WEBVTT 1 00:00:26.423 --> 00:00:34.493 Hey, good afternoon class. So this is engineering probability class 9 on 25th 2021 give or take and. 2 00:00:37.770 --> 00:00:43.710 The mice should be on oh, 1, other thing, I just have to start up. 3 00:00:43.710 --> 00:00:48.119 Um. 4 00:01:00.509 --> 00:01:09.689 Okay, we got our mirroring. 5 00:01:31.890 --> 00:01:38.250 So, let me just make sure just see if anyone hear me. 6 00:01:39.840 --> 00:01:52.829 The reason I do that there is a joke about some professor not at. Great. Thank you Curtis. There's a joke about some professor somewhere who. 7 00:01:52.829 --> 00:02:00.209 You know, gave a whole lecture on mute to no 1 told him. So okay, so. 8 00:02:04.709 --> 00:02:09.300 And then this is. 9 00:02:09.300 --> 00:02:13.860 On here. 10 00:02:13.860 --> 00:02:19.500 All right. Oh, okay. 11 00:02:19.500 --> 00:02:23.490 Oh, for fun. 12 00:02:27.599 --> 00:02:37.949 Okay, well, 1st a fun thing I applied or misapplied probability. I see. Doo, Doo, doo. 13 00:02:45.150 --> 00:02:51.180 Here we go, this is a story I told you about. Am I key? 14 00:02:51.180 --> 00:02:54.360 Where are they. 15 00:02:54.360 --> 00:02:59.909 I'll put the link on next time. Yeah. 16 00:02:59.909 --> 00:03:12.629 So, why don't we see our students doing this? Come on guys gals. Okay how they use? Probably looking I'll let you read it to see the details and so on. Okay. 17 00:03:14.370 --> 00:03:23.039 You can have fun with that. So, what we are doing today is 1st, the test coming up on. 18 00:03:26.340 --> 00:03:30.270 Monday, I mentioned it before. 19 00:03:31.979 --> 00:03:36.659 Class SEC here we go. Probably our team. 20 00:03:36.659 --> 00:03:41.009 I put a back test online for you from last year. 21 00:03:42.509 --> 00:03:51.930 And you can go through it and, you know, there is a good chance that I'll take some of the questions in this test and to recycle them. 22 00:03:51.930 --> 00:04:02.430 You know, environmental. Good. Recycling is good. So I'm going to recycle several questions from this test. Probably. So, there's the question I also have the. 23 00:04:02.430 --> 00:04:10.889 A version with answers should you like and so on? So you can look at it with how it answers to try yourself if you wish. 24 00:04:11.455 --> 00:04:24.985 And which has just publicly available, you're perfectly welcome to put it in exam banks and so on. Just you're not allowed to charge for it. Since I'm making it available for the whole Internet universe for free. 25 00:04:25.524 --> 00:04:39.894 Now, the material is somewhat different. This year this year. What with the pandemic and everything and teaching remotely? I'm going a little more slowly. I'd rather go and touch more slowly and present the material more thoroughly. 26 00:04:40.495 --> 00:04:45.834 So that's why the last exam went into chapter 3. and this exam is only up to chapter 2. 27 00:04:46.108 --> 00:04:59.579 So, it will be grade scope. We'll open the test at 3 o'clock Eastern Standard time on Monday, and goes for the 80 minutes for the class students with accommodations. So. 28 00:04:59.874 --> 00:05:08.783 Run late started 3 and run late students who are 12 hours away in China and several places. 29 00:05:09.053 --> 00:05:18.533 I will give you an option to start to test at 3 am Eastern Standard time if you wish, but if you want that option, then you have to tell me. 30 00:05:19.283 --> 00:05:31.254 Okay, the format of the exam will be generally multiple choice, but there may also be some short answers and where you might even write something out and then photographed and scan it. 31 00:05:31.254 --> 00:05:38.244 And I ask a class on Monday, if that was a problem and was told, that was not a problem. So. 32 00:05:38.579 --> 00:05:44.548 Okay, real world thing and. 33 00:05:44.548 --> 00:05:49.949 So, you can have fun reading about real world here. 34 00:05:49.949 --> 00:05:55.798 This relates to the thing I, the blurb I had. 35 00:05:55.798 --> 00:06:09.778 In class aid on my blog about false positives, false negatives. And so we'll talk about it more later. I just wanted to mention the terms. Well, we thought of things about vaccinations and everything being the news. I wanted to give you some real world problems. 36 00:06:09.778 --> 00:06:13.108 So, the moon tricks are radar so. 37 00:06:14.158 --> 00:06:18.749 And from probability things oh, okay. 38 00:06:18.749 --> 00:06:23.098 So, new stuff. 39 00:06:23.098 --> 00:06:28.978 We are in chapter 3 and not me. 40 00:06:28.978 --> 00:06:32.788 Chapter 3 here. 41 00:06:32.788 --> 00:06:41.908 Discreet random variables. Okay. So they are finite or they're countably infinite. I finite. 42 00:06:41.908 --> 00:06:52.108 Would be discreet would be Costa coin a 100 times and the number the number of heads would be the random variable. 43 00:06:52.108 --> 00:07:00.298 Some 0T to 100 countably infinite would be things like the metric distribution where. 44 00:07:00.298 --> 00:07:04.048 The rent the outcome. 45 00:07:04.048 --> 00:07:11.788 Is the number of tosses until you get your 1st head can be anywhere in 1 and up a natural number. 46 00:07:11.788 --> 00:07:23.819 Those are discreet the next chapter we'll see continuous random variables, which I mentioned briefly before they are uncomfortably infinite, which means the probability of any specific. 47 00:07:23.843 --> 00:07:34.283 Outcome is 0T so would be the time radioactive decay occurred the real time and the probability of any precise time is 0T. 48 00:07:34.283 --> 00:07:45.774 So we have to talk about an interval the probability that this radioactive Adam decade in the next. 2nd perhaps that would be an interval. Okay. So, discreet, that's 1 thing. 49 00:07:46.584 --> 00:08:00.744 Another new thing is this notion of a random variable what we've talked about before outcomes and events outcome would be some specific thing. Like we toss a, and again, these things are all defined. 50 00:08:01.704 --> 00:08:03.324 Well, 1st, let me answer the question. 51 00:08:04.139 --> 00:08:10.588 Similar difficulty to the homeworks. Yes, that is correct. We're going to recycle some homework questions. 