Introduction to Linear Combinations
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[{"Name":"Introduction to Linear Combinations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"1m 18s","ChapterTopicVideoID":12761,"CourseChapterTopicPlaylistID":245055,"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":"In this chapter, we\u0027ll be talking about"},{"Start":"00:01.680 ","End":"00:04.980","Text":"Linear Combinations of Normal Probability Distributions."},{"Start":"00:04.980 ","End":"00:07.890","Text":"Now, all linear combinations of variables with"},{"Start":"00:07.890 ","End":"00:13.395","Text":"normal probability distributions themselves have a normal probability distribution."},{"Start":"00:13.395 ","End":"00:17.880","Text":"That means that if we have a random variable x and it has"},{"Start":"00:17.880 ","End":"00:23.775","Text":"a normal probability distribution with parameter Mu and Sigma squared,"},{"Start":"00:23.775 ","End":"00:27.960","Text":"then what\u0027s a linear combination of x?"},{"Start":"00:27.960 ","End":"00:30.450","Text":"Linear combination."},{"Start":"00:30.450 ","End":"00:32.460","Text":"Now, that would be,"},{"Start":"00:32.460 ","End":"00:35.925","Text":"let\u0027s say taking x and adding a constant to it,"},{"Start":"00:35.925 ","End":"00:38.630","Text":"or subtracting a constant to it,"},{"Start":"00:38.630 ","End":"00:47.520","Text":"or maybe multiplying a constant by x or dividing a constant."},{"Start":"00:47.520 ","End":"00:51.135","Text":"Those are examples of linear combinations."},{"Start":"00:51.135 ","End":"00:56.750","Text":"These linear combinations would also have a normal probability distribution."},{"Start":"00:56.750 ","End":"00:59.840","Text":"Now, what\u0027s not a linear combination?"},{"Start":"00:59.840 ","End":"01:03.125","Text":"Well, if we have x and we square it, for example,"},{"Start":"01:03.125 ","End":"01:07.295","Text":"or we take the cube root of x, for example."},{"Start":"01:07.295 ","End":"01:13.275","Text":"These then would not have a normal probability distribution."},{"Start":"01:13.275 ","End":"01:18.210","Text":"Let\u0027s just take a look at an example to see how things work."}],"ID":13240},{"Watched":false,"Name":"Example","Duration":"7m 38s","ChapterTopicVideoID":12762,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this example, we\u0027re given that the height of men in"},{"Start":"00:02.340 ","End":"00:04.800","Text":"England has a normal probability distribution"},{"Start":"00:04.800 ","End":"00:09.885","Text":"with an expectation of 175 centimeters and a standard deviation of 10 centimeters."},{"Start":"00:09.885 ","End":"00:14.160","Text":"The height of women in England has a normal probability distribution with"},{"Start":"00:14.160 ","End":"00:19.370","Text":"an expectation of 165 centimeters and a standard deviation of 8 centimeters."},{"Start":"00:19.370 ","End":"00:21.900","Text":"We\u0027re asked, what are the chances that"},{"Start":"00:21.900 ","End":"00:26.880","Text":"a randomly selected man will be taller than a randomly selected woman?"},{"Start":"00:26.880 ","End":"00:30.990","Text":"The first thing that we want to do is to define our variables."},{"Start":"00:30.990 ","End":"00:38.030","Text":"Let\u0027s define x then as the height of men."},{"Start":"00:38.030 ","End":"00:43.505","Text":"Now, that will have a normal probability distribution where the expectation"},{"Start":"00:43.505 ","End":"00:49.290","Text":"is 175 centimeters and the standard deviation is 10,"},{"Start":"00:49.290 ","End":"00:53.000","Text":"so Sigma squared then would be 10 squared."},{"Start":"00:53.000 ","End":"00:55.265","Text":"Now, what about y?"},{"Start":"00:55.265 ","End":"01:00.530","Text":"Well, y would be the height of women."},{"Start":"01:00.530 ","End":"01:05.825","Text":"Now, that would have a normal probability distribution that\u0027s given to us,"},{"Start":"01:05.825 ","End":"01:08.030","Text":"and the expectation now would be"},{"Start":"01:08.030 ","End":"01:13.850","Text":"165 centimeters and the standard deviation is 8 centimeters."},{"Start":"01:13.850 ","End":"01:15.575","Text":"That means that Sigma squared,"},{"Start":"01:15.575 ","End":"01:18.625","Text":"that\u0027ll be equal to 8 squared."},{"Start":"01:18.625 ","End":"01:21.350","Text":"Again, we\u0027re asked, what are the chances that"},{"Start":"01:21.350 ","End":"01:26.435","Text":"a randomly selected man will be taller than a randomly selected woman?"},{"Start":"01:26.435 ","End":"01:35.050","Text":"We\u0027re looking then for the probability of x being greater than y."},{"Start":"01:36.170 ","End":"01:40.430","Text":"Let\u0027s take this probability and write this out like this."},{"Start":"01:40.430 ","End":"01:46.790","Text":"That\u0027ll be the probability of x minus y being greater than 0."},{"Start":"01:46.790 ","End":"01:48.830","Text":"Now, what\u0027s x minus y?"},{"Start":"01:48.830 ","End":"01:51.675","Text":"Well, let\u0027s define a new variable D,"},{"Start":"01:51.675 ","End":"01:55.535","Text":"that\u0027ll be equal to x minus y, D for difference."},{"Start":"01:55.535 ","End":"02:01.190","Text":"Now, we said that if x has a normal probability distribution and y,"},{"Start":"02:01.190 ","End":"02:04.400","Text":"if that also has a normal probability distribution,"},{"Start":"02:04.400 ","End":"02:07.010","Text":"then the linear combination of"},{"Start":"02:07.010 ","End":"02:10.490","Text":"these variables will also have a normal probability distribution."},{"Start":"02:10.490 ","End":"02:11.825","Text":"Well, this is what we have here."},{"Start":"02:11.825 ","End":"02:17.000","Text":"Here, D is a linear combination of x and y."},{"Start":"02:17.000 ","End":"02:22.145","Text":"Therefore, D also has a normal probability distribution with"},{"Start":"02:22.145 ","End":"02:26.325","Text":"expectation Mu D and"},{"Start":"02:26.325 ","End":"02:32.360","Text":"variance Sigma squared D. If that\u0027s the case,"},{"Start":"02:32.360 ","End":"02:35.510","Text":"what\u0027s the expectation and variance of D?"},{"Start":"02:35.510 ","End":"02:37.115","Text":"Well, let\u0027s figure that out."},{"Start":"02:37.115 ","End":"02:39.290","Text":"That\u0027s the expectation of D,"},{"Start":"02:39.290 ","End":"02:43.040","Text":"that equals to the expectation of x minus y."},{"Start":"02:43.040 ","End":"02:49.075","Text":"That equals the expectation of x minus the expectation of y."},{"Start":"02:49.075 ","End":"02:50.970","Text":"Now, the expectation of x,"},{"Start":"02:50.970 ","End":"02:54.915","Text":"that\u0027s 175 minus 165,"},{"Start":"02:54.915 ","End":"02:56.795","Text":"that\u0027s the expectation of y,"},{"Start":"02:56.795 ","End":"03:00.890","Text":"and that equals to 10 centimeters."},{"Start":"03:00.890 ","End":"03:04.235","Text":"Now, what about the variance of D?"},{"Start":"03:04.235 ","End":"03:08.045","Text":"That equals to the variance of x minus y?"},{"Start":"03:08.045 ","End":"03:12.530","Text":"Well, that equals to the variance of x plus the variance of"},{"Start":"03:12.530 ","End":"03:18.695","Text":"y minus 2 times the covariance of x and y."},{"Start":"03:18.695 ","End":"03:21.500","Text":"Now, x and y are independent variables,"},{"Start":"03:21.500 ","End":"03:24.530","Text":"so that means that the covariance is equal to 0."},{"Start":"03:24.530 ","End":"03:27.770","Text":"Now, that means that the variance of D, well,"},{"Start":"03:27.770 ","End":"03:31.055","Text":"that equals to the variance of x plus the variance of y."},{"Start":"03:31.055 ","End":"03:34.790","Text":"Well, that\u0027s 10 squared plus 8 squared,"},{"Start":"03:34.790 ","End":"03:40.410","Text":"and that equals to 164 centimeter squared."},{"Start":"03:40.910 ","End":"03:47.209","Text":"Therefore, D then now has a normal probability distribution."},{"Start":"03:47.209 ","End":"03:48.890","Text":"Now, we know the parameters,"},{"Start":"03:48.890 ","End":"03:50.185","Text":"Mu of D,"},{"Start":"03:50.185 ","End":"03:52.860","Text":"and that equals to 10 centimeters."},{"Start":"03:52.860 ","End":"03:59.685","Text":"Sigma squared, the variance of D, that equals 164."},{"Start":"03:59.685 ","End":"04:03.785","Text":"Excellent. Now, we can calculate"},{"Start":"04:03.785 ","End":"04:10.385","Text":"the probability of D being greater than 0. Let\u0027s do that."},{"Start":"04:10.385 ","End":"04:15.185","Text":"We\u0027re looking for the probability of D being greater than 0."},{"Start":"04:15.185 ","End":"04:18.620","Text":"Now, how do we calculate this probability?"},{"Start":"04:18.620 ","End":"04:24.335","Text":"Well, we have to standardize D in order to calculate this probability. Let\u0027s do that."},{"Start":"04:24.335 ","End":"04:29.690","Text":"That\u0027ll be the probability of D minus Mu divided by Sigma,"},{"Start":"04:29.690 ","End":"04:34.295","Text":"that has to be greater than 0 minus Mu divided by Sigma."},{"Start":"04:34.295 ","End":"04:36.875","Text":"Now, what does that equal to?"},{"Start":"04:36.875 ","End":"04:38.270","Text":"That\u0027s the probability."},{"Start":"04:38.270 ","End":"04:40.940","Text":"With this expression right here, well, that\u0027s z."},{"Start":"04:40.940 ","End":"04:43.100","Text":"That has to be greater than this."},{"Start":"04:43.100 ","End":"04:44.975","Text":"Well, that\u0027s 0 minus 10,"},{"Start":"04:44.975 ","End":"04:50.330","Text":"Mu is 10 divided by the square root of 164."},{"Start":"04:50.330 ","End":"04:55.705","Text":"That\u0027ll be 0 minus 10 divided by the square root of 164,"},{"Start":"04:55.705 ","End":"05:04.450","Text":"and that equals to the probability of z being greater than minus 0.78."},{"Start":"05:04.450 ","End":"05:08.720","Text":"Let\u0027s see how this thing looks on a graph."},{"Start":"05:08.720 ","End":"05:14.910","Text":"Here, we have the normal density function for D,"},{"Start":"05:14.910 ","End":"05:19.355","Text":"this is a D-axis and the expectation here is 10."},{"Start":"05:19.355 ","End":"05:25.500","Text":"We want to know the probability of D being greater than 0."},{"Start":"05:25.500 ","End":"05:31.085","Text":"That means that we\u0027re looking to calculate this area in yellow under the graph."},{"Start":"05:31.085 ","End":"05:34.790","Text":"Now, when we go to the z-axis, well,"},{"Start":"05:34.790 ","End":"05:40.385","Text":"0 becomes minus 0.78."},{"Start":"05:40.385 ","End":"05:47.960","Text":"Now, when we\u0027re looking at the normal standard table,"},{"Start":"05:47.960 ","End":"05:50.600","Text":"in order to calculate these probabilities,"},{"Start":"05:50.600 ","End":"05:57.650","Text":"we\u0027re given the probability of z being less than some value right here, well,"},{"Start":"05:57.650 ","End":"06:01.740","Text":"that equals to Phi of x."},{"Start":"06:02.090 ","End":"06:08.810","Text":"Phi of x gives me this area right here from minus infinity until the value,"},{"Start":"06:08.810 ","End":"06:10.640","Text":"and we don\u0027t want that."},{"Start":"06:10.640 ","End":"06:12.515","Text":"What are we looking for?"},{"Start":"06:12.515 ","End":"06:18.055","Text":"We\u0027re looking for 1 minus the probability of"},{"Start":"06:18.055 ","End":"06:25.110","Text":"z being greater than some value x."},{"Start":"06:25.110 ","End":"06:28.755","Text":"1 minus this right here, well,"},{"Start":"06:28.755 ","End":"06:34.320","Text":"that means that that equals to Phi of x."},{"Start":"06:34.320 ","End":"06:38.975","Text":"Now, in our case, x is this guy right here,"},{"Start":"06:38.975 ","End":"06:40.220","Text":"but with a plus sign."},{"Start":"06:40.220 ","End":"06:45.260","Text":"So we\u0027re looking for Phi of 0.78."},{"Start":"06:45.260 ","End":"06:47.615","Text":"Now, what does that equal to?"},{"Start":"06:47.615 ","End":"06:52.160","Text":"Well, let\u0027s take a look at our normal table."},{"Start":"06:52.160 ","End":"06:54.245","Text":"Here\u0027s a normal table,"},{"Start":"06:54.245 ","End":"07:00.060","Text":"and we\u0027re looking for Phi of 0.78."},{"Start":"07:00.060 ","End":"07:02.630","Text":"That means that we\u0027re looking for 0.7,"},{"Start":"07:02.630 ","End":"07:04.385","Text":"that\u0027s the row right here,"},{"Start":"07:04.385 ","End":"07:08.225","Text":"and 0.08, that\u0027s the column right here."},{"Start":"07:08.225 ","End":"07:13.720","Text":"We\u0027re looking at this value right here, 0.7823."},{"Start":"07:13.720 ","End":"07:20.745","Text":"This means that that equals to 0.7823,"},{"Start":"07:20.745 ","End":"07:27.845","Text":"so this then is the probability of D being greater than 0,"},{"Start":"07:27.845 ","End":"07:32.120","Text":"or this then is the probability of a"},{"Start":"07:32.120 ","End":"07:38.430","Text":"randomly selected male being taller than a randomly selected female."}],"ID":13241},{"Watched":false,"Name":"Exercise 1","Duration":"6m 50s","ChapterTopicVideoID":12763,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this question, we\u0027re given that the weight of men in"},{"Start":"00:02.310 ","End":"00:04.769","Text":"England has a normal probability distribution"},{"Start":"00:04.769 ","End":"00:09.540","Text":"with an expectation of 75 kilograms and a standard deviation of 10 kilograms."},{"Start":"00:09.540 ","End":"00:13.530","Text":"The weight of women in England has a normal probability distribution with"},{"Start":"00:13.530 ","End":"00:18.120","Text":"an expectation of 65 kilograms and a standard deviation of 8 kilograms,"},{"Start":"00:18.120 ","End":"00:20.300","Text":"and we\u0027re asked what are the chances that"},{"Start":"00:20.300 ","End":"00:25.325","Text":"a randomly selected woman weighs more than a randomly selected man?"},{"Start":"00:25.325 ","End":"00:28.190","Text":"First of all, let\u0027s define our variables."},{"Start":"00:28.190 ","End":"00:33.760","Text":"Let\u0027s define X as the weight of a man."},{"Start":"00:33.760 ","End":"00:39.200","Text":"Now, that has a normal probability distribution with"},{"Start":"00:39.200 ","End":"00:46.580","Text":"an expectation of 75 kilograms and a standard deviation of 10 kilograms,"},{"Start":"00:46.580 ","End":"00:50.080","Text":"that means that Sigma squared equals to 10 squared."},{"Start":"00:50.080 ","End":"00:52.740","Text":"What about Y? Well, Y,"},{"Start":"00:52.740 ","End":"00:57.298","Text":"we\u0027ll define as the weight of a woman."},{"Start":"00:57.298 ","End":"01:01.670","Text":"That has a normal probability distribution where"},{"Start":"01:01.670 ","End":"01:07.758","Text":"the expectation is 65 kilograms and the standard deviation is 8 kilograms."},{"Start":"01:07.758 ","End":"01:11.735","Text":"That means that Sigma squared here equals to 8 squared."},{"Start":"01:11.735 ","End":"01:14.030","Text":"We\u0027re asked about the chances that"},{"Start":"01:14.030 ","End":"01:19.930","Text":"a randomly selected woman weighs more than a randomly selected man."},{"Start":"01:19.930 ","End":"01:27.405","Text":"We\u0027re looking at the probability of Y being greater than X."},{"Start":"01:27.405 ","End":"01:30.410","Text":"Let\u0027s just write this down a little bit better."},{"Start":"01:30.410 ","End":"01:36.725","Text":"That\u0027ll be the probability now of X minus Y,"},{"Start":"01:36.725 ","End":"01:39.800","Text":"that has to be less than 0."},{"Start":"01:39.800 ","End":"01:46.555","Text":"The weight of a man minus the weight of a woman,"},{"Start":"01:46.555 ","End":"01:49.670","Text":"that has to be less than 0."},{"Start":"01:50.120 ","End":"01:56.200","Text":"Let\u0027s now take a look at this difference right here."},{"Start":"01:56.200 ","End":"02:03.365","Text":"Let\u0027s define the variable D as X minus Y."},{"Start":"02:03.365 ","End":"02:08.500","Text":"We see that D is a linear combination of X and Y."},{"Start":"02:08.500 ","End":"02:10.825","Text":"That means that D then has"},{"Start":"02:10.825 ","End":"02:14.080","Text":"a normal probability distribution with"},{"Start":"02:14.080 ","End":"02:18.895","Text":"an expectation of D and a standard deviation of D as well."},{"Start":"02:18.895 ","End":"02:25.480","Text":"Now, how do we calculate then the expectation of D and the standard deviation of D?"},{"Start":"02:25.480 ","End":"02:28.580","Text":"Well, the expectation of D,"},{"Start":"02:28.580 ","End":"02:32.540","Text":"that equals to the expectation of X minus Y."},{"Start":"02:32.540 ","End":"02:37.610","Text":"Now, that equals to the expectation of X minus the expectation of"},{"Start":"02:37.610 ","End":"02:44.570","Text":"Y and that equals the expectation of X,"},{"Start":"02:44.570 ","End":"02:46.580","Text":"well, that\u0027s 75,"},{"Start":"02:46.580 ","End":"02:50.890","Text":"minus the expectation of Y, that\u0027s 65."},{"Start":"02:50.890 ","End":"02:53.090","Text":"That means that equals to 10."},{"Start":"02:53.090 ","End":"02:55.310","Text":"Now, what about the variance of D?"},{"Start":"02:55.310 ","End":"02:59.745","Text":"Well, that\u0027s the variance of X minus Y."},{"Start":"02:59.745 ","End":"03:05.060","Text":"That means that this equals to the variance of X plus the variance of"},{"Start":"03:05.060 ","End":"03:10.810","Text":"Y minus 2 times the covariance of X and Y."},{"Start":"03:10.810 ","End":"03:14.450","Text":"Now, here we assume independence between X and Y"},{"Start":"03:14.450 ","End":"03:19.070","Text":"between the weight of men and women so that means that this equals to 0."},{"Start":"03:19.070 ","End":"03:20.960","Text":"Now, that equals, as we said,"},{"Start":"03:20.960 ","End":"03:22.838","Text":"to the sum of the variances."},{"Start":"03:22.838 ","End":"03:27.248","Text":"That means that we\u0027re looking at 10 squared plus 8 squared,"},{"Start":"03:27.248 ","End":"03:30.405","Text":"and that equals to 164."},{"Start":"03:30.405 ","End":"03:36.515","Text":"D has a normal probability distribution"},{"Start":"03:36.515 ","End":"03:45.220","Text":"where Mu equals to 10 and Sigma squared equals to 164."},{"Start":"03:45.220 ","End":"03:49.370","Text":"Now that we know what the distribution is of D,"},{"Start":"03:49.370 ","End":"03:54.530","Text":"and what we want to know is to calculate the probability of D,"},{"Start":"03:54.530 ","End":"03:58.685","Text":"that\u0027s x minus y being less than 0."},{"Start":"03:58.685 ","End":"04:05.360","Text":"Let\u0027s do that. The probability of D being less than 0,"},{"Start":"04:05.360 ","End":"04:10.220","Text":"well, we have to normalize this in order to calculate the probability."},{"Start":"04:10.220 ","End":"04:16.400","Text":"That means the probability of D minus Mu divided by Sigma,"},{"Start":"04:16.400 ","End":"04:21.665","Text":"well, that has to be less than 0 minus Mu divided by Sigma."