation of e^ty."},{"Start":"04:19.115 ","End":"04:26.045","Text":"Now, that is the moment generating function of x at point t,"},{"Start":"04:26.045 ","End":"04:34.200","Text":"and this is the moment generating function of y at point t. So here we go,"},{"Start":"04:34.200 ","End":"04:37.770","Text":"we\u0027ve proven this characteristic as well."}],"ID":13175},{"Watched":false,"Name":"Example","Duration":"5m 10s","ChapterTopicVideoID":12697,"CourseChapterTopicPlaylistID":245054,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this example, we assume that X has a Poisson distribution with Lambda equals 4 and Y"},{"Start":"00:05.250 ","End":"00:07.410","Text":"has a Poisson distribution where Lambda equals"},{"Start":"00:07.410 ","End":"00:10.725","Text":"2 and both X and Y are independent and we\u0027re asked,"},{"Start":"00:10.725 ","End":"00:14.010","Text":"what\u0027s the moment-generating function of 5X minus 3?"},{"Start":"00:14.010 ","End":"00:16.080","Text":"That\u0027s the transformation."},{"Start":"00:16.080 ","End":"00:19.680","Text":"Let\u0027s define W as that transformation."},{"Start":"00:19.680 ","End":"00:22.440","Text":"That\u0027ll be 5X minus 3."},{"Start":"00:22.440 ","End":"00:28.935","Text":"Now, if W follows the linear transformation template aX plus b,"},{"Start":"00:28.935 ","End":"00:35.230","Text":"then a, in our case will be equal to 5 and b will be equal to minus 3."},{"Start":"00:35.230 ","End":"00:37.010","Text":"Now, what else do we know?"},{"Start":"00:37.010 ","End":"00:41.060","Text":"Well, since X has a Poisson distribution,"},{"Start":"00:41.060 ","End":"00:45.275","Text":"then the moment generating function of X at point T. Well,"},{"Start":"00:45.275 ","End":"00:52.370","Text":"that\u0027ll be equal to e^Lambda times e^t minus 1."},{"Start":"00:52.370 ","End":"00:54.110","Text":"Now, how do I know that?"},{"Start":"00:54.110 ","End":"00:56.930","Text":"Well, I looked it up in the table that I just gave you."},{"Start":"00:56.930 ","End":"00:59.530","Text":"Now, what else?"},{"Start":"00:59.530 ","End":"01:07.370","Text":"M, the moment-generating function of X linear transformation aX plus b at point T. Well,"},{"Start":"01:07.370 ","End":"01:17.365","Text":"that equals to e^bt times the moment-generating function of X, at point at."},{"Start":"01:17.365 ","End":"01:21.555","Text":"Now, let\u0027s get back to this side."},{"Start":"01:21.555 ","End":"01:23.870","Text":"Now, what else do we know?"},{"Start":"01:23.870 ","End":"01:26.480","Text":"Well, since Lambda equals 4,"},{"Start":"01:26.480 ","End":"01:32.970","Text":"then that equals to e^4 times e^t minus 1."},{"Start":"01:33.190 ","End":"01:39.620","Text":"What\u0027s the moment-generating function of W at point T?"},{"Start":"01:39.620 ","End":"01:43.700","Text":"Well, that equals to e^bt."},{"Start":"01:43.700 ","End":"01:45.560","Text":"Now b is minus 3,"},{"Start":"01:45.560 ","End":"01:52.535","Text":"so that\u0027ll be e^minus 3t times the moment-generating function of X at at."},{"Start":"01:52.535 ","End":"01:55.745","Text":"Anytime I see t over here,"},{"Start":"01:55.745 ","End":"01:57.245","Text":"I\u0027ll plug in at,"},{"Start":"01:57.245 ","End":"01:59.690","Text":"now a is 5 so I\u0027ll plug in 5t."},{"Start":"01:59.690 ","End":"02:03.230","Text":"Let\u0027s just substitute for this,"},{"Start":"02:03.230 ","End":"02:05.300","Text":"this expression right here."},{"Start":"02:05.300 ","End":"02:14.250","Text":"That\u0027ll be e^4 times e to the power now of 5t minus 1."},{"Start":"02:14.250 ","End":"02:19.055","Text":"Now, we can simplify that a little bit,"},{"Start":"02:19.055 ","End":"02:22.430","Text":"not much to look like this,"},{"Start":"02:22.430 ","End":"02:28.660","Text":"it\u0027ll be e^minus 3t plus"},{"Start":"02:28.660 ","End":"02:36.135","Text":"4e^5t minus 4 but that\u0027s the best that we can do."},{"Start":"02:36.135 ","End":"02:40.580","Text":"In any case, these 2 expressions are identical"},{"Start":"02:40.580 ","End":"02:46.495","Text":"and they\u0027re the moment-generating function of the transformation 5X minus 3."},{"Start":"02:46.495 ","End":"02:51.030","Text":"In this section, we let T equal X plus"},{"Start":"02:51.030 ","End":"02:56.310","Text":"Y and we\u0027re asking what\u0027s the probability distribution of T?"},{"Start":"02:56.500 ","End":"03:01.085","Text":"The moment-generating function of X plus y,"},{"Start":"03:01.085 ","End":"03:04.955","Text":"we know that to be equal to the expectation of"},{"Start":"03:04.955 ","End":"03:12.305","Text":"e^tx times the expectation of e^ty."},{"Start":"03:12.305 ","End":"03:16.200","Text":"Now that equals to the moment generating function of"},{"Start":"03:16.200 ","End":"03:20.060","Text":"X at point T times the moment-generating function of"},{"Start":"03:20.060 ","End":"03:27.245","Text":"Y at point T. What\u0027s the moment-generating function of X at point T?"},{"Start":"03:27.245 ","End":"03:30.365","Text":"Well, we\u0027ve calculated that in the last section."},{"Start":"03:30.365 ","End":"03:35.990","Text":"That\u0027s e^4 times e^t minus 1."},{"Start":"03:35.990 ","End":"03:40.610","Text":"Now what about the moment-generating function of Y at point T?"},{"Start":"03:40.610 ","End":"03:44.180","Text":"Well, Y also has a Poisson distribution,"},{"Start":"03:44.180 ","End":"03:47.180","Text":"but with Lambda equaling 2."},{"Start":"03:47.180 ","End":"03:56.250","Text":"This moment-generating function equals to e to the power now of 2 times e^t minus 1."},{"Start":"03:56.930 ","End":"04:02.300","Text":"Now what\u0027s the moment-generating function of T at point T?"},{"Start":"04:02.300 ","End":"04:07.880","Text":"Well, that equals to the multiplication of the moment-generating functions of X and y."},{"Start":"04:07.880 ","End":"04:09.850","Text":"Let\u0027s just write that down."},{"Start":"04:09.850 ","End":"04:15.080","Text":"That\u0027ll be e^4 times e^t minus 1"},{"Start":"04:15.080 ","End":"04:21.590","Text":"times e^2 times e^t minus 1."},{"Start":"04:21.590 ","End":"04:29.390","Text":"Now that equals to e^6 times e^t minus 1."},{"Start":"04:29.390 ","End":"04:32.735","Text":"Now, this expression right here is"},{"Start":"04:32.735 ","End":"04:36.995","Text":"a moment-generating function of the Poisson distribution."},{"Start":"04:36.995 ","End":"04:40.590","Text":"That\u0027s the meaning of"},{"Start":"04:40.590 ","End":"04:46.130","Text":"moment-generating function that is correlated with the distribution."