52 00:08:10.588 --> 00:08:13.588 So that by the way. 53 00:08:13.588 --> 00:08:26.278 Before I get back up to answering your question. This is the reason that I'm not putting today's lecture on the test on Monday is because we haven't had time to hit you with homework questions and so on. 54 00:08:26.278 --> 00:08:31.408 So, there's a lag in the system. Okay back to here. Um, the. 55 00:08:31.408 --> 00:08:40.078 So we have in the past, we've had outcomes, which might be the sequence of 2 heads. If you toss a coin twice. So, this for. 56 00:08:40.078 --> 00:08:43.619 Outcomes and had telltale tail tail. 57 00:08:43.619 --> 00:08:49.948 And then events would be sets of interesting outcomes. The set of there was 1 head. 58 00:08:49.948 --> 00:08:57.479 And the to tosses. Okay now, now we're getting we're attaching numbers to think now a random variable. 59 00:08:57.479 --> 00:09:02.129 As the, and Garcia does it is we attach numbers to the outcomes. 60 00:09:02.129 --> 00:09:07.408 And it could be the obvious attachment like. 61 00:09:07.408 --> 00:09:12.839 We tossed a coin twice and the random variable is the number of heads we see 0, 1 or 2. 62 00:09:14.219 --> 00:09:18.298 And the thing is every time we do the experiment. 63 00:09:18.298 --> 00:09:23.278 Of tossing the coin twice. We get a different Tom. 64 00:09:23.278 --> 00:09:27.418 Random variable perhaps much random. 65 00:09:27.418 --> 00:09:31.438 Okay, so we have the notation down here. 66 00:09:31.438 --> 00:09:34.859 This is the outcome. 67 00:09:34.859 --> 00:09:38.068 You know, or whatever. 68 00:09:38.068 --> 00:09:43.288 And then the random variable is a capital letter. 69 00:09:43.288 --> 00:09:49.078 Starting capital s capital Y and that talks about that's the random variable. 70 00:09:49.078 --> 00:09:54.658 In general terms, and each particular value of it would be a lower case X. 71 00:09:54.658 --> 00:09:58.078 So, we're attaching a numeric a number to each. 72 00:09:58.078 --> 00:10:02.099 Help come. Okay. 73 00:10:04.139 --> 00:10:12.749 Oh, in the sample space, just all the outcomes. So here's an example coin toss 3 1 here. 74 00:10:12.749 --> 00:10:15.958 And we toss. 75 00:10:15.958 --> 00:10:23.849 Coin 3 times and the random variables in number. So these are the outcomes are 8 outcomes. 76 00:10:23.849 --> 00:10:27.389 And the random variable is the number of. 77 00:10:27.389 --> 00:10:35.879 And said, we say here is an outcome. This is a random variable. Okay. Now you might think that that's so simple that. 78 00:10:35.879 --> 00:10:40.798 What's the point? Well, it could get a little tricky. Like, let's say. 79 00:10:40.798 --> 00:10:44.339 You know, we're playing some sort of game. 80 00:10:44.339 --> 00:10:53.009 And, you know, you get a pay off and the pay off isn't linear. So we tossed. 81 00:10:53.009 --> 00:11:00.389 Here's a game we tossed a coin 3 times and probably we lose. 82 00:11:00.389 --> 00:11:09.119 We receive nothing however, if there's at least 2 heads, we get a dollar and if there's 3 heads. 83 00:11:09.119 --> 00:11:13.048 We get 8 dollars. 84 00:11:14.158 --> 00:11:20.969 We actually have 2 random variables here. So the set of outcomes are these 8 outcomes here. 85 00:11:25.048 --> 00:11:29.038 But the 1st, random variable is the number of heads. 86 00:11:29.038 --> 00:11:33.089 And the 2nd, random variable is the pay off. 87 00:11:33.089 --> 00:11:38.458 Now, we can define whatever we want. So, that's what we define that random variable. Why is. 88 00:11:38.458 --> 00:11:44.548 And capital Y, capital letter talks about the random variable in. 89 00:11:44.548 --> 00:11:51.178 Uh, in general as a variable, and so we'll use it for scripts and things like that. And these are the 3 possible. 90 00:11:51.178 --> 00:11:55.769 Values for the random variable and it's non linear. 91 00:11:56.849 --> 00:12:04.379 Do you define anything you wanted for random variable? Or? The other thing I went light on is the random variable to say is a real number. 92 00:12:04.379 --> 00:12:13.168 I don't know if there's anything really, that depends on being a real number. So well, the thing is this. 93 00:12:13.168 --> 00:12:24.688 We want to do a rhythm check on the random barrel we're going to do means and statistics and stuff like that. And if it was a letter or something, it has to be a number to do this numerical calculations. So. 94 00:12:24.688 --> 00:12:30.599 And could it be a complex number? Well. 95 00:12:30.599 --> 00:12:39.808 There's a problem with complex numbers is you cannot compare them. There's no way to say it 1 complex number is less than a greater than a 2nd complex number. So. 96 00:12:39.808 --> 00:12:43.499 That's that and we're not using that immediately, but that's a. 97 00:12:43.499 --> 00:12:52.828 That's something that complex numbers lack compared to real numbers. 1 thing they gain is it you can do square roots of every complex number. 98 00:12:52.828 --> 00:12:56.788 Even, but what you cannot do is compare them. 99 00:12:57.808 --> 00:13:09.418 And that's a deep statement. It gets way beyond this course, but you might think you've got a way to compare to complex numbers, like Super epic comparison. Well, any way you think you might have, I can show you. 100 00:13:09.418 --> 00:13:17.399 Has problems that lacks properties that a comparison test would recently want to have. Okay. So. 101 00:13:17.399 --> 00:13:29.339 Back up here now you can start doing probabilities on this so probably get some random variable. 102 00:13:29.339 --> 00:13:32.969 That you got 2 heads out of 3 toss is nothing is new here. 103 00:13:32.969 --> 00:13:42.028 Okay, but we're finding probabilities on random variables variable X, which is a number of heads. Okay. 104 00:13:45.359 --> 00:13:52.979 And nothing interesting here and formal definition I'm skipping over. 105 00:13:52.979 --> 00:14:04.889 Engineers don't work with the formal math as much as say, mathematicians what we can get into debate on that sort of thing. 106 00:14:04.889 --> 00:14:10.528 Okay, so we have discrete random variables. 107 00:14:12.208 --> 00:14:16.318 The probability mass function. Okay. This. 108 00:14:16.318 --> 00:14:25.379 Is a new idea we're going to be seeing using this for the whole rest of the semester. It's just the probability of each value. 109 00:14:25.379 --> 00:14:28.469 Of the random variable occurring, so. 110 00:14:28.469 --> 00:14:34.318 It's not a deep idea that it's, it's just a name and we're going to use this name. 111 00:14:34.318 --> 00:14:38.849 All the time, and the thing is this. 112 00:14:38.849 --> 00:14:43.619 So, if we have some random distribution, like we seem say. 113 00:14:43.619 --> 00:14:46.649 Um, binomial tossed the coin. Well. 114 00:14:46.649 --> 00:14:51.089 If you give the probability mass function for that, then you've defined. 