},{"Start":"04:21.665 ","End":"04:24.290","Text":"Now, that equals to the following,"},{"Start":"04:24.290 ","End":"04:26.474","Text":"that\u0027s the probability of."},{"Start":"04:26.474 ","End":"04:28.010","Text":"This expression right here, well,"},{"Start":"04:28.010 ","End":"04:30.905","Text":"that\u0027s Z and that has to be less than."},{"Start":"04:30.905 ","End":"04:36.080","Text":"Now, Mu is 10 and Sigma is the square root of 164,"},{"Start":"04:36.080 ","End":"04:41.645","Text":"so that\u0027ll be 0 minus 10 divided by the square root of 164."},{"Start":"04:41.645 ","End":"04:49.475","Text":"Now, that equals to the probability of Z being less than minus 0.78."},{"Start":"04:49.475 ","End":"04:53.200","Text":"Let\u0027s see how this looks on a graph."},{"Start":"04:53.200 ","End":"04:58.580","Text":"Here\u0027s our density function and this is the D axis."},{"Start":"04:58.580 ","End":"05:00.980","Text":"The expectation of D is 10,"},{"Start":"05:00.980 ","End":"05:03.825","Text":"so right here, that\u0027ll be 10."},{"Start":"05:03.825 ","End":"05:07.640","Text":"We\u0027re looking for the probability of D being less than 0,"},{"Start":"05:07.640 ","End":"05:10.503","Text":"so here would be 0,"},{"Start":"05:10.503 ","End":"05:13.820","Text":"and we\u0027re looking for this area right here."},{"Start":"05:13.820 ","End":"05:20.030","Text":"We want to calculate this area right here from minus infinity to 0 and the D axis."},{"Start":"05:20.030 ","End":"05:21.950","Text":"Well, on the Z axis,"},{"Start":"05:21.950 ","End":"05:23.750","Text":"the normalized axis,"},{"Start":"05:23.750 ","End":"05:28.285","Text":"then 0 becomes minus 0.78."},{"Start":"05:28.285 ","End":"05:33.740","Text":"We\u0027re looking on the normalized scale to calculate"},{"Start":"05:33.740 ","End":"05:39.480","Text":"the probability from minus infinity to minus 0.78."},{"Start":"05:39.480 ","End":"05:43.925","Text":"Again, we\u0027re looking for the probability of D being less than 0,"},{"Start":"05:43.925 ","End":"05:50.860","Text":"that equals to the probability of Z being less than minus 0.78."},{"Start":"05:50.860 ","End":"05:56.345","Text":"That means that we\u0027re looking for Phi of minus 0.78."},{"Start":"05:56.345 ","End":"06:01.265","Text":"Now, since we don\u0027t have any negative numbers in our table,"},{"Start":"06:01.265 ","End":"06:09.345","Text":"well, that equals to 1 minus Phi of 0.78."},{"Start":"06:09.345 ","End":"06:12.165","Text":"Now, what does that equal to?"},{"Start":"06:12.165 ","End":"06:18.880","Text":"Well, the value of Phi at 0.78,"},{"Start":"06:18.880 ","End":"06:23.330","Text":"that equals to 0.7823."},{"Start":"06:23.330 ","End":"06:27.920","Text":"Now, I urge you to go to the table and look up this value."},{"Start":"06:27.920 ","End":"06:30.399","Text":"You\u0027ll do by yourselves."},{"Start":"06:30.399 ","End":"06:34.950","Text":"This then equals to 1 minus 0.7823,"},{"Start":"06:34.950 ","End":"06:40.462","Text":"and that equals to 0.2177."},{"Start":"06:40.462 ","End":"06:44.060","Text":"This then is the probability of"},{"Start":"06:44.060 ","End":"06:50.670","Text":"a randomly selected man weighing less than a randomly selected woman."}],"ID":13242},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"5m 36s","ChapterTopicVideoID":12764,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this question, we\u0027re given that a person\u0027s annual spending on clothing has"},{"Start":"00:03.810 ","End":"00:06.570","Text":"a normal probability distribution with an expectation of"},{"Start":"00:06.570 ","End":"00:10.590","Text":"$3,000 and a standard deviation of $1,000."},{"Start":"00:10.590 ","End":"00:15.060","Text":"The annual spendings on entertainment has a normal probability distribution with"},{"Start":"00:15.060 ","End":"00:20.205","Text":"an expectation of $4,000 and a standard deviation of $1,500."},{"Start":"00:20.205 ","End":"00:23.130","Text":"The correlation coefficient of annual spending on"},{"Start":"00:23.130 ","End":"00:27.915","Text":"clothing and annual spending on entertainment is 0.6."},{"Start":"00:27.915 ","End":"00:31.565","Text":"Now, we\u0027re asked, what are the expectation and standard deviation"},{"Start":"00:31.565 ","End":"00:35.660","Text":"of the total annual spending on clothing and entertainment?"},{"Start":"00:35.660 ","End":"00:38.629","Text":"First of all, let\u0027s define our variables."},{"Start":"00:38.629 ","End":"00:44.395","Text":"Let\u0027s define x as the annual spending on clothes."},{"Start":"00:44.395 ","End":"00:49.550","Text":"Now, that has a normal probability distribution where"},{"Start":"00:49.550 ","End":"00:55.570","Text":"the expectation of that would be equal to $3,000,"},{"Start":"00:55.570 ","End":"01:00.120","Text":"and the standard deviation, well that\u0027s $1,000."},{"Start":"01:00.120 ","End":"01:04.770","Text":"Sigma squared x would be equal to 1,000 squared."},{"Start":"01:04.770 ","End":"01:10.520","Text":"What about y? Y would be the annual spending on entertainment."},{"Start":"01:10.520 ","End":"01:16.355","Text":"Now, that has a normal probability distribution where the expectation of y,"},{"Start":"01:16.355 ","End":"01:23.400","Text":"that equals to $4,000 and the standard deviation is $1,500,"},{"Start":"01:23.400 ","End":"01:29.140","Text":"so Sigma squared y would be 1,500 squared."},{"Start":"01:29.780 ","End":"01:32.280","Text":"Let\u0027s see what we\u0027re asked."},{"Start":"01:32.280 ","End":"01:35.180","Text":"We\u0027re asked about the expectation and standard deviation of"},{"Start":"01:35.180 ","End":"01:39.185","Text":"the total annual spending on clothing and entertainment."},{"Start":"01:39.185 ","End":"01:44.925","Text":"That means that we\u0027re asked about x plus y."},{"Start":"01:44.925 ","End":"01:47.510","Text":"Let\u0027s just define a new random variable."},{"Start":"01:47.510 ","End":"01:52.040","Text":"We\u0027ll call that t, and that would be equal to x plus y."},{"Start":"01:52.040 ","End":"01:55.550","Text":"Now, here we have a linear combination."},{"Start":"01:55.550 ","End":"01:58.400","Text":"T is a linear combination of x and y,"},{"Start":"01:58.400 ","End":"02:04.685","Text":"so we know now that t has a normal probability distribution with"},{"Start":"02:04.685 ","End":"02:07.685","Text":"an expectation of Mu of t and"},{"Start":"02:07.685 ","End":"02:13.150","Text":"a standard deviation or a variance of Sigma squared of t. Now,"},{"Start":"02:13.150 ","End":"02:16.970","Text":"we need to calculate these guys right here."},{"Start":"02:16.970 ","End":"02:20.345","Text":"Now, what are we looking for then?"},{"Start":"02:20.345 ","End":"02:25.765","Text":"Well, we\u0027re looking for the expectation of t, this guy right here."},{"Start":"02:25.765 ","End":"02:30.330","Text":"That\u0027s the expectation of x plus y,"},{"Start":"02:30.330 ","End":"02:35.935","Text":"and that equals to the expectation of x plus the expectation of y."},{"Start":"02:35.935 ","End":"02:41.740","Text":"That equals to 3,000 plus 4,000,"},{"Start":"02:42.170 ","End":"02:46.380","Text":"and that equals to 7,000."},{"Start":"02:46.380 ","End":"02:52.780","Text":"So $7,000 is the expectation of the total annual spendings on clothes and entertainment."},{"Start":"02:52.780 ","End":"02:55.160","Text":"Now what about the variance of t?"},{"Start":"02:55.160 ","End":"02:58.705","Text":"Well, that equals to the variance of x plus y."},{"Start":"02:58.705 ","End":"03:03.660","Text":"That equals to the variance of x plus the variance of"},{"Start":"03:03.660 ","End":"03:09.155","Text":"y plus 2 times the covariance of x and y."},{"Start":"03:09.155 ","End":"03:11.740","Text":"Now, we have the variance of x."},{"Start":"03:11.740 ","End":"03:14.190","Text":"That\u0027s 1,000 squared,"},{"Start":"03:14.190 ","End":"03:17.410","Text":"and we have the variance of y, that\u0027s 1,500 squared."},{"Start":"03:17.410 ","End":"03:20.450","Text":"What we don\u0027t have is the covariance of x and y,"},{"Start":"03:20.450 ","End":"03:24.815","Text":"but we do have the correlation coefficient."},{"Start":"03:24.815 ","End":"03:28.970","Text":"We were given that the correlation coefficient between x and y,"},{"Start":"03:28.970 ","End":"03:32.095","Text":"well that equal to 0.6."},{"Start":"03:32.095 ","End":"03:37.190","Text":"Now, what\u0027s the definition of the correlation coefficient?"},{"Start":"03:37.190 ","End":"03:40.525","Text":"That\u0027s the covariance of x and y"},{"Start":"03:40.525 ","End":"03:46.400","Text":"divided by the standard deviation of x times the standard deviation of y."},{"Start":"03:46.400 ","End":"03:53.390","Text":"Now, that means that here we have this standard deviation of x,"},{"Start":"03:53.390 ","End":"03:55.723","Text":"well that\u0027s 1,000,"},{"Start":"03:55.723 ","End":"03:59.835","Text":"the standard deviation of y, that\u0027s 1,500."},{"Start":"03:59.835 ","End":"04:04.990","Text":"That\u0027s the covariance of x and y."},{"Start":"04:05.570 ","End":"04:12.170","Text":"Here we have it, now we can calculate the covariance of x and y."},{"Start":"04:12.170 ","End":"04:15.065","Text":"The covariance of x and y,"},{"Start":"04:15.065 ","End":"04:19.875","Text":"that equals to 900,000."},{"Start":"04:19.875 ","End":"04:22.400","Text":"Now that we have the covariance,"},{"Start":"04:22.400 ","End":"04:32.210","Text":"we can go back and calculate the variance of t. The variance of t then, well,"},{"Start":"04:32.210 ","End":"04:34.580","Text":"that equals to the variance of x,"},{"Start":"04:34.580 ","End":"04:42.840","Text":"and the variance of x is 1,000 squared plus the variance of y,"},{"Start":"04:42.840 ","End":"04:51.990","Text":"that\u0027s 1,500 squared plus 2 times the covariance, which is 900,000."},{"Start":"04:52.540 ","End":"04:59.310","Text":"That comes out to 5,050,000."},{"Start":"05:03.280 ","End":"05:08.210","Text":"The standard deviation of t,"},{"Start":"05:08.210 ","End":"05:11.060","Text":"that equals to the square root of the variance of"},{"Start":"05:11.060 ","End":"05:14.390","Text":"t. That means that we\u0027re looking at the square root of"},{"Start":"05:14.390 ","End":"05:15.800","Text":"5,050,000"},{"Start":"05:15.800 ","End":"05:24.810","Text":"and that equals to 2247,"},{"Start":"05:24.810 ","End":"05:27.450","Text":"and that\u0027s in units of dollars."},{"Start":"05:27.450 ","End":"05:32.450","Text":"Here we go. This then is the expectation of t,"},{"Start":"05:32.450 ","End":"05:37.050","Text":"and this is the standard deviation of t."}],"ID":13243},{"Watched":false,"Name":"Exercise 2 - Parts b-c","Duration":"6m 28s","ChapterTopicVideoID":12765,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In this section we\u0027re asked,"},{"Start":"00:01.260 ","End":"00:03.630","Text":"what is the chances that the total annual spending on"},{"Start":"00:03.630 ","End":"00:07.320","Text":"clothing and entertainment is more than $8,000."},{"Start":"00:07.320 ","End":"00:10.530","Text":"Let\u0027s just recall from the last section,"},{"Start":"00:10.530 ","End":"00:17.535","Text":"T had a normal probability distribution where the expectation was 7,000,"},{"Start":"00:17.535 ","End":"00:20.610","Text":"and the standard deviation,"},{"Start":"00:20.610 ","End":"00:25.515","Text":"that equal to 2247."},{"Start":"00:25.515 ","End":"00:31.535","Text":"Now we\u0027re asked for the probability of the total annual spendings,"},{"Start":"00:31.535 ","End":"00:36.095","Text":"of clothing and entertainment to be greater than 8,000."},{"Start":"00:36.095 ","End":"00:41.915","Text":"Now, let\u0027s just take a look at this on the graph."},{"Start":"00:41.915 ","End":"00:44.690","Text":"Here we have the density function of T,"},{"Start":"00:44.690 ","End":"00:46.205","Text":"this is a t-axis,"},{"Start":"00:46.205 ","End":"00:51.590","Text":"and the expectation that\u0027s right here, that\u0027s $7,000."},{"Start":"00:51.590 ","End":"00:55.895","Text":"Now we\u0027re looking for the probability of T being greater than $8,000."},{"Start":"00:55.895 ","End":"00:59.275","Text":"Let\u0027s say this right here is 8,000."},{"Start":"00:59.275 ","End":"01:03.550","Text":"We\u0027re looking to calculate this area right here under the curve,"},{"Start":"01:03.550 ","End":"01:06.055","Text":"from 8000 to plus infinity."},{"Start":"01:06.055 ","End":"01:07.390","Text":"Now how do we do that?"},{"Start":"01:07.390 ","End":"01:10.240","Text":"Well, we need to standardize T, so let\u0027s do that."},{"Start":"01:10.240 ","End":"01:15.760","Text":"That\u0027s the probability now of T minus Mu divided by Sigma that has to"},{"Start":"01:15.760 ","End":"01:21.685","Text":"be greater than 8,000 minus Mu divided by Sigma."},{"Start":"01:21.685 ","End":"01:25.380","Text":"Well, what\u0027s this thing right here?"},{"Start":"01:25.380 ","End":"01:31.090","Text":"Well, that\u0027s Z. That equals to the probability of Z being greater than what?"},{"Start":"01:31.090 ","End":"01:40.125","Text":"Well, that\u0027s 8,000 minus Mu minus 7,000 divided by Sigma,"},{"Start":"01:40.125 ","End":"01:42.700","Text":"that\u0027s divided by 2247."},{"Start":"01:43.370 ","End":"01:51.380","Text":"That means that this is the probability of Z being greater than 0.45."},{"Start":"01:51.380 ","End":"01:55.580","Text":"Here, on the z-axis right here,"},{"Start":"01:55.580 ","End":"02:01.800","Text":"$8,000 that\u0027s translated to 0.45."},{"Start":"02:01.800 ","End":"02:06.920","Text":"Now we can go through the standard table and calculate this probability."},{"Start":"02:06.920 ","End":"02:08.195","Text":"Now, I\u0027m not going to go through"},{"Start":"02:08.195 ","End":"02:13.715","Text":"all the other steps of calculating this probability right here."},{"Start":"02:13.715 ","End":"02:20.055","Text":"You already know this from the previous chapters. This thing now."},{"Start":"02:20.055 ","End":"02:27.410","Text":"That equals to 1 minus Phi of 0.45,"},{"Start":"02:27.410 ","End":"02:32.930","Text":"that equals to 1 minus 0.6736."},{"Start":"02:32.930 ","End":"02:37.275","Text":"Again, I urge you to go to the table and calculate"},{"Start":"02:37.275 ","End":"02:42.340","Text":"Phi of 0.45 and make sure that you have this value right here."},{"Start":"02:42.340 ","End":"02:49.160","Text":"This then equals to 0.3264."},{"Start":"02:49.160 ","End":"02:53.030","Text":"This then is the probability that"},{"Start":"02:53.030 ","End":"03:00.980","Text":"the total annual expenditure for clothing and entertainment is greater than $8,000."},{"Start":"03:00.980 ","End":"03:03.290","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"03:03.290 ","End":"03:07.715","Text":"the top 10th percentile of that total annual spending on clothing and entertainment."},{"Start":"03:07.715 ","End":"03:12.710","Text":"Well, T, that\u0027s the total annual spending and clothing and entertainment that has"},{"Start":"03:12.710 ","End":"03:20.015","Text":"a normal probability distribution where Mu equals to $7,000 and the standard deviation,"},{"Start":"03:20.015 ","End":"03:24.370","Text":"that equals to $2,247."},{"Start":"03:24.500 ","End":"03:27.795","Text":"We\u0027re looking for the top 10th percentile."},{"Start":"03:27.795 ","End":"03:31.669","Text":"Well, the top 10th percentile is actually the 90th percentile."},{"Start":"03:31.669 ","End":"03:36.230","Text":"We\u0027re looking for the value of T,"},{"Start":"03:36.230 ","End":"03:42.995","Text":"where the probability of T being less than or equal to this value T_0.9 we\u0027ll call it,"},{"Start":"03:42.995 ","End":"03:45.905","Text":"that has to equal to 0.9."},{"Start":"03:45.905 ","End":"03:49.090","Text":"Let\u0027s just see how this look on the graph."},{"Start":"03:49.090 ","End":"03:53.220","Text":"Here\u0027s our distribution, the density function."},{"Start":"03:53.220 ","End":"03:55.085","Text":"This is our t-axis."},{"Start":"03:55.085 ","End":"03:59.290","Text":"Here\u0027s the expectation of T, that\u0027s $7,000."},{"Start":"03:59.290 ","End":"04:02.250","Text":"We\u0027re looking for some value of T,"},{"Start":"04:02.250 ","End":"04:04.585","Text":"let\u0027s call this T_0.9,"},{"Start":"04:04.585 ","End":"04:08.060","Text":"where from this value onwards,"},{"Start":"04:08.060 ","End":"04:09.634","Text":"right to plus infinity,"},{"Start":"04:09.634 ","End":"04:14.330","Text":"the area under the density function that would be equal to 0.1."},{"Start":"04:14.330 ","End":"04:19.075","Text":"That means that from minus infinity to this value right here,"},{"Start":"04:19.075 ","End":"04:22.670","Text":"the area under the density function would be 0.9."},{"Start":"04:22.670 ","End":"04:25.565","Text":"This is exactly what this thing means right here."},{"Start":"04:25.565 ","End":"04:29.810","Text":"Now, how do we calculate this value right here?"},{"Start":"04:29.810 ","End":"04:31.400","Text":"Well, first of all,"},{"Start":"04:31.400 ","End":"04:37.685","Text":"we have to see what the normalized value of T is."},{"Start":"04:37.685 ","End":"04:40.170","Text":"That\u0027s our Z_0.9."},{"Start":"04:40.870 ","End":"04:47.030","Text":"Now, if we go to our standard table and we look at"},{"Start":"04:47.030 ","End":"04:54.800","Text":"the probability where Z is less than or equal to this value right here,"},{"Start":"04:54.800 ","End":"04:59.810","Text":"Z_0.9, and we want that to be equal to 0.9."},{"Start":"04:59.810 ","End":"05:05.725","Text":"In essence, we\u0027re looking for Phi of Z_0.9."},{"Start":"05:05.725 ","End":"05:08.480","Text":"We want that to be equal to 0.9."},{"Start":"05:08.480 ","End":"05:13.820","Text":"Now I\u0027m not going to go through all the steps of finding out what this value is."},{"Start":"05:13.820 ","End":"05:16.280","Text":"You\u0027re going to have to do that by yourself."},{"Start":"05:16.280 ","End":"05:18.605","Text":"You already know how to do this."},{"Start":"05:18.605 ","End":"05:21.199","Text":"We\u0027ve looked at this in previous chapters."},{"Start":"05:21.199 ","End":"05:24.410","Text":"We know then that this value right here,"},{"Start":"05:24.410 ","End":"05:31.105","Text":"Z_0.9, that equals to 1.282."},{"Start":"05:31.105 ","End":"05:33.785","Text":"Now that we know this,"},{"Start":"05:33.785 ","End":"05:36.390","Text":"that equals to 1.282,"},{"Start":"05:37.370 ","End":"05:43.865","Text":"we can see through the standardization process what this value is."},{"Start":"05:43.865 ","End":"05:47.465","Text":"How did we standardized T_0.9?"