WEBVTT 1 00:00:02.488 --> 00:00:05.759 Oh. 2 00:00:06.929 --> 00:00:11.458 But that's a good question. Everyone will want to know 1. 3 00:00:15.509 --> 00:00:24.480 Well, what I'm trying to do class. 4 00:00:24.480 --> 00:00:27.780 Is I like gadgets. 5 00:00:27.780 --> 00:00:33.060 The latest gadget I got was an iPhone, 13 Pro. 6 00:00:33.060 --> 00:00:37.079 So, I'm trying to record the Webex meeting off it. 7 00:00:38.219 --> 00:00:42.090 And we'll see how this works, so. 8 00:00:46.200 --> 00:00:53.009 And. 9 00:00:54.659 --> 00:01:03.869 13 pro probably has the best graphics of, um. 10 00:01:08.609 --> 00:01:11.700 Many of my devices here I mean, the best video. 11 00:01:11.700 --> 00:01:15.599 So, but I will still project from. 12 00:01:15.599 --> 00:01:19.439 The iPad, so. 13 00:01:19.439 --> 00:01:25.469 That's the theory. 14 00:01:25.469 --> 00:01:32.340 So, we will see what happens here and I'm holding on a question about last year. So. 15 00:01:35.340 --> 00:01:45.030 So actually last come to think of it. 16 00:01:51.900 --> 00:01:56.790 Okay. 17 00:01:58.739 --> 00:02:03.480 A, 2nd, here. 18 00:02:19.199 --> 00:02:26.669 Okay. 19 00:02:33.240 --> 00:02:40.020 Okay. 20 00:02:41.909 --> 00:02:47.159 Oh, okay. That works. 21 00:02:51.330 --> 00:02:59.370 This works. 22 00:03:00.479 --> 00:03:07.139 Here. 23 00:03:11.340 --> 00:03:15.330 That works. 24 00:03:15.330 --> 00:03:26.520 Work. 25 00:03:47.189 --> 00:03:51.030 Where. 26 00:03:51.030 --> 00:03:54.090 That works. 27 00:03:54.090 --> 00:03:58.979 That works. 28 00:03:58.979 --> 00:04:05.430 Okay. 29 00:04:05.430 --> 00:04:18.569 Hello. 30 00:04:18.569 --> 00:04:21.750 Okay, um, previous exams. 31 00:04:21.750 --> 00:04:25.709 Last year's exam was on grade scope. 32 00:04:25.709 --> 00:04:29.038 Remember we haven't released at the previous exams. 33 00:04:29.038 --> 00:04:34.588 Are up on the website and. 34 00:04:34.588 --> 00:04:37.978 Let's see what we can do here. 35 00:04:39.928 --> 00:04:47.488 Um, come on. 36 00:04:47.488 --> 00:04:50.639 Okay. 37 00:04:52.139 --> 00:04:55.439 So, I'm going back here to 2020. 38 00:04:55.439 --> 00:04:58.918 And here is shows up good. 39 00:04:58.918 --> 00:05:01.949 And what we can see is. 40 00:05:01.949 --> 00:05:06.028 The exam 1 and things have. 41 00:05:09.778 --> 00:05:16.559 Things, um, with solutions and so on, I'll go through some of that also if people want. 42 00:05:16.559 --> 00:05:20.249 But and what I will do is. 43 00:05:20.249 --> 00:05:25.769 To recycle many of the old questions. 44 00:05:25.769 --> 00:05:32.218 They tell me recycling is good so we'll recycle many of the old questions. 45 00:05:33.298 --> 00:05:38.968 So, um, would people like. 46 00:05:38.968 --> 00:05:45.598 Me to go through some of these questions and answers right now or not. 47 00:05:47.338 --> 00:05:54.119 Opinions either way. Let me do I actually just do 1 our, um. 48 00:05:54.119 --> 00:06:00.149 Let me do 1 or a few of them right now. Look at this. Um. 49 00:06:00.149 --> 00:06:04.559 Bring the thing up here and then I'll. 50 00:06:04.559 --> 00:06:09.478 Works through some of them to show you a 2nd here. 51 00:06:12.629 --> 00:06:17.488 Uh. 52 00:06:17.488 --> 00:06:23.519 This would be. 53 00:06:47.184 --> 00:06:48.353 Okay. 54 00:06:48.838 --> 00:06:56.189 So, if we're looking at this is from the, um. 55 00:07:01.829 --> 00:07:08.608 A chance to review a few things. Okay so we've got, um, s, there's a small. 56 00:07:08.608 --> 00:07:13.559 Occurs the numbers are fictitious. Okay. Um. 57 00:07:14.579 --> 00:07:18.059 And, um, C is cancer. 58 00:07:19.949 --> 00:07:26.819 Okay, um, just the point is numbers made up. Okay. 59 00:07:28.139 --> 00:07:33.028 Okay, so if I know say that, um. 60 00:07:36.509 --> 00:07:40.918 If you smoke, say, 10% chance, you've got cancer. 61 00:07:41.963 --> 00:07:55.434 And there was a time generation ago when half the adult population smoke so it was awful. Smells. Okay. Um. 62 00:07:55.949 --> 00:08:04.678 Now, let's say that, um, of people who got lung cancer, say, point 990% of them smoke. 63 00:08:04.678 --> 00:08:11.908 Occasion like, get lung cancer from radiation or something breathing rate on. And let's assume that say. 64 00:08:13.889 --> 00:08:25.588 Um, I'll put it up here. Actually let's assume, let's say 20% of people smoke adult population, got to Vegas. 65 00:08:25.588 --> 00:08:29.848 That's how I so now. Okay, so that we're given that. Okay. 66 00:08:31.348 --> 00:08:35.038 Now, we want the questions we want the probability. 67 00:08:35.038 --> 00:08:43.048 Uh, that people got cancer. Okay well, we could use a total probability theorem for this. Um, so this is the question. 68 00:08:43.048 --> 00:08:47.068 We could say probably the people got cancer. There's people who got, um. 69 00:08:47.068 --> 00:08:55.048 Let me say, um. 70 00:08:55.048 --> 00:08:58.528 Say probability say people. 71 00:08:58.528 --> 00:09:04.499 Who got cancer given a smoke terms of probability they smoked, or something like that. 72 00:09:04.499 --> 00:09:08.938 And we happen to have the cancer given smoke 2.1. 73 00:09:10.019 --> 00:09:15.719 Point to say cause, I don't know. Am I doing this right here? 74 00:09:15.719 --> 00:09:23.999 I said, I want the probability. They got cancer problems. They've got cancer given they smoke if we know the probability that they smoke. 75 00:09:23.999 --> 00:09:29.759 So, cancer given smoke oops. Is. 76 00:09:29.759 --> 00:09:34.918 Point 2 times the probability they smoke. 77 00:09:38.849 --> 00:09:49.979 If I did this, right? Um, so something like that, perhaps I, I want to check this. Um, so I could check card. 78 00:09:51.629 --> 00:09:55.619 Okay, um, the example, let's say, um. 79 00:10:10.649 --> 00:10:13.828 Now, we might go to the next question was. 80 00:10:15.568 --> 00:10:18.629 You got cancer giving you did not smoke. 81 00:10:20.369 --> 00:10:24.899 Um, how can we do that? Um, let's see. 82 00:10:34.318 --> 00:10:37.798 And I want to go back to the previous 1, um. 83 00:10:40.229 --> 00:10:43.769 Yeah, I got this wrong. Yeah. Is it about the previous 1? 84 00:10:45.538 --> 00:10:52.649 Yeah, yeah, you're right. Um, let me. 85 00:10:55.889 --> 00:10:59.009 Okay, so okay. Um. 86 00:11:00.719 --> 00:11:03.899 Probably have cancer that's what we want. Um. 87 00:11:07.019 --> 00:11:15.028 Um, oh, we probably say cancer and smoke divided by. 88 00:11:15.028 --> 00:11:18.028 I did. 89 00:11:21.719 --> 00:11:26.698 Um, just basically, my mind is going completely at the moment. Um. 90 00:11:26.698 --> 00:11:31.649 All right again. Okay. Um. 91 00:11:31.649 --> 00:11:41.818 How could we do this? So we're going to say probability of cancer and smoke. 92 00:11:41.818 --> 00:11:49.259 Is the probability of cancer given smoke kind of the probability of smoke. 93 00:11:49.259 --> 00:11:52.859 Okay, and. 94 00:11:55.889 --> 00:12:01.048 Also equal to the probability smoke, given cancer in terms of probability of cancer. 95 00:12:03.389 --> 00:12:07.678 And now we can say the, probably, you've cancer is. 96 00:12:09.089 --> 00:12:15.178 The probability of cancer given smoke trends of probability of smoke, divided by, um. 97 00:12:15.178 --> 00:12:18.208 Probability a smoke given cancer. 