},{"Start":"04:46.130 ","End":"04:47.660","Text":"It\u0027s a 1-1 correlation."},{"Start":"04:47.660 ","End":"04:53.150","Text":"Whenever we recognize this format of a moment-generating function."},{"Start":"04:53.150 ","End":"04:58.220","Text":"Well, that means that this format belongs to the Poisson distribution."},{"Start":"04:58.220 ","End":"05:04.100","Text":"Now, that means then that T equaling X plus Y,"},{"Start":"05:04.100 ","End":"05:10.500","Text":"well, that has a Poisson distribution where Lambda equals to 6."}],"ID":13176},{"Watched":false,"Name":"Exercise 1","Duration":"4m 19s","ChapterTopicVideoID":12698,"CourseChapterTopicPlaylistID":245054,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"In this question, we assume that X_i has"},{"Start":"00:03.285 ","End":"00:08.100","Text":"a Poisson distribution with parameter Lambda and all the X_i\u0027s are independent."},{"Start":"00:08.100 ","End":"00:12.390","Text":"That means that X_1 has a Poisson distribution with parameter Lambda,"},{"Start":"00:12.390 ","End":"00:14.250","Text":"same for X_2, same for X_3,"},{"Start":"00:14.250 ","End":"00:15.705","Text":"and so on and so forth."},{"Start":"00:15.705 ","End":"00:21.180","Text":"We\u0027re asked in section a to find the moment generating function of the sum of the"},{"Start":"00:21.180 ","End":"00:26.940","Text":"X_i\u0027s where i goes from 1 to n. First of all,"},{"Start":"00:26.940 ","End":"00:31.110","Text":"the moment generating function of X,"},{"Start":"00:31.110 ","End":"00:32.790","Text":"and it doesn\u0027t matter which X,"},{"Start":"00:32.790 ","End":"00:36.150","Text":"as long as it has a Poisson distribution."},{"Start":"00:36.150 ","End":"00:43.095","Text":"Well, that equals to e^Lambda times e^t minus 1."},{"Start":"00:43.095 ","End":"00:45.065","Text":"What else do we know?"},{"Start":"00:45.065 ","End":"00:51.680","Text":"Well, the moment generating function of X plus y at point t,"},{"Start":"00:51.680 ","End":"01:02.000","Text":"that equals to the expectation of e^tX times the expectation of e^ty."},{"Start":"01:02.000 ","End":"01:05.480","Text":"Now that equals to the moment generating function of X at"},{"Start":"01:05.480 ","End":"01:09.290","Text":"point t times the moment generating function of y at"},{"Start":"01:09.290 ","End":"01:19.165","Text":"point t. Now let\u0027s define T as the sum of X_i,"},{"Start":"01:19.165 ","End":"01:22.865","Text":"where i goes from 1 to n. Now,"},{"Start":"01:22.865 ","End":"01:24.230","Text":"what do we want to know?"},{"Start":"01:24.230 ","End":"01:31.175","Text":"We want to know the moment generating function of T. So let\u0027s calculate that."},{"Start":"01:31.175 ","End":"01:41.420","Text":"That will be M_T at point t. That\u0027ll be equal to the moment generating function"},{"Start":"01:41.420 ","End":"01:46.150","Text":"for X_1 at t times the moment"},{"Start":"01:46.150 ","End":"01:51.745","Text":"generating function of X_2 at point t,"},{"Start":"01:51.745 ","End":"01:53.135","Text":"and so on and so forth,"},{"Start":"01:53.135 ","End":"01:58.610","Text":"times the moment generating function of X_n at point t. Now,"},{"Start":"01:58.610 ","End":"02:02.855","Text":"for each one of these moment generating functions, well,"},{"Start":"02:02.855 ","End":"02:06.500","Text":"they have the same moment generating function because they\u0027re all"},{"Start":"02:06.500 ","End":"02:11.045","Text":"distributed with the Poisson distribution with the same parameter."},{"Start":"02:11.045 ","End":"02:19.825","Text":"That will equal to e^Lambda times e^t minus 1."},{"Start":"02:19.825 ","End":"02:22.295","Text":"Now, how many of these do we have?"},{"Start":"02:22.295 ","End":"02:24.305","Text":"Well, we have n of them."},{"Start":"02:24.305 ","End":"02:32.285","Text":"Now that equals to e^nLambda times e^t minus 1."},{"Start":"02:32.285 ","End":"02:38.020","Text":"This then is the moment generating function of the sum of"},{"Start":"02:38.020 ","End":"02:42.190","Text":"X_i random variables where each X_i is"},{"Start":"02:42.190 ","End":"02:48.775","Text":"distributed with a Poisson distribution with parameter Lambda."},{"Start":"02:48.775 ","End":"02:55.640","Text":"In section b, we\u0027re asked to prove that the sum of X_i where i goes from 1 to n,"},{"Start":"02:55.640 ","End":"03:01.325","Text":"that has a Poisson distribution with parameter n times Lambda."},{"Start":"03:01.325 ","End":"03:07.085","Text":"Well, let\u0027s take a look at the moment generating function of the sum of X_i,"},{"Start":"03:07.085 ","End":"03:16.895","Text":"the moment generating function of t that equals to e^nLambda times e^t minus 1."},{"Start":"03:16.895 ","End":"03:21.545","Text":"Now, this has the same format"},{"Start":"03:21.545 ","End":"03:25.729","Text":"as the moment generating function of a Poisson distribution."},{"Start":"03:25.729 ","End":"03:28.760","Text":"Now instead of the parameter Lambda here,"},{"Start":"03:28.760 ","End":"03:31.580","Text":"we have n times Lambda here."},{"Start":"03:31.580 ","End":"03:33.530","Text":"All the rest is the same."},{"Start":"03:33.530 ","End":"03:42.680","Text":"That means that T is distributed with a Poisson distribution having parameter n Lambda."},{"Start":"03:42.680 ","End":"03:44.570","Text":"Now T, as we know,"},{"Start":"03:44.570 ","End":"03:45.950","Text":"is the sum of X_i,"},{"Start":"03:45.950 ","End":"03:51.620","Text":"i goes from 1 to n. Because this"},{"Start":"03:51.620 ","End":"03:58.955","Text":"has the same format as the moment generating function of a Poisson distribution,"},{"Start":"03:58.955 ","End":"04:03.097","Text":"and we know that there\u0027s a 1-1 correlation between the moment"},{"Start":"04:03.097 ","End":"04:08.067","Text":"generating function and the random variable,"},{"Start":"04:08.067 ","End":"04:12.230","Text":"then we can state very"},{"Start":"04:12.230 ","End":"04:19.440","Text":"definitively that T has a Poisson distribution with parameter nLambda."}],"ID":13177},{"Watched":false,"Name":"Exercise 2 - Parts a-b","Duration":"4m 5s","ChapterTopicVideoID":12700,"CourseChapterTopicPlaylistID":245054,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In this question, we assume that X has"},{"Start":"00:02.160 ","End":"00:05.895","Text":"a Poisson distribution with parameter Lambda equals 10,"},{"Start":"00:05.895 ","End":"00:09.375","Text":"Y has a Poisson distribution where Lambda equals 2,"},{"Start":"00:09.375 ","End":"00:11.700","Text":"X and Y are independent."