115 00:14:51.089 --> 00:15:00.538 The binomial distribution, so any distribution discrete distribution we so far we've seen we're normally binomial geometric. We'll see more. 116 00:15:00.538 --> 00:15:06.058 Each 1 has a probability mass function, which is the probability and so. 117 00:15:06.058 --> 00:15:12.599 What's new? Here's the term. It's a way of defining. 118 00:15:12.599 --> 00:15:17.729 Any probability, distribution bites PMF, you'll see lots of examples. So. 119 00:15:18.749 --> 00:15:23.129 You do, it has some axioms, which will skip. 120 00:15:23.129 --> 00:15:29.818 And 4, okay, coming back to binomial and coin tosses. 121 00:15:29.818 --> 00:15:29.969 So, 122 00:15:29.964 --> 00:15:34.313 this coin is not fair probability of a head is P, 123 00:15:34.644 --> 00:15:36.953 what we do assume is as always, 124 00:15:36.953 --> 00:15:37.344 I guess, 125 00:15:37.344 --> 00:15:40.134 is the tosses do not affect each other if, 126 00:15:40.134 --> 00:15:40.494 you know, 127 00:15:40.494 --> 00:15:46.193 the result of the 1st toss it does not help you knowing what will happen on the 2nd toss they are independent. 128 00:15:46.528 --> 00:15:58.558 So, here, what we've got is the mass function of just the probability of each value, their additive variables number of heads value 3 and here are the 3 probability for probabilities. 129 00:15:58.558 --> 00:16:06.839 Nothing new there. I'm just trying to nail down the ideas before we get to stuff. That's new. Okay. 130 00:16:06.839 --> 00:16:20.999 It was something new we got this betting game now and then, remember, the betting game is we talked to coin 3 times and if it got 3 heads, we got paid 8. if I got 2 heads, we got paid 1. and if it got 0T had we got paid nothing. 131 00:16:22.048 --> 00:16:32.609 So, we can talk about the probability so the 3 possible values to the random variables 0, 1 and 8 and now we can talk to the probability of getting a 0T is a half. 132 00:16:32.609 --> 00:16:40.918 Probably making a 1 is 3 eights. The problem making, uh, to annate is 1, 8 and those are the only 3 outcomes. 133 00:16:40.918 --> 00:16:47.038 And they add up to 1. okay. Nothing interesting. There. 134 00:16:47.038 --> 00:16:55.918 Okay, here's another. Here's a 4th friend. Just scrape distribution uniform. 135 00:16:55.918 --> 00:16:58.918 Get around number generator generates. 136 00:16:58.918 --> 00:17:03.958 Impossible values 0, 2 and minus 1 in New Jersey. 137 00:17:03.958 --> 00:17:07.409 And each of them has a probability 1 over and. 138 00:17:07.409 --> 00:17:13.798 So, the outcome, it's the thing, it's around a variable it's a real number. 139 00:17:13.798 --> 00:17:16.888 Um, and. 140 00:17:16.888 --> 00:17:31.409 So this being the probability mass function, the probability of any given value is 1 over. M. and if we care, we put a subscript on P, the capital X because that's the name of the random variable and all the probabilities out up to 1. 141 00:17:31.409 --> 00:17:39.898 And this is called the uniform random variable. It's got a parameter M, because these are the values. So that's our 4th. 142 00:17:39.898 --> 00:17:46.648 Random discrete random distribution. 143 00:17:46.648 --> 00:17:51.118 Now, you can look at this stuff graphically. 144 00:17:52.288 --> 00:17:55.618 And so what we have here is for some. 145 00:17:55.618 --> 00:18:05.489 Probably vast function or quickly some discrete distribution. The X axis are possible values for the random variable. And the Y, axis is the probability. 146 00:18:05.489 --> 00:18:16.229 So, if it's a coin 3 times, and the outcome is the number of heads around and variables number heads, then we got 4 possible values. 147 00:18:16.229 --> 00:18:29.308 For the random variable in the ears approximate 0T has happened. The nature of the time 1 or 2 heads are each at 3 eights and 2 had today. So we can graphically show the probably mass function. 148 00:18:30.568 --> 00:18:36.689 This is our bet a little betting game where we get paid, depending what happens. 149 00:18:36.689 --> 00:18:43.469 With the 3 coins, and again, it's if you get 3 heads, you get paid 8. 150 00:18:43.469 --> 00:18:50.219 Probably 1, 8 and 2, you never get paid 23456 or 7. you might get paid 1. 151 00:18:50.219 --> 00:18:56.939 Coin or whatever dollar, or something happens 3 each of the time and you'll get paid nothing half the time. 152 00:18:56.939 --> 00:19:09.209 Okay, so again, random variable is Tulsa coin once and it may not be fair. 153 00:19:09.209 --> 00:19:19.078 So, the PMF is just values are 0T or 1. and the probability of a 1 is the. 154 00:19:19.078 --> 00:19:23.759 Okay, nothing new there. Okay. 155 00:19:24.868 --> 00:19:28.769 Now, we get to something a touch more complicated. Well, this is our. 156 00:19:28.769 --> 00:19:33.028 Transmission thing or it's coin. 157 00:19:33.028 --> 00:19:37.679 And times, but communication is big. 158 00:19:38.788 --> 00:19:49.858 And so we'll talk about put it in a communication thing. So instead of tossing a coin and times we're going to transmit. 159 00:19:51.538 --> 00:20:02.068 We're going to transmit and packets and each packet may arrive wrong or it may arrive. Right? 160 00:20:02.068 --> 00:20:06.778 Um, and the random variable. 161 00:20:06.778 --> 00:20:10.798 Oh, so we're assuming that there's a way to tell if there's an error. 162 00:20:10.798 --> 00:20:20.999 Okay, their checks summed or something. This is not obvious, but we're assuming that gives a good check some and so. 163 00:20:20.999 --> 00:20:25.588 So, again, the experiment is we transmit it and packets and each packet arise. 164 00:20:25.588 --> 00:20:31.828 Correctly or arise with an error I ran unbearable as a number of packets that arrived. 165 00:20:31.828 --> 00:20:35.009 With error. 166 00:20:35.009 --> 00:20:44.669 So, the random variable, big X has N, plus 1 possible values, all the packets might be good or all the packets might be bad or anything in between. 167 00:20:44.669 --> 00:20:49.048 And so this is our binomial again and this is. 168 00:20:51.058 --> 00:20:56.519 And just look into that 1 means there's an error here. So this is our probability. 169 00:20:56.519 --> 00:20:59.788 For the random variable acts of having errors. 170 00:20:59.788 --> 00:21:06.328 And this is this expression you've seen before and this is binomial random variable. 171 00:21:06.328 --> 00:21:12.719 Okay, as a PM half of this expression here that you've seen before. 