WEBVTT 1 00:00:49.950 --> 00:00:57.960 Hello. 2 00:01:08.424 --> 00:01:30.025 Okay. 3 00:01:41.519 --> 00:01:54.209 Okay. 4 00:01:54.209 --> 00:01:58.409 Right. 5 00:02:00.810 --> 00:02:09.990 Okay. 6 00:02:56.969 --> 00:03:06.270 Okay, good afternoon. This is probability. Class 17, I guess on March 17th coincidentally. 7 00:03:08.009 --> 00:03:12.780 And I think I am broadcasting it um. 8 00:03:14.580 --> 00:03:17.639 Give myself a check here. 9 00:03:20.129 --> 00:03:30.180 And, um. 10 00:03:32.460 --> 00:03:38.759 Okay. 11 00:03:38.759 --> 00:03:49.110 Can you do in the room here in just a 2nd. 12 00:04:02.129 --> 00:04:08.460 See, if this works. 13 00:04:11.939 --> 00:04:21.420 Oops, so. 14 00:04:28.829 --> 00:04:36.119 I want to do 1 of these crazy things here. I'm just trying to see if. 15 00:04:36.119 --> 00:04:43.288 Okay, can you hear me when I'm saying this if you're remote. 16 00:04:45.809 --> 00:04:48.838 And thank you. 17 00:04:48.838 --> 00:04:53.278 What I was specifically checking for is I've. 18 00:04:53.278 --> 00:04:57.238 2 machines in front of me and even if I. 19 00:04:57.238 --> 00:05:08.848 For some settings, it causes feedback for other settings. I'm completely muted and so it's impossible impossible for me to exactly see what other people see. 20 00:05:08.848 --> 00:05:14.009 In any case. Oh, beautiful. 21 00:05:14.009 --> 00:05:19.858 Okay, so we're continuing on in chapter 52 variables. 22 00:05:19.858 --> 00:05:28.829 And thought the way to do this is to do lots more examples from chapter 5. 23 00:05:28.829 --> 00:05:31.918 A chapter 6 will get into a vector. 24 00:05:31.918 --> 00:05:39.238 And random variables, but so I'm going to do some of the examples and this on the left here, I've got the, um. 25 00:05:41.309 --> 00:05:47.459 I've got the blog. No. By the way it's another homework out on great scope. 26 00:05:47.459 --> 00:05:55.199 So, and this is a good way to teach it. So, um. 27 00:05:55.199 --> 00:06:03.238 What makes it? It's weird here. Um. 28 00:06:09.298 --> 00:06:23.968 I'm sorry. Oh, thank you. 29 00:06:23.968 --> 00:06:30.809 Oh, okay, good. Keep the suggestions coming. So, um. 30 00:06:30.809 --> 00:06:38.428 Great. So, um. 31 00:06:40.379 --> 00:06:43.918 Oh, that's better. Um, let's delay. 32 00:06:43.918 --> 00:06:48.629 And so what we have is we have a too variable thing here, and just the. 33 00:06:48.629 --> 00:06:53.788 You'll have the page number over there. I see what we needed to get it put it in um. 34 00:06:53.788 --> 00:07:00.449 So, what we have is a 2 variable thing. 35 00:07:01.978 --> 00:07:12.178 And it just, um. 36 00:07:12.178 --> 00:07:21.269 Some see, and, um, basically. 37 00:07:24.569 --> 00:07:37.319 So it's non 0 in a region Mr. X and this is why it requires that at that. Why be less than X? So, here on the screen here. 38 00:07:37.319 --> 00:07:41.189 Okay, another way outside here it's 0. 0. 0. 39 00:07:41.189 --> 00:07:49.619 Okay, so so the 1st question is what is C. okay. Um. 40 00:07:53.728 --> 00:07:59.158 So, we're going to integrate basically, so we've got the undergo f of X and Y. 41 00:08:01.619 --> 00:08:06.449 Was 1 cause the probability density function and, um. 42 00:08:06.449 --> 00:08:12.478 So, X would be say 0 to infinity and why it would be 0 to X. 43 00:08:12.478 --> 00:08:16.649 No, and so it's it'd be, um. 44 00:08:27.389 --> 00:08:40.889 Now, um, yes. 45 00:08:46.708 --> 00:08:52.408 Sure oh, they commute but okay. That's That'll make it neater then for you. 46 00:09:00.328 --> 00:09:10.318 Okay, so, um, in the goal of either the minus, why from 0 to X? 47 00:09:10.318 --> 00:09:19.528 Thing here will be something like, um. 48 00:09:24.448 --> 00:09:35.698 Probably give or take. Um, okay, so now what we have is. 49 00:09:38.903 --> 00:09:55.734 Okay. 50 00:09:58.828 --> 00:10:03.599 Okay, so, um. 51 00:10:05.818 --> 00:10:09.269 And either the minus X integrating that is. 52 00:10:12.719 --> 00:10:15.808 I see you the minus X. 53 00:10:15.808 --> 00:10:19.589 And just saying here will be. 54 00:10:21.953 --> 00:10:41.813 Okay, 55 00:10:42.563 --> 00:10:43.374 let me see. 56 00:10:48.389 --> 00:10:59.969 What am I doing? Um, it'd be something like 1. 57 00:11:03.538 --> 00:11:07.078 Or minus a half or something, give or take, um. 58 00:11:07.078 --> 00:11:13.019 And so this 1 that out, and the whole thing, by the way it goes 1, of course. So we've got. 59 00:11:24.359 --> 00:11:27.448 So, when you go see over 2. 60 00:11:27.448 --> 00:11:31.649 This too. Okay, so the density function. 61 00:11:33.568 --> 00:11:42.509 Um, okay. 62 00:11:42.509 --> 00:11:47.908 Okay, so now, um. 63 00:11:47.908 --> 00:11:51.269 What a, now, let's do a marginal, um. 64 00:11:55.499 --> 00:11:58.558 So this is X and Y, so, um. 65 00:12:00.719 --> 00:12:05.969 Of X equals the integral integral over Y. 66 00:12:11.188 --> 00:12:16.259 All right, we'll say 0, 2 X. okay. Um. 67 00:12:16.259 --> 00:12:27.599 Okay, um. 68 00:12:31.469 --> 00:12:35.609 And this thing here will be 1-- 6. 69 00:12:40.739 --> 00:12:43.948 Okay, so then what we will get is, um. 70 00:12:53.099 --> 00:12:56.879 So, if ax, this is a marginal is, um. 71 00:13:04.109 --> 00:13:08.038 And if we integrate it, we should get 1. okay. 72 00:13:08.038 --> 00:13:16.408 Um, and then for why it would be, um, a similar sort of thing. Okay. 73 00:13:33.448 --> 00:13:36.599 Okay, and we could work it through, um. 74 00:13:50.219 --> 00:13:54.089 And this, um, okay. 75 00:13:56.369 --> 00:13:59.908 I see it's 2 here, so he can plug this in here. 76 00:13:59.908 --> 00:14:06.178 And this thing here is going to be. 77 00:14:11.698 --> 00:14:19.318 Okay, so for why is up or something. 78 00:14:20.818 --> 00:14:25.528 Actually, sorry, we have a 2 in here I dropped, um. 79 00:14:25.528 --> 00:14:28.769 Where did I drop it? Um. 80 00:14:43.109 --> 00:14:53.369 I should have a 2 in here, um, because this thing here is wrong. So. 81 00:14:53.369 --> 00:14:58.678 Whatever waving my hand so little and so on. 82 00:14:58.678 --> 00:15:07.318 Okay, okay so we can do stuff like that. Um. 83 00:15:08.519 --> 00:15:11.729 2nd, here. 84 00:15:26.158 --> 00:15:32.219 And we could do something like, um. 85 00:15:32.219 --> 00:15:38.999 Giving out details, so waving hands a little. 