},{"Start":"00:11.700 ","End":"00:14.925","Text":"Now, let T be equal to X plus Y,"},{"Start":"00:14.925 ","End":"00:20.250","Text":"and we\u0027re asked to find the moment generating function of T. We"},{"Start":"00:20.250 ","End":"00:25.905","Text":"know that the moment generating function of X plus Y at point T,"},{"Start":"00:25.905 ","End":"00:30.090","Text":"that equals to the expectation of e to the power of t times"},{"Start":"00:30.090 ","End":"00:36.725","Text":"x times the expectation of e to the power of t times y."},{"Start":"00:36.725 ","End":"00:42.770","Text":"Now, that equals to the moment generating function of x at point t,"},{"Start":"00:42.770 ","End":"00:48.650","Text":"times the moment generating function of y at point t. Now, what else do we know?"},{"Start":"00:48.650 ","End":"00:52.810","Text":"Well, the moment generating function of x at t,"},{"Start":"00:52.810 ","End":"00:56.335","Text":"where X has a Poisson distribution, well,"},{"Start":"00:56.335 ","End":"01:01.640","Text":"that equals to e to the power of Lambda times e to the power of t minus 1."},{"Start":"01:01.640 ","End":"01:02.960","Text":"Now, how do I know that?"},{"Start":"01:02.960 ","End":"01:06.580","Text":"Well, I looked it up in the table that I gave you."},{"Start":"01:06.580 ","End":"01:09.645","Text":"Now, in our case,"},{"Start":"01:09.645 ","End":"01:11.070","Text":"Lambda equals 10,"},{"Start":"01:11.070 ","End":"01:18.915","Text":"so that equals to e to the power of 10 times e to the power of t minus 1."},{"Start":"01:18.915 ","End":"01:22.760","Text":"What about the moment generating function of Y?"},{"Start":"01:22.760 ","End":"01:26.615","Text":"Well, M sub y of t, well,"},{"Start":"01:26.615 ","End":"01:30.800","Text":"that equals to e to the power of, again,"},{"Start":"01:30.800 ","End":"01:38.570","Text":"since this has a Poisson distribution with parameter Lambda equals 2,"},{"Start":"01:38.570 ","End":"01:44.830","Text":"then that\u0027ll be e to the power of 2 times e to the power of t minus 1."},{"Start":"01:44.870 ","End":"01:53.390","Text":"Now, we\u0027re ready to find the moment generating function of T. What would that be?"},{"Start":"01:53.390 ","End":"01:59.150","Text":"Well, the moment generating function of T at point t, well, as we said,"},{"Start":"01:59.150 ","End":"02:05.350","Text":"that\u0027s a multiplication of the moment generating functions of X and Y, and we have that."},{"Start":"02:05.350 ","End":"02:07.740","Text":"This is for X and this is for Y,"},{"Start":"02:07.740 ","End":"02:09.065","Text":"so let\u0027s just multiply them."},{"Start":"02:09.065 ","End":"02:14.615","Text":"That\u0027ll be e to the power of 10 times e to the power of t minus 1,"},{"Start":"02:14.615 ","End":"02:20.600","Text":"times e to the power of 2 times e to the power of t minus 1."},{"Start":"02:20.600 ","End":"02:26.100","Text":"Now, that equals to e to the power of 10 times e to the power of"},{"Start":"02:26.100 ","End":"02:32.250","Text":"t minus 1 plus 2 times e to the power of t minus 1,"},{"Start":"02:32.250 ","End":"02:40.415","Text":"and that equals to e to the power of 12 times e to the power of t minus 1."},{"Start":"02:40.415 ","End":"02:47.940","Text":"This then is the moment generating function of T. In section B,"},{"Start":"02:47.940 ","End":"02:52.920","Text":"we want to prove that T has a Poisson distribution where Lambda equals 12."},{"Start":"02:52.920 ","End":"02:59.025","Text":"Well, we know that the moment generating function of T is this thing right here,"},{"Start":"02:59.025 ","End":"03:02.915","Text":"e to the power of 12 times e to the power of t minus 1."},{"Start":"03:02.915 ","End":"03:08.120","Text":"Now, whenever we see this moment generating function,"},{"Start":"03:08.120 ","End":"03:13.760","Text":"we automatically recognize this as that of a Poisson distribution."},{"Start":"03:13.760 ","End":"03:17.750","Text":"Why is that? Because there\u0027s a 1 to 1 correlation between the moment"},{"Start":"03:17.750 ","End":"03:22.460","Text":"generating function and the distribution or the random variable."},{"Start":"03:22.460 ","End":"03:25.650","Text":"That means that T here,"},{"Start":"03:25.650 ","End":"03:28.020","Text":"which equals to X plus Y,"},{"Start":"03:28.020 ","End":"03:31.250","Text":"well, that has a Poisson distribution."},{"Start":"03:31.250 ","End":"03:32.735","Text":"Now, with what parameter?"},{"Start":"03:32.735 ","End":"03:36.940","Text":"Well, since this thing right here,"},{"Start":"03:36.940 ","End":"03:42.020","Text":"that\u0027s the moment generating function of a Poisson distribution."},{"Start":"03:42.020 ","End":"03:43.970","Text":"Let\u0027s look at this Lambda right here,"},{"Start":"03:43.970 ","End":"03:45.440","Text":"that\u0027s the parameter Lambda."},{"Start":"03:45.440 ","End":"03:47.150","Text":"Well, here it\u0027s 12,"},{"Start":"03:47.150 ","End":"03:50.430","Text":"so that means that X plus Y,"},{"Start":"03:50.430 ","End":"03:51.685","Text":"which equals to T,"},{"Start":"03:51.685 ","End":"03:57.090","Text":"is distributed with a Poisson distribution where Lambda equals 12."},{"Start":"03:57.090 ","End":"04:00.970","Text":"Now, that\u0027s the 1 to 1 correlation again"},{"Start":"04:00.970 ","End":"04:06.180","Text":"between the random variable and the moment generating function."}],"ID":13178},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"7m 56s","ChapterTopicVideoID":12699,"CourseChapterTopicPlaylistID":245054,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this section, we want to prove that X given T equals 8 has"},{"Start":"00:04.170 ","End":"00:09.555","Text":"a binomial distribution where n equals 8 and p equals to 5/6."},{"Start":"00:09.555 ","End":"00:12.300","Text":"This is a conditional probability,"},{"Start":"00:12.300 ","End":"00:19.440","Text":"so that\u0027s the probability of X being equal to some value k,"},{"Start":"00:19.440 ","End":"00:22.665","Text":"given that T equals to 8."},{"Start":"00:22.665 ","End":"00:26.055","Text":"Now, how do we solve conditional probabilities?"},{"Start":"00:26.055 ","End":"00:28.080","Text":"Well, in the denominator,"},{"Start":"00:28.080 ","End":"00:30.840","Text":"we have what\u0027s the probability, what\u0027s given."},{"Start":"00:30.840 ","End":"00:34.470","Text":"That\u0027s the probability of T being equal to 8."},{"Start":"00:34.470 ","End":"00:38.288","Text":"In the numerator, we have the probability of the intersect,"},{"Start":"00:38.288 ","End":"00:46.220","Text":"that\u0027s the probability of X equaling k intersect, T equaling 8."