98 00:12:18.208 --> 00:12:29.068 Cancer given small call saves point 1. um, smoking was 2.2 probably smoking given cancer was point 9. okay. 99 00:12:29.068 --> 00:12:33.538 And that equals, um, to to say. 100 00:12:34.708 --> 00:12:39.359 Something like that that looks better. Okay. 101 00:12:42.239 --> 00:12:47.639 And these are just correlations and so there's nothing in this mass about causality. Of course. 102 00:12:47.639 --> 00:12:55.109 Hello. 103 00:12:59.188 --> 00:13:04.198 Okay, so the next question was cancer, given that you don't smoke. 104 00:13:04.198 --> 00:13:09.869 Um, what we do here. 105 00:13:12.089 --> 00:13:16.318 Again, we could say, well, probability of cats are giving you don't smoke. 106 00:13:16.318 --> 00:13:20.999 And the probability that you don't smoke with the probability of cancer, and. 107 00:13:20.999 --> 00:13:25.019 You don't smoke. Okay. 108 00:13:25.019 --> 00:13:28.979 So. 109 00:13:28.979 --> 00:13:36.538 Um, but it's also equals the probability. 110 00:13:36.538 --> 00:13:46.379 That you, um, you don't smoke given you had cancer as a probability of cancer. 111 00:13:46.379 --> 00:13:50.129 So, the probability of cancer giving, you don't smoke. 112 00:13:50.129 --> 00:14:01.139 We'll be, I'll bet you, if you don't smoke given cats, there are times of probability of cancer divided by probability. You don't smoke. 113 00:14:01.139 --> 00:14:05.068 Okay, now what do we have here? Um. 114 00:14:06.778 --> 00:14:16.948 So, I had that the probability or do you smoke given cancer? 115 00:14:16.948 --> 00:14:24.509 Which point 9, so the probability you don't smoke given cancerous point 1. okay. So. 116 00:14:24.509 --> 00:14:30.119 Thing here is point 1 probably of cancer. I said. 117 00:14:30.119 --> 00:14:36.328 Was, um, we calculated it back in the previous page. 118 00:14:36.328 --> 00:14:39.839 As, um, point 0 to 2. 119 00:14:42.568 --> 00:14:47.339 How will you don't smoke? Well, the probability that we had smoke was. 120 00:14:47.339 --> 00:14:50.729 Point too is the problem. You don't smoke. 1.8. 121 00:14:52.048 --> 00:14:57.418 So then, so they'll probably have cancer giving you don't smoke. 122 00:14:59.369 --> 00:15:05.308 Was point 1 time point 022 divided by point 8. 123 00:15:05.308 --> 00:15:10.259 Oh, and, um. 124 00:15:10.259 --> 00:15:13.979 Whatever the bullet points. 125 00:15:13.979 --> 00:15:17.249 Oh, 3 give or take so. 126 00:15:19.438 --> 00:15:22.619 2nd question 3rd question. 127 00:15:22.619 --> 00:15:33.178 Um, so, so if you didn't smoke was 3 and a 1000 answer, if you did smoke. 128 00:15:33.178 --> 00:15:36.869 Um, of course, point 1, so. 129 00:15:36.869 --> 00:15:41.519 Is the tobacco companies would say it was a correlation. 130 00:15:42.778 --> 00:15:47.099 Um, there are. 131 00:15:47.099 --> 00:15:53.698 In fact, in the 950 s, there were doctors who said smoking was good for you. Okay. 132 00:15:58.918 --> 00:16:03.089 Um, next question. 133 00:16:05.548 --> 00:16:12.839 Is here it's the next is the next 1 is like a geometric called probability. 134 00:16:17.339 --> 00:16:26.278 You're passing a difficult class. Um, not that. All right so this would not be, of course. Um, but I and. 135 00:16:30.178 --> 00:16:35.999 Taking a hard class MIT. 136 00:16:38.548 --> 00:16:47.399 Probability of passing was 1 half and you retake it. It's independent, so you. 137 00:16:49.229 --> 00:16:52.438 And passing is independent of the previous time. 138 00:16:58.139 --> 00:17:06.179 Of the previous attempt. Okay. 139 00:17:06.179 --> 00:17:16.108 So oh, the 1st answer is what is the relevant distribution? The relevant distribution I wrote it up here would be geometric. So. 140 00:17:17.489 --> 00:17:23.578 And, um, so. 141 00:17:31.618 --> 00:17:36.719 The probability pass. 142 00:17:39.028 --> 00:17:46.618 On the case fry is going to be over here. We've got 1 half. 143 00:17:46.618 --> 00:17:51.419 Qa was 1-peak was 1 half and the probability. Okay. 144 00:17:51.419 --> 00:17:57.509 On the case try is going to be, um, P, times Q2 the K. -1. 145 00:17:57.509 --> 00:18:00.898 Which, in this case, so we went over to to the K. 146 00:18:00.898 --> 00:18:05.338 So, 50% of the 1st attempt 25% on the 2nd attempt and so on. 147 00:18:05.338 --> 00:18:08.759 So so the mean, um. 148 00:18:10.259 --> 00:18:17.548 Expect the number of K that's someone over P. which in this case, it'd be 2 the expected number. 149 00:18:17.548 --> 00:18:28.618 Would be Kate times and if you forgot that formula, you could sum up K times P times Q. to the K. -1 equals 1 to infinity. 150 00:18:29.969 --> 00:18:36.328 And as a reminder you would, um. 151 00:18:38.459 --> 00:18:42.058 P. 152 00:18:45.628 --> 00:18:48.719 Um, you could do this by derivative. 153 00:18:48.719 --> 00:18:53.878 So, um, could use a direct. 154 00:18:58.138 --> 00:19:04.078 To show that, so, um, basically. 155 00:19:08.939 --> 00:19:12.118 And it was D. 156 00:19:12.118 --> 00:19:17.128 Some of all that acute is okay. Um. 157 00:19:19.348 --> 00:19:23.009 Some of the queue to the case 1 over. 158 00:19:26.009 --> 00:19:29.729 1-queue if I have it, right? 159 00:19:29.729 --> 00:19:37.169 And that happens to be, um. 160 00:19:40.499 --> 00:19:43.679 Whatever, and that would work its way through. So. 161 00:19:43.679 --> 00:19:49.348 Okay, um, the next question is is what is the variance of K. 162 00:19:49.348 --> 00:19:55.078 And we had the formula and 1 of the form, you could work it out or you could also summit. So. 163 00:19:58.288 --> 00:20:02.999 There was another question about, um. 164 00:20:06.449 --> 00:20:11.578 Again, this is the. 165 00:20:11.578 --> 00:20:16.588 2020 exam 1 continued. 166 00:20:16.588 --> 00:20:21.298 Um, 3. 167 00:20:21.298 --> 00:20:27.778 20 side to die that costs it regular cost. So he dread. 168 00:20:27.778 --> 00:20:37.679 Um, okay, um. 169 00:20:38.909 --> 00:20:43.979 Utah, and you see K, that's your random variable. 170 00:20:43.979 --> 00:20:49.078 This is your random experiment. 171 00:20:52.318 --> 00:21:00.689 Okay, random variable or also the outcome. It doesn't matter. Okay so, um. 172 00:21:02.219 --> 00:21:08.308 So That'll be good events. A, is that K. 173 00:21:10.048 --> 00:21:14.848 Is messed up to 10 let's say it could be. 174 00:21:14.848 --> 00:21:21.808 Is that K? It is in the set 135 up to 19. 175 00:21:21.808 --> 00:21:26.038 Solid in other words and C is the event. 176 00:21:26.038 --> 00:21:29.608 I write down here, um. 177 00:21:29.608 --> 00:21:35.159 It's in. 178 00:21:35.159 --> 00:21:39.749 151,719. 179 00:21:39.749 --> 00:21:46.409 Okay um, and then the question is. 180 00:21:48.179 --> 00:21:57.959 Are a, and be independent. Well, they'd be independent is the probability of a title probability would be. 181 00:21:57.959 --> 00:22:03.419 Is a probability they N. B. A. hey. I could use the. 182 00:22:03.419 --> 00:22:13.618 Intersection thing, or the end thing it'd be the same thing. Probability of is 1 half cause it's 10 things. Probably B, is 1 half. 183 00:22:13.618 --> 00:22:20.368 How will you see is 12345678910is1 half. 184 00:22:20.368 --> 00:22:23.969 I'll probably they end it'd be okay. 185 00:22:23.969 --> 00:22:28.979 Um, so a, and B is the set. 186 00:22:28.979 --> 00:22:35.788 Less up to 10 and they're odd. So 13579. 187 00:22:35.788 --> 00:22:39.058 There's 5 of them, so that's 1 quarter. 