},{"Start":"05:47.465 ","End":"05:53.300","Text":"Well, we know that this value right here, 1.282,"},{"Start":"05:53.300 ","End":"05:58.290","Text":"that equals to T_0.9 minus Mu,"},{"Start":"05:58.290 ","End":"06:03.380","Text":"minus 7,000, divided by the standard deviation."},{"Start":"06:03.380 ","End":"06:06.350","Text":"Well that\u0027s 2247."},{"Start":"06:06.350 ","End":"06:09.950","Text":"That means that T_0.9,"},{"Start":"06:09.950 ","End":"06:13.430","Text":"well, that equals to what?"},{"Start":"06:13.430 ","End":"06:16.410","Text":"That equals to 9881."},{"Start":"06:16.580 ","End":"06:28.440","Text":"This then is the 90th percentile of the total expenditures on clothing and entertainment."}],"ID":13244},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"6m 10s","ChapterTopicVideoID":12767,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.350","Text":"In this question, we\u0027re given that"},{"Start":"00:01.350 ","End":"00:03.870","Text":"the daily consumption of vegetables in a restaurant has"},{"Start":"00:03.870 ","End":"00:06.660","Text":"a normal probability distribution with an expectation of"},{"Start":"00:06.660 ","End":"00:10.530","Text":"50 kilograms and a standard deviation of 4 kilograms."},{"Start":"00:10.530 ","End":"00:14.925","Text":"Now, we\u0027re also given that the price of vegetables is $6 per kilogram."},{"Start":"00:14.925 ","End":"00:19.070","Text":"We\u0027re asked, what are the expectation and variance daily cost"},{"Start":"00:19.070 ","End":"00:21.355","Text":"of vegetables for the restaurant?"},{"Start":"00:21.355 ","End":"00:27.215","Text":"Let\u0027s first define x as our daily consumption."},{"Start":"00:27.215 ","End":"00:31.178","Text":"This is the consumption,"},{"Start":"00:31.178 ","End":"00:36.140","Text":"and that has a normal probability distribution,"},{"Start":"00:36.140 ","End":"00:38.870","Text":"where the expectation is 50 kilograms,"},{"Start":"00:38.870 ","End":"00:43.280","Text":"that\u0027s right here, and the standard deviation is 4 kilograms."},{"Start":"00:43.280 ","End":"00:45.695","Text":"That\u0027ll be 4 squared right here."},{"Start":"00:45.695 ","End":"00:50.285","Text":"Now, we\u0027re asked about the expectation variance of the daily cost."},{"Start":"00:50.285 ","End":"00:53.840","Text":"Now, let\u0027s define y as our cost."},{"Start":"00:53.840 ","End":"00:55.670","Text":"Now, that equals to what?"},{"Start":"00:55.670 ","End":"01:00.200","Text":"That equals to $6 per kilograms"},{"Start":"01:00.200 ","End":"01:06.500","Text":"times the daily consumption, times x."},{"Start":"01:06.500 ","End":"01:09.395","Text":"That\u0027s the daily consumption in kilograms."},{"Start":"01:09.395 ","End":"01:11.435","Text":"Now, if we take a look at this,"},{"Start":"01:11.435 ","End":"01:13.235","Text":"this is a linear transformation."},{"Start":"01:13.235 ","End":"01:15.470","Text":"Y is a linear transformation of x."},{"Start":"01:15.470 ","End":"01:17.495","Text":"Let\u0027s just look at a template."},{"Start":"01:17.495 ","End":"01:20.415","Text":"Y equals to ax plus b."},{"Start":"01:20.415 ","End":"01:25.565","Text":"In our case, a would be equal to 6 and b is equal to 0."},{"Start":"01:25.565 ","End":"01:30.350","Text":"Now, when we take a look at the expectation of y, well,"},{"Start":"01:30.350 ","End":"01:35.510","Text":"that was a times the expectation of x plus b."},{"Start":"01:35.510 ","End":"01:37.580","Text":"The variance of y,"},{"Start":"01:37.580 ","End":"01:44.910","Text":"that equals to the variance of ax plus b,"},{"Start":"01:44.910 ","End":"01:49.835","Text":"and that equals to a squared times the variance of x."},{"Start":"01:49.835 ","End":"01:53.420","Text":"In our case, what would that be?"},{"Start":"01:53.420 ","End":"01:59.640","Text":"Well, let\u0027s take a look at the expectation of y."},{"Start":"01:59.640 ","End":"02:04.070","Text":"That\u0027s the expectation of 6 times x,"},{"Start":"02:04.070 ","End":"02:10.260","Text":"and that equals to 6 times the expectation of x plus 0,"},{"Start":"02:10.260 ","End":"02:12.120","Text":"b is equal to 0."},{"Start":"02:12.120 ","End":"02:19.445","Text":"That means that that equals to 6 times 50 and that equals to 300."},{"Start":"02:19.445 ","End":"02:21.830","Text":"What about the variance of y?"},{"Start":"02:21.830 ","End":"02:26.810","Text":"Well, that equals to the variance of 6 times x."},{"Start":"02:26.810 ","End":"02:32.900","Text":"Now, that equals to 6 squared times the variance of x."},{"Start":"02:32.900 ","End":"02:37.595","Text":"Now that equals to 6 squared times 4 squared,"},{"Start":"02:37.595 ","End":"02:42.680","Text":"and that equals to 576."},{"Start":"02:42.680 ","End":"02:46.760","Text":"There we have it. This is the expectation and this"},{"Start":"02:46.760 ","End":"02:51.420","Text":"is the variance of the daily cost of vegetables for the restaurant."},{"Start":"02:51.420 ","End":"02:55.150","Text":"In this section, we\u0027re asked what\u0027s the probability of the daily cost"},{"Start":"02:55.150 ","End":"02:58.375","Text":"of vegetables being less than $290?"},{"Start":"02:58.375 ","End":"03:01.210","Text":"Well, if we recall from the last section,"},{"Start":"03:01.210 ","End":"03:08.349","Text":"y had a normal probability distribution with the expectation equal to 300,"},{"Start":"03:08.349 ","End":"03:12.129","Text":"and the standard deviation or the variance,"},{"Start":"03:12.129 ","End":"03:14.635","Text":"if you recall, was 576."},{"Start":"03:14.635 ","End":"03:20.600","Text":"The standard deviation that equal to 24."},{"Start":"03:20.600 ","End":"03:25.375","Text":"This then is the distribution with its parameters."},{"Start":"03:25.375 ","End":"03:34.820","Text":"Now, we\u0027re asked about the probability that y is less than $290 right here."},{"Start":"03:34.820 ","End":"03:36.830","Text":"Now, how do we do that?"},{"Start":"03:36.830 ","End":"03:40.620","Text":"Well, let\u0027s first of all take a look at the distribution."},{"Start":"03:42.350 ","End":"03:45.090","Text":"This is our density function."},{"Start":"03:45.090 ","End":"03:47.565","Text":"Here\u0027s our y-axis."},{"Start":"03:47.565 ","End":"03:50.550","Text":"We know that the expectation of y is 300,"},{"Start":"03:50.550 ","End":"03:53.010","Text":"so here that\u0027ll be 300."},{"Start":"03:53.010 ","End":"03:57.820","Text":"We\u0027re looking for the probability of y being less than 290."},{"Start":"03:57.820 ","End":"04:01.480","Text":"Let\u0027s say this right here is 290,"},{"Start":"04:01.480 ","End":"04:05.255","Text":"so we\u0027re looking to calculate the area under the graph,"},{"Start":"04:05.255 ","End":"04:09.800","Text":"under the curve from minus infinity to 290."},{"Start":"04:09.800 ","End":"04:11.690","Text":"Now, in order to do that,"},{"Start":"04:11.690 ","End":"04:13.730","Text":"we need to standardize."},{"Start":"04:13.730 ","End":"04:17.930","Text":"What would be our z value right here?"},{"Start":"04:17.930 ","End":"04:20.360","Text":"Z of y, well,"},{"Start":"04:20.360 ","End":"04:28.240","Text":"that would be equal to y minus mu divided by Sigma."},{"Start":"04:28.240 ","End":"04:35.645","Text":"Now that\u0027ll be equal to 290 minus 300 divided by 24,"},{"Start":"04:35.645 ","End":"04:40.680","Text":"and that equals to minus 0.42."},{"Start":"04:40.680 ","End":"04:49.530","Text":"So 290 on the y-axis is comparable to minus 0.42 on the z axis."},{"Start":"04:49.760 ","End":"04:55.837","Text":"The probability of y being less than 290,"},{"Start":"04:55.837 ","End":"05:04.040","Text":"that equals to phi of minus 0.42."},{"Start":"05:04.040 ","End":"05:11.930","Text":"Now that equals to 1 minus phi of 0.42 and that"},{"Start":"05:11.930 ","End":"05:21.990","Text":"equals to 1 minus 0.6628 and that equals to 0.3372."},{"Start":"05:21.990 ","End":"05:24.990","Text":"Now again, how did I do this?"},{"Start":"05:24.990 ","End":"05:27.610","Text":"Well again, I\u0027m not going to go through all the steps,"},{"Start":"05:27.610 ","End":"05:30.895","Text":"you should have already be familiar with the steps."},{"Start":"05:30.895 ","End":"05:39.510","Text":"Phi of a negative number that equals to 1 minus Phi of the same number."},{"Start":"05:39.510 ","End":"05:42.320","Text":"Now, phi of a number, well,"},{"Start":"05:42.320 ","End":"05:48.135","Text":"we should go to the standard table and see what that is, here it is."},{"Start":"05:48.135 ","End":"05:57.780","Text":"I urge you to do this to make sure that phi of 0.42 equals to 0.6628 minus this number,"},{"Start":"05:57.780 ","End":"06:01.155","Text":"well, that equals to 0.3372."},{"Start":"06:01.155 ","End":"06:11.100","Text":"This then is the probability of the daily cost of vegetables being less than $290."}],"ID":13245},{"Watched":false,"Name":"Exercise 3 - Part c","Duration":"5m 44s","ChapterTopicVideoID":12766,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this section, we\u0027re asked what\u0027s the 40th percentile of"},{"Start":"00:02.940 ","End":"00:07.595","Text":"the probability distribution of the constant vegetables for the restaurant?"},{"Start":"00:07.595 ","End":"00:11.940","Text":"So let\u0027s just remind ourselves Y the cost of the vegetables that has"},{"Start":"00:11.940 ","End":"00:14.220","Text":"a normal probability distribution where"},{"Start":"00:14.220 ","End":"00:18.825","Text":"the expectation is 300 and the standard deviation,"},{"Start":"00:18.825 ","End":"00:20.925","Text":"well, that\u0027s 24,"},{"Start":"00:20.925 ","End":"00:24.375","Text":"and we\u0027re asked about the 40th percentile."},{"Start":"00:24.375 ","End":"00:28.015","Text":"We want a value of Y,"},{"Start":"00:28.015 ","End":"00:32.630","Text":"where the probability of Y being less than this value,"},{"Start":"00:32.630 ","End":"00:35.090","Text":"we call this y_0.4,"},{"Start":"00:35.090 ","End":"00:39.185","Text":"then has to equal to 0.4, 40 percent."},{"Start":"00:39.185 ","End":"00:43.120","Text":"Let\u0027s just see what this looks like on a graph."},{"Start":"00:43.120 ","End":"00:46.245","Text":"Here\u0027s a density function for y,"},{"Start":"00:46.245 ","End":"00:48.680","Text":"that\u0027s the y-axis, that\u0027s the density function."},{"Start":"00:48.680 ","End":"00:51.455","Text":"The expectation right here is 300."},{"Start":"00:51.455 ","End":"00:56.165","Text":"We\u0027re looking for some value of Y,"},{"Start":"00:56.165 ","End":"00:58.955","Text":"y_0.4, that\u0027ll be here."},{"Start":"00:58.955 ","End":"01:03.890","Text":"Y_0.4 where the area under the density function from"},{"Start":"01:03.890 ","End":"01:08.690","Text":"minus infinity till this value that has to equal to 0.4."},{"Start":"01:08.690 ","End":"01:13.040","Text":"That means that from this point onwards to plus infinity,"},{"Start":"01:13.040 ","End":"01:17.030","Text":"the area under the curve has to be equal to 0.6."},{"Start":"01:17.030 ","End":"01:20.615","Text":"Now, how do we calculate this value right here for y?"},{"Start":"01:20.615 ","End":"01:24.410","Text":"Well, what we need to do is we need to go to"},{"Start":"01:24.410 ","End":"01:29.660","Text":"the standard table and look up a z-value right here,"},{"Start":"01:29.660 ","End":"01:33.830","Text":"which is comparable to the y-value of y_0.4."},{"Start":"01:33.830 ","End":"01:38.430","Text":"We call this z_0.4."},{"Start":"01:38.430 ","End":"01:45.600","Text":"What we need to do is we want to calculate Phi of z_0.4,"},{"Start":"01:45.600 ","End":"01:49.005","Text":"and that has to be equal to 0.4."},{"Start":"01:49.005 ","End":"01:56.930","Text":"But within the table,"},{"Start":"01:56.930 ","End":"02:01.280","Text":"it doesn\u0027t give us any probabilities below 0.5."},{"Start":"02:01.280 ","End":"02:05.720","Text":"What we need to do is we need to take the mirror image of this thing right"},{"Start":"02:05.720 ","End":"02:11.810","Text":"here and calculate the mirror value of 0.4."},{"Start":"02:11.810 ","End":"02:14.665","Text":"Now, what do I mean by that?"},{"Start":"02:14.665 ","End":"02:17.940","Text":"Here we have the same distribution."},{"Start":"02:17.940 ","End":"02:19.790","Text":"The same density function for Y,"},{"Start":"02:19.790 ","End":"02:24.544","Text":"that\u0027s the y-axis, that\u0027s the expectation of Y, that\u0027s 300."},{"Start":"02:24.544 ","End":"02:29.570","Text":"Now, we wanted the area under the graph here under"},{"Start":"02:29.570 ","End":"02:35.600","Text":"the density function from minus infinity to this value to be 0.4."},{"Start":"02:35.600 ","End":"02:37.865","Text":"Well, if we take the mirror image,"},{"Start":"02:37.865 ","End":"02:41.315","Text":"then here in this density function,"},{"Start":"02:41.315 ","End":"02:46.670","Text":"we want this area from some value here to plus infinity."},{"Start":"02:46.670 ","End":"02:48.920","Text":"We want that to be 0.4."},{"Start":"02:48.920 ","End":"02:51.380","Text":"That means that the value here,"},{"Start":"02:51.380 ","End":"02:57.395","Text":"the area under the graph from minus infinity to this value will then has to be 0.6."},{"Start":"02:57.395 ","End":"02:59.870","Text":"Well, in order for that to be 0.6,"},{"Start":"02:59.870 ","End":"03:03.740","Text":"we\u0027re looking then for y_0.6."},{"Start":"03:03.740 ","End":"03:06.995","Text":"Now, in order to utilize the standard table,"},{"Start":"03:06.995 ","End":"03:13.260","Text":"we need to transform this to get the comparable value of this on the z-axis."},{"Start":"03:15.890 ","End":"03:19.355","Text":"Since we can\u0027t calculate this,"},{"Start":"03:19.355 ","End":"03:24.875","Text":"we can calculate Phi of z_0.6."},{"Start":"03:24.875 ","End":"03:29.065","Text":"Because that\u0027s the comparable value here."},{"Start":"03:29.065 ","End":"03:32.755","Text":"Now, what is that equal to?"},{"Start":"03:32.755 ","End":"03:38.675","Text":"Well, we said that this thing has to be equal to 0.6."},{"Start":"03:38.675 ","End":"03:41.825","Text":"Let\u0027s now go to the table."},{"Start":"03:41.825 ","End":"03:44.285","Text":"Well, if we look at the table,"},{"Start":"03:44.285 ","End":"03:51.670","Text":"we\u0027re looking in the table right here for value that\u0027s closest to 0.6."},{"Start":"03:51.670 ","End":"03:58.320","Text":"We see here 2 values 0.5987 and 0.6026,"},{"Start":"03:58.320 ","End":"03:59.930","Text":"well this seems closest,"},{"Start":"03:59.930 ","End":"04:02.060","Text":"so we\u0027ll use this number right here."},{"Start":"04:02.060 ","End":"04:03.995","Text":"Now, if that\u0027s the case,"},{"Start":"04:03.995 ","End":"04:11.440","Text":"then the appropriate z-value would be 0.25."},{"Start":"04:11.440 ","End":"04:14.410","Text":"That\u0027s what we\u0027ll use."},{"Start":"04:14.410 ","End":"04:22.170","Text":"That means that this thing right here z_0.6,"},{"Start":"04:22.170 ","End":"04:26.055","Text":"so that equals to 0.25."},{"Start":"04:26.055 ","End":"04:33.640","Text":"Now, what\u0027s the mirror image of z_0.6 where we said that was z_0.4,"},{"Start":"04:33.640 ","End":"04:37.040","Text":"but with the opposite sign here."},{"Start":"04:37.040 ","End":"04:45.140","Text":"So z_0.4, that will be equal to minus 0.25."},{"Start":"04:45.140 ","End":"04:54.930","Text":"Now that we have this value we can calculate the comparable value of y."},{"Start":"04:55.070 ","End":"04:57.285","Text":"How do we do that?"},{"Start":"04:57.285 ","End":"05:05.190","Text":"Well, we know that z equals to y minus Mu divided by Sigma,"},{"Start":"05:05.190 ","End":"05:07.470","Text":"that\u0027s the standardization process."},{"Start":"05:07.470 ","End":"05:10.015","Text":"All we have to do is go backwards,"},{"Start":"05:10.015 ","End":"05:17.260","Text":"minus 0.25, that equals to y_0.4."},{"Start":"05:17.260 ","End":"05:20.770","Text":"That\u0027s the value that we\u0027re looking for, minus Mu,"},{"Start":"05:20.770 ","End":"05:24.930","Text":"our Mu is 300 divided by Sigma,"},{"Start":"05:24.930 ","End":"05:27.180","Text":"where Sigma is 24."},{"Start":"05:27.180 ","End":"05:31.490","Text":"That means that y_0.4,"},{"Start":"05:31.490 ","End":"05:36.250","Text":"that comes out to 294."},{"Start":"05:36.250 ","End":"05:44.210","Text":"So this value right here is the 40th percentile of the distribution."}],"ID":13246},{"Watched":false,"Name":"Exercise 4","Duration":"6m 36s","ChapterTopicVideoID":12768,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"In this question, we\u0027re given that a bottle of wine has a normal probability distribution"},{"Start":"00:05.310 ","End":"00:10.725","Text":"with an expectation of 750 milliliters and a standard deviation of 20 milliliters."},{"Start":"00:10.725 ","End":"00:14.340","Text":"A person buys a box containing 4 bottles of wine."},{"Start":"00:14.340 ","End":"00:16.370","Text":"We\u0027re asked, what are the expectation and"},{"Start":"00:16.370 ","End":"00:19.995","Text":"standard deviation of the volume of wine in the box?"},{"Start":"00:19.995 ","End":"00:22.499","Text":"First of all, let\u0027s define our variables."},{"Start":"00:22.499 ","End":"00:29.790","Text":"Let\u0027s define X_i as the volume in bottle i."},{"Start":"00:29.790 ","End":"00:34.040","Text":"Now that has a normal probability distribution where"},{"Start":"00:34.040 ","End":"00:39.605","Text":"the expectation is 750 milliliters and the standard deviation,"},{"Start":"00:39.605 ","End":"00:42.780","Text":"well, that equals to 20 milliliters."},{"Start":"00:42.780 ","End":"00:47.390","Text":"We\u0027re asked about the expectation standard deviation of the volume of wine in the box."},{"Start":"00:47.390 ","End":"00:53.525","Text":"That means that we have to add up the volume of wine for each of the bottles."},{"Start":"00:53.525 ","End":"00:59.255","Text":"Now, that means that we have to define a new variable, random variable T,"},{"Start":"00:59.255 ","End":"01:03.855","Text":"which is equal to the sum of X_i,"},{"Start":"01:03.855 ","End":"01:06.705","Text":"i goes from 1-4."},{"Start":"01:06.705 ","End":"01:11.123","Text":"We have 4 bottles of wine and i,"},{"Start":"01:11.123 ","End":"01:13.710","Text":"our index, goes from 1-4."},{"Start":"01:13.710 ","End":"01:18.830","Text":"We\u0027re asked about the expectation of T and the standard deviation."},{"Start":"01:18.830 ","End":"01:21.184","Text":"Well, in order to calculate the standard deviation,"},{"Start":"01:21.184 ","End":"01:26.680","Text":"we first have to calculate the variance of T. Let\u0027s get to work."},{"Start":"01:26.680 ","End":"01:31.460","Text":"Now the expectation of T, well,"},{"Start":"01:31.460 ","End":"01:35.960","Text":"that equals to the expectation of the sum of X_i,"},{"Start":"01:35.960 ","End":"01:38.360","Text":"i goes from 1-4."