172 00:21:12.719 --> 00:21:20.459 Oh, okay, okay. 173 00:21:20.459 --> 00:21:23.699 Here's something new now. 174 00:21:24.838 --> 00:21:34.259 Expected value we have a definition. I mean, there's an intuitive definition for expected value, but here's the. 175 00:21:35.308 --> 00:21:44.848 The formal definition, and so if we have any discreet probability distribution, we have around unbearable. We can find the mean of that random variable. 176 00:21:45.898 --> 00:21:50.459 And and it's just the, some here. 177 00:21:50.459 --> 00:21:54.868 And we'll start working through examples now, but. 178 00:21:57.028 --> 00:22:00.689 There is a little detailed point that. 179 00:22:02.308 --> 00:22:10.739 There are random variables. There are discreet distributions for which there is no expected value. 180 00:22:10.739 --> 00:22:16.439 What that means is that this would be a case where it's an infinite number. 181 00:22:16.439 --> 00:22:21.479 Possible outcomes and you do the, some, the, some diverges. 182 00:22:21.479 --> 00:22:24.838 or the sum goes up to infinity . 183 00:22:24.838 --> 00:22:32.098 You may be able to sum the probabilities they sum to 1, but when you some X times of probabilities. 184 00:22:32.098 --> 00:22:39.929 It does not exist so that distribution, it could be a perfectly reasonable distribution. 185 00:22:39.929 --> 00:22:46.499 It's well defined, but it does not have a mean. 186 00:22:46.499 --> 00:22:50.999 I'll give you an example for continuous at a minute or 2, but. 187 00:22:50.999 --> 00:22:54.298 Things that go weird. Like this. 188 00:22:54.298 --> 00:22:59.669 Now, oh, how this affects the real world here. 189 00:22:59.669 --> 00:23:04.439 Well, for example, for the stock market, well, that, um. 190 00:23:04.439 --> 00:23:10.229 There are good argument that the distribution of stock prices. 191 00:23:10.229 --> 00:23:18.269 That the best way to model them are with distributions that do not have means. 192 00:23:18.269 --> 00:23:21.598 This is not a new ideas. 193 00:23:21.598 --> 00:23:28.318 Idea was floating around when I was a student in some respected stock market mathematicians. 194 00:23:28.318 --> 00:23:31.709 Have said that the distribution of stock prices. 195 00:23:31.709 --> 00:23:36.719 The best model for it does not have I mean, now. 196 00:23:36.719 --> 00:23:42.179 What does that mean? Intuitively? Is that the prices jump around too much. 197 00:23:42.179 --> 00:23:48.118 There are too many black swans that tails are too fat or too broad. So you cannot. 198 00:23:48.118 --> 00:23:53.128 Compute if these models for prices are correct, you cannot compute. 199 00:23:53.128 --> 00:24:02.699 Means now, what's the implication of that if you cannot if expected values and things do not exist then what this means is that most. 200 00:24:02.699 --> 00:24:06.838 Analysis methods are invalid. They do not work. 201 00:24:06.838 --> 00:24:11.669 In other words, so there are good arguments theoretical arguments that most of the. 202 00:24:11.669 --> 00:24:15.659 Analysis methods used on the stock market. 203 00:24:15.659 --> 00:24:20.368 Actually, don't work they're invalid. They produce garbage. 204 00:24:21.838 --> 00:24:29.969 And this idea has been acknowledged for many decades but the thing is, if it's true if you can't use any of these nice mathematical techniques and. 205 00:24:29.969 --> 00:24:34.798 What can you do? So, the analyst solution is to ignore it. 206 00:24:34.798 --> 00:24:43.949 Okay, and what this means is that every so often some intelligent person, like. 207 00:24:43.949 --> 00:24:51.509 Gave me diamond the head of a major bank or something? Well, we'll observe. Yeah the stock market is varying more widely than the models. 208 00:24:51.509 --> 00:25:01.199 Say it should, and every so often somebody who pets a lot of money on these models loses a lot of money. 209 00:25:01.199 --> 00:25:11.608 Is a case some years ago with a hedge fund called long term capital that was actually run by a Nobel laureate, using these nice mathematical models. The hedge fund did. 210 00:25:11.608 --> 00:25:20.249 It's such a bad well, the hedge fund failed, but it was spending so much money that it nearly brought down the economy. It needed a government intervention. 211 00:25:20.249 --> 00:25:23.729 So, okay, but getting back to the local thing. 212 00:25:23.729 --> 00:25:27.989 Let me just show you an example here. Um. 213 00:25:29.038 --> 00:25:33.118 Chilly come on. 214 00:25:34.229 --> 00:25:39.358 Okay. 215 00:25:43.229 --> 00:25:47.249 Maybe a continuous example here. 216 00:26:31.739 --> 00:26:42.088 Okay, this continues what this means is I'll probably continue so so for any range, we would. 217 00:26:42.088 --> 00:26:53.848 It'll give relative probabilities. I'm going to ignore some details with the concept here and see is a normalized onset. So she's just used to make it equal to 1. so the integral. 218 00:26:57.239 --> 00:27:00.509 And put in occupant plan. 219 00:27:03.449 --> 00:27:07.318 That's okay, but the interval of X f of X. 220 00:27:11.009 --> 00:27:18.568 and that just had it cannot be calculated so it's because infinity . 221 00:27:18.568 --> 00:27:27.838 So no means, okay. Okay any case. 222 00:27:27.838 --> 00:27:31.378 Back to here, so we have the mean and. 223 00:27:34.679 --> 00:27:38.788 Show you some simple examples hopes and. 224 00:27:40.199 --> 00:27:49.858 Okay, so discrete pains, you know. 225 00:27:51.209 --> 00:27:54.868 Got it right here so. 226 00:27:54.868 --> 00:27:58.588 You know, it's going to be the sum of, um. 227 00:27:59.788 --> 00:28:03.358 So probability of a. 228 00:28:03.358 --> 00:28:12.898 Of a 0T equals 1 for 1 minus P. probably 1 equals pay and then the mean expected value of X. 229 00:28:14.759 --> 00:28:20.818 Would be 0, 1 minus P plus 1 it was paid. 230 00:28:20.818 --> 00:28:24.568 They have there, for example. Okay. Um. 231 00:28:24.568 --> 00:28:32.909 3 2 costs the coin costs the coin 3 times, or we got our probabilities. 232 00:28:32.909 --> 00:28:42.898 1, 8, 3, 8, and so on, we've got the value of the random variables. 0, 1, 2 and 3 we sum up we have 1.5, for example. 233 00:28:44.098 --> 00:28:50.759 Our uniform discreet. Let me do 313 with specific value. Let's say. 234 00:28:52.259 --> 00:28:58.558 Say M, equals 3 or something. 235 00:28:58.558 --> 00:29:06.209 So our possible values, so the probability of a 0T equals 4th 1 equals 4th. 236 00:29:09.269 --> 00:29:12.568 Professor's screen is partially blocking. We're there you go. 237 00:29:14.038 --> 00:29:18.509 So the expected value of X equals, um. 238 00:29:19.949 --> 00:29:30.808 0T to 4th, plus 1 to 4th plus 2 4th, plus 3 times. 