188 00:22:41.308 --> 00:22:44.308 And the answer is yes. Okay. 189 00:22:45.419 --> 00:22:52.348 Um, oh. 190 00:22:55.138 --> 00:22:58.949 And then we'd ask our independent and so on. 191 00:23:00.028 --> 00:23:05.999 And see well, and they set a intersect C. 192 00:23:05.999 --> 00:23:11.608 Is, um, 2. 193 00:23:11.608 --> 00:23:16.949 There's 5 of them here, so it'll probably only cause 1 quarter. So, the answer is yes. So so. 194 00:23:16.949 --> 00:23:27.838 Um, because probability you see is a half also, um, because it's got 10 things in it. Um, 1234512345. 195 00:23:27.838 --> 00:23:32.098 Agency independent, and also be in C. 196 00:23:33.358 --> 00:23:38.939 Yes, why is the, why is it would change. 197 00:23:40.318 --> 00:23:44.548 Well, the definition of independence, if we've got 2 overlapping events. 198 00:23:44.548 --> 00:23:52.499 Is our independent the probability of a times the probability equals or probability they and be together. 199 00:23:54.328 --> 00:23:59.038 So, and, um. 200 00:24:00.628 --> 00:24:04.648 Cannot hear you here. 201 00:24:04.648 --> 00:24:11.308 I stopped working through that, but that's the start of how that would work. 202 00:24:11.308 --> 00:24:16.019 It she, she should include these sorts of things so. 203 00:24:16.019 --> 00:24:22.348 Well, you make it up, so. 204 00:24:22.348 --> 00:24:26.249 Um, okay, well, what there is. 205 00:24:26.249 --> 00:24:30.269 Okay, something you want to put a bookmark or in the textbook. 206 00:24:30.269 --> 00:24:35.729 There's 2 pages on discrete distributions and pages on continuous distributions. 207 00:24:35.729 --> 00:24:43.409 And you want to put them on your, um, put a note for them, and maybe put them on your cheat sheet. So. 208 00:24:44.429 --> 00:24:53.338 Um, well, let me show you then, since you brought that up the 2nd here. 209 00:24:59.759 --> 00:25:09.628 Um. 210 00:25:09.628 --> 00:25:18.298 Hello. 211 00:25:32.729 --> 00:25:36.419 Okay. 212 00:25:43.259 --> 00:25:48.209 The speed readers in the audience. 213 00:26:38.759 --> 00:26:46.378 Okay, cable 3.1, that's in the textbook on page. We'll. 214 00:26:46.378 --> 00:26:52.588 115 and 116 is an important table you want a bookmark and make a copy so. 215 00:26:52.588 --> 00:26:58.288 So, it lists your common, random variables. These are discreet and, um. 216 00:26:58.288 --> 00:27:01.648 Means and variances and so on. 217 00:27:03.419 --> 00:27:08.249 So that is page 115. 218 00:27:09.538 --> 00:27:13.979 All right, remind me, I'll write it back on the blog when I switch back to the blog so. 219 00:27:15.118 --> 00:27:22.828 Uh, let me put it down now. Let me just put aside here, come on. 220 00:27:36.989 --> 00:27:42.479 Is important. Okay. 221 00:27:44.098 --> 00:27:49.739 Um. 222 00:27:54.118 --> 00:28:02.249 Let it go. Um, okay, so for geometric, um. 223 00:28:03.929 --> 00:28:11.459 Geometric had to versions here, depending on if you count. Um, the 2nd version is the 1 that I. 224 00:28:11.459 --> 00:28:17.098 Was using so it's the probability of, um, success. 225 00:28:17.098 --> 00:28:24.179 On the case attempt K counting from 1. so you see the expected values 1 over, repeat on the variance. 226 00:28:24.179 --> 00:28:29.189 You write it down, we went over it, so that would give you that for that exam question. So. 227 00:28:30.624 --> 00:28:44.213 Okay, um, we haven't seen generating functions. Yeah but now there's another table for continuous random variables that make find that since we're at it. I think it's about page 135, give or take, um. 228 00:28:45.719 --> 00:28:54.269 If I go through this the more, um, we've seen by a negative binomial, we haven't seen, um. 229 00:28:54.269 --> 00:29:01.409 Maybe I should talk about that. Um, Hassan, we've seen uniform we've seen, is this. 230 00:29:01.409 --> 00:29:06.959 Yeah, I don't know it gets the popular thing. I don't like it as much, so. 231 00:29:09.568 --> 00:29:15.659 The next thing is the continuous to speed reading here. 232 00:29:28.169 --> 00:29:35.368 Hello. 233 00:29:41.788 --> 00:29:51.598 Okay, this is a table 4.1 on page 164. 234 00:29:52.104 --> 00:30:06.864 Um, write this down. Okay. Also. 235 00:30:07.769 --> 00:30:12.598 Okay, um, this would be. 236 00:30:14.699 --> 00:30:20.368 Should be discreet continues. 237 00:30:22.409 --> 00:30:29.429 Okay, so these are, if I go back to that table, um. 238 00:30:31.199 --> 00:30:34.588 So, we seen uniform to smooth, um. 239 00:30:34.588 --> 00:30:49.528 I haven't worked out the variance means obvious and generating function exponential. We've seen so if you have like, crosstalk process, this would be the random variables the time between adjacent events between adjacent radioactive. 240 00:30:49.528 --> 00:30:53.729 D. K. Gaussian we saw and again. 241 00:30:55.199 --> 00:30:59.278 Most of all the other distributions that are reasonable. 242 00:30:59.278 --> 00:31:06.778 And then gets larger, and the limit look like a Gaussian and that limit often happens very quickly like an equals 5 and. 243 00:31:06.778 --> 00:31:19.288 Some cases, so, um, also normal or Gaussian called Gaussian cause Carl goes switch the mathematician who invented it and it's called normal because it's used so much. 244 00:31:19.288 --> 00:31:24.358 Gamma may talk about the flash and I don't know rally. 245 00:31:24.358 --> 00:31:33.209 Talk about these, um, the koshi random variable is notable, because it can easily occur. 246 00:31:33.209 --> 00:31:38.788 And but it does not have a mean, and it does not have a variance. 247 00:31:39.594 --> 00:31:51.354 What, I mean, when I say that, it does not have a mean is that if you apply the formula, it's integrate X times, half of X X sequence minus if any to infinity effects, it's a density function. 248 00:31:52.013 --> 00:31:57.084 You do that for the koshi then the integral diverges. The article does not exist. 249 00:31:57.778 --> 00:32:01.348 And Ditto for any moment mean and so on. So. 250 00:32:01.348 --> 00:32:11.068 But it is a perfectly reasonable, random variable. I mean, are you, in a minute how it happens how it can happen? So. 251 00:32:11.068 --> 00:32:20.308 So this is a case where the mathematics breaks down, you might say so you kind of talk about things, like means and stuff like that. Um. 252 00:32:20.308 --> 00:32:23.818 It's relevant to the real world. 253 00:32:23.818 --> 00:32:29.308 Because many people think that random variables for the stock market sometimes. 254 00:32:29.308 --> 00:32:35.729 I would like the koshi random variable to her just log distributions of the. 255 00:32:36.959 --> 00:32:40.229 Of variables and so on Saturday or whatever. 256 00:32:40.229 --> 00:32:43.259 And if these people are correct. 257 00:32:43.259 --> 00:32:51.328 Then none of your standard techniques for analyzing the stock market are valid. They are all mathematically invalid. 258 00:32:51.328 --> 00:32:56.159 Um, now that is very interesting because people use these techniques. 259 00:32:56.159 --> 00:32:59.429 Um, so. 260 00:32:59.429 --> 00:33:05.489 They use these techniques, as you say, why would we, you know, what else would we use? So. 261 00:33:06.628 --> 00:33:14.548 And if you like to look at the stock market, somewhat, when you should be studying, and you see people talk about. 262 00:33:14.548 --> 00:33:18.028 Black swans and abnormally large. 263 00:33:18.028 --> 00:33:21.328 Jumps in values and so on. 264 00:33:21.328 --> 00:33:26.098 That is an informal way of saying that the mathematical techniques failed. 