86 00:15:38.999 --> 00:15:43.379 Okay, okay. 87 00:15:43.379 --> 00:15:51.389 And then if you want to, um. 88 00:15:54.688 --> 00:15:57.719 Get something a little fancier, um. 89 00:16:06.208 --> 00:16:19.139 I'll say, 5.17. okay. How, how would we, how would we handle something like that? 90 00:16:24.958 --> 00:16:28.739 Again, we got the thing here. 91 00:16:28.739 --> 00:16:32.038 We got this now. 92 00:16:34.918 --> 00:16:39.509 So, X, plus Y, less than 1 would be, um. 93 00:16:41.999 --> 00:16:52.349 That'd be this region here, so okay. Um. 94 00:16:58.109 --> 00:17:05.368 Okay, so we'd have to, um, and we could integrate our way. Um. 95 00:17:05.368 --> 00:17:09.479 We, we could integrate our way around this, um. 96 00:17:09.479 --> 00:17:13.769 Slack. 97 00:17:17.159 --> 00:17:20.939 So, we could do it, we could integrate over the region actually. 98 00:17:29.159 --> 00:17:33.689 So, we have something like, and go back to black. It's easier to see. 99 00:17:33.689 --> 00:17:44.249 So. 100 00:17:53.759 --> 00:18:02.669 Mike, 0 to 1 and why would call 0 up to the minimum. 101 00:18:02.669 --> 00:18:14.818 Of, um, X and 1-X actually to be that region and so we might have to integrate it in 2 parts or something. So. 102 00:18:27.594 --> 00:18:40.614 0, to say 1, half 02, X of, say, 0210 to 1 half or the 1-X. 103 00:18:45.328 --> 00:18:52.259 Why do you want, DX? Because you see the region we're integrating over is this complicated. 104 00:18:52.259 --> 00:19:06.834 Um, it's complicated region up here, cause we want X plus Y, less than 1 and so the left side is triangle, wireless and access to controlling thing for the right side of the triangle. Why less than 1-X? 105 00:19:06.834 --> 00:19:15.683 Otherwise X plus wise bigger than 1. so you can do the 2 hearts and integrate now. Oh, drop it at that point but that's how we, that's how we can do that sort of thing. 106 00:19:16.378 --> 00:19:19.919 Okay. 107 00:19:25.078 --> 00:19:37.348 So now. 108 00:19:42.568 --> 00:19:50.398 The next question I want to have listed here is on, um, 519 on page page to 55. um. 109 00:19:54.479 --> 00:20:02.519 And it's a little more complicated mess here. So, um. 110 00:20:03.749 --> 00:20:15.898 How do I what to what I dropped it at that point it's just getting into simple integration, but the problem. Okay. So it's the thing where we have the, um. 111 00:20:15.898 --> 00:20:21.058 The function, or the density here here is. 112 00:20:22.499 --> 00:20:26.159 Different color. 113 00:20:26.159 --> 00:20:31.169 X and Y equals 2-X minus Y. 114 00:20:31.169 --> 00:20:42.148 Okay, and but I want the probability that X plus Y, is able to 1. 115 00:20:42.148 --> 00:20:49.679 So, it's in this, um, in this region here. So what I want to do is integrate the, um. 116 00:20:49.679 --> 00:20:54.179 The problem of the density function over that triangle. 117 00:20:55.348 --> 00:21:00.239 Um, the left side is because this left vertical thing here is because. 118 00:21:00.239 --> 00:21:10.439 Why is less than X and this side here is because we want the interval where X plus Y, slash equal to 1. so I want to integrate over that triangle. 119 00:21:10.439 --> 00:21:19.048 And it's what I'm saying, it's easiest to integrate over the last half and then the right half. So so that's what I did here. I'm integrating. 120 00:21:19.048 --> 00:21:24.689 So, at this point, here is 1 half. Okay. So integrating X equals 0 to a half. 121 00:21:24.689 --> 00:21:29.398 And, oops, this thing, he may have caught me on an error here. Um. 122 00:21:33.209 --> 00:21:39.058 That would be, um. 123 00:21:39.058 --> 00:21:45.628 To 1, so I, we could integrate those 2 things. So. 124 00:21:45.628 --> 00:21:49.858 Okay, let me get to the various calculus. 125 00:21:49.858 --> 00:21:55.618 Actually, tried him the examples is sort of as simple as possible, but. 126 00:21:55.618 --> 00:22:00.328 Those limits. Okay. So, um. 127 00:22:11.398 --> 00:22:16.648 Now, um, the okay 519. 128 00:22:16.648 --> 00:22:20.729 Let me see, I go back up to here. Um. 129 00:22:26.638 --> 00:22:29.638 I'll catch up what we have here. 130 00:22:39.808 --> 00:22:45.148 Okay, starting a page. 131 00:22:46.648 --> 00:22:52.888 Now, it refers to a case of 2 dice. 132 00:22:54.419 --> 00:22:58.828 But the probability mass function, here's the probability of mass function. 133 00:23:07.648 --> 00:23:12.808 Um, basic all right, so the probability of. 134 00:23:21.719 --> 00:23:31.824 So, tell us the 2 dies, and it's the probability that the 2 number, like, 2 ones come up is to 40 seconds probability to 23242566orwhatever come up 240 seconds. 135 00:23:31.824 --> 00:23:37.163 The probability say that a 1 and a 4 comes up however, is 142nd or. 136 00:23:41.128 --> 00:23:48.058 Whatever over here would be 142nd let's say so we talked the 2 dice. 137 00:23:48.058 --> 00:23:56.788 The probability that any particular number comes up twice a 140 seconds, the probability that a particular pair. 138 00:23:56.788 --> 00:24:08.578 Of different numbers, like 5 and 6 come up 5 on the 1st die 6 on the 2nd day. Being specific is 147 6 and 5 would be another 140 seconds. So. 139 00:24:08.578 --> 00:24:12.239 How you'd physically make a pair of dice that did that. 140 00:24:12.239 --> 00:24:18.929 I don't know, of course, slot machines and casinos and so on. Now are, um. 141 00:24:20.969 --> 00:24:24.449 They're all computers, so you could do anything um. 142 00:24:25.709 --> 00:24:33.239 And the sort of things you can do is to have access to the computers and so on some years ago, there was a multi state lottery. 143 00:24:33.239 --> 00:24:39.209 Um, drawing balls and the security director. 144 00:24:39.209 --> 00:24:43.528 For the law for the multi state lottery hacked into it. 145 00:24:43.528 --> 00:24:52.888 And was causing particular numbers to come up sometimes. So the security director, he got trouble defending against that, except with good auditing. Um. 146 00:24:52.888 --> 00:25:04.469 So, he constructed it, so he could actually make some big winnings. The problem was, he couldn't figure out how to cash the winning tickets and that's how they caught him. But okay. So. 147 00:25:04.469 --> 00:25:08.759 Any case, so we have this odd sort of thing. Um. 148 00:25:08.759 --> 00:25:15.298 And the question for 519 is, are the 2 dice independent. 