},{"Start":"01:38.360 ","End":"01:43.560","Text":"Now that equals to the sum of the expectation of X_i,"},{"Start":"01:43.560 ","End":"01:45.880","Text":"i goes from 1-4."},{"Start":"01:45.880 ","End":"01:49.505","Text":"That means, what\u0027s the expectation of X_i?"},{"Start":"01:49.505 ","End":"01:53.225","Text":"Well, that\u0027s 750."},{"Start":"01:53.225 ","End":"01:58.335","Text":"Now we have to add that up 4 times for each 1 of the bottles."},{"Start":"01:58.335 ","End":"02:06.795","Text":"So we\u0027ll just multiply it by 4 and that comes out to 3,000 units or in milliliters."},{"Start":"02:06.795 ","End":"02:09.350","Text":"What about the variance of T?"},{"Start":"02:09.350 ","End":"02:13.850","Text":"Well, that\u0027s the variance of the sum of X_i,"},{"Start":"02:13.850 ","End":"02:17.330","Text":"i goes from 1-4."},{"Start":"02:17.330 ","End":"02:19.410","Text":"Now, that equals to what?"},{"Start":"02:19.410 ","End":"02:28.755","Text":"Well, we assume that X_i are independent of each other so the covariances must equal 0."},{"Start":"02:28.755 ","End":"02:38.445","Text":"That means that the variance of the sum equals the sum of the variance of X_i\u0027s,"},{"Start":"02:38.445 ","End":"02:40.965","Text":"i goes from 1-4."},{"Start":"02:40.965 ","End":"02:43.860","Text":"Now, what\u0027s the variance of X_i?"},{"Start":"02:43.860 ","End":"02:45.885","Text":"Well, for each X_i,"},{"Start":"02:45.885 ","End":"02:48.440","Text":"the variance is 20 squared."},{"Start":"02:48.440 ","End":"02:52.785","Text":"That\u0027ll be 20 squared plus,"},{"Start":"02:52.785 ","End":"02:54.240","Text":"plus 20 squared, plus 20 squared,"},{"Start":"02:54.240 ","End":"02:55.830","Text":"plus 20 squared, 4 times."},{"Start":"02:55.830 ","End":"02:58.495","Text":"Let\u0027s just multiply that by 4,"},{"Start":"02:58.495 ","End":"03:02.120","Text":"and that comes out to 1,600."},{"Start":"03:02.120 ","End":"03:05.920","Text":"Now the units here are in milliliters squared."},{"Start":"03:05.920 ","End":"03:09.830","Text":"What\u0027s the standard deviation of T?"},{"Start":"03:09.830 ","End":"03:12.665","Text":"Well, that\u0027s the square root of the variance of"},{"Start":"03:12.665 ","End":"03:15.770","Text":"T. That means that we have the square root of"},{"Start":"03:15.770 ","End":"03:22.575","Text":"1,600 and that equals to 40 and the units here are milliliters."},{"Start":"03:22.575 ","End":"03:29.180","Text":"There we go. This is the standard deviation of T and this is the expectation of"},{"Start":"03:29.180 ","End":"03:32.870","Text":"T. In this section we\u0027re given that the person"},{"Start":"03:32.870 ","End":"03:37.490","Text":"pours the wine in the box into a container with a capacity of 3.1 liters."},{"Start":"03:37.490 ","End":"03:42.835","Text":"We\u0027re asked what\u0027s the probability that the wine will overflow from the container?"},{"Start":"03:42.835 ","End":"03:45.560","Text":"Let\u0027s just recall what we did in the last section."},{"Start":"03:45.560 ","End":"03:52.410","Text":"We defined T. That was equal to the sum of X_i, i equaling 1-4."},{"Start":"03:52.410 ","End":"03:55.610","Text":"T is the total volume in the box."},{"Start":"03:55.610 ","End":"04:01.430","Text":"Now that has a normal probability distribution where Mu,"},{"Start":"04:01.430 ","End":"04:08.530","Text":"the expectation equals to 3,000 and the standard deviation that equals to 40."},{"Start":"04:08.530 ","End":"04:17.600","Text":"We\u0027re asked what\u0027s the probability then of T being greater than 3,100?"},{"Start":"04:17.600 ","End":"04:22.760","Text":"If T is greater than 3,100 or 3.1 liter,"},{"Start":"04:22.760 ","End":"04:26.425","Text":"then the wine will overflow from the container."},{"Start":"04:26.425 ","End":"04:30.160","Text":"Let\u0027s see how this looks on a graph."},{"Start":"04:30.770 ","End":"04:35.840","Text":"Here we have a density function here we have our T-axis."},{"Start":"04:35.840 ","End":"04:40.720","Text":"This is the expectation for T right here, that\u0027s 3,000."},{"Start":"04:40.720 ","End":"04:45.185","Text":"We\u0027re looking for the probability of T being greater than 3,100."},{"Start":"04:45.185 ","End":"04:51.050","Text":"Let\u0027s say right here that\u0027s 3,100 and we\u0027re looking then to calculate"},{"Start":"04:51.050 ","End":"04:58.460","Text":"the area under the density function from 3,100 to plus infinity on the T axis."},{"Start":"04:58.460 ","End":"05:00.200","Text":"Well, in order to do that,"},{"Start":"05:00.200 ","End":"05:02.870","Text":"we need to standardize this number right here,"},{"Start":"05:02.870 ","End":"05:04.925","Text":"3,100. How do we do that?"},{"Start":"05:04.925 ","End":"05:11.509","Text":"Well, we define Z as T minus Mu divided by Sigma."},{"Start":"05:11.509 ","End":"05:12.935","Text":"Now in our case,"},{"Start":"05:12.935 ","End":"05:15.920","Text":"that\u0027ll be 3,100 minus Mu,"},{"Start":"05:15.920 ","End":"05:20.100","Text":"where Mu is 3,000 divided by Sigma,"},{"Start":"05:20.100 ","End":"05:21.765","Text":"where Sigma is 40."},{"Start":"05:21.765 ","End":"05:26.050","Text":"That equals to 2.5."},{"Start":"05:26.720 ","End":"05:33.140","Text":"So the probability of T being greater than 3,100, well,"},{"Start":"05:33.140 ","End":"05:39.365","Text":"that equals to the probability of Z being greater than 2.5."},{"Start":"05:39.365 ","End":"05:44.135","Text":"Here that\u0027s 2.5 on the Z scale."},{"Start":"05:44.135 ","End":"05:51.665","Text":"Now, that equals to 1 minus Phi of 2.5."},{"Start":"05:51.665 ","End":"05:55.460","Text":"Now, I hope this transition is well understood."},{"Start":"05:55.460 ","End":"05:57.935","Text":"You should understand this by now."},{"Start":"05:57.935 ","End":"06:02.150","Text":"If you don\u0027t, please refer back to previous chapters."},{"Start":"06:02.150 ","End":"06:06.630","Text":"That equals to 1 minus,"},{"Start":"06:06.630 ","End":"06:15.035","Text":"now Phi of 2.5 let\u0027s go to the standard table and see that that equals to 0.9938."},{"Start":"06:15.035 ","End":"06:17.285","Text":"I urge you to do this."},{"Start":"06:17.285 ","End":"06:21.450","Text":"That equals to 0.0062."},{"Start":"06:21.500 ","End":"06:28.260","Text":"This then is the probability of the wine"},{"Start":"06:28.260 ","End":"06:36.420","Text":"overflowing when it\u0027s being poured into a container with a capacity of 3.1 liters."}],"ID":13247},{"Watched":false,"Name":"Exercise 5 - Part a","Duration":"8m 28s","ChapterTopicVideoID":12769,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.455","Text":"In this question, we\u0027re given that a farmer has a farm with a cow and a goat."},{"Start":"00:04.455 ","End":"00:08.610","Text":"The daily yield of cow\u0027s milk has a normal probability distribution with"},{"Start":"00:08.610 ","End":"00:12.915","Text":"an average of 20 liters per day and a standard deviation of 5 liters."},{"Start":"00:12.915 ","End":"00:17.220","Text":"The daily yield of goat\u0027s milk has a normal probability distribution with"},{"Start":"00:17.220 ","End":"00:21.480","Text":"an average of 10 liters per day and the standard deviation of 2 liters."},{"Start":"00:21.480 ","End":"00:25.065","Text":"Now, the cow\u0027s milk is sold for $2 per liter,"},{"Start":"00:25.065 ","End":"00:28.230","Text":"and the goat\u0027s milk is sold for $3 per liter."},{"Start":"00:28.230 ","End":"00:31.710","Text":"We\u0027re asked what are the chances that the farmers daily"},{"Start":"00:31.710 ","End":"00:35.840","Text":"proceeds from milk will be at least $62?"},{"Start":"00:35.840 ","End":"00:38.570","Text":"Let\u0027s define our variables."},{"Start":"00:38.570 ","End":"00:43.465","Text":"X will be the daily yield of cow\u0027s milk."},{"Start":"00:43.465 ","End":"00:50.465","Text":"Now, that has a normal probability distribution with the expectation"},{"Start":"00:50.465 ","End":"00:57.519","Text":"equaling 20 liters and the standard deviation equaling 5 liters."},{"Start":"00:57.519 ","End":"01:00.005","Text":"It\u0027s right here and right here."},{"Start":"01:00.005 ","End":"01:01.430","Text":"Now, what about the goat?"},{"Start":"01:01.430 ","End":"01:06.365","Text":"Well, let\u0027s define y as the daily yield of goat\u0027s milk."},{"Start":"01:06.365 ","End":"01:11.090","Text":"Now, that has a normal probability distribution,"},{"Start":"01:11.090 ","End":"01:12.725","Text":"now what\u0027s the expectation?"},{"Start":"01:12.725 ","End":"01:18.290","Text":"The expectation is 10 liters per day and the standard deviation,"},{"Start":"01:18.290 ","End":"01:22.640","Text":"well, that equals to 2 liters per day."},{"Start":"01:22.640 ","End":"01:25.070","Text":"Now, what else are we given?"},{"Start":"01:25.070 ","End":"01:28.835","Text":"Well, we\u0027re given that the cow\u0027s milk is sold for $2 per liters so"},{"Start":"01:28.835 ","End":"01:33.780","Text":"the price of cow\u0027s milk, well,"},{"Start":"01:33.780 ","End":"01:44.430","Text":"that equals to $2 per liter and the price of goat\u0027s milk,"},{"Start":"01:44.430 ","End":"01:49.510","Text":"that equals to $3 per liter."},{"Start":"01:49.520 ","End":"01:53.000","Text":"Now that we have all the information,"},{"Start":"01:53.000 ","End":"01:55.460","Text":"again we\u0027re asked what is the chances that"},{"Start":"01:55.460 ","End":"01:58.450","Text":"the farmer\u0027s daily proceeds from all the milk,"},{"Start":"01:58.450 ","End":"02:00.315","Text":"from the cow and the goat,"},{"Start":"02:00.315 ","End":"02:03.345","Text":"will be at least $62?"},{"Start":"02:03.345 ","End":"02:10.095","Text":"Let\u0027s get to work. Let\u0027s first define variable,"},{"Start":"02:10.095 ","End":"02:14.600","Text":"we\u0027ll call this w, and that\u0027ll be the daily proceeds of milk."},{"Start":"02:14.600 ","End":"02:18.050","Text":"Now, what is the daily proceeds of milk?"},{"Start":"02:18.050 ","End":"02:21.230","Text":"Well, that\u0027s the proceeds from the cow\u0027s milk and the goat\u0027s milk."},{"Start":"02:21.230 ","End":"02:26.210","Text":"The proceeds of the cow\u0027s milk would be $2 per liter times x,"},{"Start":"02:26.210 ","End":"02:28.430","Text":"the daily yield of cow\u0027s milk,"},{"Start":"02:28.430 ","End":"02:33.500","Text":"plus $3 per liter times y,"},{"Start":"02:33.500 ","End":"02:36.860","Text":"the daily yield of milk from the goat."},{"Start":"02:36.860 ","End":"02:38.345","Text":"Now, as we can see,"},{"Start":"02:38.345 ","End":"02:40.795","Text":"this is a linear transformation of x and y,"},{"Start":"02:40.795 ","End":"02:46.175","Text":"that\u0027s w. Now, let\u0027s take a look first at a template of a linear transformation."},{"Start":"02:46.175 ","End":"02:50.690","Text":"If we define w is ax plus by,"},{"Start":"02:50.690 ","End":"02:57.915","Text":"well then a would be equal to 2 and b would be equal to 3."},{"Start":"02:57.915 ","End":"02:59.960","Text":"What else do we know about this?"},{"Start":"02:59.960 ","End":"03:02.855","Text":"Well, we know that the expectation of w,"},{"Start":"03:02.855 ","End":"03:10.530","Text":"that equals to a times the expectation of x plus b times the expectation of y."},{"Start":"03:10.530 ","End":"03:13.435","Text":"What about the variance of w?"},{"Start":"03:13.435 ","End":"03:17.630","Text":"Well, that equals to a squared times"},{"Start":"03:17.630 ","End":"03:22.250","Text":"the variance of x plus b squared times the variance of"},{"Start":"03:22.250 ","End":"03:32.610","Text":"y plus 2 times a times b times the covariance of x and y."},{"Start":"03:32.610 ","End":"03:34.805","Text":"Now that we have that,"},{"Start":"03:34.805 ","End":"03:41.275","Text":"let\u0027s just calculate the expectation of w. The expectation of w,"},{"Start":"03:41.275 ","End":"03:43.400","Text":"that equals to a,"},{"Start":"03:43.400 ","End":"03:46.170","Text":"2 times the expectation of x,"},{"Start":"03:46.170 ","End":"03:50.025","Text":"well of the expectation of x is 20 plus b,"},{"Start":"03:50.025 ","End":"03:54.870","Text":"that\u0027s 3 times the expectation of y, that\u0027s 10."},{"Start":"03:54.870 ","End":"03:58.730","Text":"That equals to 2 times 20, that\u0027s 40,"},{"Start":"03:58.730 ","End":"04:00.350","Text":"and 3 times 10, that\u0027s 30,"},{"Start":"04:00.350 ","End":"04:03.634","Text":"so 40 plus 30, that equals to 70."},{"Start":"04:03.634 ","End":"04:06.640","Text":"Now, what about the variance of w?"},{"Start":"04:06.640 ","End":"04:09.240","Text":"Well, that equals to a squared,"},{"Start":"04:09.240 ","End":"04:13.235","Text":"that\u0027s 2 squared times the variance of x, well,"},{"Start":"04:13.235 ","End":"04:17.639","Text":"that\u0027s 5 squared plus b squared,"},{"Start":"04:17.639 ","End":"04:21.245","Text":"that\u0027ll be 3 squared times the variance of y,"},{"Start":"04:21.245 ","End":"04:23.090","Text":"that\u0027s 2 squared,"},{"Start":"04:23.090 ","End":"04:28.585","Text":"plus 2 times a times b times the covariance of x and y."},{"Start":"04:28.585 ","End":"04:34.955","Text":"The daily yield of milk from a cow and from a goat, they\u0027re independent."},{"Start":"04:34.955 ","End":"04:37.429","Text":"We can assume that x and y are independent,"},{"Start":"04:37.429 ","End":"04:40.235","Text":"so the covariance must equal to 0."},{"Start":"04:40.235 ","End":"04:45.785","Text":"Therefore, the variance of w would be just these guys right here."},{"Start":"04:45.785 ","End":"04:54.365","Text":"That means that\u0027s 4 times 25 plus 9 times 4 and that equals to 136."},{"Start":"04:54.365 ","End":"04:57.560","Text":"Now, we know that w has"},{"Start":"04:57.560 ","End":"05:05.135","Text":"a normal distribution where the expectation equals to 70 and the variance now,"},{"Start":"05:05.135 ","End":"05:09.150","Text":"that equals to 136."},{"Start":"05:09.170 ","End":"05:15.950","Text":"After calculating the values of the parameters of the distribution for w,"},{"Start":"05:15.950 ","End":"05:19.670","Text":"only then can we calculate the chances that"},{"Start":"05:19.670 ","End":"05:24.455","Text":"the farmer\u0027s daily proceeds from milk will be at least $62."},{"Start":"05:24.455 ","End":"05:28.165","Text":"Let\u0027s get to work on how to calculate this."},{"Start":"05:28.165 ","End":"05:35.905","Text":"We\u0027re looking for the probability of w being greater than 62."},{"Start":"05:35.905 ","End":"05:39.405","Text":"Well, how would that look on a graph?"},{"Start":"05:39.405 ","End":"05:43.771","Text":"Here\u0027s the density function for w. Here we see the w axis,"},{"Start":"05:43.771 ","End":"05:49.000","Text":"this is a density function for w. Here we have the expectation of w, that\u0027s 70."},{"Start":"05:49.000 ","End":"05:54.560","Text":"Now, we\u0027re asked about the probability of w being greater than 62."},{"Start":"05:54.560 ","End":"05:57.890","Text":"Let\u0027s say this right here, that\u0027s 62."},{"Start":"05:57.890 ","End":"06:01.730","Text":"The probability of w being greater than 62,"},{"Start":"06:01.730 ","End":"06:09.905","Text":"that\u0027s the area under the density function from 62 on to plus infinity."},{"Start":"06:09.905 ","End":"06:12.875","Text":"That\u0027s this area in yellow right here."},{"Start":"06:12.875 ","End":"06:15.125","Text":"Now, how do we do that?"},{"Start":"06:15.125 ","End":"06:17.600","Text":"Well, we have to go to a standard table,"},{"Start":"06:17.600 ","End":"06:19.280","Text":"we have to define a standard score,"},{"Start":"06:19.280 ","End":"06:23.725","Text":"go to the standard table and calculate that probability."},{"Start":"06:23.725 ","End":"06:27.615","Text":"What we need to do is, we need to standardize w,"},{"Start":"06:27.615 ","End":"06:33.920","Text":"with this value of w. That means that this is the probability of w"},{"Start":"06:33.920 ","End":"06:41.180","Text":"minus Mu divided by Sigma that has to be greater than 62 minus Mu divided by Sigma."},{"Start":"06:41.180 ","End":"06:44.510","Text":"Now, that equals to the probability of what?"},{"Start":"06:44.510 ","End":"06:46.340","Text":"Well, this expression right here,"},{"Start":"06:46.340 ","End":"06:53.074","Text":"well that\u0027s z and that has to be greater than 62 minus Mu, Mu is 70."},{"Start":"06:53.074 ","End":"07:00.755","Text":"That\u0027ll be 62 minus 70 divided by the square root of 136."},{"Start":"07:00.755 ","End":"07:09.840","Text":"Now, that equals to the probability of z being greater than minus 0.69."},{"Start":"07:09.840 ","End":"07:16.440","Text":"Here, the comparable value of 62 on the z axis,"},{"Start":"07:16.440 ","End":"07:19.335","Text":"that\u0027ll be minus 0.69."},{"Start":"07:19.335 ","End":"07:21.495","Text":"Now that we have that,"},{"Start":"07:21.495 ","End":"07:29.090","Text":"we can go on to the normal probability table."},{"Start":"07:29.090 ","End":"07:31.520","Text":"Now, let\u0027s just write this out again."},{"Start":"07:31.520 ","End":"07:35.735","Text":"The probability of w being greater than 62,"},{"Start":"07:35.735 ","End":"07:42.410","Text":"that equals to the probability of z being greater than minus 0.69."},{"Start":"07:42.410 ","End":"07:49.625","Text":"Now, that equals to Phi of 0.69."},{"Start":"07:49.625 ","End":"07:53.240","Text":"Now, if this transition is unclear to you,"},{"Start":"07:53.240 ","End":"07:57.635","Text":"then I urge you to go back and review the previous lessons."},{"Start":"07:57.635 ","End":"08:00.070","Text":"Now, this then,"},{"Start":"08:00.070 ","End":"08:04.685","Text":"after looking 0.69 on the standard table,"},{"Start":"08:04.685 ","End":"08:11.590","Text":"Phi of 0.69 comes out to 0.7549."},{"Start":"08:11.590 ","End":"08:17.000","Text":"This then is the probability of w being greater than 62."},{"Start":"08:17.000 ","End":"08:21.605","Text":"That means that this is the probability of getting"},{"Start":"08:21.605 ","End":"08:27.540","Text":"at least $62 from the proceeds from selling the milk."}],"ID":13248},{"Watched":false,"Name":"Exercise 5 - Part b","Duration":"6m 34s","ChapterTopicVideoID":12770,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this section, we\u0027re asked what is the chances that"},{"Start":"00:02.490 ","End":"00:04.850","Text":"the combined milk yield of the cow and"},{"Start":"00:04.850 ","End":"00:10.895","Text":"the goat will be less than 30 liters on at least 4 out of 5 consecutive days."},{"Start":"00:10.895 ","End":"00:14.160","Text":"Let\u0027s just recall our variables."},{"Start":"00:14.160 ","End":"00:17.580","Text":"X was a daily yield of milk for a cow."