4th and that is going to be. 239 00:29:33.538 --> 00:29:44.878 What did I do? Ron? 0. 1. 240 00:29:44.878 --> 00:29:50.219 Ms. 4 here. Sorry. 241 00:29:54.568 --> 00:30:02.038 So, I'm going to leave at 3, but okay, so. 242 00:30:02.038 --> 00:30:06.778 Okay, 8 calls. 243 00:30:08.608 --> 00:30:13.798 Okay, so we need over here in the form. 244 00:30:13.798 --> 00:30:20.459 Okay, so it's the average. 245 00:30:20.459 --> 00:30:29.278 Okay, we do the mean for our bedding game. Now here's where or it's useful at 314. 246 00:30:30.358 --> 00:30:33.568 So you got the betting game. 247 00:30:33.568 --> 00:30:43.588 We could talk about maybe you have to pay a dollar 50 to play the game. You pay a dollar 50 you toss the coin. 248 00:30:44.759 --> 00:30:55.138 3 times, and you get paid 8 dollars if it's got 3 has a dollar if it's got 2 heads and otherwise nothing. So, are you winning or are you losing. 249 00:30:55.138 --> 00:30:59.489 Well, does the calculation here. 250 00:30:59.489 --> 00:31:10.618 Why is the random variable for your pay off its mean? Is this the possible value side of their probabilities? You're paying a buck 50 so, the mean is 11 8. 251 00:31:10.618 --> 00:31:18.628 Add this up, you get 11, 8, but you're paying 12 8. so your expected gain is minus and 8th year losing. 252 00:31:18.628 --> 00:31:25.739 Okay, okay. Is interesting. Now. 253 00:31:25.739 --> 00:31:33.598 There are 2 slightly different definitions for geometric. 254 00:31:33.598 --> 00:31:39.088 It just makes a nice thing about standards. There is there are so many of them. 255 00:31:42.538 --> 00:32:02.219 Silence. 256 00:32:14.308 --> 00:32:26.338 I mean, how the outcomes 1 up are the outcomes from 0T op. So if we toss the coin and times. 257 00:32:27.898 --> 00:32:35.278 Just a 2nd, here. 258 00:32:41.729 --> 00:32:45.269 Okay. 259 00:32:47.368 --> 00:32:51.479 Okay, if we toss, you know. 260 00:32:51.479 --> 00:33:04.828 We talked the coin until we get ahead version. 1 is the number of tosses, including that 1st head when we stop version 2 is a number of tosses before the 1st head. 261 00:33:04.828 --> 00:33:08.788 Okay, that just makes life. Interesting. So. 262 00:33:11.398 --> 00:33:15.388 So, what we have here is version 1. 263 00:33:16.588 --> 00:33:21.358 And we can do expected value. So if I give you an example. 264 00:33:30.388 --> 00:33:37.169 And X is, um. 265 00:33:44.788 --> 00:33:49.769 1st head, including. 266 00:33:52.019 --> 00:34:02.429 Tell us. Okay. So the probability the 1st head is on task 1 is 1 app. So the 2nd head is 1 corridor. 267 00:34:02.429 --> 00:34:11.849 third head is one eight and the sum of all the probability of k one to infinity is one okay . 268 00:34:11.849 --> 00:34:14.878 Expected value of X. 269 00:34:17.489 --> 00:34:26.458 Is 1 times a half plus 2 times 1 quarter plus 3 times 1, 8 plus and so on. 270 00:34:34.498 --> 00:34:39.298 Okay, now. 271 00:34:40.829 --> 00:34:44.608 How we would do that. 272 00:34:46.168 --> 00:34:59.489 oh and by the way note here that in this particular case that's also the sum of cake was zero to infinity and k or two okay because okay was zero zero okay . 273 00:35:00.838 --> 00:35:04.829 How to find that. 274 00:35:07.378 --> 00:35:10.739 Well, they give the way to compute it here. 275 00:35:11.759 --> 00:35:15.599 And I'll work it through because this is a useful. 276 00:35:15.599 --> 00:35:19.858 Technique, um. 277 00:35:21.599 --> 00:35:24.929 How to find it. 278 00:35:26.458 --> 00:35:34.798 A new page it's because my mirroring program can't mirror 2 pages at once. Okay. Well. 279 00:35:34.798 --> 00:35:42.418 I'll work through, so if we have if we want over 1 minus X, then. 280 00:35:42.418 --> 00:35:47.398 You work it out as an expanding fraction and that is. 281 00:35:49.889 --> 00:35:57.088 Okay. Okay. Cool. Sarah. Okay. 282 00:35:58.619 --> 00:36:01.858 You can work it out. 283 00:36:01.858 --> 00:36:07.559 How would I prove that? Well, suppose I took 1 minus X. 284 00:36:08.728 --> 00:36:21.268 times the sum of all the x to the k okay that's going to be the sum of all the zero to infinity zero to infinity x to the k minus . 285 00:36:21.268 --> 00:36:25.139 the sum of all the one to infinity x to the k . 286 00:36:26.789 --> 00:36:29.998 And that's going to be. 287 00:36:31.588 --> 00:36:35.759 Next to the a minute here. 288 00:36:39.838 --> 00:36:45.329 x0T equals 1. 289 00:36:45.329 --> 00:36:49.949 Okay, so therefore. 290 00:36:56.099 --> 00:37:01.588 Okay, you see how that worked out here. 291 00:37:04.193 --> 00:37:18.804 So, what I just did here was I showed you how you can find the sum of all the 0T. But if any X to the, this assumes that X is absolutely less than 1. okay. The next derivation is it's good to do a little math from time to time. 292 00:37:20.273 --> 00:37:23.213 Okay. So now suppose we want to find. 293 00:37:27.509 --> 00:37:30.958 How to do that? Okay. 294 00:37:30.958 --> 00:37:35.998 Well. 295 00:37:35.998 --> 00:37:39.449 Do we want to do that? 296 00:37:39.449 --> 00:37:45.838 We used a test. 297 00:37:45.838 --> 00:37:49.168 So we know that. 298 00:37:56.429 --> 00:38:06.268 Okay, well we take the derivative on the left on the left. It's going to be. 299 00:38:08.938 --> 00:38:16.559 Eva. 300 00:38:16.559 --> 00:38:23.400 I can work that out. That's freshmen math and on the right. 301 00:38:29.159 --> 00:38:35.880 Okay, so we have the answer right there. 302 00:38:35.880 --> 00:38:40.949 Okay, so so the expected value of X. 303 00:38:42.030 --> 00:38:46.440 Which is a sum of all the K, um. 304 00:38:46.440 --> 00:38:56.070 Peter the K. P. I'm going to do it for 1 half to make life easier. Let's say. 305 00:39:17.280 --> 00:39:20.340 Okay, then. 306 00:39:22.079 --> 00:39:26.610 And by the formula, then that is. 307 00:39:29.039 --> 00:39:33.599 Oh, I forgot to take it back here. 1 more thing. So. 308 00:39:36.989 --> 00:39:40.139 Okay, 1 more step here. 309 00:39:40.139 --> 00:39:46.920 So so the sum of all the K X to the K not K minus 1. 310 00:39:46.920 --> 00:39:52.619 It goes X minus X squared. Okay. 311 00:39:52.619 --> 00:39:58.380 Okay, so over here, then this is going to be, um. 312 00:39:59.460 --> 00:40:03.599 X is a half and that's going to be. 313 00:40:08.820 --> 00:40:15.809 Okay, so. 314 00:40:15.809 --> 00:40:18.929 So, in other words. 315 00:40:18.929 --> 00:40:22.440 The expected number of. 316 00:40:24.059 --> 00:40:32.159 It's a probability of success as a half hour as a half on average you and you're transmitting until you get. 317 00:40:32.159 --> 00:40:36.300 A success on the average yield transmit. 