149 00:25:15.298 --> 00:25:20.519 Using my definition well, 1st, we can get the, um. 150 00:25:20.519 --> 00:25:27.689 The marginal probability no, it's not. 151 00:25:27.689 --> 00:25:36.989 So, and I'm not encouraging to cheat a gambling and so on, but we're constructing examples for this. Course. So. 152 00:25:36.989 --> 00:25:47.788 Okay, I'm not expressing opinions on the games is real world games. We're just looking at the mathematical aspects of it. So. 153 00:25:49.888 --> 00:25:54.028 So, the marginal so say. 154 00:25:54.028 --> 00:25:58.259 3, just for 3, that would be P of, um. 155 00:25:58.259 --> 00:26:03.808 13 say P of 2 3. 156 00:26:05.278 --> 00:26:10.558 3 or whatever I equals 1 to 6 and that's going to be 1 6. 157 00:26:10.558 --> 00:26:17.729 There'll be 540 seconds. +240secondsequals, 7 and 40 seconds equals 1. 6. 158 00:26:17.729 --> 00:26:24.298 So each die separately. 159 00:26:29.548 --> 00:26:35.159 Is fair okay. Was 1 6 for. 160 00:26:35.159 --> 00:26:40.108 All right close 16. okay. However. 161 00:26:40.108 --> 00:26:46.229 They're correlated the 2 days. 162 00:26:47.459 --> 00:26:54.959 Are dependent, for example, P of say. 163 00:26:54.959 --> 00:27:03.598 Um, 1 and 2 equals 142nd, but for the single day separately P1 time fee to. 164 00:27:03.598 --> 00:27:09.118 Equals 16 times 106 equals 136 equals. 165 00:27:09.118 --> 00:27:12.358 Does not equal 142nd. 166 00:27:12.358 --> 00:27:25.588 So, the definition of independence of 2 events, 1st, diving 1 and 2nd dye being 2 would be the probability of that. Combo event should be the product of the probabilities. 167 00:27:25.588 --> 00:27:31.318 Of the 2 dice separately so, by this definition, the 2 dyce are. 168 00:27:31.318 --> 00:27:38.699 Dependent on each other, so each die separately is fair, but the combo. 169 00:27:38.699 --> 00:27:47.939 Is not fair in the combos dependent weird things might have again. This is a mathematical example, but weird things can happen. So. 170 00:27:47.939 --> 00:27:51.088 That was 519 or 2. 171 00:27:51.088 --> 00:28:00.778 So, and this table up here, just for fun. 172 00:28:00.778 --> 00:28:04.078 This table is on page 240. 173 00:28:04.078 --> 00:28:07.979 Okay, is that great? Okay. 174 00:28:10.739 --> 00:28:16.078 520, which I mentioned here, that's with the, um. 175 00:28:20.969 --> 00:28:27.689 Just mention it quickly here. So this is where you split. 176 00:28:29.009 --> 00:28:37.348 A long message or file do it a lot of questions about that last time. 177 00:28:38.848 --> 00:28:44.368 And to, um, like few blocks. 178 00:28:46.588 --> 00:28:51.568 Of size M and 1 partial block. 179 00:28:56.788 --> 00:29:03.179 Of size R. okay. Um. 180 00:29:04.949 --> 00:29:10.108 And again is what we saw last time, basically, on Monday. 181 00:29:11.308 --> 00:29:14.788 So, we assume a long message, um. 182 00:29:14.788 --> 00:29:22.528 Of size, and basically probability of and we said, we say geometric. 183 00:29:23.788 --> 00:29:31.348 We're not saying it isn't the real world necessarily. We're saying the purpose of this example and then that we computed. 184 00:29:31.348 --> 00:29:34.409 A computed part way that. 185 00:29:34.409 --> 00:29:37.499 Of the number of the full blocks. 186 00:29:39.269 --> 00:29:50.909 Was also geometric and also our also geometric. So now now what we could do this with this example, is, is that we could test. 187 00:29:52.769 --> 00:29:56.608 Um, our Q and R, independent. 188 00:30:02.638 --> 00:30:10.078 And so, and again, I'll wait my hand slightly a little, but what we would do. 189 00:30:10.078 --> 00:30:15.449 Is we would look at the question is the probability. 190 00:30:15.449 --> 00:30:21.388 Of some, some total length message, and it goes in. 191 00:30:22.469 --> 00:30:27.719 Equal the probability of for some queue and that are. 192 00:30:27.719 --> 00:30:31.259 Or or any calls. 193 00:30:31.259 --> 00:30:42.058 2 M, plus R, and it turns out the answer will be yes, I won't work out the details, but you've got the techniques to work out the details yourself. I think. 194 00:30:42.058 --> 00:30:56.548 So, and 1 reason say application might be is, I mean, some file systems, they pack the little fragments. 195 00:30:56.548 --> 00:31:01.378 Of these partial blocks into 1 block and to save some space on the disk. 196 00:31:01.378 --> 00:31:08.848 So, it's particular because it might even be at a lot of files on the disk are very, very small files. 197 00:31:08.848 --> 00:31:11.969 There's no complete blogs there's just 1 fragment. 198 00:31:11.969 --> 00:31:17.098 Particularly if the block size, like, on current is 4 K bytes. 199 00:31:17.098 --> 00:31:26.338 You know, most of your files are less than 4 K bytes long then they're gonna be fractions. So you pack all these pieces together. You'll save space on the disk. 200 00:31:26.338 --> 00:31:31.229 And you also save I owe time if you're reading a lot of the files at once. 201 00:31:31.229 --> 00:31:37.199 Any case, so that was, um, 520 and so on. 202 00:31:39.808 --> 00:31:43.378 Okay, here. 203 00:31:45.269 --> 00:31:48.538 And I listed, let's see, 5, um. 204 00:31:50.009 --> 00:31:55.469 Yeah, so joint moments, joint moments I mentioned, um. 205 00:31:58.528 --> 00:32:06.898 Okay, um, yeah, this could be here. 206 00:32:06.898 --> 00:32:12.298 I'm continuing on with these examples that I listed here. 207 00:32:13.949 --> 00:32:17.308 Oh, starting a new page here. 208 00:32:21.628 --> 00:32:24.989 And this is on page 257. 209 00:32:26.098 --> 00:32:36.598 Okay, so, um, some of 2, random variables. So I've got something like. 210 00:32:38.999 --> 00:32:49.679 So, I went to Las Vegas and maybe actually see random variable. How much money I lost yesterday and why is the random variables? How much money I lost today. 211 00:32:49.679 --> 00:32:55.439 And, um, would be the random variable for how much. 212 00:32:55.439 --> 00:33:00.058 I lost in total for yesterday and today, um. 213 00:33:02.638 --> 00:33:13.108 Shouldn't be completely insulting occasionally some people when, um, few decades ago, some MIT students analyze the Massachusetts state. 214 00:33:13.108 --> 00:33:20.848 Numbers game, and they found in certain cases, occasionally the expectation actually went positive. 215 00:33:22.019 --> 00:33:27.778 And when the expectation, when they predicted the expectation would go positive. 216 00:33:27.778 --> 00:33:33.959 They, they had a syndicate and they form they just bought hundreds of thousands of tickets. 