},{"Start":"00:46.220 ","End":"00:51.725","Text":"Now, we know that T equals to X plus Y."},{"Start":"00:51.725 ","End":"00:54.250","Text":"Now, we\u0027re given that T equals to 8,"},{"Start":"00:54.250 ","End":"00:59.684","Text":"and what I want to do is I want to express this thing right here in terms of Y."},{"Start":"00:59.684 ","End":"01:05.245","Text":"Then Y would be equal to 8 minus X."},{"Start":"01:05.245 ","End":"01:10.040","Text":"Let\u0027s just put this into here."},{"Start":"01:10.040 ","End":"01:19.790","Text":"That\u0027ll be equal to the probability of X equaling k intersect Y equals to 8 minus,"},{"Start":"01:19.790 ","End":"01:21.920","Text":"now instead of X, well, X equals k,"},{"Start":"01:21.920 ","End":"01:25.516","Text":"so that\u0027ll be k,"},{"Start":"01:25.516 ","End":"01:31.265","Text":"divided by the probability of T being equal to 8."},{"Start":"01:31.265 ","End":"01:39.740","Text":"Now we have the intersect of 2 random variables that are independent of each other,"},{"Start":"01:39.740 ","End":"01:44.735","Text":"where we\u0027re given that X and Y are independent of each other."},{"Start":"01:44.735 ","End":"01:48.860","Text":"That means that we can write this probability like this, as a multiplication."},{"Start":"01:48.860 ","End":"01:56.340","Text":"That\u0027s the probability of X equaling k times the probability of Y being equal"},{"Start":"01:56.340 ","End":"02:04.910","Text":"to 8 minus k divided by the probability of T being equal to 8."},{"Start":"02:04.910 ","End":"02:11.075","Text":"Here we have the basis"},{"Start":"02:11.075 ","End":"02:16.130","Text":"for solving these probabilities because we know what the distribution is of X,"},{"Start":"02:16.130 ","End":"02:20.344","Text":"Y, and T. All of them have a Poisson distribution,"},{"Start":"02:20.344 ","End":"02:22.940","Text":"but for each random variable,"},{"Start":"02:22.940 ","End":"02:25.730","Text":"the parameter Lambda is different."},{"Start":"02:25.730 ","End":"02:29.045","Text":"For X, the parameter Lambda is 10;"},{"Start":"02:29.045 ","End":"02:30.464","Text":"for Y, it\u0027s 2;"},{"Start":"02:30.464 ","End":"02:33.365","Text":"and for T, it\u0027s 12."},{"Start":"02:33.365 ","End":"02:39.787","Text":"Let\u0027s just figure out or calculate the probabilities for each 1 of these guys."},{"Start":"02:39.787 ","End":"02:44.770","Text":"That will be equal to the probability of X being equal to k,"},{"Start":"02:44.770 ","End":"02:51.755","Text":"where X has a Poisson distribution that\u0027s e to the power of minus 10,"},{"Start":"02:51.755 ","End":"02:53.865","Text":"Lambda was 10 here,"},{"Start":"02:53.865 ","End":"02:58.305","Text":"times 10 to the power of k divided by k factorial,"},{"Start":"02:58.305 ","End":"03:00.140","Text":"that\u0027s this guy right here."},{"Start":"03:00.140 ","End":"03:02.435","Text":"Now, what\u0027s this guy?"},{"Start":"03:02.435 ","End":"03:06.073","Text":"For Y, that\u0027ll be e to the power of minus 2,"},{"Start":"03:06.073 ","End":"03:08.485","Text":"Lambda for Y was 2,"},{"Start":"03:08.485 ","End":"03:11.285","Text":"times 2 to the power of k. Now,"},{"Start":"03:11.285 ","End":"03:19.970","Text":"k here is 8 minus k divided by 8 minus k factorial,"},{"Start":"03:19.970 ","End":"03:26.045","Text":"and all that\u0027s divided by e to the power of minus 12,"},{"Start":"03:26.045 ","End":"03:29.560","Text":"Lambda for T was 12,"},{"Start":"03:29.560 ","End":"03:34.175","Text":"times 12 to the power of"},{"Start":"03:34.175 ","End":"03:41.430","Text":"8 divided by 8 factorial."},{"Start":"03:41.430 ","End":"03:42.865","Text":"Let\u0027s simplify that."},{"Start":"03:42.865 ","End":"03:46.490","Text":"Let\u0027s just bring this ratio up here as a multiplication,"},{"Start":"03:46.490 ","End":"03:48.175","Text":"and let\u0027s write this out."},{"Start":"03:48.175 ","End":"03:52.460","Text":"That\u0027ll be e to the power of minus 10 times 10"},{"Start":"03:52.460 ","End":"03:57.605","Text":"to the power of k divided by k factorial times"},{"Start":"03:57.605 ","End":"04:02.450","Text":"e to the power of minus 2 times 2 to the power of"},{"Start":"04:02.450 ","End":"04:08.240","Text":"8 minus k divided by 8 minus k factorial."},{"Start":"04:08.240 ","End":"04:09.925","Text":"Now, we\u0027re bringing this up,"},{"Start":"04:09.925 ","End":"04:15.950","Text":"so that means that\u0027s times 8 factorial divided by e"},{"Start":"04:15.950 ","End":"04:22.704","Text":"to the power of minus 12 times 12 to the power of 8."},{"Start":"04:22.704 ","End":"04:27.830","Text":"Let\u0027s go ahead and clean this up a little bit better."},{"Start":"04:27.830 ","End":"04:29.695","Text":"Well, that equals to,"},{"Start":"04:29.695 ","End":"04:32.671","Text":"now let\u0027s collect like elements,"},{"Start":"04:32.671 ","End":"04:35.255","Text":"that\u0027ll be e to the power of minus 12,"},{"Start":"04:35.255 ","End":"04:38.435","Text":"that\u0027s e to the power of minus 10 and e to the power minus 2,"},{"Start":"04:38.435 ","End":"04:45.705","Text":"times 10 to the power of k times 2 to the power of 8 minus k,"},{"Start":"04:45.705 ","End":"04:52.918","Text":"divided by k factorial 8 minus k factorial,"},{"Start":"04:52.918 ","End":"05:03.090","Text":"times 8 factorial here divided by e to the power of minus 12 times 12 to the power of 8."},{"Start":"05:03.090 ","End":"05:05.690","Text":"This expression equals to this expression."},{"Start":"05:05.690 ","End":"05:09.430","Text":"Now, we can cancel these guys out right here."},{"Start":"05:09.430 ","End":"05:11.930","Text":"We\u0027re left with the following."},{"Start":"05:11.930 ","End":"05:14.150","Text":"Let\u0027s just rearrange this a little bit."},{"Start":"05:14.150 ","End":"05:20.120","Text":"That\u0027ll be 8 factorial divided by k factorial times"},{"Start":"05:20.120 ","End":"05:27.290","Text":"8 minus k factorial times 10 to"},{"Start":"05:27.290 ","End":"05:34.830","Text":"the power of k times 2 to the power of 8 minus k,"},{"Start":"05:34.830 ","End":"05:36.645","Text":"that\u0027s 2 to the power of 8,"},{"Start":"05:36.645 ","End":"05:45.790","Text":"divided by 2 to the power of k times 12 to the power of 8."},{"Start":"05:45.790 ","End":"05:50.840","Text":"This thing then equals to this thing, but what\u0027s this?"