318 00:40:36.300 --> 00:40:44.849 Twice because again, the probability of transmitting at once was a half of a transfer twice as a corridor. 319 00:40:47.034 --> 00:41:02.005 And but it nets out that the expected value of all of this is 2. so, 1, half plus 2 quarters plus 3, 8 equals 2. okay. Mean geometric, random variables. So, a new concept here is. 320 00:41:02.280 --> 00:41:06.690 Finding these functions of random variables means. 321 00:41:06.690 --> 00:41:11.730 Okay, there is a general case here. I just did a special case. 322 00:41:12.989 --> 00:41:19.079 I just mentioned what's happening here. 323 00:41:19.079 --> 00:41:26.460 It's something called the St Peter St Petersburg paradox or the gamblers room. 324 00:41:26.460 --> 00:41:30.929 The concept is that you're going to go to Las Vegas. 325 00:41:30.929 --> 00:41:39.329 Like city whatever, and every time you lose, you keep adding and every time I lose, he'll increase your back. Assuming you'll eventually when. 326 00:41:39.329 --> 00:41:42.599 And then you'll make enough money to pay all your losses. 327 00:41:45.360 --> 00:41:52.199 The problem in the real world is, he seems to house limit for for you win perhaps. 328 00:41:52.199 --> 00:41:57.420 And then the expected number of. 329 00:41:57.420 --> 00:42:02.010 And then you get a divergence here. Okay. 330 00:42:05.550 --> 00:42:09.510 Expected value functions of a random variable. 331 00:42:10.619 --> 00:42:14.159 Um. 332 00:42:14.159 --> 00:42:19.710 Again, so what we have our gambling game, when he talked to the coin 3 times, and so on. 333 00:42:19.710 --> 00:42:25.349 So, you can find a spective over and above, you can have a function, and you find the expected value of the function. 334 00:42:26.730 --> 00:42:31.380 He's a nice example here. 335 00:42:31.380 --> 00:42:39.750 And I square law divisive so, let me get back here. Come on come on. 336 00:42:39.750 --> 00:42:45.420 Okay. 337 00:42:45.420 --> 00:42:53.639 You got voltage now you have no luck for your software types. If you got a voltage going through a 1 on resistance. 338 00:42:53.639 --> 00:43:00.329 The power, the how much the resistor heats up depends on the square of the voltage. 339 00:43:00.329 --> 00:43:12.570 So that, let's just assume to make this artificial example that the voltage is minus 1 minus 3 minus 1 plus 1 plus 3. 340 00:43:12.570 --> 00:43:18.329 Each 1 is uniform we want to find how much the resistor heats up. 341 00:43:18.329 --> 00:43:26.519 And it's a voltage is minus 3. the receptor heats up at 9 walks is a voltage is minus 1 then. 342 00:43:26.519 --> 00:43:33.719 1 walk, so plus or minus doesn't matter the heat depends on the square of the voltage and we'd want to find out. 343 00:43:33.719 --> 00:43:37.050 But the voltage keeps changing, so we want to find out what's the average heat. 344 00:43:37.050 --> 00:43:43.829 Generated in the resistor and so that we need to know the expected value of where. 345 00:43:43.829 --> 00:43:49.679 Of the voltage and so what we do is that. 346 00:43:49.679 --> 00:43:53.849 It's a voltage is minus 3 or 3. 347 00:43:53.849 --> 00:44:01.559 They were going to get 9 Watts and the problem is that happening is a half of the voltages as plus or minus 1. we get 1. what probably is a half. 348 00:44:01.559 --> 00:44:06.960 So, we can see the expected value for the power is going to be fine. 349 00:44:08.610 --> 00:44:21.059 Yeah, I'm interested in things like powers and so on because it I've got solar panels on my route that potentially. 350 00:44:21.059 --> 00:44:32.760 Are generating up to 8.16 kilowatts and I'm looking at my app right now as I'm talking, my solar panels are generating 3 kilowatts of heat right now as. 351 00:44:32.760 --> 00:44:36.599 As I'm talking, I was very nice test the app on my phone. 352 00:44:36.599 --> 00:44:45.869 And, um, there, um, let me. 353 00:44:45.869 --> 00:44:53.099 Yeah, you can sort of see it 3 kilowatts coming from the sun and. 354 00:44:53.099 --> 00:45:04.710 Point 6 kilowatt, going to the house and it just went down. Yeah. And then the rest going to the grid is a battery, but batteries fully charged. So I might just sit in things like power. 355 00:45:04.710 --> 00:45:10.559 I like gimmicks and as I mentioned my. 356 00:45:10.559 --> 00:45:18.599 27 kilowatt hours of batteries in the crash and then I got the 8 kilowatts speak. 357 00:45:18.599 --> 00:45:21.989 On the roof. Okay. 358 00:45:24.030 --> 00:45:38.489 Next thing I'm waiting for is the Tesla car, but I'm waiting to see how the tax situation comes out. Okay. So we random expected values, the functions of random variables, and they do obvious combinations. We'll do an example. 359 00:45:38.489 --> 00:45:48.239 And so on okay, 3.19 is is an example that's a little harder. And it's a real thing. 360 00:45:50.400 --> 00:45:54.000 Okay, we got our. 361 00:45:55.530 --> 00:45:58.920 Cable company or whatever, or phone company. 362 00:45:58.920 --> 00:46:03.480 And fries and wiring, or whatever you want to call them. 363 00:46:04.675 --> 00:46:17.635 And they don't provision enough cable for everyone to talk at the same time because it's expensive. And because probably, we don't have everyone talking at the same time. 364 00:46:17.844 --> 00:46:21.385 So they're trying to minimize how much money they spend on infrastructure. 365 00:46:21.690 --> 00:46:26.130 But not making in our infrastructure, so crappy that all the customers leave. 366 00:46:26.130 --> 00:46:31.739 Find they teach us in business school, but okay, so. 367 00:46:31.739 --> 00:46:38.940 Numbers we got 48 speakers say we're using this to encode cell phones. 368 00:46:38.940 --> 00:46:46.019 We got 48 speakers and the probability of any speaker talking at this current. 369 00:46:46.019 --> 00:46:49.019 Time slot is 4th. 370 00:46:49.019 --> 00:46:53.760 And we have 20 channels available. 371 00:46:53.760 --> 00:47:05.460 So you look at it right here there's so there's 48 speakers a 3rd of chance. Anyone is talking. So, therefore, the expected number of simultaneous speakers is 48 over 3. 372 00:47:05.460 --> 00:47:09.119 Which is 16 so on the. 373 00:47:09.119 --> 00:47:13.679 The main number of speakers and 16, and were providing 20 channels. 374 00:47:13.679 --> 00:47:21.690 Is that good enough? So, any thing any speaker more than 20. 375 00:47:21.690 --> 00:47:25.710 His packets will be discarded. 