217 00:33:33.959 --> 00:33:39.269 There's another syndicate formed by some people from Ohio also doing this in Massachusetts state lottery. 218 00:33:39.269 --> 00:33:45.269 And they actually, I think the expectation was actually positive by about 20%. So. 219 00:33:45.269 --> 00:33:50.878 They bought 100,000 dollars and they made on say, 20,000 dollars or something. 220 00:33:50.878 --> 00:34:00.028 And I think at 1 point, the syndicates were buying, I think a half of all the tickets sold in the whole state of Massachusetts. 221 00:34:00.028 --> 00:34:08.068 And they were just going to some small stores they'd have arrangements with the owner of the store, let's say, and they would just run the ticket. 222 00:34:08.068 --> 00:34:17.759 Machines and they store these little convenience stores from when the store opened until the store closed or something, or until the machine burned out I guess. 223 00:34:17.759 --> 00:34:22.469 And, um, they were making money and, um. 224 00:34:22.469 --> 00:34:26.188 In more detail, what happened is that. 225 00:34:26.188 --> 00:34:31.588 Like, if he got 6 numbers right? Get the guest 6 numbers. Have you got them? Right? You'd win the jackpot. 226 00:34:31.588 --> 00:34:38.639 If no 1 got it, right they would roll over the jackpot to the next week, which is what they do today for some of them. 227 00:34:38.639 --> 00:34:46.318 Which would make the jackpot bigger now normally, of course, everyone knows that, but there's a little messy complication. 228 00:34:46.318 --> 00:34:56.219 Is that they would roll the jackpot only if the next week with so much was bad and it's almost that much is going to be better than they'd buy more tickets, push it over the bump. And then bet a lot. 229 00:34:56.219 --> 00:35:06.329 And there was there was an official investigation into this, um, MIT students, and the reports online. 230 00:35:06.329 --> 00:35:09.599 And the report is that they were completely legal. 231 00:35:09.599 --> 00:35:17.608 It did not break any rules at all. However, they did change the rules to stop to some happening in the future. So. 232 00:35:17.608 --> 00:35:27.449 I can give you the link to the report if you want, but it goes into detail about what they were doing. So, in any case, you get things such as variable. Um. 233 00:35:27.449 --> 00:35:38.818 I think I suspect the lottery people knew what was happening, um, fairly quickly, because there's more tickets being sold and sun they're watching the tickets that are being sold. 234 00:35:38.818 --> 00:35:44.759 But they're making, you know, the more tickets are bought the more money the state makes. So. 235 00:35:44.759 --> 00:35:51.809 I think the state was happy with selling more tickets to the MIT students, and also the Ohio syndicate. So. 236 00:35:51.809 --> 00:35:57.059 But in any case. 237 00:35:57.059 --> 00:36:00.478 See, you think courses aren't useful in the real world. 238 00:36:00.478 --> 00:36:06.958 They are some, not that particular thing you got to figure out new stuff to a new generation. 239 00:36:06.958 --> 00:36:12.329 Um, you can actually. 240 00:36:12.329 --> 00:36:15.509 It's just for fun here. Um. 241 00:36:17.998 --> 00:36:21.659 Just curious, um. 242 00:36:29.693 --> 00:37:01.074 Okay. 243 00:37:01.648 --> 00:37:05.188 Time magazine back 10 years ago so. 244 00:37:07.168 --> 00:37:10.889 Okay. 245 00:37:20.248 --> 00:37:23.759 Is it cash rent and fall game and so on. 246 00:37:23.759 --> 00:37:28.679 Okay, as I said, they're not the only, um. 247 00:37:29.938 --> 00:37:42.329 That's the only people to do that. So, and I mentioned there was a, uh. 248 00:37:42.329 --> 00:37:46.079 There was a geological statistician up in Ontario. 249 00:37:46.079 --> 00:37:55.498 Who, um, his day job was computing, where to do test drilled bores to to find gold and other hard rock minerals. 250 00:37:55.498 --> 00:38:06.389 And he's very sophisticated probability techniques called Latin squares and so on. And he realized that the scratch off game in the province of Ontario, Canada. 251 00:38:06.389 --> 00:38:10.648 Seemed to use the same mathematics as he was using in a state job. 252 00:38:11.273 --> 00:38:25.704 Same probability stuff and at that time you could look so you'd get a ticket, you'd scratch off some squares but at that time, the stories would let you look at the tickets before you bought them cause to see if the park you could see was lucky or something. 253 00:38:26.460 --> 00:38:30.329 And he could look at the tickets, do some metal calculation in his head. 254 00:38:30.329 --> 00:38:33.719 And determined, most of the time as the ticket would be a winner. 255 00:38:33.719 --> 00:38:43.289 So, he could just buy winning tickets. So he bought a pile of tickets. Did not scratch them off and sent them with an accompanying letter to the provincial lottery commission. 256 00:38:43.289 --> 00:38:55.739 And the 1st time, they ignored him, so he did it again. He said, like, this bundle of tickets are probably all winners. This bundle of tickets is not notice I haven't scratched them off. And the story goes that they, uh, cancel the game the next day. 257 00:38:55.739 --> 00:39:02.369 And then he was written up, you could you get enough information to Google him and, um. 258 00:39:03.900 --> 00:39:15.030 So, in any case, um, the obvious question to him is, why didn't you keep it secret and. 259 00:39:15.030 --> 00:39:26.250 Make money off of it and his answer was that it wasn't worth it and that. So, you know, you go into the convenience store. You look at a pile of tickets, you buy a couple of them at 2 dollars each. 260 00:39:26.250 --> 00:39:34.230 You scratch them off and you make I don't know what 5 bucks or some things and he said his day job paid more money. 261 00:39:34.230 --> 00:39:37.769 So any case, um. 262 00:39:37.769 --> 00:39:42.719 You know, so real world thing, look that 1 up for fun. Um. 263 00:39:54.059 --> 00:40:02.070 I don't see if that we'll find it. 264 00:40:02.070 --> 00:40:09.389 Here we go. No, that's not the 1. sorry. 