},{"Start":"00:17.580 ","End":"00:21.870","Text":"That had a normal probability distribution with"},{"Start":"00:21.870 ","End":"00:28.350","Text":"an expectation of 20 and a standard deviation of 5."},{"Start":"00:28.350 ","End":"00:30.795","Text":"What about the goat?"},{"Start":"00:30.795 ","End":"00:36.010","Text":"Y was the daily yield of milk from a goat,"},{"Start":"00:36.010 ","End":"00:40.335","Text":"and that had a normal probability distribution with an expectation of"},{"Start":"00:40.335 ","End":"00:44.925","Text":"10 and a standard deviation of 2."},{"Start":"00:44.925 ","End":"00:51.824","Text":"Let\u0027s define t as the total daily yield of milk from the cow and from the goat,"},{"Start":"00:51.824 ","End":"00:53.525","Text":"that\u0027s x plus y."},{"Start":"00:53.525 ","End":"01:00.565","Text":"That means that t also has a normal probability distribution with parameters."},{"Start":"01:00.565 ","End":"01:05.465","Text":"The expectation of t and the standard deviation or the variance of"},{"Start":"01:05.465 ","End":"01:11.255","Text":"t. Let\u0027s calculate these parameters right now."},{"Start":"01:11.255 ","End":"01:13.615","Text":"The expectation of t,"},{"Start":"01:13.615 ","End":"01:18.210","Text":"that equals to the expectation of x plus y and that equals"},{"Start":"01:18.210 ","End":"01:23.335","Text":"to the expectation of x plus the expectation of y."},{"Start":"01:23.335 ","End":"01:27.080","Text":"That equals to 20 plus 10,"},{"Start":"01:27.080 ","End":"01:29.045","Text":"that equals to 30."},{"Start":"01:29.045 ","End":"01:31.235","Text":"What about the variance of t?"},{"Start":"01:31.235 ","End":"01:37.740","Text":"That equals to the variance of x plus y and that equals to the variance of"},{"Start":"01:37.740 ","End":"01:46.405","Text":"x plus the variance of y plus 2 times the covariance of x and y."},{"Start":"01:46.405 ","End":"01:50.210","Text":"Let\u0027s look at the covariance of x and y."},{"Start":"01:50.210 ","End":"01:55.070","Text":"X is defined as the daily yield of milk from a cow,"},{"Start":"01:55.070 ","End":"01:57.740","Text":"and y is the daily yield of milk from a goat."},{"Start":"01:57.740 ","End":"02:02.505","Text":"We can assume that x and y are independent."},{"Start":"02:02.505 ","End":"02:07.475","Text":"If that\u0027s the case, then the covariance of x and y is equal to 0."},{"Start":"02:07.475 ","End":"02:11.750","Text":"This then equals to the variance of x,"},{"Start":"02:11.750 ","End":"02:14.690","Text":"which is 5 squared, plus the variance of y,"},{"Start":"02:14.690 ","End":"02:15.800","Text":"which is 2 squared,"},{"Start":"02:15.800 ","End":"02:18.485","Text":"and that equals to 29."},{"Start":"02:18.485 ","End":"02:28.340","Text":"That would be the variance of t. That means that t has a normal probability distribution,"},{"Start":"02:28.340 ","End":"02:32.435","Text":"but now we know what the values of the parameters are."},{"Start":"02:32.435 ","End":"02:37.205","Text":"Mu equals to 30 and Sigma squared,"},{"Start":"02:37.205 ","End":"02:40.800","Text":"the variance that equals to 29."},{"Start":"02:41.230 ","End":"02:44.360","Text":"Now that we\u0027ve calculated the values of"},{"Start":"02:44.360 ","End":"02:48.130","Text":"the parameters from the normal distribution for t,"},{"Start":"02:48.130 ","End":"02:50.960","Text":"we want to calculate the chances that"},{"Start":"02:50.960 ","End":"02:55.820","Text":"the combined milk yield of the cow and the goat will be less than 30 liters."},{"Start":"02:55.820 ","End":"03:04.730","Text":"Let\u0027s do that. That\u0027ll be the probability of t being less than 30."},{"Start":"03:04.730 ","End":"03:06.770","Text":"What does that equal to?"},{"Start":"03:06.770 ","End":"03:10.375","Text":"Let\u0027s take a look at this thing on a graph."},{"Start":"03:10.375 ","End":"03:13.305","Text":"Here\u0027s our density function."},{"Start":"03:13.305 ","End":"03:17.965","Text":"We have our t-axis and this expectation of t, that\u0027s 30."},{"Start":"03:17.965 ","End":"03:21.940","Text":"The probability of t being less than or equal to 30,"},{"Start":"03:21.940 ","End":"03:28.600","Text":"that\u0027s the area under the density function from minus infinity until t equaling 30."},{"Start":"03:28.600 ","End":"03:31.610","Text":"That;s this area right here,"},{"Start":"03:31.610 ","End":"03:33.925","Text":"as we can easily see,"},{"Start":"03:33.925 ","End":"03:37.160","Text":"that equals to 0.5."},{"Start":"03:37.200 ","End":"03:42.260","Text":"Having calculated that, let\u0027s go on to the next step."},{"Start":"03:42.360 ","End":"03:44.860","Text":"We\u0027re looking for the chances that"},{"Start":"03:44.860 ","End":"03:47.620","Text":"the combined milk yield of the cow and the goat will be"},{"Start":"03:47.620 ","End":"03:53.375","Text":"less than 30 liters on at least 4 of 5 consecutive days."},{"Start":"03:53.375 ","End":"03:56.200","Text":"Let\u0027s do that now."},{"Start":"03:56.390 ","End":"03:59.340","Text":"We\u0027re looking at 5 days."},{"Start":"03:59.340 ","End":"04:01.890","Text":"N here will be equal to 5."},{"Start":"04:01.890 ","End":"04:06.965","Text":"We have 5 consecutive days and we want that"},{"Start":"04:06.965 ","End":"04:14.115","Text":"at least 4 of them would be less than 30 liters per day."},{"Start":"04:14.115 ","End":"04:16.980","Text":"That\u0027s the yield that we\u0027re looking for."},{"Start":"04:16.980 ","End":"04:23.415","Text":"Let\u0027s define success as"},{"Start":"04:23.415 ","End":"04:30.180","Text":"less than 30 liters, that\u0027s our success."},{"Start":"04:30.180 ","End":"04:33.195","Text":"What\u0027s the probability of success?"},{"Start":"04:33.195 ","End":"04:35.450","Text":"That equals to 0.5."},{"Start":"04:35.450 ","End":"04:37.580","Text":"We just calculated that."},{"Start":"04:37.580 ","End":"04:41.990","Text":"Let\u0027s define a new random variable r. That\u0027ll"},{"Start":"04:41.990 ","End":"04:46.320","Text":"be the number of days that we have success,"},{"Start":"04:46.320 ","End":"04:51.225","Text":"that we have less than 30 liters."},{"Start":"04:51.225 ","End":"04:53.595","Text":"When we define r like this,"},{"Start":"04:53.595 ","End":"05:01.595","Text":"then r has a binomial distribution where n equals to 5 and p equals to 0.5."},{"Start":"05:01.595 ","End":"05:06.275","Text":"That means that the probability of r equaling some value k,"},{"Start":"05:06.275 ","End":"05:08.945","Text":"that equals to n over k,"},{"Start":"05:08.945 ","End":"05:16.160","Text":"p^k times 1 minus p^n minus k. We\u0027re"},{"Start":"05:16.160 ","End":"05:23.120","Text":"interested now in the probability where r is at least 4 days,"},{"Start":"05:23.120 ","End":"05:27.235","Text":"that means that r is greater or equal to 4."},{"Start":"05:27.235 ","End":"05:32.480","Text":"That means that we\u0027re looking for the probability of r equaling"},{"Start":"05:32.480 ","End":"05:38.760","Text":"4 plus the probability of r equaling 5."},{"Start":"05:39.590 ","End":"05:42.330","Text":"That equals to the following."},{"Start":"05:42.330 ","End":"05:46.715","Text":"Let\u0027s just plug in the numbers into this equation right here."},{"Start":"05:46.715 ","End":"05:50.585","Text":"The probability of r equaling 4 while n equals 5."},{"Start":"05:50.585 ","End":"05:53.455","Text":"It\u0027ll be 5 over 4,"},{"Start":"05:53.455 ","End":"05:59.825","Text":"0.50 that\u0027s the probability to the power of 4 times 0.5^1."},{"Start":"05:59.825 ","End":"06:05.495","Text":"That\u0027s the probability of r equaling 4 plus the probability of r equaling 5,"},{"Start":"06:05.495 ","End":"06:07.255","Text":"that\u0027ll be 5 over 5,"},{"Start":"06:07.255 ","End":"06:14.515","Text":"0.5^ 5 times 0.5^0."},{"Start":"06:14.515 ","End":"06:20.265","Text":"All this comes out to 0.1875."},{"Start":"06:20.265 ","End":"06:25.260","Text":"This then is the probability of getting"},{"Start":"06:25.260 ","End":"06:35.230","Text":"less than 30 liters on at least 4 days in 5 consecutive days."}],"ID":13249},{"Watched":false,"Name":"Exercise 5 - Part c","Duration":"5m 43s","ChapterTopicVideoID":12771,"CourseChapterTopicPlaylistID":245055,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.350","Text":"In this section we\u0027re asked,"},{"Start":"00:01.350 ","End":"00:03.570","Text":"what are the chances that the cow\u0027s yield will be"},{"Start":"00:03.570 ","End":"00:06.885","Text":"lower than the goat\u0027s yield on a given day?"},{"Start":"00:06.885 ","End":"00:13.650","Text":"We\u0027re asking about the probability of x being less than y."},{"Start":"00:13.650 ","End":"00:14.925","Text":"Now if we recall,"},{"Start":"00:14.925 ","End":"00:17.753","Text":"x would be the yield of the cow,"},{"Start":"00:17.753 ","End":"00:25.035","Text":"and that had a normal probability distribution when Mu equaled 20,"},{"Start":"00:25.035 ","End":"00:28.200","Text":"and Sigma squared equals to 5 squared,"},{"Start":"00:28.200 ","End":"00:31.860","Text":"and y was the yield of the goat,"},{"Start":"00:31.860 ","End":"00:34.770","Text":"and that also had a normal distribution,"},{"Start":"00:34.770 ","End":"00:36.788","Text":"but Mu equaled 10,"},{"Start":"00:36.788 ","End":"00:40.230","Text":"and Sigma squared equaled 2 squared."},{"Start":"00:40.230 ","End":"00:43.410","Text":"This is what we\u0027re looking for."},{"Start":"00:43.410 ","End":"00:51.635","Text":"We\u0027re looking also, we can write this out as the probability of x minus y is less than 0."},{"Start":"00:51.635 ","End":"00:53.810","Text":"Now if we take a look at this,"},{"Start":"00:53.810 ","End":"00:55.190","Text":"we define D,"},{"Start":"00:55.190 ","End":"01:00.050","Text":"the difference, that will be equal to x minus y."},{"Start":"01:00.050 ","End":"01:04.970","Text":"Well, we know that that also has a normal probability distribution"},{"Start":"01:04.970 ","End":"01:11.075","Text":"with its expectation and its variance."},{"Start":"01:11.075 ","End":"01:15.935","Text":"Now let\u0027s calculate this expectation and variance."},{"Start":"01:15.935 ","End":"01:19.370","Text":"The expectation of D. Well,"},{"Start":"01:19.370 ","End":"01:23.413","Text":"that equals to the expectation of x minus y,"},{"Start":"01:23.413 ","End":"01:28.550","Text":"and that equals to the expectation of x minus the expectation of y,"},{"Start":"01:28.550 ","End":"01:33.610","Text":"and that equals to 20 minus 10, that equals to 10."},{"Start":"01:33.610 ","End":"01:35.980","Text":"What about the variance of D?"},{"Start":"01:35.980 ","End":"01:39.530","Text":"That\u0027s the variance of x minus y,"},{"Start":"01:39.530 ","End":"01:44.000","Text":"and that equals to the variance of x plus the variance of"},{"Start":"01:44.000 ","End":"01:48.995","Text":"y minus 2 times the covariance of x and y."},{"Start":"01:48.995 ","End":"01:54.320","Text":"Now we can assume that x and y are independent"},{"Start":"01:54.320 ","End":"02:01.195","Text":"because that\u0027s the daily yield of milk from a cow and from a goat."},{"Start":"02:01.195 ","End":"02:03.450","Text":"That\u0027s independent of each other."},{"Start":"02:03.450 ","End":"02:06.095","Text":"The covariance here would be equal to 0."},{"Start":"02:06.095 ","End":"02:11.045","Text":"Now that\u0027ll be equal to 5 squared then plus 2 squared,"},{"Start":"02:11.045 ","End":"02:14.065","Text":"and that equals to 29."},{"Start":"02:14.065 ","End":"02:18.500","Text":"Now we know what the distribution of D is."},{"Start":"02:18.500 ","End":"02:21.676","Text":"D has a normal distribution."},{"Start":"02:21.676 ","End":"02:27.005","Text":"We know that because D is a linear combination of x and y."},{"Start":"02:27.005 ","End":"02:30.800","Text":"Now what are the values of the distribution?"},{"Start":"02:30.800 ","End":"02:32.933","Text":"Well, Mu here well that equals to 10,"},{"Start":"02:32.933 ","End":"02:34.985","Text":"that\u0027s the expectation of D,"},{"Start":"02:34.985 ","End":"02:37.400","Text":"and the variance of D,"},{"Start":"02:37.400 ","End":"02:40.380","Text":"well that equals to 29."},{"Start":"02:40.630 ","End":"02:43.460","Text":"Now that we know what the values are for"},{"Start":"02:43.460 ","End":"02:46.400","Text":"the parameters of the normal distribution for D,"},{"Start":"02:46.400 ","End":"02:49.370","Text":"then we can go ahead and calculate the chances that"},{"Start":"02:49.370 ","End":"02:52.865","Text":"the cow\u0027s yield would be lower than the goat\u0027s yield on a given day."},{"Start":"02:52.865 ","End":"02:58.855","Text":"That means that we\u0027re looking at the probability of x minus y being less than 0,"},{"Start":"02:58.855 ","End":"03:01.335","Text":"or in terms of D,"},{"Start":"03:01.335 ","End":"03:05.775","Text":"the probability of D being less than 0."},{"Start":"03:05.775 ","End":"03:09.070","Text":"Let\u0027s calculate this probability."},{"Start":"03:09.070 ","End":"03:14.069","Text":"Now the probability of D being less than 0,"},{"Start":"03:14.069 ","End":"03:18.425","Text":"well, let\u0027s see what this looks like on a graph."},{"Start":"03:18.425 ","End":"03:21.185","Text":"Here\u0027s the density function for D,"},{"Start":"03:21.185 ","End":"03:22.430","Text":"that\u0027s the D axis,"},{"Start":"03:22.430 ","End":"03:23.711","Text":"that\u0027s the density function,"},{"Start":"03:23.711 ","End":"03:26.765","Text":"here\u0027s the expectation of D, and that\u0027s 10."},{"Start":"03:26.765 ","End":"03:30.445","Text":"We\u0027re looking for the probability of D being less than 0."},{"Start":"03:30.445 ","End":"03:32.730","Text":"Here, that\u0027ll be 0,"},{"Start":"03:32.730 ","End":"03:39.755","Text":"and we\u0027re looking for the area under the density function from minus infinity to 0."},{"Start":"03:39.755 ","End":"03:42.900","Text":"We want to calculate that area."},{"Start":"03:43.010 ","End":"03:45.020","Text":"In order to do that,"},{"Start":"03:45.020 ","End":"03:48.335","Text":"we need to standardize this value right here."},{"Start":"03:48.335 ","End":"03:53.855","Text":"That\u0027ll be equal to the probability now of D minus Mu divided by Sigma,"},{"Start":"03:53.855 ","End":"03:58.115","Text":"that has to be less than 0 minus Mu divided by Sigma."},{"Start":"03:58.115 ","End":"04:00.395","Text":"Now, that equals to what?"},{"Start":"04:00.395 ","End":"04:01.730","Text":"That\u0027s the probability."},{"Start":"04:01.730 ","End":"04:03.110","Text":"Now this expression right here,"},{"Start":"04:03.110 ","End":"04:07.670","Text":"that\u0027s Z and that has to be less than 0 minus Mu."},{"Start":"04:07.670 ","End":"04:11.365","Text":"Well Mu is 10, divided by Sigma."},{"Start":"04:11.365 ","End":"04:15.720","Text":"Sigma is the square root of 29."},{"Start":"04:15.720 ","End":"04:24.385","Text":"That equals to the probability of Z being less than minus 1.86."},{"Start":"04:24.385 ","End":"04:33.590","Text":"The comparable value of 0 on the z-axis, that\u0027s minus 1.86."},{"Start":"04:33.930 ","End":"04:36.440","Text":"How do we calculate that?"},{"Start":"04:36.440 ","End":"04:43.385","Text":"Well, this then equals to Phi of minus 1.86."},{"Start":"04:43.385 ","End":"04:51.710","Text":"Now that equals to 1 minus Phi of 1.86."},{"Start":"04:51.710 ","End":"04:54.680","Text":"Now if this transition isn\u0027t clear to you,"},{"Start":"04:54.680 ","End":"04:58.370","Text":"then I urge you to go back and review the material."},{"Start":"04:58.370 ","End":"05:00.050","Text":"Now, let\u0027s continue."},{"Start":"05:00.050 ","End":"05:03.650","Text":"1 minus Phi of 1.86, well,"},{"Start":"05:03.650 ","End":"05:11.030","Text":"we have to go to the standardized normal table to find out this value right here."},{"Start":"05:11.030 ","End":"05:13.850","Text":"Again, I\u0027ll let you do that,"},{"Start":"05:13.850 ","End":"05:22.680","Text":"but that equals to 1 minus 0.9686."},{"Start":"05:22.680 ","End":"05:28.200","Text":"That equals to 0.0314."},{"Start":"05:28.200 ","End":"05:33.650","Text":"This then is the probability of D being less than 0."},{"Start":"05:33.650 ","End":"05:36.380","Text":"That means that the probability of"},{"Start":"05:36.380 ","End":"05:43.500","Text":"a cow\u0027s daily yield being less than that of a goat\u0027s daily yield."}],"ID":13250}],"Thumbnail":null,"ID":245055},{"Name":"Log Normal Distribution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"2m 38s","ChapterTopicVideoID":12772,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12772.jpeg","UploadDate":"2019-01-20T14:34:32.8070000","DurationForVideoObject":"PT2M38S","Description":null,"MetaTitle":"Tutorial: Video + Workbook | Proprep","MetaDescription":"Linear Combinations of Normal Probability Distributions - Log Normal Distribution. 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/linear-combinations-of-normal-probability-distributions/log-normal-distribution/vid13251","VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:02.970","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:02.970 ","End":"00:06.795","Text":"a distribution which is called the log normal distribution."},{"Start":"00:06.795 ","End":"00:12.960","Text":"Now, assume that we have a variable Y and Y is distributed with the normal distribution."},{"Start":"00:12.960 ","End":"00:17.325","Text":"Y has a normal distribution with parameters Mu and Sigma squared."},{"Start":"00:17.325 ","End":"00:23.625","Text":"Now, let\u0027s define a relationship between Y and a new variable called X."},{"Start":"00:23.625 ","End":"00:25.200","Text":"This is the relationship."},{"Start":"00:25.200 ","End":"00:28.740","Text":"X would be equal to e to the power of Y."},{"Start":"00:28.740 ","End":"00:31.500","Text":"This Y is this Y right here,"},{"Start":"00:31.500 ","End":"00:32.960","Text":"which has a normal distribution."},{"Start":"00:32.960 ","End":"00:37.455","Text":"So we say that X has a log normal distribution."},{"Start":"00:37.455 ","End":"00:41.225","Text":"Now, the reason for calling this a log normal distribution is,"},{"Start":"00:41.225 ","End":"00:46.790","Text":"if we take the natural logarithm from both sides of this relationship right here,"},{"Start":"00:46.790 ","End":"00:48.770","Text":"then we have log of X,"},{"Start":"00:48.770 ","End":"00:52.030","Text":"and that\u0027ll be equal to Y."},{"Start":"00:52.030 ","End":"00:58.280","Text":"Now, the domain of Y is from minus infinity to plus infinity."},{"Start":"00:58.280 ","End":"01:03.560","Text":"Variable Y can have values from minus infinity to plus infinity."},{"Start":"01:03.560 ","End":"01:11.000","Text":"But the domain of X has values only from 0 to plus infinity, and why is that?"},{"Start":"01:11.000 ","End":"01:13.670","Text":"Well again, let\u0027s look at this relationship."