},{"Start":"05:50.840 ","End":"05:55.800","Text":"I hope you recognize that this is 8 over k,"},{"Start":"05:55.800 ","End":"05:57.640","Text":"and we\u0027re left with, well,"},{"Start":"05:57.640 ","End":"06:02.060","Text":"10 to the power of k divided by 2 to the power of k. Well,"},{"Start":"06:02.060 ","End":"06:09.140","Text":"that\u0027s 5 to the power of k times 2 divided by 12 to the power of 8."},{"Start":"06:09.140 ","End":"06:13.215","Text":"That\u0027s 1/6 to the power of 8."},{"Start":"06:13.215 ","End":"06:16.545","Text":"Now, we\u0027re nearly there."},{"Start":"06:16.545 ","End":"06:23.336","Text":"If we want to prove that this is a binomial distribution with parameters n equals 8,"},{"Start":"06:23.336 ","End":"06:27.900","Text":"now, p has to be equal to 5/6."},{"Start":"06:27.900 ","End":"06:30.080","Text":"We don\u0027t have that, we have only 5,"},{"Start":"06:30.080 ","End":"06:35.735","Text":"so how can we transfer this or translate this into 5/6?"},{"Start":"06:35.735 ","End":"06:41.135","Text":"Well, we can convert it by multiplying this by 1."},{"Start":"06:41.135 ","End":"06:43.955","Text":"Now, what type of 1 am I thinking of?"},{"Start":"06:43.955 ","End":"06:46.375","Text":"Well, that\u0027s 8/k."},{"Start":"06:46.375 ","End":"06:52.035","Text":"Now, here we have 5 to the power of k. Now,"},{"Start":"06:52.035 ","End":"06:58.620","Text":"I want to divide by 6 to the power of k times 1 divided by 6 to the power of 8."},{"Start":"06:58.620 ","End":"07:01.560","Text":"Now, I divided here by 6 to the power of k so I have to"},{"Start":"07:01.560 ","End":"07:05.700","Text":"multiply by 6 to the power of k. Now,"},{"Start":"07:05.700 ","End":"07:08.925","Text":"that equals to 8/k."},{"Start":"07:08.925 ","End":"07:15.353","Text":"This is 5/6 to the power of k,"},{"Start":"07:15.353 ","End":"07:17.080","Text":"and right here,"},{"Start":"07:17.080 ","End":"07:23.730","Text":"we have 1/6 to the power of 8 times, now,"},{"Start":"07:23.730 ","End":"07:27.620","Text":"I can write 6 to the power of k as 1 divided by 6 to the power of"},{"Start":"07:27.620 ","End":"07:32.485","Text":"minus k. That equals to 8 over k,"},{"Start":"07:32.485 ","End":"07:37.148","Text":"5/6 to the power of k,"},{"Start":"07:37.148 ","End":"07:44.225","Text":"and 1/6 to the power of 8 minus k. This has"},{"Start":"07:44.225 ","End":"07:52.970","Text":"a binomial distribution where n equals to 8 and p equals to 5/6."},{"Start":"07:52.970 ","End":"07:56.100","Text":"Now, this is what we\u0027re asked to prove."}],"ID":13179},{"Watched":false,"Name":"Exercise 3","Duration":"10m 22s","ChapterTopicVideoID":12701,"CourseChapterTopicPlaylistID":245054,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In this question we assume that x_i has"},{"Start":"00:02.580 ","End":"00:07.560","Text":"an exponential distribution with parameter 1 for all i equaling 1,"},{"Start":"00:07.560 ","End":"00:11.985","Text":"2, and so on and so forth until n. Now the variables are independent,"},{"Start":"00:11.985 ","End":"00:15.330","Text":"x_1 is independent of x_2 and so on and so forth."},{"Start":"00:15.330 ","End":"00:20.305","Text":"We define t as the sum of all the excise."},{"Start":"00:20.305 ","End":"00:25.110","Text":"We\u0027re asked to find the moment generating function of t. Let\u0027s just"},{"Start":"00:25.110 ","End":"00:31.215","Text":"recall that the moment generating function of x plus y at point t,"},{"Start":"00:31.215 ","End":"00:34.730","Text":"that equals to the expectation of e to the power of"},{"Start":"00:34.730 ","End":"00:40.940","Text":"tx times the expectation of e to the power of ty."},{"Start":"00:40.940 ","End":"00:44.510","Text":"That equals to the moment generating function of"},{"Start":"00:44.510 ","End":"00:50.720","Text":"x times the moment generating function of y at t.."},{"Start":"00:50.720 ","End":"00:54.975","Text":"The moment generating function of x at t,"},{"Start":"00:54.975 ","End":"00:58.200","Text":"since x has an exponential distribution,"},{"Start":"00:58.200 ","End":"01:03.935","Text":"that equals to lambda divided by lambda minus t. How do I know that?"},{"Start":"01:03.935 ","End":"01:06.670","Text":"I went to the table that I gave you."},{"Start":"01:06.670 ","End":"01:10.140","Text":"In our case lambda equals to 1,"},{"Start":"01:10.140 ","End":"01:11.805","Text":"that\u0027s given right here."},{"Start":"01:11.805 ","End":"01:19.040","Text":"That means that for each x_i the moment generating function is 1"},{"Start":"01:19.040 ","End":"01:26.315","Text":"divided by 1 minus t. A moment generating function of T at point t,"},{"Start":"01:26.315 ","End":"01:35.435","Text":"that equals to m_x_1 at t times m_x_2 at t,"},{"Start":"01:35.435 ","End":"01:42.000","Text":"and so on and so forth until m_x_n at"},{"Start":"01:42.000 ","End":"01:49.490","Text":"t. That equals to 1 divided by 1 minus t times 1 divided by 1 minus t,"},{"Start":"01:49.490 ","End":"01:53.960","Text":"and so on and so forth until n by 1 divided by"},{"Start":"01:53.960 ","End":"01:59.255","Text":"1 minus t. That equals to 1 divided by 1 minus t"},{"Start":"01:59.255 ","End":"02:04.048","Text":"to the power of n. This then is the moment"},{"Start":"02:04.048 ","End":"02:09.110","Text":"generating function of T. In this section we\u0027re asked to calculate"},{"Start":"02:09.110 ","End":"02:13.115","Text":"the expectation and variance of T. Let\u0027s just recall"},{"Start":"02:13.115 ","End":"02:17.390","Text":"that the moment generating function of t equal to"},{"Start":"02:17.390 ","End":"02:25.810","Text":"1 divided by 1 minus t all to the power of n. The expectation if we recall,"},{"Start":"02:25.810 ","End":"02:30.200","Text":"the expectation of T is the first moment."},{"Start":"02:30.200 ","End":"02:32.370","Text":"What\u0027s the first moment?"},{"Start":"02:32.370 ","End":"02:36.300","Text":"We have to take the derivative of the moment generating function."},{"Start":"02:36.300 ","End":"02:39.150","Text":"That\u0027s the moment generating function of T at"},{"Start":"02:39.150 ","End":"02:42.864","Text":"point t. We have to take the derivative of that,"},{"Start":"02:42.864 ","End":"02:45.555","Text":"set t to 0,"},{"Start":"02:45.555 ","End":"02:50.385","Text":"and that would be our first moment or the expectation."},{"Start":"02:50.385 ","End":"02:59.415","Text":"That will equal to 1 divided by 1 minus t to the power of n,"},{"Start":"02:59.415 ","End":"03:03.625","Text":"and we have to take the derivative of that."