265 00:40:27.420 --> 00:40:35.550 Here we go. Okay. 266 00:40:38.429 --> 00:40:43.530 And that was again, 10 year. Okay. So you'll probably, he can actually be useful at times. So. 267 00:40:48.420 --> 00:40:57.329 Okay, any case. 268 00:40:57.329 --> 00:41:00.360 Let's go back to, um. 269 00:41:02.190 --> 00:41:13.440 Come on and let me, um, just a 2nd here. 270 00:41:14.789 --> 00:41:16.344 Back to the page. 271 00:41:46.380 --> 00:41:51.420 Oh, okay good. So we're talking joint moments and so on. 272 00:41:57.750 --> 00:42:10.050 Okay, so we've got 2 random variables. I have a probably density function for them, and I would like to compute the probability density function for the, some of the 2 of them. Okay. 273 00:42:11.789 --> 00:42:17.039 Well, even just the expectation. Um, so. 274 00:42:30.840 --> 00:42:36.059 Anticipating I talked a long time ago about that should be the sum of the expectations. 275 00:42:36.059 --> 00:42:39.599 But let's work it out um. 276 00:42:43.349 --> 00:42:48.750 So, expectation defines the next prime swipe prime Time's app. Um. 277 00:42:53.820 --> 00:42:59.369 And it doesn't matter what order I put this in D. Y. or something on. Um. 278 00:43:05.159 --> 00:43:08.190 We can, you can commute them and so on. Okay. 279 00:43:10.800 --> 00:43:14.130 Okay, so. 280 00:43:17.190 --> 00:43:19.885 And I can just split it apart here. Um. 281 00:43:42.210 --> 00:43:50.159 Say that again, I mean, here for the inner goal limits, you need. 282 00:43:52.769 --> 00:43:57.960 I'm leaving out the why that's what you're asking, but, um. 283 00:43:59.280 --> 00:44:06.989 It gives me a 2nd, 1 there here. Um, here. 284 00:44:14.460 --> 00:44:17.699 Here here. 285 00:44:23.519 --> 00:44:34.829 Can you describe it in more detail? Yes. 286 00:44:34.829 --> 00:44:42.090 Okay here. Yeah. So, um, let me just erase them into the extra marks. I put it in. 287 00:44:42.090 --> 00:44:53.190 Um, yeah, so well, I'm saying, because it says equals X plus Y. 288 00:44:53.190 --> 00:44:58.530 So the expected value of Z equals younger girls the say. 289 00:45:02.039 --> 00:45:05.039 This is the definition of expected value. 290 00:45:05.039 --> 00:45:08.909 What was there. 291 00:45:14.070 --> 00:45:24.179 Doesn't know well, it's expected well, CS equals X plus Y, expected value C, suspected value of X. plus Y. 292 00:45:24.179 --> 00:45:31.619 By definition, and then the definition of expected value is like, we look down here. 293 00:45:31.619 --> 00:45:40.440 Let's say, um, that's like the definition I integrate the, it's a variable times of density. 294 00:45:42.809 --> 00:45:48.929 And so when I have 2 lines above, so is experts wise, I just expand it out. 295 00:45:48.929 --> 00:45:55.380 And, um, um. 296 00:45:58.800 --> 00:46:04.289 So, is it a problem that I'm using the FX and Y, here? 297 00:46:08.610 --> 00:46:12.840 Little. 298 00:46:12.840 --> 00:46:24.690 Just using little prime 2 different names for the variables, because we may want outside the undergo. We may want to use X and Y, so inside, we'll say X, Prime and Y, Prime, just sort of different names. 299 00:46:27.719 --> 00:46:31.679 Well, Z is X plus Y, Z, Prime because X prime um. 300 00:46:31.679 --> 00:46:36.659 It just, it, it just me. 301 00:46:36.659 --> 00:46:41.190 Means I have a different name for the variable. Now I get XX, Prime to 2 different things. 302 00:46:41.190 --> 00:46:49.349 So, I'm actually fairly well following what's in the book. So. 303 00:46:49.349 --> 00:46:54.840 And again, that's on page, um, 257 up there. So. 304 00:46:54.840 --> 00:47:04.409 That could be helpful, but in any case, we expected values the so double clinical X this up here. And, um. 305 00:47:06.630 --> 00:47:10.559 Okay, so what I'm doing is that I'm looking at, um. 306 00:47:10.559 --> 00:47:17.940 This thing here okay now, um. 307 00:47:19.079 --> 00:47:23.760 Keep us the red, so this 1st part next prime. 308 00:47:26.730 --> 00:47:33.480 It doesn't matter what order here. Um, I could put it. 309 00:47:33.480 --> 00:47:37.679 The or something now, um. 310 00:47:37.679 --> 00:47:41.190 Oh, I could do, is I could I could move this. 311 00:47:41.190 --> 00:47:45.690 I met a girl for a Y, inside. 312 00:47:45.690 --> 00:47:51.150 Like that, and that's the X and that. So why? Because what I've got inside here. 313 00:47:52.739 --> 00:47:56.730 That will be the marginal probability for X. 314 00:47:58.559 --> 00:48:09.000 I integrate out, I got the 2 variable density function to integrate out 1 of the variables minus to infinity. What I have is the marginal distribution to the other variable. 315 00:48:09.000 --> 00:48:12.480 So, that thing equals. 316 00:48:14.369 --> 00:48:20.400 Integral of ex, Prime, half of ex, Prime DX Prime. 317 00:48:20.400 --> 00:48:23.880 And that is the expected value effects. 318 00:48:25.230 --> 00:48:37.320 So, as a thing up above where I that I boxed and read the 1st part is expected value facts. And then similarly, the 2nd part will be the expected value of why and it will narrow to. 319 00:48:43.320 --> 00:48:48.599 And what you should do is work out these examples on your own somewhat, but that, um. 320 00:48:48.599 --> 00:48:57.449 How we could do some for 2 and this is, this does not matter. 321 00:48:57.449 --> 00:49:05.550 If the variables are correlated or not so let me write that down. It's a big thing actually. Um. 322 00:49:05.550 --> 00:49:20.489 Regardless. 323 00:49:23.760 --> 00:49:31.889 Of any dependence are not. 324 00:49:31.889 --> 00:49:37.860 Always true. Um, that's 524. um. 325 00:49:39.780 --> 00:49:49.019 And also, it would extend it expected value of the some will say oh, lots of random variables is the sum of all the expected value. 326 00:49:49.019 --> 00:49:53.250 Okay, um. 327 00:49:58.469 --> 00:50:02.130 Her product is true if they're independent. 328 00:50:02.130 --> 00:50:11.699 So, they're independent. 329 00:50:15.210 --> 00:50:21.539 If there, if they're dependent, it may be true or it may be false but, um. 330 00:50:23.190 --> 00:50:29.730 So, um, this would be sort of like example. 331 00:50:29.730 --> 00:50:32.940 525 on page. 332 00:50:32.940 --> 00:50:37.110 258 a, um. 333 00:50:38.849 --> 00:50:42.630 I'll do a simple thing so suppose equals X times Y. 334 00:50:42.630 --> 00:50:50.760 Um, the expected value Zee. 335 00:50:50.760 --> 00:50:54.389 I would say the prime D. 336 00:50:54.389 --> 00:50:58.110 F of Z prime GC Prime. 337 00:50:58.110 --> 00:51:03.900 Which and nosy, so but this will be double integral of, um. 338 00:51:05.760 --> 00:51:10.469 X prime Y, Prime, um. 339 00:51:14.579 --> 00:51:28.139 I just need her here. This thing here. 340 00:51:35.099 --> 00:51:41.460 If independent okay. 341 00:51:41.460 --> 00:51:44.730 So, this, um, 8 close. 