265 00:33:26.098 --> 00:33:30.929 So, um, or they work until they don't work. 266 00:33:30.929 --> 00:33:35.489 And people like, say, Jamie diamond, who's the head of Chase bank? 267 00:33:35.489 --> 00:33:43.229 Has talked about stuff like this so this is applications in the real world where the nice mathematical formulas. 268 00:33:43.229 --> 00:33:47.278 Possibly don't work, so. 269 00:33:47.278 --> 00:33:52.078 But okay, any case so, this is the table 4.1. 270 00:33:52.078 --> 00:33:56.548 Uh, notable, random variables. We'll see someone's all so. 271 00:33:56.548 --> 00:34:07.828 Okay, um, questions on new fables since I mentioned, Kelsey, let me go back to that. So, this is in the real probability in the real world. 272 00:34:10.438 --> 00:34:15.688 Okay, um, sure. 273 00:34:21.059 --> 00:34:24.358 So, let me just show you what it is again. Um. 274 00:34:26.309 --> 00:34:29.938 koshi is. 275 00:34:31.349 --> 00:34:39.599 Do we have it here? 1 over 100+X squared. Okay. Um. 276 00:34:54.869 --> 00:35:01.438 Okay, um, so when it. 277 00:35:01.438 --> 00:35:06.599 Happens, um, but of how it happens um. 278 00:35:16.559 --> 00:35:20.068 It might have been a point here. 279 00:35:22.139 --> 00:35:27.389 At a point to do to do. Okay. 280 00:35:27.389 --> 00:35:34.378 And, um, and what we do is that. 281 00:35:34.378 --> 00:35:39.659 The access here and we look at where. 282 00:35:39.659 --> 00:35:43.018 It hits the X access. 283 00:35:49.829 --> 00:35:53.009 See access. Okay. 284 00:35:53.009 --> 00:35:56.699 Maybe the point here is perhaps centered at, um. 285 00:36:01.228 --> 00:36:11.878 Standard, I'd say 0 over in 1 up, like here. Okay look at where it gets the X axis and the pointer is uniform. Okay. So that is, um. 286 00:36:11.878 --> 00:36:20.759 That value of X there so where that's the random variable. Okay. 287 00:36:22.889 --> 00:36:25.978 And that's the probability distribution up there. 288 00:36:25.978 --> 00:36:37.318 Now, um, so it's, you know, perfectly legitimate physical experiment that K who's probably density function. 289 00:36:37.318 --> 00:36:41.338 On the outcome of this experiment is what's called. 290 00:36:41.338 --> 00:36:44.728 And you could integrate at 100 dollars. 291 00:36:44.728 --> 00:36:49.259 Dx equals 1. that's okay. Um. 292 00:36:50.369 --> 00:36:55.498 The problem is that if we try to compute oh and if I plot the thing. 293 00:36:55.498 --> 00:37:01.798 Um, if I plot it, it looks something like this. Um. 294 00:37:04.768 --> 00:37:14.338 X, and then the but the tails are wider the tail. 295 00:37:14.338 --> 00:37:18.418 Are thicker than, um. 296 00:37:18.418 --> 00:37:22.528 And for the go see, okay. 297 00:37:25.378 --> 00:37:31.648 Okay, it doesn't go to 0 as fast. Um, but the problem is the expected value of X. 298 00:37:31.648 --> 00:37:36.119 Which is defined as the undergrowth X. Equifax. 299 00:37:36.119 --> 00:37:40.798 Dx over pie the. 300 00:37:40.798 --> 00:37:46.679 Square DX not. 301 00:37:49.378 --> 00:37:54.389 You cannot compute the interval doesn't go to 0, fast enough. So. 302 00:37:55.768 --> 00:38:06.329 I can does not exist. So what this means, um, apart from mathematics. 303 00:38:06.329 --> 00:38:13.739 Is that so, let me give you what a practical meaning of expected value? Um. 304 00:38:15.929 --> 00:38:23.039 So, generally if you sample. 305 00:38:24.059 --> 00:38:28.349 So, say, sample whatever outcomes. 306 00:38:29.460 --> 00:38:36.179 From some random variable and the samples thing and outcomes. 307 00:38:37.829 --> 00:38:42.119 And maybe outcomes X1 X2 X up to N. 308 00:38:43.500 --> 00:38:51.329 Well, and then you compute the sample mean so of all. 309 00:38:51.329 --> 00:38:55.260 I equals 1 and divided by N. 310 00:38:55.260 --> 00:39:02.130 This limit here, the sample means starts converging. 311 00:39:06.510 --> 00:39:13.050 I'm getting into statistics now, but this is there's some content in what I'm saying, which is why I'm spending some time on it. 312 00:39:13.050 --> 00:39:19.559 Um, so you're sampling, um. 313 00:39:19.559 --> 00:39:24.719 You're running some random experiment, you're sampling outcomes. Um. 314 00:39:25.769 --> 00:39:30.030 Whatever you're looking at the stock market averages day by day, perhaps. 315 00:39:30.030 --> 00:39:37.860 And each day is a different experiment, and you take it over many days and you get the average value or something, or we sample temperatures or. 316 00:39:37.860 --> 00:39:43.679 Whatever we sample koshi variable so, as you take a larger and larger sample. 317 00:39:45.030 --> 00:39:58.710 Which means more and more random numbers, and you take the average of those and random numbers as then gets larger this what's called the sample mean? It settles down. It stops bouncing around. 318 00:39:58.710 --> 00:40:02.550 Or we talk, let's say we talk the coin. 319 00:40:02.550 --> 00:40:08.909 If we and the random variables there, we're 1 for a tails your heads. 320 00:40:08.909 --> 00:40:17.610 As we tossed the coin more times and look at the average number of heads we got in our sample. 321 00:40:17.610 --> 00:40:22.980 As we tossed the coin more, the average the number of heads in the sample. 322 00:40:22.980 --> 00:40:37.980 Settles down around 50% and the percentage different from 50% shrinks as then gross. I showed that to you. 2 classes of glow because the standard deviation is like the spread around the expected value. 323 00:40:37.980 --> 00:40:42.000 Okay, so that's no, that's. 324 00:40:42.000 --> 00:40:53.909 So, both random variables do that the Cosi distribution does not do that. If I sample numbers from the calcium distribution and I take the mean of the sample. 325 00:40:53.909 --> 00:40:59.849 Even as an growth that sample mean continues to bounce around. 326 00:40:59.849 --> 00:41:05.010 It does not shrink down and converge on 1 point as end gets larger. 327 00:41:05.010 --> 00:41:10.110 So, that's what the practical meaning of the expected value does not. 328 00:41:10.110 --> 00:41:19.619 Exist for the calcium, so let me maybe I'm going to write this down because this is an important point here. Like I said, that's. 329 00:41:19.619 --> 00:41:23.940 Well, I am doing this divergence into a new area, so I'm. 330 00:41:25.289 --> 00:41:31.019 So, use your Wally let let me talk about the meaning of. 331 00:41:38.070 --> 00:41:43.409 Okay, for most distribute probability distributions. 332 00:41:53.519 --> 00:42:01.079 You can take a sample and. 333 00:42:01.079 --> 00:42:05.789 Random variables and compute. 334 00:42:08.039 --> 00:42:15.929 The sample mean and, um. 335 00:42:18.780 --> 00:42:23.909 So, let's say, I don't know, or I can't my way. 336 00:42:23.909 --> 00:42:27.329 And, um, as end grows. 337 00:42:29.130 --> 00:42:37.289 A sample mean just to settle down. 338 00:42:38.550 --> 00:42:43.590 By which, I mean, it stopped bouncing, um, E. G. 339 00:42:45.719 --> 00:42:49.800 Cost the coin and times. 340 00:42:51.510 --> 00:42:58.590 Then, um, basically coin, um, the sample mean is, um. 341 00:42:58.590 --> 00:43:05.190 You know, it's, you know, 51.5+or . 342 00:43:05.190 --> 00:43:10.889 The standard deviation 2 thirds of the time the standard deviation is, um. 343 00:43:10.889 --> 00:43:15.900 Square root of Van over 22 thirds of the time. Okay. 344 00:43:15.900 --> 00:43:19.769 So, it's it settles down. Okay. 345 00:43:19.769 --> 00:43:25.320 The, the, um, the fractional spread gets less and then gets larger. 