},{"Start":"03:03.625 ","End":"03:11.210","Text":"That would be equal to n times 1"},{"Start":"03:11.210 ","End":"03:18.200","Text":"divided by 1 minus t to the power of n minus 1 times the internal derivative."},{"Start":"03:18.200 ","End":"03:23.200","Text":"That\u0027s 1 divided by 1 minus t squared."},{"Start":"03:23.200 ","End":"03:25.710","Text":"Let\u0027s just simplify that."},{"Start":"03:25.710 ","End":"03:33.883","Text":"That\u0027ll be n times 1 divided by 1 minus t. Now,"},{"Start":"03:33.883 ","End":"03:36.590","Text":"all we have to do is we have to add the exponents"},{"Start":"03:36.590 ","End":"03:40.010","Text":"here and that means that will be n minus 1 plus 2,"},{"Start":"03:40.010 ","End":"03:43.105","Text":"that\u0027ll be n plus 1."},{"Start":"03:43.105 ","End":"03:49.040","Text":"This is the first derivative of the moment generating function."},{"Start":"03:49.040 ","End":"03:53.290","Text":"Now let\u0027s just set t to be equal to 0."},{"Start":"03:53.290 ","End":"03:57.930","Text":"We have n times 1 divided by 1."},{"Start":"03:57.930 ","End":"03:59.670","Text":"1 minus 0 is 1,"},{"Start":"03:59.670 ","End":"04:01.020","Text":"to the power of n minus 1."},{"Start":"04:01.020 ","End":"04:07.650","Text":"That\u0027s all this expression now becomes 1 and so this then becomes n."},{"Start":"04:07.650 ","End":"04:15.300","Text":"This then is the expectation of T. What about the variance?"},{"Start":"04:15.300 ","End":"04:17.274","Text":"In order to calculate the variance,"},{"Start":"04:17.274 ","End":"04:20.490","Text":"we have to first calculate the second moment."},{"Start":"04:20.490 ","End":"04:24.385","Text":"The expectation of T squared."},{"Start":"04:24.385 ","End":"04:30.550","Text":"The second moment is taking the second derivative of the moment"},{"Start":"04:30.550 ","End":"04:36.990","Text":"generating function and setting that to t equaling 0."},{"Start":"04:36.990 ","End":"04:39.660","Text":"The first derivative was this guy right here,"},{"Start":"04:39.660 ","End":"04:45.625","Text":"so let\u0027s just take the derivative of this expression to get the second derivative."},{"Start":"04:45.625 ","End":"04:49.360","Text":"That\u0027ll be equal to n,"},{"Start":"04:49.360 ","End":"04:55.915","Text":"this guy times n plus 1 times"},{"Start":"04:55.915 ","End":"05:03.630","Text":"1 divided by 1 minus t to the power of n times the internal derivative."},{"Start":"05:03.630 ","End":"05:08.085","Text":"That\u0027s 1 divided by 1 minus t squared."},{"Start":"05:08.085 ","End":"05:12.695","Text":"Again, we have to set t to be equal to 0."},{"Start":"05:12.695 ","End":"05:18.525","Text":"Well, when we set t to equal 0 then this expression becomes 1."},{"Start":"05:18.525 ","End":"05:21.495","Text":"This expression becomes 1."},{"Start":"05:21.495 ","End":"05:26.150","Text":"We have here n times n plus 1,"},{"Start":"05:26.150 ","End":"05:27.815","Text":"that\u0027s the second moment."},{"Start":"05:27.815 ","End":"05:30.710","Text":"That\u0027s the expectation of T squared."},{"Start":"05:30.710 ","End":"05:33.310","Text":"What\u0027s the variance of T?"},{"Start":"05:33.310 ","End":"05:35.130","Text":"In all cases,"},{"Start":"05:35.130 ","End":"05:41.945","Text":"the variance of T equals to the expectation of T squared minus"},{"Start":"05:41.945 ","End":"05:49.980","Text":"the expectation squared of T. In our case that\u0027s n times n plus 1,"},{"Start":"05:49.980 ","End":"05:52.185","Text":"that\u0027s the expectation of T-squared"},{"Start":"05:52.185 ","End":"05:57.225","Text":"minus the expectation squared of T. The expectation of T,"},{"Start":"05:57.225 ","End":"05:59.625","Text":"that\u0027s n, so that\u0027ll be n squared."},{"Start":"05:59.625 ","End":"06:05.670","Text":"That equals to n squared plus n minus n squared,"},{"Start":"06:05.670 ","End":"06:12.975","Text":"so that will be n. The variance of T is also n."},{"Start":"06:12.975 ","End":"06:16.580","Text":"We have the expectation of T equaling n and the variance of"},{"Start":"06:16.580 ","End":"06:20.785","Text":"T equaling n. In this section,"},{"Start":"06:20.785 ","End":"06:23.870","Text":"we let z be equal to t minus the expectation of"},{"Start":"06:23.870 ","End":"06:27.080","Text":"T divided by standard deviation of T. That means that"},{"Start":"06:27.080 ","End":"06:35.000","Text":"z is the standardization of T. We\u0027re asked to find the moment generating function of z."},{"Start":"06:35.000 ","End":"06:38.250","Text":"What do we know? Again,"},{"Start":"06:38.250 ","End":"06:41.390","Text":"we\u0027re given that z equals to T minus"},{"Start":"06:41.390 ","End":"06:45.650","Text":"the expectation of T divided by the standard deviation of T."},{"Start":"06:45.650 ","End":"06:48.830","Text":"We know from previous sections that"},{"Start":"06:48.830 ","End":"06:53.588","Text":"the expectation of T that equals to n. The variance of T,"},{"Start":"06:53.588 ","End":"06:57.830","Text":"that equals to n. That means that the standard deviation of T,"},{"Start":"06:57.830 ","End":"07:04.425","Text":"that equals to the square root of n. Let\u0027s just plug this values in."},{"Start":"07:04.425 ","End":"07:09.919","Text":"That equals z equals to T minus n,"},{"Start":"07:09.919 ","End":"07:13.630","Text":"the expectation of T divided by the standard deviation of T."},{"Start":"07:13.630 ","End":"07:18.760","Text":"That\u0027s the square root of n. Let\u0027s just write this a little bit differently."},{"Start":"07:18.760 ","End":"07:22.705","Text":"That\u0027ll be T divided by the square root of n"},{"Start":"07:22.705 ","End":"07:27.730","Text":"minus n divided by the square root of n. That"},{"Start":"07:27.730 ","End":"07:32.590","Text":"equals to n to the power of minus a 1/2 times"},{"Start":"07:32.590 ","End":"07:39.910","Text":"T minus n to the power of 1/2."},{"Start":"07:39.910 ","End":"07:45.315","Text":"This is a very comfortable way of looking at z,"},{"Start":"07:45.315 ","End":"07:50.525","Text":"looking at the standardization of T. Why is that?"},{"Start":"07:50.525 ","End":"07:54.200","Text":"Because this is a linear transformation."},{"Start":"07:54.200 ","End":"07:58.130","Text":"If we take a look at the template of a linear transformation,"},{"Start":"07:58.130 ","End":"08:02.645","Text":"we see that z equals to aT plus b."},{"Start":"08:02.645 ","End":"08:05.285","Text":"That\u0027s the template of the linear transformation."},{"Start":"08:05.285 ","End":"08:07.505","Text":"In our case,"},{"Start":"08:07.505 ","End":"08:12.020","Text":"a will be equal to n to the power of minus 1/2,"},{"Start":"08:12.020 ","End":"08:20.815","Text":"and b will be equal to minus n to the power of 1/2."