342 00:51:56.724 --> 00:52:01.764 Hey, calls and then we can split it apart in our goal. Ex, Prime Equifax, Prime. 343 00:52:02.429 --> 00:52:06.210 It's fine. 344 00:52:09.389 --> 00:52:12.510 Um. 345 00:52:12.510 --> 00:52:20.579 And this isn't going to be I'm going to go to the next page. 346 00:52:22.559 --> 00:52:27.150 Equals expected value of X to the value of Y. 347 00:52:29.130 --> 00:52:37.619 If they're independent, so okay. 348 00:52:37.619 --> 00:52:41.280 Um, yes. 349 00:52:41.280 --> 00:52:49.469 Okay. 350 00:52:49.469 --> 00:52:52.739 This is the case is equals X times. Y. 351 00:52:52.739 --> 00:53:01.110 So so, plus it's always true. Times is true if they're independent. 352 00:53:01.110 --> 00:53:06.750 So, okay. 353 00:53:08.070 --> 00:53:17.760 Um, okay. 354 00:53:17.760 --> 00:53:28.170 Now, I'm in the neighborhood for people to look at the book of a section 5.6.2, which is on page. 355 00:53:28.170 --> 00:53:34.829 258, so you have high order moment. This is from the previous chapter. 356 00:53:39.630 --> 00:53:46.260 And you get things, you know, obviously the expected value of X we saw was the integral X FX. 357 00:53:46.260 --> 00:53:51.480 Except a value of X squared would be the inner goal of X squared. 358 00:53:51.480 --> 00:53:56.280 And so unexpected value of X to the K. 359 00:53:56.280 --> 00:54:02.010 The intercom. Okay. 360 00:54:02.010 --> 00:54:07.349 Um, that's okay there and we saw remember that the variance. 361 00:54:08.460 --> 00:54:12.929 Was the expected value of X squared minus expected value X. 362 00:54:12.929 --> 00:54:16.170 Squared and so on. 363 00:54:16.170 --> 00:54:25.289 And so that just use the 1st, 2 moments, um, the high order moments that capture more details about the shape of the probability density function. 364 00:54:25.289 --> 00:54:28.769 And if you have all the moments, you can compute. 365 00:54:28.769 --> 00:54:43.704 The probability density function, it's like a tailor expansion. You have all of the terms of the tailor expansion. You can compute the original function. So it's similar with the moment. So we saw this. We had talked about the moment generating function a while back. Okay. 366 00:54:43.704 --> 00:54:45.985 Now, you can get the joint moments. 367 00:54:46.230 --> 00:54:57.239 Um, and we can get say the expected value of say. 368 00:54:57.239 --> 00:55:00.840 X to the J. Y. to the K. 369 00:55:00.840 --> 00:55:11.280 2nd, here is the J. 370 00:55:11.280 --> 00:55:15.510 Is a K and that's. 371 00:55:22.110 --> 00:55:26.039 X D, Y, you can put an X and Y, there and so on. 372 00:55:26.039 --> 00:55:36.630 Okay, that's for continuous. And if they're discreet, it would be just the sum of all the extra the J wide is a K. P. 373 00:55:36.630 --> 00:55:41.130 Um. 374 00:55:41.130 --> 00:55:46.260 Whatever, um, and so on X and Y. 375 00:55:48.059 --> 00:55:52.320 And Jay, or whatever, you could put subscription there if you want and so on. 376 00:55:52.320 --> 00:55:58.769 Okay, um. 377 00:56:01.199 --> 00:56:06.300 And it's capturing the distribute the shape of the more complicated, multiple. 378 00:56:06.300 --> 00:56:14.940 Thing and I'm going to tell you hit this quickly. Uh. 379 00:56:14.940 --> 00:56:19.949 2 weeks ago now, I guess hopefully yes, so it's a chance to. 380 00:56:19.949 --> 00:56:23.400 Hit it in more detail and so on. 381 00:56:24.449 --> 00:56:29.280 In any case, so we have, we got this joint again, so we've got the joint moment. 382 00:56:31.889 --> 00:56:37.559 Which is expected value of X to the J. Y. to the. K. okay. Um. 383 00:56:42.960 --> 00:56:46.110 And there's the linear thing, the correlation. 384 00:56:48.869 --> 00:56:52.050 Would be just expected value of X and Y. 385 00:56:52.050 --> 00:56:56.250 It's not the correlation moment. It's the correlation. So. 386 00:56:57.570 --> 00:57:06.599 Okay, and you can get the, um. 387 00:57:09.269 --> 00:57:13.440 You could get central moments where we subtract out the mean. Um. 388 00:57:19.559 --> 00:57:25.679 Central moments is subtract attract the mean. 389 00:57:25.679 --> 00:57:34.170 J, you might have for 1 variable expected value of X minus, expect the value of X. 390 00:57:34.170 --> 00:57:40.949 It's clear it. Okay, so he's the fact that, I mean, before we added the squares and took the mean, and so on. 391 00:57:40.949 --> 00:57:46.019 Okay, um, that's for 1 variable. 392 00:57:49.440 --> 00:57:57.090 And then for 2 variables, um, you can get the code variants. 393 00:58:00.750 --> 00:58:06.900 And it would be equal to the expected value. 394 00:58:10.320 --> 00:58:17.670 Times expected value of, um. 395 00:58:24.449 --> 00:58:31.110 Okay, oops, what is not here? 396 00:58:33.690 --> 00:58:39.300 And we can, um, like games with how we compute it and so on. So. 397 00:58:39.300 --> 00:58:46.949 And just as a refresher on. 398 00:58:46.949 --> 00:58:56.880 Since expected value to sum, so we can, we can take this thing and we can work it out. We can multiply out. So the expected value of X Y . 399 00:58:56.880 --> 00:59:01.980 Ex, expected value of Y minus Y. ex. 400 00:59:02.545 --> 00:59:17.215 A 600 value of X. okay. So, and that 1, that, uh, because the extra security wise, a console do an expectation of that. It doesn't change this 1 that out to expected value of X. 401 00:59:17.215 --> 00:59:18.534 Y, minus X. 402 00:59:22.800 --> 00:59:27.690 So just a too variable thing. Um, and that's, um. 403 00:59:27.690 --> 00:59:34.650 It's a CO variance and, um, captures linear relations. So. 404 00:59:37.860 --> 00:59:46.559 And if they're independent, then. 405 00:59:47.820 --> 00:59:53.340 Expected value of X. Y, so he's in the covariances 0 so. 406 00:59:59.250 --> 01:00:02.849 If the reverse is not necessarily true. So. 407 01:00:03.929 --> 01:00:07.110 Okay. 408 01:00:07.110 --> 01:00:20.550 Okay, you. 409 01:00:24.210 --> 01:00:27.210 On the notes here to the previous page. 410 01:00:29.670 --> 01:00:33.420 What's the 2nd out there? Okay. 411 01:01:01.920 --> 01:01:08.309 Okay. 412 01:01:15.360 --> 01:01:24.449 Okay, um. 413 01:01:25.530 --> 01:01:29.130 No, a big point is that 2 variables. 414 01:01:29.130 --> 01:01:36.780 Could be strongly related, but not dependent by the definite definition here. 415 01:01:42.420 --> 01:01:49.199 What this means, um, just a 2nd, complaining about something, um. 416 01:01:53.784 --> 01:01:54.594 Okay, 417 01:02:14.244 --> 01:02:16.315 stop the broadcast or something. 