WEBVTT 1 00:00:08.130 --> 00:00:39.780 Silence. 2 00:00:41.670 --> 00:00:47.369 Okay, good afternoon probability people. So this is. 3 00:00:47.369 --> 00:00:50.820 Last 13 and. 4 00:00:50.820 --> 00:00:56.280 Thursday, March 11th, 2021. 5 00:00:56.280 --> 00:01:01.350 And 1st, my question is, can you hear me. 6 00:01:01.350 --> 00:01:13.859 I'm looking over, I see the chat window. Thank you. Thank you Curtis and Nicholas. So we're in chapter 4. what I'm going to do today is some review stuff and. 7 00:01:13.859 --> 00:01:23.730 And so on, rather than new material to make it a somewhat lightweight day also, I said it every so often. I'd have a lightweight day. Can I ask what an exam question Michael? Yes, you may. 8 00:01:23.730 --> 00:01:34.620 Make your Mike, if you like, can you hear me? Yes, so I had a question about what are the answer choices for. 9 00:01:34.620 --> 00:01:38.700 It was a question regarding history in random variables. 10 00:01:38.700 --> 00:01:43.079 I was the answer that I picked him was Mark wrong for. 11 00:01:43.079 --> 00:01:50.129 Was saying that only discrete variables have a finite amount of outcomes. 12 00:01:50.129 --> 00:01:57.870 And I talked to 1 of the ta's and they're, they also think that that could be a correct answer given the wording of the question. 13 00:01:57.870 --> 00:02:00.959 So, I just wanted to know, like, your your thoughts on it. 14 00:02:00.959 --> 00:02:04.079 Yeah, you're right. 15 00:02:04.079 --> 00:02:08.580 So, submit a regrading request on that. 16 00:02:08.580 --> 00:02:16.289 Okay, yeah, and anyone else was listening submitter, rewriting and quest on that. Yes. 17 00:02:16.289 --> 00:02:27.120 I could've worded it a little more clear clearly what I wanted to say, right? I discreet could be finite or could be countably. Infinite. Continuous is only uncomfortably infinite. Yes. 18 00:02:27.120 --> 00:02:32.280 Thank you. Okay. Thanks so much. You're welcome more questions from the test. 19 00:02:32.280 --> 00:02:39.360 If you hadn't noticed, I, I published the exam things, so. 20 00:02:40.740 --> 00:02:48.150 Excuse me if you're curious about version 1 exam versus version version 2 was for. 21 00:02:48.150 --> 00:02:54.930 The middle of the night task version 1 was for the Western hemisphere cast. 22 00:02:54.930 --> 00:03:00.449 Okay, so 1st, I'd like to show you. 23 00:03:00.449 --> 00:03:04.020 Just a math lab for a few minutes and. 24 00:03:04.020 --> 00:03:08.969 Okay, Matlab window here. Did you do. 25 00:03:10.080 --> 00:03:14.009 Windows Windows Windows. Okay. So. 26 00:03:14.009 --> 00:03:19.409 Matlab is a major commercial. 27 00:03:19.409 --> 00:03:31.974 System it works with numbers, and it works with matrices arrays. I defined a matrix is an array plus operations on it, like modifying and so on. It's very good with that. 28 00:03:32.215 --> 00:03:36.235 It has many tool kits for control systems and that sort of thing. 29 00:03:36.539 --> 00:03:47.430 So, engineers like it a lot, we don't have it formally in classes at so much as much as we used to because we figure, I think you can. 30 00:03:47.430 --> 00:03:51.810 I do it on your own somewhat just a couple of quick things here. 31 00:03:51.810 --> 00:03:57.810 Obviously, it does, the rest of my tech is type. You get an answer. 32 00:03:59.039 --> 00:04:03.360 You can work with a raise and factors. 33 00:04:04.979 --> 00:04:08.490 Oops. 34 00:04:08.490 --> 00:04:14.639 Sorry. 35 00:04:14.639 --> 00:04:19.949 That sort of thing. 36 00:04:19.949 --> 00:04:23.040 As an inverse. 37 00:04:23.040 --> 00:04:35.908 Guys now, okay work on 1st, you can work with a raise and some element by element. So we can do things like. 38 00:04:35.908 --> 00:04:39.899 8 times 10. 39 00:04:39.899 --> 00:04:44.098 Fall by 10, for example. 