346 00:43:26.789 --> 00:43:29.909 Not true for the calcium. 347 00:43:35.880 --> 00:43:47.219 Domain of a large sample you can find the meat of a sample, even with the meat of distribution of the samples, a finite number numbers you can find to me. 348 00:43:47.219 --> 00:43:53.579 So, the meaning of a large sample does not settled down. 349 00:43:58.860 --> 00:44:03.210 Yeah, and so in mathematical terms, expected, value of X does not exist. 350 00:44:05.070 --> 00:44:09.329 The 2nd value for the just to be I'm not either. 351 00:44:09.329 --> 00:44:12.329 A variance and so on. 352 00:44:13.769 --> 00:44:21.300 Or any other distribution, so limits of mass, the limits of all these beautiful mathematical techniques. So so. 353 00:44:22.769 --> 00:44:28.409 Um. 354 00:44:28.409 --> 00:44:36.150 Real world is people use fancy math to analyze, for example, stock options and price them and in the world of it. 355 00:44:36.150 --> 00:44:44.789 In economics, or kinda metric um, there was a model developed a few a few decades ago called the black shoals model. 356 00:44:44.789 --> 00:44:53.340 For assigning a fair price to an option for an option would be say the right to buy Tesla in the next year for 1000 dollars. 357 00:44:53.340 --> 00:44:56.789 Test this 1 now, 800 bucks or some things. So. 358 00:44:56.789 --> 00:45:06.809 If the value of tested in 2 years is under a 1000 dollars, your option is worth list. That's the value of Tesla 2 years at 1100 dollars. Your options worth 100 dollars and so on. 359 00:45:06.809 --> 00:45:12.690 I'm going to simplify if you'd like to, you know, you'd like to find expected value for it. 360 00:45:12.690 --> 00:45:15.809 The options, so they develop this technique called black. 361 00:45:15.809 --> 00:45:27.179 And then built an economics company out of it and 1, or 2 of the people involved, got Nobel Prize in economics and so on for this. So, it was very. 362 00:45:27.179 --> 00:45:35.730 Famous thing, but the trouble is that this formula underneath all the beautiful mathematics had a couple of assumptions. 363 00:45:35.730 --> 00:45:44.489 1 of which was that this underlying probability distributions in fact, had moments like I'm showing right here. 364 00:45:44.489 --> 00:45:55.409 And I said some people think the stock market does not and it also assumed that there was a fair price at which people would be willing to buy or sell equally. And if the market is crashing. 365 00:45:55.409 --> 00:46:01.619 No, 1 wants to buy so they developed this formula that they built it into. 366 00:46:01.619 --> 00:46:09.840 The hedge fund and so on, and then a dozen years ago, it's all fell apart and then he took the economy down with them. So. 367 00:46:09.840 --> 00:46:13.469 The beautiful math was useful until it failed. 368 00:46:13.469 --> 00:46:16.860 Oh, and, um. 369 00:46:16.860 --> 00:46:21.449 Let me write down, so let's see. 370 00:46:21.449 --> 00:46:28.559 Um, so. 371 00:46:30.690 --> 00:46:34.980 In the real world, maybe. 372 00:46:38.340 --> 00:46:43.380 Some, maybe some economics models. 373 00:46:45.420 --> 00:46:53.579 Also are invalid and let me show you, um. 374 00:46:58.045 --> 00:47:10.764 Okay. 375 00:47:23.190 --> 00:47:29.039 Yeah, um. 376 00:47:31.559 --> 00:47:46.289 So so you see, the board of long term capital management is the 1 to put a director to 2 Nobel Prize in economics people on it. They develop the black shoals model. I just told you about. 377 00:47:46.289 --> 00:47:50.969 Um, let's see or pricing options. 378 00:47:50.969 --> 00:47:54.329 Very good. Very nicely for a few years. 379 00:47:55.469 --> 00:48:02.940 And then collapsed and needed to be bailed out by the federal government, which would be. 380 00:48:02.940 --> 00:48:06.150 You and me the taxpayers, so. 381 00:48:06.150 --> 00:48:13.590 So, you know, it worked until it didn't, I'd say, okay um. 382 00:48:14.760 --> 00:48:22.199 Yeah, riskier investments and analysis. Um, yeah. 383 00:48:23.610 --> 00:48:31.860 They underestimated and over us, so there are other things I could have done better analysis and so on. But their underlying math was so. 384 00:48:49.320 --> 00:48:56.760 Okay, the reason I tell you this look, you guys are your students who your smart people you'll figure out the math. 385 00:48:56.760 --> 00:49:06.750 Most of you, but once you figure out the math, then the next important problem is which math is relevant to the problem you're trying to solve. 386 00:49:06.750 --> 00:49:10.619 And you can have math, which is very beautiful. 387 00:49:10.619 --> 00:49:13.710 It's just wrong. 388 00:49:13.710 --> 00:49:20.489 Okay, so so if I go back to the, uh, um. 389 00:49:22.260 --> 00:49:26.429 Back to the, um, 2020. 390 00:49:27.449 --> 00:49:33.000 There's more questions on that, um, real world stuff up after the 2020. 391 00:49:33.000 --> 00:49:37.650 Thanks Sam. 1. um. 392 00:49:37.650 --> 00:49:48.659 So, I had these 3 sets, I'm testing the I cost the Hydro dye with 20 faces. 393 00:49:48.659 --> 00:49:52.980 And the question was. 394 00:49:52.980 --> 00:49:56.039 Each pair of events was independent. 395 00:49:57.119 --> 00:50:01.619 So event a was asked equal to attend he was on. 396 00:50:01.619 --> 00:50:05.670 And see, what's this complicated thing I'm not going to rewrite. 397 00:50:05.670 --> 00:50:19.199 The probability of a was 1 half, because it's 1020, was the probability of B to C and B are independent cause a, and B would be odd numbers up to 10 and there were 5 of them and so on. 398 00:50:19.199 --> 00:50:27.480 Um, but a B and C the set a intersect B intersect C. 399 00:50:27.480 --> 00:50:33.570 But that was, would be or not, um, would be the empty set. In fact. 400 00:50:35.610 --> 00:50:44.550 He said, so probably did a intersect. Be interesting to see would be 0, which was not equal to the probability of, hey, probably it would be. 401 00:50:44.550 --> 00:50:51.719 Oh, yes, so the 3 events, those would be 1 of the 3 sets in is giving. 402 00:50:51.719 --> 00:50:56.940 Comes from the events, any pair of the. 403 00:50:56.940 --> 00:51:00.659 Events was independent, but all 3. 404 00:51:00.659 --> 00:51:04.409 We're dependent now, what does this mean in English? 405 00:51:04.409 --> 00:51:08.070 If you knew that event a had happened. 406 00:51:08.070 --> 00:51:13.619 That did not tell you anything about whether it be event B, had happened or not. 407 00:51:13.619 --> 00:51:27.300 If you knew that it happened, it did not tell you anything about whether C had happened or not. The probability was the same. However, if you knew that event a, and event B, had both happened. 408 00:51:27.300 --> 00:51:30.750 Then that would tell you something about. 409 00:51:30.750 --> 00:51:34.079 Whether or not C had happened. 410 00:51:34.079 --> 00:51:39.449 In fact, if you knew that event a, and event B had both happened. 411 00:51:39.449 --> 00:51:44.340 Then, what you would know is that C could not have happened. 412 00:51:44.340 --> 00:51:52.590 So, knowing a, or knowing be told you nothing about C, knowing a, and B, together told you everything about C. 413 00:51:52.590 --> 00:51:57.480 That's what this 3, this complicated form of independence. 414 00:51:57.480 --> 00:52:00.929 Let me, let me write this down. Cause this is an important thing. 