418 01:02:27.119 --> 01:02:30.119 Okay, got it we're working, um. 419 01:02:34.380 --> 01:02:41.579 Okay, um, what I was doing is I got a message that it recording. Okay. So the big point is here. 420 01:02:47.010 --> 01:03:00.059 May be strongly related, but still. 421 01:03:02.820 --> 01:03:16.019 But still independent by the definition, by our definition. 422 01:03:20.699 --> 01:03:26.250 Independent. 423 01:03:26.250 --> 01:03:34.860 Um, and then give an example 5.27 on page. 424 01:03:34.860 --> 01:03:37.980 260. 425 01:03:39.630 --> 01:03:47.730 Um, basically, the data is a random variables uniform from 0 to 2 pie. 426 01:03:47.730 --> 01:03:54.780 Now, we've got some functions of data X, equals coast data and, like, science data. 427 01:03:54.780 --> 01:04:00.269 So X and Y are strongly related. Okay so given. 428 01:04:00.269 --> 01:04:04.920 X there's only 2 possible. There's only 2 choices. 429 01:04:07.079 --> 01:04:11.159 Or why? Okay um. 430 01:04:11.159 --> 01:04:19.710 However, um, they're linearly. 431 01:04:22.440 --> 01:04:26.969 Um, they're uncorrelated because. 432 01:04:30.119 --> 01:04:40.469 Um, check your X times, why is going to be expected value sign say it a. 433 01:04:40.469 --> 01:04:45.570 Oh, so, um. 434 01:04:49.409 --> 01:04:53.880 Sign whatever I say to say, do you say it. 435 01:04:53.880 --> 01:05:06.750 Um, and that's going to net out to, um, so science data coach data is sign of tooth data divided by 2. okay. So, this, um. 436 01:05:11.219 --> 01:05:15.059 Um, so. 437 01:05:18.210 --> 01:05:21.599 Undergo sign. 438 01:05:23.219 --> 01:05:31.530 0, um, so, X and Y are uncorrelated. 439 01:05:34.469 --> 01:05:37.949 Or they're independent, they're also independent. Same thing. 440 01:05:37.949 --> 01:05:42.960 But they're very strongly related, but just by our linear definition. 441 01:05:42.960 --> 01:05:50.579 Um, so the thing is that our definitions just capture linear relationships. 442 01:05:50.579 --> 01:05:56.760 So, um, so that's a limitation in them. Um. 443 01:05:56.760 --> 01:06:00.659 So, the mathematics that the statisticians use. 444 01:06:00.659 --> 01:06:06.630 They're related, they're looking for linear relationships and if there's a more complicated relationship. 445 01:06:06.630 --> 01:06:14.039 Then, um, you know, then they won't capture it so. 446 01:06:14.039 --> 01:06:17.429 Which, um. 447 01:06:21.059 --> 01:06:26.699 Can be an issue, you see people they make statements about nutrients and so on um. 448 01:06:26.699 --> 01:06:33.090 In a linear relationship between water. How much water you drink and are you healthy? 449 01:06:33.090 --> 01:06:36.300 You know, just so many glasses of water a day or whatever. 450 01:06:36.300 --> 01:06:41.070 You know, more water is healthier well, but too much water and you drown. 451 01:06:41.070 --> 01:06:47.730 You know, so there's a more complicated or too little water and you diet thirst. So you see, it's not a linear relationship. 452 01:06:47.730 --> 01:06:51.119 And so that's an issue with, um. 453 01:06:51.119 --> 01:06:56.519 Some sorts of things here. Okay. 454 01:07:03.239 --> 01:07:13.230 Any questions about that, so we were looking at a quote and then the correlation coefficient I showed you the formula before early. It's this day. 455 01:07:13.230 --> 01:07:23.639 Take the variants and they, they still divide out the effect of the scale factor for the particular units. And, um. 456 01:07:23.639 --> 01:07:30.210 In, like, if I random variables are length, why his units of lane squared. 457 01:07:30.210 --> 01:07:37.949 The correlation coefficient will divide through by the standard deviations of X and Y, and remove this dependence. So, the result is a dimension list. 458 01:07:37.949 --> 01:07:49.650 Okay um, and then we get conditional probabilities. 459 01:07:57.210 --> 01:08:01.230 And this is on page 261. 460 01:08:02.880 --> 01:08:10.769 And, um. 461 01:08:13.230 --> 01:08:18.600 And just the definition, and we saw this a long time ago, but this is just bring it back. Um, say. 462 01:08:24.239 --> 01:08:30.149 Probably something why given some X I'll go to that Y index. 463 01:08:31.260 --> 01:08:35.520 Over the probability index, let's say, um. 464 01:08:38.579 --> 01:08:47.579 And some examples back, um, let me look at that all the example of, um. 465 01:08:49.979 --> 01:08:58.619 Of the dice thing that that's used by point 6 example. Um, so we had the 2 linked dice. 466 01:09:00.960 --> 01:09:09.840 And again, so the probability thing to go up the diagonal, it was 240 seconds anywhere else. It was 142nd. Okay. 467 01:09:09.840 --> 01:09:13.800 Probably okay. K. it would be 240 seconds. 468 01:09:13.800 --> 01:09:21.630 Probably J. K. it goes 140 secondary. Not equal to K. okay. 469 01:09:23.880 --> 01:09:31.710 So, um, let's say, I don't know, it's notation. 470 01:09:31.710 --> 01:09:36.270 X is the 1st die oh, is the 2nd. 471 01:09:39.479 --> 01:09:43.439 So, we could say the probability that the, let the, um. 472 01:09:47.310 --> 01:09:50.850 2nd day is 1 given the 1st dye is 1. 473 01:09:51.899 --> 01:09:57.600 With the probability that it's both are 1 divided by the probability of X equals 1. 474 01:09:57.600 --> 01:10:01.229 The problem is, is they're both 1 is 240 seconds. 475 01:10:01.229 --> 01:10:08.159 The problem remember any 1 die independently is fair 16 so that will be. 476 01:10:08.159 --> 01:10:11.189 Um, 240 seconds. 477 01:10:11.189 --> 01:10:15.029 Which would be 27 so, let's say. 478 01:10:20.039 --> 01:10:25.680 All right try to make an error. 479 01:10:30.659 --> 01:10:35.760 So I want the problem is the 2nd guy comes up 1, given the 1st guy came up 1. 480 01:10:35.760 --> 01:10:39.210 The problem is both coming up 142nd it's. 481 01:10:39.210 --> 01:10:50.189 Part of the definition of the problem. The problem is the 1st 1 on its own coming up 1 each day looking at on its own is fair. So probably the 1st time its own coming up 1 is 1 6. 482 01:10:50.189 --> 01:10:53.430 So, 240 seconds divided by 1 6. 483 01:10:53.430 --> 01:10:59.250 Is 27th I think so. Conditional probability. So. 484 01:11:08.939 --> 01:11:15.390 And the other ones that probability so I was 2 given X equals 1 would be. 485 01:11:18.840 --> 01:11:26.220 17, so, and if you add it all up, it's correct. So the sum of all the. 486 01:11:27.239 --> 01:11:32.970 Say, why it goes 1 to 6 probability Y, or some Y, given. 