},{"Start":"01:13.670 ","End":"01:16.490","Text":"As Y goes to minus infinity, well,"},{"Start":"01:16.490 ","End":"01:18.385","Text":"then X goes to 0,"},{"Start":"01:18.385 ","End":"01:20.910","Text":"and as Y goes to plus infinity,"},{"Start":"01:20.910 ","End":"01:23.925","Text":"well, then X goes to plus infinity."},{"Start":"01:23.925 ","End":"01:31.190","Text":"We say that X has a log normal distribution with parameters Mu and Sigma squared."},{"Start":"01:31.190 ","End":"01:34.010","Text":"Now, we have to be careful here because"},{"Start":"01:34.010 ","End":"01:37.400","Text":"this Mu and Sigma squared are not actually perimeters of X,"},{"Start":"01:37.400 ","End":"01:38.900","Text":"but they are parameters of Y."},{"Start":"01:38.900 ","End":"01:41.980","Text":"These parameters right here,"},{"Start":"01:41.980 ","End":"01:47.530","Text":"so Mu and Sigma squared are the expectation and variance of the variable Y."},{"Start":"01:47.530 ","End":"01:52.550","Text":"How do we calculate the expectation and variance of X?"},{"Start":"01:52.550 ","End":"01:57.350","Text":"Well, the expectation of X is equal to e to"},{"Start":"01:57.350 ","End":"02:02.690","Text":"the power of Mu plus 1/2 Sigma squared and the variance of X,"},{"Start":"02:02.690 ","End":"02:07.250","Text":"well that\u0027s equal to e to the power of Sigma squared minus 1"},{"Start":"02:07.250 ","End":"02:12.305","Text":"times e to the power of 2 times Mu plus Sigma squared."},{"Start":"02:12.305 ","End":"02:17.300","Text":"Also, the shape of the log normal distribution."},{"Start":"02:17.300 ","End":"02:22.985","Text":"Well, the log normal distribution has a right asymmetry."},{"Start":"02:22.985 ","End":"02:26.395","Text":"It has a long tail going off to the right."},{"Start":"02:26.395 ","End":"02:30.350","Text":"Now that we know about the log normal distribution,"},{"Start":"02:30.350 ","End":"02:35.210","Text":"we know how to calculate the expectation and variance and we know what it looks like,"},{"Start":"02:35.210 ","End":"02:38.100","Text":"let\u0027s take a look at an example."}],"ID":13251},{"Watched":false,"Name":"Example","Duration":"2m 50s","ChapterTopicVideoID":12773,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.025","Text":"In this example, we\u0027re given that X has a Log Normal distribution with"},{"Start":"00:05.025 ","End":"00:09.675","Text":"Mu equal to 10 and Sigma squared equals 2 squared,"},{"Start":"00:09.675 ","End":"00:14.925","Text":"and we\u0027re asked to find the 90th percentile of this distribution right here."},{"Start":"00:14.925 ","End":"00:21.810","Text":"So let\u0027s just take a look at the relationship between Y and X."},{"Start":"00:21.810 ","End":"00:25.560","Text":"This is it. Y equals to ln X."},{"Start":"00:25.560 ","End":"00:28.365","Text":"This is Y and this is X right here."},{"Start":"00:28.365 ","End":"00:33.705","Text":"We can see that Y can go from minus infinity to plus infinity."},{"Start":"00:33.705 ","End":"00:36.090","Text":"Y, this line right here,"},{"Start":"00:36.090 ","End":"00:43.115","Text":"can go from Y equals to minus infinity up to plus infinity right here."},{"Start":"00:43.115 ","End":"00:49.134","Text":"On the other hand, X can only go from 0 to plus infinity."},{"Start":"00:49.134 ","End":"00:52.250","Text":"So this is the relationship between X and Y."},{"Start":"00:52.250 ","End":"00:59.090","Text":"Now, we\u0027re asked to find the 90th percentile of the Log Normal distribution here."},{"Start":"00:59.090 ","End":"01:03.745","Text":"We know that in a normal distribution,"},{"Start":"01:03.745 ","End":"01:08.100","Text":"we know that Z_0.9, well,"},{"Start":"01:08.100 ","End":"01:15.890","Text":"that equals to 1.282 and I invite you to go to the tables and check this number."},{"Start":"01:15.890 ","End":"01:19.265","Text":"We also know that Z, well,"},{"Start":"01:19.265 ","End":"01:24.375","Text":"that equals to Y minus Mu divided by Sigma."},{"Start":"01:24.375 ","End":"01:27.315","Text":"Let\u0027s just plug in the numbers here."},{"Start":"01:27.315 ","End":"01:31.065","Text":"So Y or Z,"},{"Start":"01:31.065 ","End":"01:35.130","Text":"that\u0027s 1.282, well,"},{"Start":"01:35.130 ","End":"01:38.310","Text":"that equals to Y minus Mu,"},{"Start":"01:38.310 ","End":"01:41.640","Text":"our Mu is 10, divided by Sigma,"},{"Start":"01:41.640 ","End":"01:43.250","Text":"where Sigma squared is 2 squared."},{"Start":"01:43.250 ","End":"01:45.440","Text":"So Sigma equals to 2."},{"Start":"01:45.440 ","End":"01:48.715","Text":"Let\u0027s just solve for Y."},{"Start":"01:48.715 ","End":"01:51.350","Text":"Now, we know that by solving for Y,"},{"Start":"01:51.350 ","End":"01:55.100","Text":"this will give me the value of the 90th percentile."},{"Start":"01:55.100 ","End":"02:04.080","Text":"So that\u0027ll be Y_0.9 and that equals to 12.564."},{"Start":"02:04.080 ","End":"02:06.320","Text":"We\u0027re not done yet. Why is that?"},{"Start":"02:06.320 ","End":"02:11.120","Text":"Because we know that Y equals to ln of X."},{"Start":"02:11.120 ","End":"02:15.140","Text":"So if we take e of both sides,"},{"Start":"02:15.140 ","End":"02:18.765","Text":"then we have e^Y,"},{"Start":"02:18.765 ","End":"02:23.010","Text":"which is 12.564,"},{"Start":"02:23.010 ","End":"02:26.685","Text":"and that\u0027ll be equal to X."},{"Start":"02:26.685 ","End":"02:36.455","Text":"That means that X is equal to approximately 286,072."},{"Start":"02:36.455 ","End":"02:45.335","Text":"So this then would be the 90th percentile of the Log Normal distribution,"},{"Start":"02:45.335 ","End":"02:49.950","Text":"where Mu equals 10 and Sigma squared equals 2 squared."}],"ID":13252},{"Watched":false,"Name":"Exercise 1","Duration":"4m 13s","ChapterTopicVideoID":12774,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"In this question, we\u0027re given that X has"},{"Start":"00:02.520 ","End":"00:07.784","Text":"a Log Normal distribution with parameters 0 and 1 squared."},{"Start":"00:07.784 ","End":"00:13.110","Text":"Instead, mu equals to 0 and the sigma squared equals to 1 squared."},{"Start":"00:13.110 ","End":"00:15.330","Text":"We\u0027re asked, what\u0027s the distribution of Y,"},{"Start":"00:15.330 ","End":"00:17.430","Text":"which equals to In of X?"},{"Start":"00:17.430 ","End":"00:26.535","Text":"Well, we know that if X has a Log Normal distribution, then Y,"},{"Start":"00:26.535 ","End":"00:30.510","Text":"which is equals to ln of x, well,"},{"Start":"00:30.510 ","End":"00:35.595","Text":"that has a normal distribution with the same parameters because mu"},{"Start":"00:35.595 ","End":"00:42.040","Text":"equals to 0 and sigma squared that equals to 1 squared."},{"Start":"00:42.040 ","End":"00:47.285","Text":"This is called the Standard Normal Distribution."},{"Start":"00:47.285 ","End":"00:52.040","Text":"Now, we know that a standard normal distribution"},{"Start":"00:52.040 ","End":"00:59.075","Text":"has an expectation of 0 and a standard deviation or variance of 1."},{"Start":"00:59.075 ","End":"01:03.230","Text":"This then would be the answer for Section A."},{"Start":"01:03.230 ","End":"01:05.030","Text":"In this section,"},{"Start":"01:05.030 ","End":"01:07.160","Text":"we\u0027re asked what\u0027s the median of X?"},{"Start":"01:07.160 ","End":"01:12.810","Text":"We know that Y equals to ln of x"},{"Start":"01:12.810 ","End":"01:18.695","Text":"and that has a normal distribution with parameters 0 and 1."},{"Start":"01:18.695 ","End":"01:21.410","Text":"This is a standard normal distribution."},{"Start":"01:21.410 ","End":"01:24.305","Text":"So let\u0027s take a look at this distribution."},{"Start":"01:24.305 ","End":"01:29.474","Text":"Here it is and we see here that mu here that equals to 0."},{"Start":"01:29.474 ","End":"01:35.449","Text":"Now, because this is a symmetrical distribution that also equals to the median."},{"Start":"01:35.449 ","End":"01:39.185","Text":"So the median of ln of x,"},{"Start":"01:39.185 ","End":"01:41.360","Text":"well, that equals to the median of Y,"},{"Start":"01:41.360 ","End":"01:43.265","Text":"which is equal to 0."},{"Start":"01:43.265 ","End":"01:50.360","Text":"We can say that the In of the median of x,"},{"Start":"01:50.360 ","End":"01:52.519","Text":"well, that equals to 0."},{"Start":"01:52.519 ","End":"01:54.900","Text":"Now, what\u0027s ln?"},{"Start":"01:54.900 ","End":"02:02.336","Text":"In is the natural logarithm or log base e of the median of x."},{"Start":"02:02.336 ","End":"02:05.570","Text":"Now, we know that equals to 0, right?"},{"Start":"02:05.570 ","End":"02:07.835","Text":"So what does that equal to?"},{"Start":"02:07.835 ","End":"02:10.910","Text":"Well, let\u0027s take e of both sides."},{"Start":"02:10.910 ","End":"02:15.065","Text":"That means that\u0027ll be the median of x,"},{"Start":"02:15.065 ","End":"02:18.920","Text":"that equals to e to the power of 0."},{"Start":"02:18.920 ","End":"02:21.350","Text":"Now, that equals to 1."},{"Start":"02:21.350 ","End":"02:25.385","Text":"So the median of x is equal to 1."},{"Start":"02:25.385 ","End":"02:27.050","Text":"In this section,"},{"Start":"02:27.050 ","End":"02:33.290","Text":"we\u0027re asked to calculate the probability where X is greater than e. Let\u0027s get to work."},{"Start":"02:33.290 ","End":"02:37.400","Text":"Probability of X greater than e. Well,"},{"Start":"02:37.400 ","End":"02:41.720","Text":"let\u0027s take ln of both sides of this equation."},{"Start":"02:41.720 ","End":"02:50.115","Text":"That\u0027ll be the probability of In of x being greater than ln of e. Now,"},{"Start":"02:50.115 ","End":"02:52.650","Text":"what\u0027s ln of e?"},{"Start":"02:52.650 ","End":"02:54.840","Text":"Well, In of e,"},{"Start":"02:54.840 ","End":"02:56.213","Text":"well, that equals to 1."},{"Start":"02:56.213 ","End":"02:58.725","Text":"What\u0027s ln of x?"},{"Start":"02:58.725 ","End":"03:00.835","Text":"Well, that equals to y."},{"Start":"03:00.835 ","End":"03:03.740","Text":"Now, we know that y has"},{"Start":"03:03.740 ","End":"03:10.190","Text":"a normal distribution where mu equals to 0 and sigma squared equals to 1."},{"Start":"03:10.190 ","End":"03:16.821","Text":"Let\u0027s take a look at this distribution right here. Here we go."},{"Start":"03:16.821 ","End":"03:21.260","Text":"This is a standard normal distribution where mu equals to 0."},{"Start":"03:21.260 ","End":"03:23.930","Text":"Now, assume that this in here is equal to 1."},{"Start":"03:23.930 ","End":"03:28.310","Text":"We\u0027re looking for the probability where y is greater than 1."},{"Start":"03:28.310 ","End":"03:31.880","Text":"So we\u0027re looking at this area right here."},{"Start":"03:31.880 ","End":"03:34.205","Text":"Now, what is that equal to?"},{"Start":"03:34.205 ","End":"03:41.840","Text":"Well, that equals to 1 minus the probability of y being less than or equal to 1."},{"Start":"03:41.840 ","End":"03:43.970","Text":"Well, what\u0027s this in right here?"},{"Start":"03:43.970 ","End":"03:45.935","Text":"That\u0027s Phi of 1."},{"Start":"03:45.935 ","End":"03:49.710","Text":"So we\u0027re looking at 1 minus Phi of 1,"},{"Start":"03:49.710 ","End":"03:52.485","Text":"which is equal to 1 minus"},{"Start":"03:52.485 ","End":"03:59.675","Text":"0.8413 and I invite you to go to the tables and check this number."},{"Start":"03:59.675 ","End":"04:05.150","Text":"Now, this is equal to 0.1587."},{"Start":"04:05.150 ","End":"04:09.275","Text":"So the probability of x being greater than e,"},{"Start":"04:09.275 ","End":"04:13.650","Text":"that equals to 0.1587."}],"ID":13253},{"Watched":false,"Name":"Exercise 2","Duration":"6m 38s","ChapterTopicVideoID":12775,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this question, we\u0027re given that the salary of a certain market has"},{"Start":"00:03.900 ","End":"00:07.920","Text":"a Log Normal distribution with an average of 10 and a variance of 2."},{"Start":"00:07.920 ","End":"00:13.140","Text":"Now, let\u0027s take a look at the ln of the salary, ln of x."},{"Start":"00:13.140 ","End":"00:16.410","Text":"Now, how\u0027s the ln of the salary distributed?"},{"Start":"00:16.410 ","End":"00:21.675","Text":"We\u0027re asked what are the values of Mu and Sigma squared?"},{"Start":"00:21.675 ","End":"00:30.660","Text":"We\u0027re given that X has a Log Normal distribution with parameters Mu and Sigma squared."},{"Start":"00:30.660 ","End":"00:32.700","Text":"Now Mu and Sigma squared,"},{"Start":"00:32.700 ","End":"00:35.704","Text":"they don\u0027t belong to X."},{"Start":"00:35.704 ","End":"00:39.540","Text":"They belong to ln of X."},{"Start":"00:39.540 ","End":"00:41.115","Text":"We\u0027ll call that Y."},{"Start":"00:41.115 ","End":"00:49.020","Text":"We know that ln of X has a normal distribution with Mu and Sigma squared."},{"Start":"00:49.250 ","End":"00:52.610","Text":"This answers the first question."},{"Start":"00:52.610 ","End":"00:55.070","Text":"How\u0027s the ln of the salary distributed?"},{"Start":"00:55.070 ","End":"00:59.600","Text":"Well, again, if X has a Log Normal distribution, well,"},{"Start":"00:59.600 ","End":"01:07.250","Text":"ln of X has a normal distribution with the parameters Mu and Sigma squared."},{"Start":"01:07.250 ","End":"01:12.770","Text":"Now, what about the values of Mu and Sigma squared?"},{"Start":"01:12.770 ","End":"01:15.365","Text":"This is what we want to calculate right now."},{"Start":"01:15.365 ","End":"01:23.270","Text":"Well, we know that the expectation of X and the variance of X,"},{"Start":"01:23.270 ","End":"01:26.185","Text":"they have a specific formula."},{"Start":"01:26.185 ","End":"01:33.005","Text":"These are the formulas for the expectation of X and the variance of X."},{"Start":"01:33.005 ","End":"01:35.930","Text":"Now, we know what these are equal to,"},{"Start":"01:35.930 ","End":"01:38.615","Text":"the expectation of X that equals to 10,"},{"Start":"01:38.615 ","End":"01:41.135","Text":"and the variance of X that equals to 2."},{"Start":"01:41.135 ","End":"01:44.180","Text":"Let\u0027s just plug in the numbers here."},{"Start":"01:44.180 ","End":"01:47.850","Text":"We know the 2, the variance of X,"},{"Start":"01:47.850 ","End":"01:53.090","Text":"that equals to e^Sigma squared minus"},{"Start":"01:53.090 ","End":"02:00.230","Text":"1 times e^2 Mu plus Sigma squared."},{"Start":"02:00.230 ","End":"02:02.510","Text":"Now, what about the expectation?"},{"Start":"02:02.510 ","End":"02:04.010","Text":"That equals to 10."},{"Start":"02:04.010 ","End":"02:05.570","Text":"It\u0027s given right here."},{"Start":"02:05.570 ","End":"02:15.510","Text":"That equals to e^Mu plus 1/2 Sigma squared."},{"Start":"02:15.950 ","End":"02:21.560","Text":"Well, then let\u0027s take a look at these 2 expressions right here,"},{"Start":"02:21.560 ","End":"02:23.540","Text":"this expression right here,"},{"Start":"02:23.540 ","End":"02:26.525","Text":"and this expression right here."},{"Start":"02:26.525 ","End":"02:31.910","Text":"We see that if we take this expression and we square it,"},{"Start":"02:31.910 ","End":"02:34.100","Text":"then we get this expression."},{"Start":"02:34.100 ","End":"02:35.570","Text":"We take all this expression,"},{"Start":"02:35.570 ","End":"02:39.035","Text":"we square it, then we have this expression."},{"Start":"02:39.035 ","End":"02:41.660","Text":"Let\u0027s do that. What do we have?"},{"Start":"02:41.660 ","End":"02:43.955","Text":"We have 10 squared,"},{"Start":"02:43.955 ","End":"02:54.890","Text":"that equals to e^2 Mu plus Sigma squared."},{"Start":"02:54.890 ","End":"02:58.500","Text":"I hope you remember your algebra."},{"Start":"02:58.910 ","End":"03:02.060","Text":"We know that this expression, well,"},{"Start":"03:02.060 ","End":"03:06.740","Text":"that equals to 10 squared and that equals to 100."},{"Start":"03:06.740 ","End":"03:11.525","Text":"Let\u0027s just plug in 100 into this expression."},{"Start":"03:11.525 ","End":"03:21.245","Text":"We have 2 and that equals to e^Sigma squared minus 1 times,"},{"Start":"03:21.245 ","End":"03:24.505","Text":"well, again, instead of this expression,"},{"Start":"03:24.505 ","End":"03:26.060","Text":"we\u0027ll be putting in 100,"},{"Start":"03:26.060 ","End":"03:28.070","Text":"so that\u0027ll be a 100."},{"Start":"03:28.070 ","End":"03:36.240","Text":"All we have to do now is to extract Sigma squared. Let\u0027s do that."},{"Start":"03:36.890 ","End":"03:41.260","Text":"The first thing that we want to do is divide everything by a 100."},{"Start":"03:41.260 ","End":"03:49.310","Text":"We have 0.02 and that equals to e^Sigma squared minus 1."},{"Start":"03:49.310 ","End":"03:51.429","Text":"Let\u0027s move the minus 1 over,"},{"Start":"03:51.429 ","End":"03:56.915","Text":"so we have 1.02 and that equals to e^Sigma squared."},{"Start":"03:56.915 ","End":"04:00.925","Text":"Let\u0027s now take ln of both sides."},{"Start":"04:00.925 ","End":"04:11.000","Text":"We have ln of 1.02 and that\u0027ll be equal to ln of e^Sigma"},{"Start":"04:11.000 ","End":"04:15.690","Text":"squared and that means that we have here ln"},{"Start":"04:15.690 ","End":"04:21.855","Text":"of 1.02 and that equals to Sigma squared."},{"Start":"04:21.855 ","End":"04:26.390","Text":"Now, we have a value for Sigma squared."},{"Start":"04:26.390 ","End":"04:31.385","Text":"Now, let\u0027s take this value and plug it back"},{"Start":"04:31.385 ","End":"04:37.280","Text":"right here into this formula so we can extract Mu."},{"Start":"04:37.280 ","End":"04:39.305","Text":"What do we have here?"},{"Start":"04:39.305 ","End":"04:43.621","Text":"We have here 100,"},{"Start":"04:43.621 ","End":"04:51.035","Text":"that equals to e^2 Mu plus Sigma squared."},{"Start":"04:51.035 ","End":"04:53.180","Text":"Now, Sigma squared is this value,"},{"Start":"04:53.180 ","End":"04:55.520","Text":"so let\u0027s plugin this value over here."},{"Start":"04:55.520 ","End":"05:00.950","Text":"That\u0027ll be 100 and that\u0027ll be equal to"},{"Start":"05:00.950 ","End":"05:12.740","Text":"e^2 Mu times e^ln of 1.02."},{"Start":"05:12.740 ","End":"05:16.440","Text":"I hope you still remember your algebra."},{"Start":"05:16.710 ","End":"05:19.400","Text":"What do we have here?"},{"Start":"05:19.400 ","End":"05:24.225","Text":"Well, let\u0027s just calculate for Mu."},{"Start":"05:24.225 ","End":"05:29.465","Text":"Now, e^ln of 1.02,"},{"Start":"05:29.465 ","End":"05:33.425","Text":"well, that equals to 1.02."},{"Start":"05:33.425 ","End":"05:39.260","Text":"Here, we still have e^2 Mu and this equals to 100."},{"Start":"05:39.260 ","End":"05:42.815","Text":"Let\u0027s divide everything by 1.02."