415 00:52:00.929 --> 00:52:04.590 Um, it's again, that's why I'm spending time on the review. 416 00:52:04.590 --> 00:52:12.179 Because these are important points so knowing a. 417 00:52:13.590 --> 00:52:16.590 Tells you nothing. 418 00:52:18.360 --> 00:52:22.590 About see about whether it happened, knowing bee. 419 00:52:22.590 --> 00:52:29.190 It's nothing by knowing, I mean, knowing that it happened okay. 420 00:52:31.590 --> 00:52:34.980 Knowing that both. 421 00:52:34.980 --> 00:52:38.820 A, and B be. 422 00:52:38.820 --> 00:52:45.960 Happened tells you that. 423 00:52:47.699 --> 00:52:59.280 He is impossible in fact, so she is not independent. So the 3 of them, a B, and C together are dependent on each other. 424 00:52:59.280 --> 00:53:03.329 You see this complicated things that can go on. 425 00:53:06.510 --> 00:53:11.550 Yes, all of you. 426 00:53:14.340 --> 00:53:25.380 Nothing, let me rewrite that. 427 00:53:25.380 --> 00:53:28.889 Oh. 428 00:53:30.119 --> 00:53:33.150 Because see is a 50, 50 chance of happening. 429 00:53:33.150 --> 00:53:37.619 10 out comes out as 20 possible for the pace of the day. 430 00:53:39.179 --> 00:53:42.630 If you know, that a has happened. 431 00:53:44.489 --> 00:53:48.360 Are still see is 50, 50, so. 432 00:53:48.360 --> 00:53:56.940 See versus not C, so see, given a, is a half not see given a, is still a half. So. 433 00:54:07.710 --> 00:54:16.170 1 other questions. 434 00:54:20.519 --> 00:54:26.340 Okay. 435 00:54:28.079 --> 00:54:32.280 And there's more similar, um, examples here. So. 436 00:54:32.280 --> 00:54:36.960 These are base rules, sorts of things. Um. 437 00:54:36.960 --> 00:54:45.630 Phase and there's more based type questions um. 438 00:54:47.699 --> 00:54:55.650 Oh, okay. Let's do another 1. um, so this is a display. 439 00:54:58.860 --> 00:55:02.610 With Excel. 440 00:55:03.630 --> 00:55:11.400 Okay, um, now you buy a display display some more reliable now. 441 00:55:11.400 --> 00:55:16.380 At 1 time, um, the, the seller said that, um. 442 00:55:16.380 --> 00:55:19.980 No, I was in the manufacturer would say. 443 00:55:27.630 --> 00:55:35.969 To find a good display as say. 444 00:55:35.969 --> 00:55:39.570 Things with 10 bad pixels. 445 00:55:39.570 --> 00:55:48.989 Okay, in other words, if they delivered a display to you and it had 5 bad pixels, you they would not accept a return. 446 00:55:48.989 --> 00:55:55.110 If they delivered it display that had 15 bad pixels, they would accept a return a warranty return. For example. 447 00:55:55.110 --> 00:55:59.070 And let's say the probability of. 448 00:56:00.690 --> 00:56:12.900 A, given pixel being bad, let's say was, I don't know um. 449 00:56:15.659 --> 00:56:19.829 And to the -6, let's say, okay, so. 450 00:56:22.019 --> 00:56:28.050 And let's assume the pixels are independent, um, which I always have to say, because in the real world. 451 00:56:28.050 --> 00:56:33.929 Um, it might not be true. Okay. Um. 452 00:56:35.519 --> 00:56:39.510 You make it a bad run. The production line was contaminated. 453 00:56:39.510 --> 00:56:45.179 Or, maybe if a pixel is, I mean, pixels are addressed and rows and columns, just like, you know. 454 00:56:45.179 --> 00:56:59.130 Some types of memory, so it's possible for pixel some sort of row control, or could be bad. So there might be many bad pixels on that role perhaps but in any case independent so, the 1st question is. 455 00:56:59.130 --> 00:57:02.730 What's the, um, what's the appropriate. 456 00:57:04.980 --> 00:57:13.079 Probability distribution and what would you say. 457 00:57:14.429 --> 00:57:18.989 Did you talk? Yes correct. 458 00:57:18.989 --> 00:57:25.889 That's the right 1 to use. You could you could use binomial, but the numbers are horrible. 459 00:57:25.889 --> 00:57:28.889 And you could maybe use normal. 460 00:57:28.889 --> 00:57:34.739 But, um, it might not be so good for these numbers plus on is right 1. okay. 461 00:57:34.739 --> 00:57:42.119 Um, you know, so now you wanna say the probability. 462 00:57:43.650 --> 00:57:48.210 You know okay, bad. Well, 1st, um. 463 00:57:48.210 --> 00:57:51.539 For is the problem. So the probably the 1, but. 464 00:57:51.539 --> 00:57:55.829 It's probably a K, bad using binomial would be like, you know, what? 465 00:57:55.829 --> 00:58:10.170 And equals 4Million, so you'd have and 4,000,002 something that's 4Million factorial and all that stuff. And then you would have P, which is 10 to -6. 466 00:58:10.170 --> 00:58:17.280 Um, so he was kind of the -6. so now you've got Peter the K. 467 00:58:19.590 --> 00:58:27.030 So, 10 of the -62. okay, well, the thing is 1-if you had to evaluate this on a computer. It's the 1-P. that's. 468 00:58:27.030 --> 00:58:32.730 Point 999999, you're taking it to a very large number like 3,000,000. 469 00:58:32.730 --> 00:58:36.360 999,995 or something. 470 00:58:36.360 --> 00:58:42.570 And so this would be not so fun to evaluate. So you want to use crosstalk. 471 00:58:42.570 --> 00:58:45.869 Um, let's say plus on. 472 00:58:47.340 --> 00:58:50.730 And the average number is. 473 00:58:50.730 --> 00:58:55.530 There's 4Million pixels probably with any 1 being bad is 1 in a 1,000,000. 474 00:58:55.530 --> 00:58:59.429 So the average number of bad pixels is 4. okay. 475 00:58:59.429 --> 00:59:03.210 So the expected number of K was full, it would be 4. 476 00:59:03.210 --> 00:59:06.480 The variance would also be for. 477 00:59:07.590 --> 00:59:13.139 The standard deviation segments 2 so basically, 2 thirds of the time. 478 00:59:16.679 --> 00:59:22.619 You know, we have 226 bad pixels. Okay. 479 00:59:25.230 --> 00:59:35.579 Okay, um, this is an approximation. Now it's, you're getting to the point where normal would actually work, but maybe you're not quite there. So. 480 00:59:36.630 --> 00:59:40.349 Um, okay, so. 481 00:59:43.139 --> 00:59:46.469 So the formula for K, bad pixels that being. 482 00:59:46.469 --> 00:59:53.940 Hassan is, um, okay bad pixels is. 483 00:59:57.000 --> 01:00:01.590 A K over K factorial unit, the minus alpha. 484 01:00:01.590 --> 01:00:09.929 Okay. Um, so the probability is 0, bad pixels. 485 01:00:09.929 --> 01:00:14.579 Would be 1 over 1 either the . 486 01:00:14.579 --> 01:00:18.989 For whatever that happens to be, so. 487 01:00:18.989 --> 01:00:22.050 Okay. 488 01:00:25.739 --> 01:00:31.079 And. 489 01:00:31.079 --> 01:00:34.829 What fraction of displays are acceptable. 490 01:00:45.510 --> 01:00:53.400 Are acceptable well, that would be the sum of all. 491 01:00:54.420 --> 01:01:01.800 Okay, equal to 0 to 10. um, if we say up to 10 are acceptable so. 492 01:01:05.190 --> 01:01:08.730 Okay. 493 01:01:08.730 --> 01:01:12.840 So, whatever you do, the minus for some. 494 01:01:18.750 --> 01:01:24.539 And if you had this on an exam, I would not expect you to use calculators. We could stop right there. 495 01:01:24.539 --> 01:01:30.119 If you work out something to the point where you need a calculator and write that down there. 496 01:01:31.139 --> 01:01:38.909 Or I might make a multiple choice or something. Okay. 497 01:01:40.500 --> 01:01:45.989 More questions about test, so. 498 01:01:51.030 --> 01:02:01.050 Okay, um. 499 01:02:10.019 --> 01:02:14.340 Okay, so if I go back to chapter 4 for a minute. 500 01:02:18.900 --> 01:02:26.190 This is a quick reminder. Um, also called normal. 