40 00:04:45.718 --> 00:04:52.619 We could have you could do matrix by Matrix if I said B equaled. 41 00:04:55.678 --> 00:04:58.978 And then. 42 00:04:58.978 --> 00:05:02.728 A, plus B, it will go element by element. 43 00:05:04.019 --> 00:05:10.408 A times be, this would do it in theory. 44 00:05:10.408 --> 00:05:14.699 A proper matrix multiplication let's take a look at. Here's a again. 45 00:05:14.699 --> 00:05:23.848 I put a and B together now. Okay. So a Times B, it would take 1 times 5 plus 2 times 7. that's 19. 46 00:05:23.848 --> 00:05:36.718 12, how do we get to 22? 1? It's 12 times 68 so 1 time 6 is 6 Plus 2 times. 8 is 16 is 22. so, 8 times B here did matrix modification for example. 47 00:05:36.718 --> 00:05:41.999 And you could also modification if you wanted. 48 00:05:43.048 --> 00:05:48.059 What else could we do? I could have a vector V equaled. 49 00:06:00.149 --> 00:06:05.038 And if we look at, then that. 50 00:06:05.038 --> 00:06:10.199 Times that day a Times B times be, for example. 51 00:06:10.199 --> 00:06:14.908 Um, you can. 52 00:06:15.988 --> 00:06:21.238 So, it works as matrix is that works as actors I think I can values. 53 00:06:21.238 --> 00:06:26.369 Those are the values of a, we can also find the conductors of a. 54 00:06:26.369 --> 00:06:36.149 We could solve again if we have a linear system 8 times X equals B. 55 00:06:36.149 --> 00:06:40.288 We can solve quite simply. 56 00:06:43.918 --> 00:06:49.738 Could be so long to do what is what's happening here. 57 00:06:55.918 --> 00:06:59.309 Off to think what that is. Actually, I've forgotten as I was talking. 58 00:06:59.309 --> 00:07:05.639 Okay, so you can do simple stuff like that. You're going you could have special or factors like. 59 00:07:05.639 --> 00:07:12.059 This is just a simple 1 by 1 factor. For example. 60 00:07:12.059 --> 00:07:18.149 These arrows factor the basics, then you, you can build stuff up. 61 00:07:18.149 --> 00:07:21.749 Solve equations. 62 00:07:23.639 --> 00:07:29.278 So that's the basic thing and, as I say, it does lots of more powerful things, but. 63 00:07:29.278 --> 00:07:33.629 You can also write programs to, um, do something so. 64 00:07:35.879 --> 00:07:46.829 Um, I don't know. 65 00:07:52.019 --> 00:07:58.858 I forgot how you can define functions and big thing. Now. 66 00:07:58.858 --> 00:08:07.918 The advantage of this, I'll show you more stuff with probabilities later. Just wants to introduce so anything involving numbers and matrices. 67 00:08:07.918 --> 00:08:11.428 It works out, it works out nicely. 68 00:08:11.428 --> 00:08:14.699 So, it's also very efficient. 69 00:08:16.343 --> 00:08:29.814 You could work with a race with, you know, millions of elements. It has good algorithms in it. It executes fast with parallel machines on. And so on some walkie, it will also do things. 70 00:08:30.389 --> 00:08:35.668 Do things quickly you can also find information about it. 71 00:08:35.668 --> 00:08:39.089 Thanks, how you do a demo. 72 00:08:41.038 --> 00:08:45.298 Okay. 73 00:08:46.073 --> 00:09:00.803 Any case you can get information you can find out about stuff to do. I'll show you more latest. It's an introduction. Here you can. Now, the trouble is that I have a high resolution display on my laptop here and. 74 00:09:03.594 --> 00:09:18.083 And the math lab was not written for a high DPI display. So it's all I can hardly read the menu items here and also, but you can save your workstation workspace and started up again, you can write functions in a separate. 75 00:09:18.808 --> 00:09:23.548 In a separate window and load them. 76 00:09:23.548 --> 00:09:27.089 Whatever else about it. 77 00:09:28.109 --> 00:09:37.288 Writing efficient functions don't lose loops so much. You have to right using a raise. The limitations are. 78 00:09:37.288 --> 00:09:40.469 That I see with this as if you get into the real world. 