},{"Start":"05:42.815 ","End":"05:45.583","Text":"So 100 divided by 1.02,"},{"Start":"05:45.583 ","End":"05:50.395","Text":"that equals to a^2 Mu."},{"Start":"05:50.395 ","End":"05:53.835","Text":"Let\u0027s take ln of both sides."},{"Start":"05:53.835 ","End":"05:59.555","Text":"So we have ln of 100 divided by 1.02,"},{"Start":"05:59.555 ","End":"06:02.950","Text":"that equals to 2 Mu."},{"Start":"06:02.950 ","End":"06:07.070","Text":"The last step is to divide everything by 2."},{"Start":"06:07.070 ","End":"06:15.185","Text":"Mu here, that equals to ln of 100 divided by 1.02,"},{"Start":"06:15.185 ","End":"06:18.570","Text":"all this divided by 2."},{"Start":"06:18.570 ","End":"06:23.210","Text":"Here we go. This then is the value for Sigma squared,"},{"Start":"06:23.210 ","End":"06:24.455","Text":"that\u0027s the variance,"},{"Start":"06:24.455 ","End":"06:31.455","Text":"and this then would be the value of the expectation of ln of X,"},{"Start":"06:31.455 ","End":"06:38.260","Text":"the expectation of the ln of the salary and this is a variance of the ln of the salary."}],"ID":13254},{"Watched":false,"Name":"Exercise 3","Duration":"1m 50s","ChapterTopicVideoID":12776,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this question, we\u0027re asked to prove that the median of x that has"},{"Start":"00:04.140 ","End":"00:10.380","Text":"a log-normal distribution with these parameters that equals to e^Mu."},{"Start":"00:10.380 ","End":"00:15.300","Text":"If x has a log-normal distribution with Mu and Sigma squared,"},{"Start":"00:15.300 ","End":"00:17.520","Text":"then lawn of x,"},{"Start":"00:17.520 ","End":"00:23.400","Text":"well that has a normal distribution with Mu and Sigma squared."},{"Start":"00:23.400 ","End":"00:25.185","Text":"We\u0027ll call that y."},{"Start":"00:25.185 ","End":"00:29.805","Text":"Let\u0027s just take a look at this distribution right here."},{"Start":"00:29.805 ","End":"00:33.480","Text":"Here we go. This is a normal distribution."},{"Start":"00:33.480 ","End":"00:35.745","Text":"Here we have Mu."},{"Start":"00:35.745 ","End":"00:39.925","Text":"Now because this is symmetrical distribution,"},{"Start":"00:39.925 ","End":"00:42.859","Text":"then this equals to the median,"},{"Start":"00:42.859 ","End":"00:45.095","Text":"now this is y right here."},{"Start":"00:45.095 ","End":"00:54.440","Text":"Now, if we plug in the value of y into this relationship right here,"},{"Start":"00:54.440 ","End":"00:57.290","Text":"then by solving for x,"},{"Start":"00:57.290 ","End":"00:59.930","Text":"we\u0027ll be able to get the median of x."},{"Start":"00:59.930 ","End":"01:03.410","Text":"Just because of this relationship right here."},{"Start":"01:03.410 ","End":"01:05.155","Text":"What\u0027s the median of y?"},{"Start":"01:05.155 ","End":"01:07.640","Text":"Well, the mean of y is Mu."},{"Start":"01:07.640 ","End":"01:13.220","Text":"Now, that equals to lawn of the median of x right here."},{"Start":"01:13.220 ","End":"01:16.970","Text":"Now, let\u0027s just solve for x. Mu here,"},{"Start":"01:16.970 ","End":"01:20.955","Text":"well that equals to log base e of x."},{"Start":"01:20.955 ","End":"01:25.005","Text":"Let\u0027s just take e of both sides here."},{"Start":"01:25.005 ","End":"01:27.230","Text":"That\u0027ll be e^Mu,"},{"Start":"01:27.230 ","End":"01:29.960","Text":"and that will be equal to x."},{"Start":"01:29.960 ","End":"01:31.880","Text":"Now, this x right here,"},{"Start":"01:31.880 ","End":"01:35.030","Text":"well that\u0027s the median of x. Why is that?"},{"Start":"01:35.030 ","End":"01:37.550","Text":"Because we\u0027ve plugged in Mu here,"},{"Start":"01:37.550 ","End":"01:41.605","Text":"the median of y into this relationship right here."},{"Start":"01:41.605 ","End":"01:47.420","Text":"There we go. Here we proved that the median of x is e^Mu,"},{"Start":"01:47.420 ","End":"01:50.670","Text":"which is exactly what we were asked to do."}],"ID":13255},{"Watched":false,"Name":"Exercise 4","Duration":"5m 34s","ChapterTopicVideoID":12777,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this question, we\u0027re given that X_i has"},{"Start":"00:03.300 ","End":"00:06.840","Text":"a log-normal distribution with Mu and Sigma squared for all i"},{"Start":"00:06.840 ","End":"00:13.140","Text":"going from 1 to n. Also given is that all the observations are independent of each other."},{"Start":"00:13.140 ","End":"00:16.395","Text":"We\u0027re asked to prove that the product of X_i,"},{"Start":"00:16.395 ","End":"00:23.715","Text":"i going from 1 to n has a log-normal distribution with nMu and nSigma squared."},{"Start":"00:23.715 ","End":"00:28.895","Text":"Let\u0027s begin. We\u0027re given that X_i has"},{"Start":"00:28.895 ","End":"00:34.580","Text":"a log normal distribution with Mu and Sigma squared,"},{"Start":"00:34.580 ","End":"00:41.090","Text":"that means that the Mu is the same for all of the X_i\u0027s and so as the variance."},{"Start":"00:41.090 ","End":"00:44.075","Text":"The variances of all the X_i\u0027s are the same."},{"Start":"00:44.075 ","End":"00:49.150","Text":"Now, we know that ln of X_i,"},{"Start":"00:49.150 ","End":"00:54.560","Text":"well that has a normal distribution with Mu and Sigma squared."},{"Start":"00:54.560 ","End":"00:57.405","Text":"Now, we\u0027ll call that y_ i."},{"Start":"00:57.405 ","End":"01:01.550","Text":"Because X_i has a log-normal distribution,"},{"Start":"01:01.550 ","End":"01:07.550","Text":"that means that ln of X_i has a normal distribution with the same parameters."},{"Start":"01:07.550 ","End":"01:12.135","Text":"Now, what do we know about y_i?"},{"Start":"01:12.135 ","End":"01:15.710","Text":"We know that the sum of y_i,"},{"Start":"01:15.710 ","End":"01:20.660","Text":"i going from 1 to n. What distribution does that have?"},{"Start":"01:20.660 ","End":"01:23.150","Text":"That has a normal distribution."},{"Start":"01:23.150 ","End":"01:27.610","Text":"Now let\u0027s take a look at the expectation."},{"Start":"01:27.610 ","End":"01:31.035","Text":"What\u0027s the expectation of the sum?"},{"Start":"01:31.035 ","End":"01:36.245","Text":"The expectation of the sum is the sum of the expectations,"},{"Start":"01:36.245 ","End":"01:38.390","Text":"but since all the expectations,"},{"Start":"01:38.390 ","End":"01:45.450","Text":"all the Mus are the same and we have n variables right here, y_i to 1n."},{"Start":"01:45.450 ","End":"01:50.545","Text":"That means that the expectation here would be n times Mu."},{"Start":"01:50.545 ","End":"01:52.725","Text":"Now, what about the variance?"},{"Start":"01:52.725 ","End":"01:58.308","Text":"Because the variables are independent,"},{"Start":"01:58.308 ","End":"02:06.145","Text":"given that here, then the variance of the sum is the sum of the variances."},{"Start":"02:06.145 ","End":"02:09.355","Text":"Now, since other variances are the same,"},{"Start":"02:09.355 ","End":"02:13.180","Text":"that means that this equals to n times Sigma squared."},{"Start":"02:13.180 ","End":"02:16.150","Text":"Now all this, we\u0027ve learned in the chapter"},{"Start":"02:16.150 ","End":"02:20.290","Text":"dealing with linear combinations of the normal distribution."},{"Start":"02:20.990 ","End":"02:28.630","Text":"Now let\u0027s take a look at the relationship between x_i and y_i."},{"Start":"02:28.630 ","End":"02:30.490","Text":"We know that y_i,"},{"Start":"02:30.490 ","End":"02:33.905","Text":"that equals to ln of x_i."},{"Start":"02:33.905 ","End":"02:38.025","Text":"But let\u0027s now convert back to x_i."},{"Start":"02:38.025 ","End":"02:44.480","Text":"X_i that equals to e^y_i."},{"Start":"02:44.480 ","End":"02:47.400","Text":"Now, why do we need that?"},{"Start":"02:47.400 ","End":"02:51.690","Text":"Let\u0027s now convert this back into x_i."},{"Start":"02:51.690 ","End":"02:58.895","Text":"Now, if we have the sum of y_i\u0027s that has a normal distribution with these parameters,"},{"Start":"02:58.895 ","End":"03:03.125","Text":"let\u0027s just take the e of this thing right here."},{"Start":"03:03.125 ","End":"03:07.910","Text":"That\u0027ll be e to the power of the sum of y_i,"},{"Start":"03:07.910 ","End":"03:13.375","Text":"i going from 1 to n. We know that that has"},{"Start":"03:13.375 ","End":"03:23.850","Text":"a log normal distribution with these parameters nMu and nSigma squared."},{"Start":"03:23.960 ","End":"03:26.918","Text":"Now, it makes sense."},{"Start":"03:26.918 ","End":"03:32.195","Text":"If x has a log-normal distribution with a specific set of parameters,"},{"Start":"03:32.195 ","End":"03:34.505","Text":"then ln of x,"},{"Start":"03:34.505 ","End":"03:38.990","Text":"which is y, has a normal distribution."},{"Start":"03:38.990 ","End":"03:42.290","Text":"Now if we started with y and we want to go back to x,"},{"Start":"03:42.290 ","End":"03:49.465","Text":"then e^y_i, that\u0027s x_i,"},{"Start":"03:49.465 ","End":"03:52.920","Text":"and that has a log normal distribution with these parameters,"},{"Start":"03:52.920 ","End":"03:57.560","Text":"and this is exactly what I\u0027ve written down here for the sum of the y_i\u0027s."},{"Start":"03:57.560 ","End":"04:02.245","Text":"Now, let\u0027s take a look at this expression right here."},{"Start":"04:02.245 ","End":"04:06.435","Text":"E to the power of the sum of y_i."},{"Start":"04:06.435 ","End":"04:08.300","Text":"What is that equal to?"},{"Start":"04:08.300 ","End":"04:11.990","Text":"We\u0027re looking at e to the power of the sum of y_i,"},{"Start":"04:11.990 ","End":"04:21.120","Text":"i goes from 1 to n. That equals to e^y_1 plus y_2 plus,"},{"Start":"04:21.120 ","End":"04:24.490","Text":"and so on and so forth plus y_n."},{"Start":"04:24.490 ","End":"04:34.300","Text":"Now, that equals to e^y_1 times e^y_2 times,"},{"Start":"04:34.300 ","End":"04:38.935","Text":"and so on and so forth times e^y_n."},{"Start":"04:38.935 ","End":"04:43.485","Text":"What does e^y_1?"},{"Start":"04:43.485 ","End":"04:44.730","Text":"What does that equal to?"},{"Start":"04:44.730 ","End":"04:47.175","Text":"Right here, that equals to x_1."},{"Start":"04:47.175 ","End":"04:50.115","Text":"E^y_i equals to x_i."},{"Start":"04:50.115 ","End":"04:56.220","Text":"Here we\u0027re dealing with x_1 times x_2,"},{"Start":"04:56.220 ","End":"05:01.895","Text":"that\u0027s e^y_2 times, and so on and so forth times x_n."},{"Start":"05:01.895 ","End":"05:08.970","Text":"That equals to the product i goes from 1 to n of x_i."},{"Start":"05:09.440 ","End":"05:14.000","Text":"We know now that this expression right here,"},{"Start":"05:14.000 ","End":"05:16.940","Text":"that equals to this expression right here."},{"Start":"05:16.940 ","End":"05:22.865","Text":"That means that this expression has a log-normal distribution"},{"Start":"05:22.865 ","End":"05:30.040","Text":"with parameters nMu and nSigma squared."},{"Start":"05:30.070 ","End":"05:34.650","Text":"This then is what we were asked to prove."}],"ID":13256},{"Watched":false,"Name":"Exercise 5 Part a","Duration":"6m 47s","ChapterTopicVideoID":15225,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this question, we\u0027re given that the life cycle of an electrical device has"},{"Start":"00:04.140 ","End":"00:06.690","Text":"a log-normal distribution with an expectation of"},{"Start":"00:06.690 ","End":"00:10.365","Text":"6 years and a standard deviation of 1.5 years."},{"Start":"00:10.365 ","End":"00:14.970","Text":"We\u0027re asked to find the upper 10th percentile of the device\u0027s life cycle."},{"Start":"00:14.970 ","End":"00:20.740","Text":"Let\u0027s define x as the life cycle of the device."},{"Start":"00:21.200 ","End":"00:24.705","Text":"This would be measured in years."},{"Start":"00:24.705 ","End":"00:32.640","Text":"Now, that has a log-normal distribution with Mu and Sigma squared."},{"Start":"00:32.640 ","End":"00:37.080","Text":"Now, we know that ln of x,"},{"Start":"00:37.080 ","End":"00:43.130","Text":"we\u0027ll call that y, that has a normal distribution with the same parameters,"},{"Start":"00:43.130 ","End":"00:45.065","Text":"Mu and Sigma squared."},{"Start":"00:45.065 ","End":"00:53.100","Text":"Now, we\u0027re given that the expectation of x is equal to 6 right here."},{"Start":"00:53.100 ","End":"00:59.970","Text":"The standard deviation of x that equals to 1.5."},{"Start":"01:00.250 ","End":"01:10.145","Text":"Now, let\u0027s see what are the expressions for the expectation of x and the variance of x."},{"Start":"01:10.145 ","End":"01:13.265","Text":"Now, the expectation of x,"},{"Start":"01:13.265 ","End":"01:21.680","Text":"that equals to e to the power of Mu plus 1/2 Sigma squared."},{"Start":"01:21.680 ","End":"01:24.515","Text":"The variance of x, well,"},{"Start":"01:24.515 ","End":"01:31.400","Text":"that equals to e to the power of Sigma squared minus 1 times"},{"Start":"01:31.400 ","End":"01:39.820","Text":"e to the power of 2 Mu plus Sigma squared."},{"Start":"01:39.820 ","End":"01:48.750","Text":"Let\u0027s just plug in these numbers here to extract the values of Mu and Sigma squared."},{"Start":"01:49.820 ","End":"01:55.035","Text":"We have here 1.5 squared,"},{"Start":"01:55.035 ","End":"01:57.675","Text":"this is the standard deviation not the variance."},{"Start":"01:57.675 ","End":"02:03.709","Text":"1.5 squared, that equals to e to the power of Sigma squared"},{"Start":"02:03.709 ","End":"02:10.115","Text":"minus 1 times e to the power of 2 Mu plus Sigma squared."},{"Start":"02:10.115 ","End":"02:12.440","Text":"Now, what about the expectation,"},{"Start":"02:12.440 ","End":"02:15.005","Text":"that 6, that\u0027s right here."},{"Start":"02:15.005 ","End":"02:21.155","Text":"That equals to e to the power of Mu plus 1/2 Sigma squared."},{"Start":"02:21.155 ","End":"02:24.530","Text":"Now, let\u0027s take a look at this expression right here."},{"Start":"02:24.530 ","End":"02:27.305","Text":"We see that if we square it,"},{"Start":"02:27.305 ","End":"02:31.415","Text":"then we\u0027ll get this expression right here. Let\u0027s do that."},{"Start":"02:31.415 ","End":"02:41.940","Text":"6 squared, that equals to 36 and that equals to e the power of 2 Mu plus Sigma squared."},{"Start":"02:41.940 ","End":"02:46.400","Text":"Now we brought this expression into this form,"},{"Start":"02:46.400 ","End":"02:49.955","Text":"they\u0027re the same expressions and we know that equals to 36."},{"Start":"02:49.955 ","End":"02:52.150","Text":"Let\u0027s just plug that in here."},{"Start":"02:52.150 ","End":"03:03.485","Text":"We have here 1.5 squared that equals to e to the power of Sigma squared minus 1 times 36."},{"Start":"03:03.485 ","End":"03:06.095","Text":"Now, let\u0027s just move things around here."},{"Start":"03:06.095 ","End":"03:12.265","Text":"1.5 squared, that\u0027s 2.25 divided by 36."},{"Start":"03:12.265 ","End":"03:14.655","Text":"We add 1,"},{"Start":"03:14.655 ","End":"03:20.050","Text":"and that\u0027ll give us e to the power of Sigma squared."},{"Start":"03:20.050 ","End":"03:24.505","Text":"All we need to do right now is just take the In of both sides."},{"Start":"03:24.505 ","End":"03:32.630","Text":"That\u0027ll be ln of 2.25 divided by 36 plus 1,"},{"Start":"03:32.630 ","End":"03:35.950","Text":"and that\u0027ll be equal to Sigma squared."},{"Start":"03:35.950 ","End":"03:38.950","Text":"Now, when we calculate this,"},{"Start":"03:38.950 ","End":"03:43.640","Text":"this comes out to 0.0606."},{"Start":"03:44.280 ","End":"03:47.725","Text":"Now we have the value of Sigma squared."},{"Start":"03:47.725 ","End":"03:51.260","Text":"Now we need to extract the value of Mu."},{"Start":"03:51.260 ","End":"03:55.945","Text":"Let\u0027s just plug in this number here into this expression."},{"Start":"03:55.945 ","End":"04:06.320","Text":"We have 36 and that equals to e to the power of 2 Mu plus 0.0606."},{"Start":"04:07.290 ","End":"04:12.660","Text":"Now, let\u0027s just move things around here."},{"Start":"04:12.660 ","End":"04:17.050","Text":"We have here 33.88"},{"Start":"04:18.680 ","End":"04:25.490","Text":"that\u0027ll be equal to e to the power of 2 Mu."},{"Start":"04:25.490 ","End":"04:28.850","Text":"Let\u0027s just take the square root of both sides."},{"Start":"04:28.850 ","End":"04:35.820","Text":"We have here e to the power of Mu that equals to 5.82."},{"Start":"04:36.370 ","End":"04:43.240","Text":"When we take the In of both sides,"},{"Start":"04:43.240 ","End":"04:46.800","Text":"In of 5.82,"},{"Start":"04:46.800 ","End":"04:49.560","Text":"we get that Mu,"},{"Start":"04:49.560 ","End":"04:53.955","Text":"that equals to 1.76."},{"Start":"04:53.955 ","End":"04:57.950","Text":"We have here the values of Mu and Sigma squared,"},{"Start":"04:57.950 ","End":"04:59.090","Text":"but we\u0027re not done yet."},{"Start":"04:59.090 ","End":"05:04.760","Text":"We still have to calculate the upper 10th percentile of x."},{"Start":"05:06.980 ","End":"05:13.370","Text":"How do we do that? Well, we know that in the standard normal distribution,"},{"Start":"05:13.370 ","End":"05:20.270","Text":"the 90th percentile, well that equals to 1.282."},{"Start":"05:20.270 ","End":"05:23.000","Text":"I invite you to look that up in the tables."},{"Start":"05:23.000 ","End":"05:24.410","Text":"Now, what does that equal to?"},{"Start":"05:24.410 ","End":"05:27.485","Text":"Well, that equals to y minus Mu,"},{"Start":"05:27.485 ","End":"05:32.390","Text":"which is 1.76 divided by the standard deviation where"},{"Start":"05:32.390 ","End":"05:38.440","Text":"the standard deviation is the square root of 0.0606."},{"Start":"05:38.440 ","End":"05:42.040","Text":"That means that y,"},{"Start":"05:42.040 ","End":"05:48.335","Text":"well that equals to 2.0766."},{"Start":"05:48.335 ","End":"05:54.310","Text":"Now let\u0027s recall the relationship between x and y."},{"Start":"05:54.310 ","End":"05:59.280","Text":"We know that y equals In of x."},{"Start":"05:59.280 ","End":"06:04.460","Text":"Now, if we want the 90th percentile or the upper 10th percentile,"},{"Start":"06:04.460 ","End":"06:07.640","Text":"then let\u0027s just plug in this number here."},{"Start":"06:07.640 ","End":"06:11.630","Text":"We have 2.0766."},{"Start":"06:11.630 ","End":"06:14.655","Text":"Well that equals to ln of x."},{"Start":"06:14.655 ","End":"06:17.460","Text":"Let\u0027s take e of both sides."},{"Start":"06:17.460 ","End":"06:22.895","Text":"We have here e the power of 2.0766,"},{"Start":"06:22.895 ","End":"06:24.890","Text":"that\u0027ll be equal to x."},{"Start":"06:24.890 ","End":"06:26.975","Text":"But this is a special x."},{"Start":"06:26.975 ","End":"06:32.870","Text":"This will be the 90th percentile or the upper 10th percentile of x,"},{"Start":"06:32.870 ","End":"06:37.150","Text":"and that equals to 7.98."