501 01:02:30.840 --> 01:02:37.829 And, um, um. 502 01:02:44.969 --> 01:02:49.079 Mine is. Okay. Okay. 503 01:02:49.079 --> 01:02:53.369 That's the PDF. That's the density function. 504 01:02:55.500 --> 01:03:00.179 Also called the PDF the. 505 01:03:00.179 --> 01:03:04.230 It'd be good to go from minus infinity up to access of acts. 506 01:03:05.280 --> 01:03:08.820 Um, so. 507 01:03:08.820 --> 01:03:12.420 And that is, um, sometimes called. 508 01:03:12.420 --> 01:03:17.369 5 X for the go see, em, so side effects. 509 01:03:17.369 --> 01:03:21.420 It goes the undergo minus infinity, 2 X. um. 510 01:03:25.829 --> 01:03:30.090 Okay, um. 511 01:03:33.059 --> 01:03:36.809 And there's the compliment of that for some reason. 512 01:03:36.809 --> 01:03:42.300 And 2 called Q1-side effects, so. 513 01:03:42.300 --> 01:03:51.960 Um, so that's the right tale for the calcium. 514 01:03:54.480 --> 01:04:00.300 And, um. 515 01:04:00.300 --> 01:04:05.639 Some random numbers I can show you um. 516 01:04:05.639 --> 01:04:17.280 Discount per minute, by the way for the last few times recording the video and. 517 01:04:17.280 --> 01:04:22.019 By handwritten notes, they've worked out, so I've uploaded them to the blog. 518 01:04:27.210 --> 01:04:30.239 Okay, so to show you a few of these things. 519 01:04:30.239 --> 01:04:34.289 Um, so this is. 520 01:04:36.599 --> 01:04:39.840 Okay, so, um. 521 01:04:41.130 --> 01:04:45.750 To 0, is this this 1 half. 522 01:04:45.750 --> 01:04:56.519 Um, 1 would be this. I saw 0, it's just under the display under display like that. Okay. 523 01:04:56.519 --> 01:05:05.309 To 106, um. 524 01:05:05.309 --> 01:05:10.019 Go to I've got 2 back here. 525 01:05:12.510 --> 01:05:16.260 It's the right tab. It's about 2%. 526 01:05:16.260 --> 01:05:20.639 And Q3, um, up here. 527 01:05:22.079 --> 01:05:25.530 There's 1 in 1000 so. 528 01:05:25.530 --> 01:05:30.780 And now you can also. 529 01:05:33.329 --> 01:05:36.539 Now, you can start using this, um. 530 01:05:36.539 --> 01:05:40.769 This is this now this is for meaningful 0 Sigma equals 1. 531 01:05:40.769 --> 01:05:44.639 And you can transform for other means. 532 01:05:48.119 --> 01:05:52.320 For other things yeah, 3rd there. So. 533 01:05:52.320 --> 01:05:58.139 So suppose you supposedly me 500. 534 01:05:58.139 --> 01:06:06.539 500, um, and Sigma equals 100. 535 01:06:07.590 --> 01:06:16.949 And suppose we want probabilty that X is greater or equal to 400 well greater for 500. 536 01:06:16.949 --> 01:06:23.130 0 equals 1, half probability expert equal to 400. 537 01:06:24.269 --> 01:06:36.239 That's, um, to -1, which is 1-Q 1, cause everything's symmetric. And that will be a half +2 6 5, 6. 538 01:06:36.239 --> 01:06:41.550 And so on, um. 539 01:06:41.550 --> 01:06:46.289 Um, 400. 540 01:06:46.289 --> 01:06:53.699 Like, 50,400, this whole thing is 1, half this chunk here. 541 01:06:55.230 --> 01:06:59.340 Is a 3rd, uh. 542 01:07:01.679 --> 01:07:10.409 So, it will be 56, I think whatever. 543 01:07:11.849 --> 01:07:15.389 My mind is mine completely something like that. So. 544 01:07:18.360 --> 01:07:23.039 Yes, so it's new minus Sigma. 545 01:07:24.210 --> 01:07:28.199 Oh, um. 546 01:07:28.199 --> 01:07:39.389 I think that's right. Sorry. Okay. 547 01:07:41.280 --> 01:07:46.619 Um, so with that. 548 01:07:56.519 --> 01:08:00.690 Okay, I just this, this is a review actually, but I wanted to get back on. 549 01:08:00.690 --> 01:08:07.260 Crack a little so, s, a. T scores are sort of like that. So. 550 01:08:08.369 --> 01:08:15.360 Okay, and for the density function. 551 01:08:15.360 --> 01:08:20.970 Um, well, it would be, um. 552 01:08:20.970 --> 01:08:29.189 Given some you in Sigma, I mentioned this last that 1 over here too high. 553 01:08:29.189 --> 01:08:35.010 Sigma is the minus X minus from you. 554 01:08:35.010 --> 01:08:41.819 Uh, squared 1. 555 01:08:41.819 --> 01:08:45.090 Okay. 556 01:08:45.090 --> 01:08:48.869 Application of that. 557 01:08:51.510 --> 01:08:54.689 Noisy channel. 558 01:08:55.770 --> 01:09:01.619 You're going to get sick of noisy communication channels cause I'm gonna use them so many times in this course. 559 01:09:01.619 --> 01:09:05.250 Because they're important, um, you transmit. 560 01:09:05.250 --> 01:09:13.739 0, or 1 probability of a 0 stays 1. half probably is. 561 01:09:13.739 --> 01:09:24.359 I make a Pi. Okay. Um. 562 01:09:24.359 --> 01:09:30.149 So, it's a transmitted signal is acts call it X. then you add noise. 563 01:09:33.149 --> 01:09:37.710 And and the noise. 564 01:09:37.710 --> 01:09:51.630 It's go see and equals 0 Sigma equals 1 and 1 way. This is representative, we say, end. 01. okay. And then the received signal. 565 01:09:55.590 --> 01:10:00.300 Y, equals X + and okay. Um. 566 01:10:00.300 --> 01:10:09.239 Track down here. Okay so. 567 01:10:09.239 --> 01:10:13.979 So, what happens is you get a transmitted signal. 568 01:10:15.989 --> 01:10:23.489 0 or 1, and then it gets blurred by the noise here. Okay. So. 569 01:10:24.750 --> 01:10:29.970 The transmit is signals that then receipt signal's going to be something like that. 570 01:10:29.970 --> 01:10:35.460 The transmitted signal is this he received signal is something like. 571 01:10:35.460 --> 01:10:42.149 Okay, totally real world. Now here's the issue. 572 01:10:42.149 --> 01:10:49.020 Is that suppose and we saw this before it's discreet things. Here's the thing with continuous things. 573 01:10:49.020 --> 01:10:56.550 Is suppose that suppose this is what you receive over here. Okay. 574 01:10:56.550 --> 01:11:06.000 Way high up then. Okay it was the 1 that was transmitted. If you receive something way down there you figured to 0 that was transmitted. 575 01:11:06.000 --> 01:11:12.000 You know, what, if you receive something like here it's a little over 1. 576 01:11:12.000 --> 01:11:19.470 Well, it was more likely to 1 was transmitted probably, but maybe it was a 0. okay. And of course, it's in the middle. 577 01:11:19.470 --> 01:11:27.899 It starts getting difficult now, it makes it, uh, as I mentioned this before, but it's important is what is. 578 01:11:27.899 --> 01:11:32.670 The 0 and 1 transmitted signals do not have equal probabilities. If. 579 01:11:32.670 --> 01:11:38.220 Um, let's suppose it was more likely that a 1 was transmitted. Now. This will. 580 01:11:38.220 --> 01:11:44.609 Bias what you should guess about the transmitted signal when you add the noise, but they were not equal probability at the start. 581 01:11:44.609 --> 01:11:52.770 So, um, so now we might want to start, um. 582 01:11:52.770 --> 01:11:58.350 Doing math on this, and this is an example from chapter 4 so you can read chapter 4. 583 01:11:58.350 --> 01:12:02.579 Um. 584 01:12:02.579 --> 01:12:05.729 So so what you want. 585 01:12:05.729 --> 01:12:09.960 Go back to black actually. 586 01:12:11.609 --> 01:12:15.899 So you want f, a. Y. A. 587 01:12:15.899 --> 01:12:24.149 1st thing you want, and then initially, and then later on what you want is then, um. 588 01:12:25.289 --> 01:12:31.319 You know, f, X given whatever, you know, given that. 589 01:12:33.029 --> 01:12:36.270 Why equals some particular value, you know. 590 01:12:36.270 --> 01:12:44.670 Some particular value of why let's say, and then you can use that to guess what was transmitted. So. 591 01:12:51.630 --> 01:13:02.579 I was trying to match it, finished the sales side. So now for so the 1st part is what's the probability density function for the receipt? 