376 00:47:25.710 --> 00:47:33.599 So we want to find out the expected so that's a function Z, the number discarded packet and we want the expected value. 377 00:47:36.179 --> 00:47:40.980 Okay, so how do we do that? 378 00:47:42.059 --> 00:47:53.849 So so m's the number of channels we actually have, it's 20 X is, is the random variable for the number of active speakers. 379 00:47:53.849 --> 00:47:58.650 And we define Z, the number of speakers whose packets get dropped. 380 00:47:58.650 --> 00:48:10.530 If it's less than or equal to 20, nothing gets dropped. If it's more than 20. what gets dropped is X minus 20. it's it's 25 speakers trying to talk 5 packets get drop. 381 00:48:10.530 --> 00:48:17.039 That's a definition here. The little Delta above equal sign means this is a definition up here. Okay. 382 00:48:18.210 --> 00:48:27.300 How to compute that? Well, we don't each value. So the expectation is the value of the. 383 00:48:27.300 --> 00:48:30.570 Times the probability of that value. 384 00:48:30.570 --> 00:48:34.139 Summed over all the possible values so. 385 00:48:34.139 --> 00:48:42.210 So the could be anything from 20 to 48 if it's less than 20 of the probability. 0. 386 00:48:42.210 --> 00:48:47.099 Um, so. 387 00:48:47.099 --> 00:48:56.039 Number of speakers is C plus 20 C. plus 18. M. plus 20. I'm sorry I'm MS. 20. it's binomial. 388 00:48:56.039 --> 00:49:02.610 48 speakers, so the probability of there being Kay, as I'm talking is binomial thing here. 389 00:49:02.610 --> 00:49:07.650 This formula works, because we only want to know the probability of. 390 00:49:07.650 --> 00:49:11.039 K. speakers we don't care which of the. 391 00:49:11.039 --> 00:49:21.030 48 are talking, we only want to know what's the probability that K of any care of them are talking but we don't care which K. so that's correct. 392 00:49:21.030 --> 00:49:25.199 And probability times the value, we sum that up. 393 00:49:25.199 --> 00:49:31.349 Okay, now real world, how do you sum that up? 394 00:49:31.349 --> 00:49:37.170 Well, you know, use your favorite toy Mathematica, math lab or something. So. 395 00:49:39.300 --> 00:49:45.329 But this suggests that point 2 packets are discarded and any time slot. 396 00:49:45.329 --> 00:49:50.699 Which is 1%, so you have to decide. 397 00:49:50.699 --> 00:49:53.909 The business person is 1% packets. 398 00:49:53.909 --> 00:50:01.679 Boss oh, you want to play probability with our API some message from the head doctor. 399 00:50:01.679 --> 00:50:10.260 Yesterday that our API had 7 positive call with tests last week, if we could 7 more, then we go all online. 400 00:50:10.260 --> 00:50:19.050 Et cetera, et cetera, et cetera. So you might say how many people are in the community what's the probability of any given 1 being positive? 401 00:50:19.050 --> 00:50:22.920 And then you could calculate the probability that 7 more will be positive. 402 00:50:22.920 --> 00:50:33.510 The only problem that would be beautiful math. It would not be real in the real world because the beautiful math assumes independence. And of course, the coping infections are very, very dependent. 403 00:50:33.510 --> 00:50:36.989 You, in fact, your friends that you went out drinking with, or something. 404 00:50:38.460 --> 00:50:41.579 I know you never go out drinking. Okay. 405 00:50:41.579 --> 00:50:45.989 So that was mean. 406 00:50:45.989 --> 00:50:52.260 It tells us, you know, if we're betting how much we win or lose. 407 00:50:52.260 --> 00:51:07.139 Oh, I got it. I think I may have told her. I had to tell you, I had the sure. Fire way. I make about 100 bucks a year on the state lottery 100 bucks a year tax free. It's no, it's compared to the average New Yorker, so it's whatever 19M New Yorkers give or take. 408 00:51:07.139 --> 00:51:12.300 And I make a 100 bucks more than the average 1 every year with don't work at all. 409 00:51:12.300 --> 00:51:19.050 I never buy a ticket. Okay because the average person who pays this is a 100 bucks. 410 00:51:19.050 --> 00:51:23.820 I call it an IQ tax, but that's that's nasty. Okay. 411 00:51:23.820 --> 00:51:34.380 So, how much you're going to win the next question is is how predictable are your winnings and that's a new function called a variance. 412 00:51:34.380 --> 00:51:40.469 So, it talks about the brand of the random variable. 413 00:51:41.639 --> 00:51:52.019 So so maybe I play a game and it's every time he plays the game, I make a. 414 00:51:53.250 --> 00:52:03.630 You know, on the average, I may make a dollar, but sometimes I actually make 100 dollars and sometimes I lose 99 dollars. It's got a big spread. 415 00:52:03.630 --> 00:52:08.039 And that's completed by the variance and that has a definition. 416 00:52:10.494 --> 00:52:25.434 Here's an example from coal mining state underground call my, it's obsolete. They don't do mountain top removal, but underground coal mining the roof of the mine is held up by half. The coal is left and pillars that holds up the roof. 417 00:52:25.829 --> 00:52:30.150 I'm ignoring long wall and stuff just for the purpose of this fun example. 418 00:52:30.150 --> 00:52:35.460 So, you might imagine a pile of coal as a stack of dollar bills. 419 00:52:35.460 --> 00:52:38.550 That's being used to hold up the roof and you're just leaving them there. 420 00:52:38.550 --> 00:52:43.889 So, let's say you might say, oh, let's remove 1% of the dollar bills from the stack and. 421 00:52:43.889 --> 00:52:48.000 Of every stack of every filler holding up the roof and so we're going to make. 422 00:52:48.000 --> 00:52:54.059 100 dollars on each each time we do that. 423 00:52:54.059 --> 00:53:03.750 The expectation is positive, but the variance is this most of the time you remove some dollar bill from the sack of bills holding, nothing happens. 424 00:53:03.750 --> 00:53:09.210 A small fraction of the time the roof falls in and you get squashed flat. 425 00:53:09.210 --> 00:53:20.880 So, there's a large that so that little game of removing some of the pillars holding up the roof as a policy of expectation, but a very large variance. 426 00:53:20.880 --> 00:53:26.250 So, okay variance here. 427 00:53:26.250 --> 00:53:31.619 Definition is that. 428 00:53:31.619 --> 00:53:37.769 It's the expected value of the of the square of the deviation from the mean. 429 00:53:37.769 --> 00:53:46.650 And the standard now, it has units of square of length or whatever. The standard deviation is a square root. So, this is the definition here. 430 00:53:46.650 --> 00:53:50.909 Ok, now. 431 00:53:50.909 --> 00:54:00.179 There's different ways to compute yet. So the definition is expected value of the random variables difference from the means squared. 