},{"Start":"06:37.910 ","End":"06:47.260","Text":"This then would be the upper 10th percentile of the life cycle of the electrical device."}],"ID":16012},{"Watched":false,"Name":"Exercise 5 Part b","Duration":"3m 45s","ChapterTopicVideoID":15226,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"In this section, we\u0027re asked,"},{"Start":"00:01.845 ","End":"00:06.150","Text":"what\u0027s the probability that the device will last more than 3 years?"},{"Start":"00:06.150 ","End":"00:11.205","Text":"We\u0027re looking for the probability that X is greater than 3."},{"Start":"00:11.205 ","End":"00:19.080","Text":"Now, we know that X has a log normal distribution where Mu"},{"Start":"00:19.080 ","End":"00:27.045","Text":"equals to 1.76 and Sigma squared that equals to 0.0606."},{"Start":"00:27.045 ","End":"00:30.555","Text":"We\u0027ve calculated that in the last section."},{"Start":"00:30.555 ","End":"00:36.490","Text":"But these parameters don\u0027t belong to the log normal distribution."},{"Start":"00:36.920 ","End":"00:43.035","Text":"They belong to the normal distribution which is ln of x,"},{"Start":"00:43.035 ","End":"00:44.920","Text":"and we\u0027ll call that y."},{"Start":"00:44.920 ","End":"00:47.255","Text":"Now, if that\u0027s the case,"},{"Start":"00:47.255 ","End":"00:56.930","Text":"then we\u0027re looking for the probability that ln x will be greater than ln 3,"},{"Start":"00:56.930 ","End":"01:03.890","Text":"or the probability of y being greater than ln 3."},{"Start":"01:03.890 ","End":"01:11.900","Text":"Now, we know that y has a normal distribution where Mu"},{"Start":"01:11.900 ","End":"01:19.910","Text":"that equals to 1.76 and Sigma squared that equals to 0.0606,"},{"Start":"01:19.910 ","End":"01:23.455","Text":"the same parameters that we have here."},{"Start":"01:23.455 ","End":"01:27.320","Text":"Now that we know that y has this distribution,"},{"Start":"01:27.320 ","End":"01:29.450","Text":"we\u0027re looking for this probability,"},{"Start":"01:29.450 ","End":"01:31.460","Text":"well then we have to standardize."},{"Start":"01:31.460 ","End":"01:37.135","Text":"We\u0027re looking for the probability of Z,"},{"Start":"01:37.135 ","End":"01:45.345","Text":"which is equal to Y minus Mu divided by Sigma,"},{"Start":"01:45.345 ","End":"01:52.770","Text":"we want that to be greater than ln 3 minus Mu, well,"},{"Start":"01:52.770 ","End":"01:55.340","Text":"Mu is 1.76,"},{"Start":"01:55.340 ","End":"02:02.725","Text":"divided by the square root of 0.0606."},{"Start":"02:02.725 ","End":"02:06.665","Text":"This is the probability that we\u0027re looking for."},{"Start":"02:06.665 ","End":"02:11.705","Text":"Now, this comes out to the probability of Z"},{"Start":"02:11.705 ","End":"02:18.525","Text":"being greater than minus 2.68."},{"Start":"02:18.525 ","End":"02:26.210","Text":"Now let\u0027s take a look at a standard normal distribution and see what we\u0027re talking about."},{"Start":"02:26.330 ","End":"02:29.760","Text":"Here\u0027s our normal distribution."},{"Start":"02:29.760 ","End":"02:32.175","Text":"This is a standard normal distribution."},{"Start":"02:32.175 ","End":"02:34.670","Text":"Here, Mu is equal to 0,"},{"Start":"02:34.670 ","End":"02:42.050","Text":"and this number right here is minus 2.68."},{"Start":"02:42.050 ","End":"02:49.810","Text":"Now we\u0027re looking for the probability that we\u0027re above this number right here."},{"Start":"02:49.810 ","End":"02:56.880","Text":"Well, this is equal to the same probability where"},{"Start":"02:56.880 ","End":"03:03.750","Text":"Z is less than plus 2.68."},{"Start":"03:03.750 ","End":"03:09.785","Text":"You have to know your normal probability in order to do this."},{"Start":"03:09.785 ","End":"03:16.155","Text":"Now, that equals to Phi of 2.68."},{"Start":"03:16.155 ","End":"03:18.165","Text":"Now, if we go to the tables,"},{"Start":"03:18.165 ","End":"03:24.180","Text":"we\u0027ll see that that comes out to 0.9963."},{"Start":"03:24.180 ","End":"03:30.935","Text":"Now, I invite you to look at the tables to make sure that Phi of 2.68,"},{"Start":"03:30.935 ","End":"03:33.860","Text":"that equals to 0.9963."},{"Start":"03:33.860 ","End":"03:37.410","Text":"But this is the answer to our question,"},{"Start":"03:37.410 ","End":"03:40.520","Text":"what\u0027s the probability that the life cycle of"},{"Start":"03:40.520 ","End":"03:45.330","Text":"the electrical device would be greater than 3 years?"}],"ID":16013},{"Watched":false,"Name":"Exercise 6","Duration":"9m 51s","ChapterTopicVideoID":12780,"CourseChapterTopicPlaylistID":245056,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this question, we\u0027re told that rabbits multiply by a factor of X_i every month,"},{"Start":"00:05.370 ","End":"00:08.100","Text":"where X_i are independent variables."},{"Start":"00:08.100 ","End":"00:13.440","Text":"We\u0027re also given that X_i has a log-normal distribution where the expectation of"},{"Start":"00:13.440 ","End":"00:18.840","Text":"X_i is the square root of e and the variance of X_i is e times e minus 1."},{"Start":"00:18.840 ","End":"00:23.355","Text":"We\u0027re asked what\u0027s the probability that at the end of 1 year,"},{"Start":"00:23.355 ","End":"00:28.320","Text":"the rabbit population will be 20 times more than at the beginning of that year."},{"Start":"00:28.320 ","End":"00:33.450","Text":"The first thing that we want to do is to write down what we\u0027re given."},{"Start":"00:33.450 ","End":"00:37.820","Text":"The first thing that we\u0027re given is that X_i has"},{"Start":"00:37.820 ","End":"00:43.624","Text":"a log-normal distribution with parameters Mu and Sigma squared."},{"Start":"00:43.624 ","End":"00:48.860","Text":"Now, we\u0027re also told that the expectation of X_i,"},{"Start":"00:48.860 ","End":"00:51.070","Text":"that equals to the square root of e,"},{"Start":"00:51.070 ","End":"00:52.469","Text":"that\u0027s right here,"},{"Start":"00:52.469 ","End":"00:55.424","Text":"and the variance of X_i, well,"},{"Start":"00:55.424 ","End":"00:59.390","Text":"that equals to e times e minus 1,"},{"Start":"00:59.390 ","End":"01:00.860","Text":"and that\u0027s right here."},{"Start":"01:00.860 ","End":"01:02.435","Text":"Now, what is X_i?"},{"Start":"01:02.435 ","End":"01:08.405","Text":"Well, we\u0027re told that X_i is a growth factor,"},{"Start":"01:08.405 ","End":"01:10.430","Text":"but it\u0027s a monthly growth factor,"},{"Start":"01:10.430 ","End":"01:13.790","Text":"and we\u0027re asked about a yearly growth factor."},{"Start":"01:13.790 ","End":"01:15.605","Text":"What do we need to do?"},{"Start":"01:15.605 ","End":"01:20.300","Text":"Well, how do we get from a monthly growth factor to a yearly growth factor?"},{"Start":"01:20.300 ","End":"01:25.880","Text":"Well, we take the growth factor in month number 1 in January, for example,"},{"Start":"01:25.880 ","End":"01:29.510","Text":"and we multiply that by the growth factor in February,"},{"Start":"01:29.510 ","End":"01:33.065","Text":"and so on and so forth until we get to December."},{"Start":"01:33.065 ","End":"01:36.995","Text":"So, the product of the monthly growth factors,"},{"Start":"01:36.995 ","End":"01:40.670","Text":"the product where i goes from 1-12 of X_i,"},{"Start":"01:40.670 ","End":"01:44.465","Text":"that would be the yearly growth factor."},{"Start":"01:44.465 ","End":"01:47.270","Text":"Now, what are we looking for?"},{"Start":"01:47.270 ","End":"01:52.310","Text":"Well, we\u0027re looking for the probability that this yearly growth factor,"},{"Start":"01:52.310 ","End":"02:00.665","Text":"the product where i goes from 1-12 of X_i is greater than 20."},{"Start":"02:00.665 ","End":"02:02.825","Text":"This is what we have to calculate."},{"Start":"02:02.825 ","End":"02:06.500","Text":"Now, what do we know about the product of"},{"Start":"02:06.500 ","End":"02:11.645","Text":"variables that all the variables have a log-normal distribution."},{"Start":"02:11.645 ","End":"02:16.895","Text":"Well, we\u0027ve calculated that in the previous question."},{"Start":"02:16.895 ","End":"02:21.905","Text":"We know that the product of variables that each 1,"},{"Start":"02:21.905 ","End":"02:24.440","Text":"each X_i has a log-normal distribution."},{"Start":"02:24.440 ","End":"02:27.950","Text":"Well, the product also has a log-normal distribution with"},{"Start":"02:27.950 ","End":"02:31.700","Text":"parameters N Mu and N Sigma squared."},{"Start":"02:31.700 ","End":"02:39.515","Text":"What\u0027s the distribution of the product of the growth factors here?"},{"Start":"02:39.515 ","End":"02:46.130","Text":"We know that the product where i goes from 1-12 of X_i, well,"},{"Start":"02:46.130 ","End":"02:50.830","Text":"that has a log-normal distribution with"},{"Start":"02:50.830 ","End":"02:56.990","Text":"12Mu and 12 Sigma squared."},{"Start":"02:56.990 ","End":"03:00.560","Text":"Now, that we know that this has"},{"Start":"03:00.560 ","End":"03:06.230","Text":"a log-normal distribution with parameters 12Mu and 12 Sigma squared,"},{"Start":"03:06.230 ","End":"03:11.240","Text":"let\u0027s calculate these values right here."},{"Start":"03:11.240 ","End":"03:16.440","Text":"Now, we know that the expectation of X_i,"},{"Start":"03:16.440 ","End":"03:17.610","Text":"that\u0027s the square root of e,"},{"Start":"03:17.610 ","End":"03:18.980","Text":"and the variance of X_i,"},{"Start":"03:18.980 ","End":"03:21.770","Text":"well, that equals to e times e minus 1."},{"Start":"03:21.770 ","End":"03:27.350","Text":"But, we also know what the equations are for these expressions,"},{"Start":"03:27.350 ","End":"03:29.570","Text":"so let\u0027s write them down."},{"Start":"03:29.570 ","End":"03:33.800","Text":"These are the expressions or the equations for"},{"Start":"03:33.800 ","End":"03:38.255","Text":"calculating the variance of X_i and the expectation of X_i."},{"Start":"03:38.255 ","End":"03:39.935","Text":"But, we know what these are;"},{"Start":"03:39.935 ","End":"03:41.570","Text":"the variance of X_i, well,"},{"Start":"03:41.570 ","End":"03:43.175","Text":"that\u0027s this thing right here,"},{"Start":"03:43.175 ","End":"03:46.055","Text":"and the expectation is this thing right here."},{"Start":"03:46.055 ","End":"03:50.900","Text":"That\u0027ll help us to calculate the values of Mu and Sigma squared."},{"Start":"03:50.900 ","End":"03:52.295","Text":"Let\u0027s get to it."},{"Start":"03:52.295 ","End":"03:54.260","Text":"Well, first of all,"},{"Start":"03:54.260 ","End":"03:55.670","Text":"this thing right here,"},{"Start":"03:55.670 ","End":"03:59.690","Text":"that\u0027ll be e times e minus 1, well,"},{"Start":"03:59.690 ","End":"04:04.265","Text":"that equals to e to the power of Sigma squared minus 1,"},{"Start":"04:04.265 ","End":"04:08.930","Text":"times e to the power of 2Mu plus Sigma squared."},{"Start":"04:08.930 ","End":"04:10.970","Text":"Here for the expectation,"},{"Start":"04:10.970 ","End":"04:12.815","Text":"well, that\u0027s the square root of e,"},{"Start":"04:12.815 ","End":"04:19.850","Text":"and that will be equal to e to the power of Mu plus 1/2 Sigma squared."},{"Start":"04:19.850 ","End":"04:24.920","Text":"Now, we know that if we square this expression right here,"},{"Start":"04:24.920 ","End":"04:27.140","Text":"that will give us this expression right here,"},{"Start":"04:27.140 ","End":"04:29.630","Text":"so let\u0027s square both sides."},{"Start":"04:29.630 ","End":"04:32.615","Text":"We get e,"},{"Start":"04:32.615 ","End":"04:38.715","Text":"and that will be equal to e to the power of 2Mu plus Sigma squared."},{"Start":"04:38.715 ","End":"04:43.525","Text":"Now, we can replace this expression with the value e,"},{"Start":"04:43.525 ","End":"04:44.800","Text":"so let\u0027s do that."},{"Start":"04:44.800 ","End":"04:49.210","Text":"That\u0027ll be e to the power of e times e minus 1,"},{"Start":"04:49.210 ","End":"04:54.475","Text":"and that\u0027ll be equal to e to the power of Sigma squared minus 1"},{"Start":"04:54.475 ","End":"05:01.330","Text":"times e. This e right here will replace this expression right here."},{"Start":"05:01.330 ","End":"05:04.315","Text":"Let\u0027s just move things around here."},{"Start":"05:04.315 ","End":"05:07.300","Text":"This e, and this e cancel each other out,"},{"Start":"05:07.300 ","End":"05:10.210","Text":"so we\u0027re left with e minus 1,"},{"Start":"05:10.210 ","End":"05:15.100","Text":"that equals to e to the power of Sigma squared minus 1."},{"Start":"05:15.100 ","End":"05:18.025","Text":"Now, minus 1 minus 1 cancel each other out,"},{"Start":"05:18.025 ","End":"05:22.750","Text":"so that means that we have here e to the power of 1,"},{"Start":"05:22.750 ","End":"05:26.240","Text":"that equals to e to the power of Sigma squared,"},{"Start":"05:26.240 ","End":"05:30.870","Text":"which means that Sigma squared has to be equal to 1."},{"Start":"05:30.870 ","End":"05:34.430","Text":"Now, that we know the value of Sigma squared,"},{"Start":"05:34.430 ","End":"05:41.870","Text":"let\u0027s just replace this value right here with the value 1."},{"Start":"05:41.870 ","End":"05:48.620","Text":"We have here e that equals to e to the power of 2Mu plus 1."},{"Start":"05:48.620 ","End":"05:53.165","Text":"We replace this Sigma squared with this value right here."},{"Start":"05:53.165 ","End":"05:55.340","Text":"That means that we have 1,"},{"Start":"05:55.340 ","End":"05:57.500","Text":"this is e to the power of 1,"},{"Start":"05:57.500 ","End":"06:02.585","Text":"that equals 2Mu plus 1."},{"Start":"06:02.585 ","End":"06:04.985","Text":"1\u0027s cancel each other out,"},{"Start":"06:04.985 ","End":"06:09.620","Text":"that means that 2Mu equals 0."},{"Start":"06:09.620 ","End":"06:11.930","Text":"That means that Mu equals to 0."},{"Start":"06:11.930 ","End":"06:15.420","Text":"Now, we have the value of Mu."},{"Start":"06:16.810 ","End":"06:23.715","Text":"This means now that the product of X_i,"},{"Start":"06:23.715 ","End":"06:26.441","Text":"where i goes from 1-12,"},{"Start":"06:26.441 ","End":"06:31.955","Text":"well, that has a log-normal distribution with 0,"},{"Start":"06:31.955 ","End":"06:36.660","Text":"Mu equals 0 and Sigma squared."},{"Start":"06:36.660 ","End":"06:39.360","Text":"Well, that\u0027ll be N Sigma squared,"},{"Start":"06:39.360 ","End":"06:41.665","Text":"so that\u0027ll be 12 times 1."},{"Start":"06:41.665 ","End":"06:48.230","Text":"This would be the parameters for the normal distribution,"},{"Start":"06:48.230 ","End":"06:49.820","Text":"not the log-normal distribution,"},{"Start":"06:49.820 ","End":"06:51.260","Text":"but this would be the parameters,"},{"Start":"06:51.260 ","End":"06:52.475","Text":"Mu and Sigma squared,"},{"Start":"06:52.475 ","End":"06:54.080","Text":"for the normal distribution."},{"Start":"06:54.080 ","End":"06:56.660","Text":"Now, what is the normal distribution?"},{"Start":"06:56.660 ","End":"06:59.150","Text":"Well, let\u0027s just recall,"},{"Start":"06:59.150 ","End":"07:04.645","Text":"we\u0027re looking for the probability that the product of X_i,"},{"Start":"07:04.645 ","End":"07:07.255","Text":"where i goes from 1-12,"},{"Start":"07:07.255 ","End":"07:11.060","Text":"well, what\u0027s the probability that that is greater than 20?"},{"Start":"07:11.060 ","End":"07:14.660","Text":"Well, if we take ln of both sides,"},{"Start":"07:14.660 ","End":"07:19.160","Text":"then we have the probability of ln of the product of X_i,"},{"Start":"07:19.160 ","End":"07:21.020","Text":"i goes from 1-12,"},{"Start":"07:21.020 ","End":"07:24.200","Text":"that\u0027ll be greater than ln of 20."},{"Start":"07:24.200 ","End":"07:27.140","Text":"This is a probability that we\u0027re looking for."},{"Start":"07:27.140 ","End":"07:28.865","Text":"Now, if we recall,"},{"Start":"07:28.865 ","End":"07:34.895","Text":"if the product of X_i has a log-normal distribution,"},{"Start":"07:34.895 ","End":"07:40.520","Text":"then ln of the product of X_i,"},{"Start":"07:40.520 ","End":"07:42.605","Text":"we\u0027ll call that y,"},{"Start":"07:42.605 ","End":"07:45.740","Text":"that has a normal distribution."},{"Start":"07:45.740 ","End":"07:46.970","Text":"Now, in our case,"},{"Start":"07:46.970 ","End":"07:52.684","Text":"where Mu equals to 0 and Sigma squared that equals to 12."},{"Start":"07:52.684 ","End":"07:57.170","Text":"We can replace this by saying, \"Well,"},{"Start":"07:57.170 ","End":"07:59.585","Text":"we\u0027re looking for the probability of y,"},{"Start":"07:59.585 ","End":"08:04.460","Text":"ln of the product of X_i being greater than ln of 20.\""},{"Start":"08:04.460 ","End":"08:05.870","Text":"Well, ln of 20 here,"},{"Start":"08:05.870 ","End":"08:07.520","Text":"that\u0027s very close to 3,"},{"Start":"08:07.520 ","End":"08:10.430","Text":"so we\u0027ll say that we\u0027re looking for"},{"Start":"08:10.430 ","End":"08:15.905","Text":"the product or the probability of y being greater than 3,"},{"Start":"08:15.905 ","End":"08:21.245","Text":"and we know that y has a normal distribution."},{"Start":"08:21.245 ","End":"08:25.955","Text":"The next thing that we need to do is we need to standardize this."},{"Start":"08:25.955 ","End":"08:29.509","Text":"Now, Z, that\u0027s a standardization,"},{"Start":"08:29.509 ","End":"08:34.760","Text":"that equals to y minus Mu divided by Sigma."},{"Start":"08:34.760 ","End":"08:39.230","Text":"In our case, that\u0027ll be 3 minus 0,"},{"Start":"08:39.230 ","End":"08:43.564","Text":"Mu is 0, divided by the square root of 12."},{"Start":"08:43.564 ","End":"08:49.985","Text":"Now, that comes out to 0.86."},{"Start":"08:49.985 ","End":"08:59.330","Text":"We\u0027re looking for the probability of z being greater than 0.86."},{"Start":"08:59.330 ","End":"09:10.100","Text":"Now, that equals to 1 minus the probability of z being less than 0.86."},{"Start":"09:10.100 ","End":"09:16.850","Text":"Well, that equals to 1 minus Phi of 0.86."},{"Start":"09:16.850 ","End":"09:23.015","Text":"Now, I urge you to go to the standard normal distribution table,"},{"Start":"09:23.015 ","End":"09:25.880","Text":"and look up this value right here."},{"Start":"09:25.880 ","End":"09:33.050","Text":"It\u0027ll come out to 0.8051,"},{"Start":"09:33.050 ","End":"09:37.820","Text":"and that equals 1 minus 0.8051,"},{"Start":"09:37.820 ","End":"09:42.320","Text":"that equals to 0.1949."},{"Start":"09:42.320 ","End":"09:51.719","Text":"This then would be the probability that the yearly growth factor is greater than 20."}],"ID":13259}],"Thumbnail":null,"ID":245056}]

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