592 01:13:02.579 --> 01:13:12.359 Signal well, what we can do is that we can split it up. We can do total probabilities there. So it's half of why given that. 593 01:13:12.359 --> 01:13:18.210 The transmitted signal was 0 times the probability of 0. 594 01:13:18.210 --> 01:13:22.439 Plus given, um. 595 01:13:25.289 --> 01:13:36.210 Kinds of probability of vaccines. +1. Okay. Now we know these 2 probabilities here. Okay. Cause that would be P and that would be 1-P. 596 01:13:36.210 --> 01:13:39.329 This thing here. 597 01:13:42.720 --> 01:13:48.989 That is just, um, Congo. This is and this is just a normal 01 distribution. 598 01:13:48.989 --> 01:13:57.720 Because we're adding the noise. Okay. So this thing here f, of why given X equals 0 would be 1 or. 599 01:13:57.720 --> 01:14:00.869 To pie . 600 01:14:00.869 --> 01:14:06.539 Cool well, for some particular value Y, and so on. 601 01:14:08.010 --> 01:14:13.979 Um, f, of why give an X equals 1. 602 01:14:13.979 --> 01:14:18.420 Well, this is normal standard on 1, so that's going to be. 603 01:14:21.689 --> 01:14:26.159 --1 squared over 2. 604 01:14:27.569 --> 01:14:31.979 Because the fax equals 1, then, you know, then because Y equals X +1. 605 01:14:33.180 --> 01:14:36.659 Sorry, Y, equals, um. 606 01:14:38.670 --> 01:14:50.010 Y, was 1+and the normal error distribution so, you know, X plus M equals 1 x plus equals 1. 607 01:14:50.010 --> 01:14:55.319 So, and up here, it's X, + and equals 0+then. Okay. 608 01:14:57.090 --> 01:15:01.890 So, we can put these together and get the density function for Y. 609 01:15:01.890 --> 01:15:06.000 And then, I think I thought was I saying here P, for X equals 1. 610 01:15:06.000 --> 01:15:10.050 So, it's 1-P. okay. 611 01:15:13.680 --> 01:15:18.119 Whatever, you know, Y, squared over 2+P. 612 01:15:19.439 --> 01:15:22.859 Either minus Y, -1 square it over to. 613 01:15:24.300 --> 01:15:27.539 And that's the density function for the receipt signal. So. 614 01:15:27.539 --> 01:15:31.649 And now we can start playing base rules. 615 01:15:31.649 --> 01:15:34.739 Which I'll do on Thursday and get. 616 01:15:34.739 --> 01:15:42.779 Go backwards and good thing. So last chapter, we did this noisy channel with a discreet noise. 617 01:15:42.779 --> 01:15:48.060 Now, we're doing or more simple. Now we're doing noisy channel with a Gaussian noise. 618 01:15:50.819 --> 01:15:54.630 Yes. 619 01:15:55.739 --> 01:16:09.510 Well, okay, well, and okay, I made my, um, my notation may have been nasty. So, what we're doing is in is the Gaussian noise that we add to the transmitted signal. 620 01:16:11.489 --> 01:16:22.500 So, let me write that down. It's a random variable. Okay. 621 01:16:24.750 --> 01:16:32.369 Added to the transmitted signal. Um, okay also. 622 01:16:32.369 --> 01:16:35.460 And new Sigma. 623 01:16:35.460 --> 01:16:39.119 Is the and random variable. 624 01:16:42.420 --> 01:16:45.539 No, it has 2 different usage, so that's. 625 01:16:45.539 --> 01:16:49.140 Maybe the what could have done things better so. 626 01:16:49.140 --> 01:16:56.159 A shorthand for a Gaussian random variables to pick the new and segments and you and segment. So. 627 01:17:02.640 --> 01:17:05.970 You don't trust in my ability to upload the handwritten notes. 628 01:17:05.970 --> 01:17:09.659 You're smart. 629 01:17:09.659 --> 01:17:15.239 Yeah. Okay. Never trust technology. So. 630 01:17:17.010 --> 01:17:21.479 By the way somebody wants to be an on from her I said, I'm perfectly fine. 631 01:17:21.479 --> 01:17:25.319 With any of you making your own recording of me talking. 632 01:17:25.319 --> 01:17:31.770 In fact, you're welcome to bring your camera up to the front of the room and pointed at the screen and pointed at me. So. 633 01:17:31.770 --> 01:17:36.359 I have no problem with that at all. So, um. 634 01:17:36.359 --> 01:17:41.609 It'd be a copy maybe, but, uh, yes question. Um. 635 01:17:41.609 --> 01:17:45.239 Yeah, the combination 1 question what. 636 01:17:45.239 --> 01:17:49.229 4 propositions where they want permutation is automatic. 637 01:17:51.569 --> 01:17:56.039 I don't remember the question is a combination of to. 638 01:17:56.039 --> 01:17:59.909 How many does that mean? 639 01:17:59.909 --> 01:18:05.699 Or in combination, so should mean the technical term yeah. 640 01:18:05.699 --> 01:18:11.850 Of course, if they're all different, then it's just a factor of, you know, K, factorial. But. 641 01:18:11.850 --> 01:18:18.149 You show me the question so I. 642 01:18:18.149 --> 01:18:23.340 Don't remember the question the ta's actually set up the questions to some extent. 643 01:18:23.340 --> 01:18:31.529 I give them some input then they work out the details. Oh, they put up another homework but for some reason they've said not to release till tomorrow. So. 644 01:18:31.529 --> 01:18:39.390 Do in a week, so yeah. 645 01:18:43.829 --> 01:18:48.779 Well, there was a homework for what. 646 01:18:49.829 --> 01:18:55.199 I think there is some counts up, um. 647 01:18:56.609 --> 01:19:02.489 Well, which homework are we talking? 3 for? Sorry? 648 01:19:05.159 --> 01:19:08.460 No, um. 649 01:19:12.359 --> 01:19:17.310 Okay, um, just a 2nd, um. 650 01:19:17.310 --> 01:19:22.590 Um. 651 01:19:26.220 --> 01:19:32.579 Okay, um, what I'm doing is, I'm f***. Good. 652 01:19:50.880 --> 01:19:54.119 Well, I'm trying to plug in Thank you. Um. 653 01:19:55.949 --> 01:20:05.909 Okay um, 3. 654 01:20:07.289 --> 01:20:13.199 We should have been released on. Okay. I'll, I'll check into that then. 655 01:20:13.199 --> 01:20:16.350 They should have been released, so. 656 01:20:17.760 --> 01:20:23.010 I don't know, but you're right. No one's submitted though. Okay, I'll check that. Yeah. 657 01:20:25.649 --> 01:20:28.859 Okay, other questions I'll stay here. 658 01:20:28.859 --> 01:20:40.439 That was the other question? Yes. Show me. Where's the. 659 01:20:40.439 --> 01:20:44.130 Back. 660 01:20:51.630 --> 01:20:58.020 Oh, just a 2nd, um. 661 01:20:59.819 --> 01:21:04.529 Spring 2020, so if you go to my. 662 01:21:06.779 --> 01:21:16.380 Hello. 663 01:21:17.640 --> 01:21:22.350 Okay. 664 01:21:23.369 --> 01:21:28.439 Hello. 665 01:21:32.760 --> 01:21:41.039 Thank you. 666 01:21:41.039 --> 01:21:44.460 I appreciate about, uh, the exams, um. 667 01:21:44.460 --> 01:21:49.859 Yeah, is it gonna be open note or we're gonna have a quick sheet you can create your own encryption. 668 01:21:49.859 --> 01:21:53.069 So 1 page from the yeah. Okay. 669 01:21:53.069 --> 01:21:59.340 And a half by 11 inches, then you're welcome to mechanically produce it. It does not have to be handwritten. Okay. 670 01:21:59.340 --> 01:22:08.520 And you're welcome to form consortium, set up a business doing Greg sheets. Give me a copy if you do I'm curious, but yeah. 671 01:22:08.520 --> 01:22:20.430 Yeah, supervisor, thank you for the extension. I messaged you for an extension for Rossi. That's something for watching for a route to events. So I remember I messaged you for an extension. You gave it to me thank you. Professor. I did. 672 01:22:20.430 --> 01:22:25.470 Okay, okay Thank you. Okay. Well, good luck. 673 01:22:25.470 --> 01:22:33.810 Hello? Hello? Quick question on 1 of the problem yeah. Okay. Let me just shut this down. And then. 674 01:22:36.090 --> 01:22:42.539 Just curious here. Mm. Hmm. 675 01:22:42.539 --> 01:22:46.619 Uh.