432 00:54:00.179 --> 00:54:09.090 Now, you can do simple algebra here. I mean, you can read the type sadder, or you see that you can read my handwriting. 433 00:54:09.090 --> 00:54:12.150 So that X minus. 434 00:54:12.150 --> 00:54:24.090 X squared you expand the square now in this case, the mean is is known, it's a constant it comes out of the expectation because expectation you can full Constance out. 435 00:54:24.090 --> 00:54:27.599 And you can full constant factors out. 436 00:54:27.599 --> 00:54:30.630 To this thing here and changed into this. 437 00:54:30.630 --> 00:54:40.380 And expected value of access. The means that's minus 2 mean squared. Plus means squared comes down to that and you got down to this. 438 00:54:40.380 --> 00:54:44.550 Okay, so that's another definition another way to compute. 439 00:54:44.550 --> 00:54:51.570 Expected values, X squared. This is called what is called the 2nd moment the expected value of the square of X. 440 00:54:51.570 --> 00:54:56.940 By us, I mean, the square the mean now, the only problem with this method. 441 00:54:56.940 --> 00:55:05.730 Is that if you random variable is not centered on 0. 442 00:55:05.730 --> 00:55:16.739 The main might be fairly high, the expected value the square. The square X might be very large. And if your computing was single precision floats and only of 7 significant digits. 443 00:55:17.880 --> 00:55:21.630 You may lose a lot of significant digits with this method. 444 00:55:21.630 --> 00:55:25.320 And, you know, if you're going to use, this is double precision and probably. 445 00:55:25.320 --> 00:55:29.820 Real World problem. Okay. Any case. 446 00:55:29.820 --> 00:55:35.519 So, there's some ways you can work with means and expected values. 447 00:55:37.139 --> 00:55:40.920 Let me write down some of the rules for. 448 00:55:40.920 --> 00:55:44.429 Means they're on a previous page, Tom. 449 00:55:44.429 --> 00:55:50.010 Well, actually, let me find them. 450 00:55:50.010 --> 00:55:53.670 Yeah, okay. Top of the page here. 451 00:55:53.670 --> 00:55:57.210 Expected values they add. 452 00:55:57.210 --> 00:56:12.000 You can scale factors, you can add to expect round expected values the 2, random variables functions, scale factor, as you can add in constantly expected. I have a constant as a constant. 453 00:56:12.000 --> 00:56:16.320 For variance. 454 00:56:18.090 --> 00:56:26.880 If you add a constant to the expected value to the to the random variable, it does not change. 455 00:56:26.880 --> 00:56:30.780 The it does not change the variance. So. 456 00:56:31.980 --> 00:56:36.090 Scaling comes out as a square, actually. 457 00:56:36.090 --> 00:56:41.789 Let me give you some simple examples here. 458 00:56:56.340 --> 00:57:00.989 Number 1, number of heads. 459 00:57:00.989 --> 00:57:05.400 Effective value of X is. 460 00:57:07.530 --> 00:57:15.780 The value of X squared. 461 00:57:17.429 --> 00:57:21.659 Equals 0T plus. 462 00:57:23.099 --> 00:57:27.840 So also on half Chile, the variance of X. 463 00:57:30.750 --> 00:57:37.739 Hey. 464 00:57:41.760 --> 00:57:45.989 Or the other way, the variance of X. 465 00:57:45.989 --> 00:57:49.409 Value of X, minus the mean. 466 00:57:50.880 --> 00:57:53.969 So, possible values, so. 467 00:57:55.050 --> 00:58:00.510 We're going to say, let Y, equals X minus the mean. 468 00:58:00.510 --> 00:58:04.349 I also write it as X. 469 00:58:04.349 --> 00:58:18.780 So that's going to be the mean, is a half. So that's 1, 1, half with probably minus half the probably about half or plus a half was probability of 1 half. Okay. So the expected value of why. 470 00:58:18.780 --> 00:58:26.010 Equals. 471 00:58:26.010 --> 00:58:33.449 Okay, well, that's got 1 more come on. Okay. Okay. So so why. 472 00:58:33.449 --> 00:58:44.460 Is X minus minus the mean of X? Y Z equal Y squared. Okay. New, random variable. So, Z is. 473 00:58:44.460 --> 00:58:47.849 So, why is minus a half or a half? Okay. 474 00:58:47.849 --> 00:58:51.059 2 minus a half or 1 half. 475 00:58:51.059 --> 00:58:54.239 And so Z. 476 00:58:54.239 --> 00:58:57.539 So, Z is always 1 quarter. 477 00:58:57.539 --> 00:59:05.070 Always okay, so basically the probability. 478 00:59:05.070 --> 00:59:10.289 Little Z equals 1 quarter. I'm sorry. 479 00:59:15.570 --> 00:59:20.340 Well, we could add you to send a simpler here. 480 00:59:24.389 --> 00:59:27.510 So, the variants of acts. 481 00:59:28.679 --> 00:59:33.960 It goes 1 now and just a 2nd. 482 00:59:36.989 --> 00:59:43.139 No, the variance of axis is 1 quarter because. 483 00:59:46.079 --> 00:59:51.599 Because the expected value of Z is 1 quarter. 484 00:59:54.269 --> 01:00:01.079 Always okay, so there's another way to find so 2 ways to find the variance doing this example, with tossing the coin once. 485 01:00:01.079 --> 01:00:10.920 So, okay, in any case to show these, let me show these variables. So, again, we're tossing the coin once. 486 01:00:15.480 --> 01:00:20.610 Us calling once number of ads. 487 01:00:23.760 --> 01:00:29.550 We're going to find a new wouldn't let's say, Dolby you equals so I don't know. 488 01:00:31.949 --> 01:00:44.789 5 X, you can do anything you want. Okay so now, what about so, basically, the probability on Dolby of getting 0T is 1 half the probability of W, of getting. 489 01:00:44.789 --> 01:00:54.929 5 equals 1 half, it's either 0T or 5. okay. And the expected value on W is going to be 2 and a half. Okay. 490 01:00:54.929 --> 01:00:58.980 And that's 5 of the expected value effects. 491 01:00:58.980 --> 01:01:05.730 So, you multiply random variable by some constant you multiply the. 492 01:01:05.730 --> 01:01:09.840 We just lost something here. 493 01:01:10.920 --> 01:01:18.719 What is going on? We are recording. We're not. 494 01:01:18.719 --> 01:01:25.019 We are still sharing. Okay. I'm sorry. We're not still sharing. 495 01:01:27.210 --> 01:01:30.210 We are still sharing. 496 01:01:32.760 --> 01:01:40.019 Um. 497 01:01:40.019 --> 01:01:44.849 I'm thinking sharing just failed. 498 01:01:44.849 --> 01:01:48.449 Then we start start sharing and start. 499 01:01:48.449 --> 01:01:49.914 We can see your screen. Okay. 500 01:02:11.579 --> 01:02:18.300 I'm still I'm. 501 01:02:18.300 --> 01:02:23.639 I can't tell if I'm sharing at the moment. 502 01:02:25.800 --> 01:02:32.579 We see your screen, I can tell you're sharing everything. 503 01:02:34.949 --> 01:02:35.844 The 2nd. 504 01:03:10.769 --> 01:03:14.010 Okay.