},{"Start":"08:20.815 ","End":"08:22.425","Text":"Now we\u0027ve gotten this far,"},{"Start":"08:22.425 ","End":"08:26.460","Text":"what about the moment generating function of z?"},{"Start":"08:26.460 ","End":"08:29.220","Text":"Let\u0027s take a look at this."},{"Start":"08:29.220 ","End":"08:35.900","Text":"We learned that the moment generating function of a linear transformation ax,"},{"Start":"08:35.900 ","End":"08:39.555","Text":"plus b at point t,"},{"Start":"08:39.555 ","End":"08:45.350","Text":"that equals to e to the power of bt times"},{"Start":"08:45.350 ","End":"08:51.390","Text":"the moment generating function of x at point at."},{"Start":"08:51.390 ","End":"09:00.540","Text":"Now we can input these numbers into here."},{"Start":"09:00.540 ","End":"09:03.960","Text":"What was our moment generating function for T?"},{"Start":"09:03.960 ","End":"09:06.525","Text":"The moment generating function for T,"},{"Start":"09:06.525 ","End":"09:16.095","Text":"if we remember was 1 divided by 1 minus t to the power of n. Let\u0027s just continue here."},{"Start":"09:16.095 ","End":"09:21.955","Text":"The moment generating function of z,"},{"Start":"09:21.955 ","End":"09:29.510","Text":"that\u0027s ax plus b or at plus b at point t. That equals to e to the power."},{"Start":"09:29.510 ","End":"09:35.020","Text":"What\u0027s b? b here is minus n to the power of 1/2,"},{"Start":"09:35.020 ","End":"09:40.125","Text":"so that\u0027s minus n to the power of 1/2 times t times"},{"Start":"09:40.125 ","End":"09:46.695","Text":"the moment generating function of x or T in our case at at."},{"Start":"09:46.695 ","End":"09:48.570","Text":"Anytime we see t here,"},{"Start":"09:48.570 ","End":"09:50.895","Text":"we just have to plug in at."},{"Start":"09:50.895 ","End":"09:55.890","Text":"That\u0027ll be equal to 1 divided by 1 minus."},{"Start":"09:55.890 ","End":"09:58.260","Text":"Instead of t we have at."},{"Start":"09:58.260 ","End":"10:01.800","Text":"What\u0027s a? A is n to the power of minus 1/2."},{"Start":"10:01.800 ","End":"10:09.750","Text":"It\u0027s n to the power of minus 1/2 times t and all this to the power"},{"Start":"10:09.750 ","End":"10:17.552","Text":"of n. This then is the moment generating function of z,"},{"Start":"10:17.552 ","End":"10:22.760","Text":"the linear transformation or the standardization of T."}],"ID":13180},{"Watched":false,"Name":"Exercise 4","Duration":"7m 26s","ChapterTopicVideoID":12702,"CourseChapterTopicPlaylistID":245054,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.680","Text":"The moment generating function of"},{"Start":"00:01.680 ","End":"00:05.310","Text":"the normal probability distribution is given by the following formula."},{"Start":"00:05.310 ","End":"00:07.125","Text":"That\u0027s M_X of T,"},{"Start":"00:07.125 ","End":"00:09.900","Text":"that equals to e to the power of Mu t"},{"Start":"00:09.900 ","End":"00:13.185","Text":"plus Sigma squared t squared divided by 2 for any t,"},{"Start":"00:13.185 ","End":"00:18.525","Text":"where x has a normal distribution with parameters Mu and Sigma squared."},{"Start":"00:18.525 ","End":"00:23.115","Text":"We\u0027re asked to prove that if y equals to 2 times x,"},{"Start":"00:23.115 ","End":"00:30.345","Text":"then y has a normal distribution with parameters 2 Mu and 4 Sigma squared."},{"Start":"00:30.345 ","End":"00:33.330","Text":"Let\u0027s first of all take a look at Y."},{"Start":"00:33.330 ","End":"00:37.200","Text":"Now, Y is a linear transformation and"},{"Start":"00:37.200 ","End":"00:42.300","Text":"the template of a linear transformation is a times X plus b."},{"Start":"00:42.300 ","End":"00:50.675","Text":"In our case, a here would be equal to 2 and b would be equal to 0. Now, what else?"},{"Start":"00:50.675 ","End":"00:55.325","Text":"Well, the moment generating function of a linear transformation,"},{"Start":"00:55.325 ","End":"00:57.260","Text":"at point t, well,"},{"Start":"00:57.260 ","End":"01:00.900","Text":"that equals to e to the power of b t"},{"Start":"01:00.900 ","End":"01:06.235","Text":"times the moment generating function of X at point at."},{"Start":"01:06.235 ","End":"01:08.540","Text":"Now, let\u0027s just plug in"},{"Start":"01:08.540 ","End":"01:15.025","Text":"these numbers and see what happens to the moment generating function of Y."},{"Start":"01:15.025 ","End":"01:20.015","Text":"Well, the moment generating function of Y at point t,"},{"Start":"01:20.015 ","End":"01:21.950","Text":"that\u0027s the linear transformation here,"},{"Start":"01:21.950 ","End":"01:25.830","Text":"so that equals to e to the power of bt."},{"Start":"01:25.830 ","End":"01:32.120","Text":"Now b equals 0, so that\u0027ll be 0 times t times the moment generating function of X."},{"Start":"01:32.120 ","End":"01:35.780","Text":"Well, that\u0027s it right here at point at."},{"Start":"01:35.780 ","End":"01:37.895","Text":"Instead of t here,"},{"Start":"01:37.895 ","End":"01:39.605","Text":"I have to plug in at."},{"Start":"01:39.605 ","End":"01:41.480","Text":"Let\u0027s get to it."},{"Start":"01:41.480 ","End":"01:44.450","Text":"That\u0027ll be e to the power of Mu,"},{"Start":"01:44.450 ","End":"01:46.535","Text":"and now instead of t,"},{"Start":"01:46.535 ","End":"01:47.720","Text":"that\u0027ll be at,"},{"Start":"01:47.720 ","End":"01:49.040","Text":"where a equals 2."},{"Start":"01:49.040 ","End":"01:54.970","Text":"There will be times 2t plus Sigma squared."},{"Start":"01:55.100 ","End":"01:59.570","Text":"Instead of t squared, I have to put in at squared."},{"Start":"01:59.570 ","End":"02:05.590","Text":"Now a is 2, so there will be 2t squared divided by 2."},{"Start":"02:05.590 ","End":"02:08.030","Text":"Let\u0027s just clean this up."},{"Start":"02:08.030 ","End":"02:14.870","Text":"That equals to 1 times now e to the power of 2 Mu t,"},{"Start":"02:14.870 ","End":"02:16.940","Text":"you just switch these guys around,"},{"Start":"02:16.940 ","End":"02:23.640","Text":"plus 4 Sigma squared t squared divided by 2."},{"Start":"02:23.640 ","End":"02:27.810","Text":"I just square this expression right here."},{"Start":"02:27.810 ","End":"02:29.790","Text":"That becomes 4t squared."},{"Start":"02:29.790 ","End":"02:32.220","Text":"Then I move the 4 over here."},{"Start":"02:32.220 ","End":"02:36.255","Text":"That\u0027ll be 4 Sigma squared t squared divided by 2."