487 01:11:32.970 --> 01:11:37.319 Equals 1 would be a 240 seconds. 488 01:11:38.789 --> 01:11:45.630 Us 5 times 142nd, which would be 240 seconds. 489 01:11:47.489 --> 01:11:53.909 No, which, I mean, 240 seconds +540secondsat 740 seconds. 490 01:11:58.260 --> 01:12:06.569 It was 16, so the condition to the point where I just showed is that the conditional thing 1, it added up to 1 all the conditional things. So. 491 01:12:06.569 --> 01:12:15.930 Okay, so so this is exactly this here, but I just showed you. 492 01:12:17.430 --> 01:12:23.520 This here was example 5.229 on page 263. so. 493 01:12:27.779 --> 01:12:33.840 Okay. 494 01:12:38.729 --> 01:12:50.699 It's okay. To scroll up. Yes, I mean, the last line here. 495 01:12:50.699 --> 01:13:03.300 Well, I, some for all 6, like, was 1 to 6 so Y equals 1. it's the same as X. that's 240 seconds. 234 and 5 it's different. So, each of those is 142nd and there's 5 of them. 496 01:13:03.300 --> 01:13:06.689 So, it's a 2+5 equals 740 seconds. 497 01:13:06.689 --> 01:13:11.220 Which is 16th, so okay. Um. 498 01:13:18.210 --> 01:13:22.710 Okay, no example. 499 01:13:26.010 --> 01:13:29.069 Did you do 530. 500 01:13:33.270 --> 01:13:39.300 Also on page 263 is a beautiful example. 501 01:13:42.210 --> 01:13:46.079 Which means you should go run screaming out of the room, but, um. 502 01:13:47.939 --> 01:13:51.119 So, it's we got defects on a chip. 503 01:13:51.119 --> 01:13:55.140 And let's see how this works here. 504 01:14:04.920 --> 01:14:08.609 Okay, big chip. 505 01:14:11.880 --> 01:14:20.220 It'll region are okay. Um. 506 01:14:21.600 --> 01:14:28.920 Defects defects. 507 01:14:31.260 --> 01:14:37.529 Okay um, so. 508 01:14:39.689 --> 01:14:42.689 X, it's a random variable. 509 01:14:43.890 --> 01:14:50.189 Total number of defects, and it's plus on. 510 01:14:51.810 --> 01:14:56.789 Which, I mean, with alpha. Okay. Um. 511 01:15:02.579 --> 01:15:07.770 He is the probability of. 512 01:15:08.064 --> 01:15:22.465 A specific effect being in the region or reach defects separately. Okay. Now, um. 513 01:15:26.670 --> 01:15:30.510 So we want to learn about. 514 01:15:36.449 --> 01:15:41.729 Defects in R. so the whole thing, um, the whole big chip. 515 01:15:41.729 --> 01:15:45.449 It's getting blasted by cosmic rays. 516 01:15:45.449 --> 01:15:52.770 Or it's getting blasted by alpha particles from the uranium in the concrete and the walls of the room. Perhaps. 517 01:15:52.770 --> 01:16:03.390 Or in the steel steel that was manufactured after 945, and the early 50 s has more. 518 01:16:03.390 --> 01:16:07.409 Radiation and and steal that was manufactured before. 519 01:16:07.409 --> 01:16:13.800 Um, because it incorporates radiation from the atmosphere from the atomic tests. 520 01:16:13.800 --> 01:16:21.149 So, in fact, what some people designing some experiments that done, they've tried to look for warships. 521 01:16:21.564 --> 01:16:35.635 That were saying that sank before 945 and were not recycled so they go and get the steel from those battleship battleship and use it to manufacture some of the equipment for their experiments. For example, in any case that's irrelevant to the course. 522 01:16:37.409 --> 01:16:42.210 We got this big chip and number of. 523 01:16:42.210 --> 01:16:47.010 Alpha particles hitting at number defects it's plus on and the mean is alpha. 524 01:16:47.010 --> 01:16:54.960 Okay, so so the distribution of, um. 525 01:16:54.960 --> 01:17:03.420 Oops, so again, so alpha, so if we remember. 526 01:17:05.460 --> 01:17:19.529 Um, I'll talk to the X over X factorial if I, if I got it right? Something like that. Okay. And access. 527 01:17:19.529 --> 01:17:30.449 3, it has to be an editor obviously. Now, any particular defect, it could hit different places on the chip. Okay. The probability of it hitting the region are. 528 01:17:30.449 --> 01:17:35.789 Is P, for each separate thing okay. And. 529 01:17:37.800 --> 01:17:44.250 I want to learn and so let's just keep doing this. 530 01:17:45.930 --> 01:17:51.449 I scrolled up too fast. So why it's the random variable for the number. 531 01:17:51.449 --> 01:17:58.229 Of defects in the region are, and the question is. 532 01:18:00.390 --> 01:18:03.569 What is it probably a mass function of why. 533 01:18:03.569 --> 01:18:08.850 Okay, the probability of each different. Y, okay. 534 01:18:10.140 --> 01:18:17.609 So, we got 2 things here, so we're combining 2 random processes. 1st, the random number of defects on the whole chip. 535 01:18:17.609 --> 01:18:23.220 And then for each separate defect, the probability is going to be in the small region are. 536 01:18:23.220 --> 01:18:32.159 Okay, so so how are we going to do that? 537 01:18:32.159 --> 01:18:38.880 And that is a good point. We're going to see the answer on Monday. 538 01:18:42.989 --> 01:18:46.619 Okay, good to start. So. 539 01:18:46.619 --> 01:18:57.239 Okay, that's enough for today. Just to remind you what I was doing is, I was continuing on in chapter 5. we've been in chapter 5 for a while. 540 01:18:57.239 --> 01:19:04.170 Looking at 2, random variables we looked at some new stuff we looked at where the. 541 01:19:04.170 --> 01:19:08.909 The probability density function, or mass function for the sum of 2, random variables. 542 01:19:08.909 --> 01:19:22.319 And the, or the expectations, add the expectations and multiply if the variables are independent. So we saw some stuff like that, um, computing, some marginal probabilities. And so on high order moments. 543 01:19:22.319 --> 01:19:26.010 And things involving this funny paradise, where. 544 01:19:26.010 --> 01:19:33.239 Probability of the 2 days being the same is higher than that. They're separate and so on. And I listed the. 545 01:19:33.239 --> 01:19:41.430 Most of the examples that I did, so what we'll do Monday is continue on and by the way there's a new homework out on grade scope. 546 01:19:41.430 --> 01:19:48.630 So, have a good weekend. Doesn't look like there's going to be much skiing. Snow is melting a lot, but. 547 01:19:48.630 --> 01:19:55.380 Hiking season, they'll get some exercise you can work past, or if you get some exercise on the weekend. 548 01:19:55.380 --> 01:19:58.560 So, yeah, okay. 549 01:20:09.479 --> 01:20:14.819 How you have a good weekend. 550 01:20:16.380 --> 01:20:21.090 Yes. 551 01:20:23.729 --> 01:20:32.760 Hello.