},{"Start":"02:36.255 ","End":"02:40.320","Text":"Then again that equals to e, without the 1,"},{"Start":"02:40.320 ","End":"02:45.975","Text":"e to the power of 2 Mu t. Let\u0027s just put that here."},{"Start":"02:45.975 ","End":"02:53.225","Text":"Plus 4 Sigma squared times t squared divided by 2."},{"Start":"02:53.225 ","End":"02:56.510","Text":"Now, let\u0027s take a look at the moment generating"},{"Start":"02:56.510 ","End":"03:00.790","Text":"function of X and the moment generating function of Y."},{"Start":"03:00.790 ","End":"03:04.970","Text":"Now, we see they\u0027re pretty much the same except for here."},{"Start":"03:04.970 ","End":"03:13.045","Text":"For X, the parameter Mu is this guy right here and instead of Mu in Y the we have 2 Mu."},{"Start":"03:13.045 ","End":"03:15.600","Text":"Instead of Sigma squared,"},{"Start":"03:15.600 ","End":"03:17.465","Text":"we have 4 Sigma squared."},{"Start":"03:17.465 ","End":"03:19.895","Text":"Everything else is pretty much the same."},{"Start":"03:19.895 ","End":"03:26.945","Text":"Now, we have 1 of the characteristics of the moment generating functions, if we recall,"},{"Start":"03:26.945 ","End":"03:30.860","Text":"is that there\u0027s a 1-to-1 correlation between the moment"},{"Start":"03:30.860 ","End":"03:35.920","Text":"generating function and the random variable or the distribution."},{"Start":"03:35.920 ","End":"03:46.170","Text":"We can say definitively that Y has a normal distribution because the format is the same."},{"Start":"03:46.170 ","End":"03:49.065","Text":"It\u0027s just that the parameters are different."},{"Start":"03:49.065 ","End":"03:51.405","Text":"What are the parameters for Y?"},{"Start":"03:51.405 ","End":"03:53.325","Text":"Well, instead of Mu here,"},{"Start":"03:53.325 ","End":"03:54.840","Text":"it\u0027s 2 Mu,"},{"Start":"03:54.840 ","End":"03:56.925","Text":"and instead of Sigma squared,"},{"Start":"03:56.925 ","End":"03:59.385","Text":"it\u0027s 4 Sigma squared."},{"Start":"03:59.385 ","End":"04:02.670","Text":"This is exactly what we had to prove."},{"Start":"04:02.670 ","End":"04:06.150","Text":"We got from here to here."},{"Start":"04:06.150 ","End":"04:13.910","Text":"Then Y, again has a normal distribution with parameters 2 Mu and 4 Sigma squared."},{"Start":"04:13.910 ","End":"04:19.430","Text":"In this section, we\u0027re asked to prove that if t equals to X_1 plus X_2 and"},{"Start":"04:19.430 ","End":"04:24.695","Text":"X_1 and X_2 are independent variables with the same normal probability distribution,"},{"Start":"04:24.695 ","End":"04:31.030","Text":"then T has a normal distribution with parameters 2 Mu and 2 Sigma squared."},{"Start":"04:31.030 ","End":"04:37.685","Text":"What do we know? Well, we know that t equals to X_1 plus X_2."},{"Start":"04:37.685 ","End":"04:44.060","Text":"We know that X_1 has a normal distribution with parameters Mu and Sigma squared,"},{"Start":"04:44.060 ","End":"04:51.910","Text":"and X_2 has the same normal distribution with parameters Mu and Sigma squared."},{"Start":"04:51.910 ","End":"05:00.615","Text":"What else? We know that the moment generating function of the sum of 2 random variables,"},{"Start":"05:00.615 ","End":"05:06.395","Text":"that equals to the expectation of e to the power"},{"Start":"05:06.395 ","End":"05:14.735","Text":"of tx times the expectation of e to the power of ty."},{"Start":"05:14.735 ","End":"05:18.840","Text":"Now that equals to the moment generating function of"},{"Start":"05:18.840 ","End":"05:23.850","Text":"X times the moment generating function of Y."},{"Start":"05:23.850 ","End":"05:29.025","Text":"Now, the moment generating function of X,"},{"Start":"05:29.025 ","End":"05:32.630","Text":"it was given to us because X has a normal distribution and it really"},{"Start":"05:32.630 ","End":"05:36.050","Text":"doesn\u0027t matter if it\u0027s X_1 or X_2 that equals to"},{"Start":"05:36.050 ","End":"05:43.930","Text":"e to the power of Mu t plus Sigma squared t squared divided by 2."},{"Start":"05:44.090 ","End":"05:47.910","Text":"What\u0027s the moment generating function of t,"},{"Start":"05:47.910 ","End":"05:53.455","Text":"which is the sum of X_1 plus X_2 at point t?"},{"Start":"05:53.455 ","End":"05:59.030","Text":"Well, that equals to the multiplication of the moment generating functions."},{"Start":"05:59.030 ","End":"06:02.990","Text":"Now, what\u0027s the moment generating function of X_1?"},{"Start":"06:02.990 ","End":"06:11.250","Text":"Well, that\u0027s e to the power of Mu t plus Sigma squared t squared divided by 2."},{"Start":"06:11.250 ","End":"06:14.520","Text":"What about the moment generating function of X_2?"},{"Start":"06:14.520 ","End":"06:15.795","Text":"Well, that\u0027s the same,"},{"Start":"06:15.795 ","End":"06:19.660","Text":"because they have the same normal probability distribution."},{"Start":"06:19.660 ","End":"06:23.770","Text":"All we have to do is take this expression and square it."},{"Start":"06:23.770 ","End":"06:25.810","Text":"Now, let\u0027s just clean that up."},{"Start":"06:25.810 ","End":"06:31.345","Text":"That equals to e to the power of 2 Mu times t,"},{"Start":"06:31.345 ","End":"06:33.235","Text":"we put brackets around here,"},{"Start":"06:33.235 ","End":"06:40.570","Text":"plus 2 Sigma squared times t squared divided by 2."},{"Start":"06:40.570 ","End":"06:47.290","Text":"Now, there\u0027s a characteristic of a moment generating function that says that there\u0027s"},{"Start":"06:47.290 ","End":"06:50.930","Text":"a 1-to-1 correlation between the moment generating"},{"Start":"06:50.930 ","End":"06:55.370","Text":"function and the random variable or the distribution."},{"Start":"06:55.370 ","End":"06:59.450","Text":"Here we see the same format of a moment"},{"Start":"06:59.450 ","End":"07:03.770","Text":"generating function of a normal probability distribution."},{"Start":"07:03.770 ","End":"07:05.585","Text":"That\u0027s right here, we see that here."},{"Start":"07:05.585 ","End":"07:07.850","Text":"Now, instead of Mu here,"},{"Start":"07:07.850 ","End":"07:09.695","Text":"we have 2 Mu,"},{"Start":"07:09.695 ","End":"07:11.660","Text":"and instead of Sigma squared,"},{"Start":"07:11.660 ","End":"07:13.745","Text":"we\u0027ll have 2 Sigma squared."},{"Start":"07:13.745 ","End":"07:17.060","Text":"That means that t has"},{"Start":"07:17.060 ","End":"07:25.360","Text":"a normal distribution with parameters 2 Mu and 2 Sigma squared."}],"ID":13181}],"Thumbnail":null,"ID":245054}]

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