79 00:09:40.469 --> 00:09:45.899 It gets very expensive. Our API, it's part of your tuition. 80 00:09:45.899 --> 00:09:50.158 And the 2nd thing is that. 81 00:09:50.158 --> 00:10:00.928 It works best for data that looks like a raise. If you want to do, say, a binary search tree was pointers or even a graph. We have got nodes and arcs. 82 00:10:02.038 --> 00:10:12.568 It gets much more difficult to do this and Matt, you can, you can do it efficiently in Matlab. So, I mean, there's no natural concept of pointers, for example. 83 00:10:12.568 --> 00:10:16.109 Now, you can force it. 84 00:10:16.109 --> 00:10:29.849 Either way you do a graph that you have something to call into jason's Matrix, which is zeros and ones that has a 1 an entry. I. K, if notes by has an arc going to notes. J. that's okay. And. 85 00:10:29.849 --> 00:10:36.538 You can do stuff like that, but it's, you know, programming becomes an art somewhat. 86 00:10:36.538 --> 00:10:43.948 And I can show you, and perhaps what, let me see if I can find a Matlab program show, you. 87 00:10:43.948 --> 00:10:51.479 Okay, we've got solutions and things and say. 88 00:10:51.479 --> 00:11:03.239 No, don't have any nice examples here for you, but I'll do more of this later. I want to get you into this gently. So it is a tool that you should I think. 89 00:11:03.239 --> 00:11:08.698 There's, there's lots of tutorial information on it. Help files. 90 00:11:15.208 --> 00:11:22.229 Okay, so we can get matrices and I can use and that sort of thing. 91 00:11:22.229 --> 00:11:27.538 So, for example, if my matrix a, I could say. 92 00:11:27.538 --> 00:11:32.489 I don't know, um. 93 00:11:37.619 --> 00:11:45.688 So D, are the values and V are the vectors. 94 00:11:45.688 --> 00:11:51.448 So, if you haven't had values, and I can factories yet. 95 00:11:51.774 --> 00:12:06.594 If you apply matrix to a position, most some vectors get scaled, but most factors get rotated, not just scaled the iCan vector. The vectors that get scaled their lanes gets changed, but they don't get rotated. 96 00:12:06.899 --> 00:12:19.889 And so, if they capture important properties of a transformation is what, a teigen vectors and I can value and the value is how much it is scaled. Okay. So that's. 97 00:12:19.889 --> 00:12:26.818 Little introduction to a Matlab I'll get it more deeper later, but. 98 00:12:26.818 --> 00:12:30.058 It's, it's again worth learning. 99 00:12:30.058 --> 00:12:35.308 Okay, what I'd like to do now is. 100 00:12:35.308 --> 00:12:48.568 Do some fun with with various stuff from chapter 3 what I've been doing before I've been walking you through chapter 3 giving my commentary on it? No, I'd like to do some. 101 00:12:48.568 --> 00:12:52.349 Some computations and so on. 102 00:12:52.349 --> 00:12:57.658 Okay, I see here today my. 103 00:12:57.658 --> 00:13:01.259 Connection to my iPad is working. 104 00:13:01.259 --> 00:13:05.219 Well, okay, cool. So. 105 00:13:05.219 --> 00:13:08.879 Hello. 106 00:13:12.928 --> 00:13:18.538 Okay. 107 00:13:20.698 --> 00:13:25.589 And pen. 108 00:13:27.269 --> 00:13:32.129 Okay. 109 00:13:46.644 --> 00:13:48.083 And that's basically. 110 00:13:51.869 --> 00:13:58.078 Okay. 111 00:13:59.698 --> 00:14:07.288 Good show you a little about normal distribution is how you can work with stuff on it. 112 00:14:10.889 --> 00:14:15.028 Silence. 113 00:14:16.499 --> 00:14:20.849 Got a black it's easier to read. 114 00:14:20.849 --> 00:14:26.788 Silence. 115 00:14:26.788 --> 00:14:30.629 Silence. 116 00:14:30.629 --> 00:14:35.788 So, a normal distribution, it's this site here called a spell shaped curve. 117 00:14:40.469 --> 00:14:44.938 And it's written as I'm going to assume Sigma equals 1. 118 00:14:53.639 --> 00:14:57.778 This assumes week was 0 on a Sigma equals 1. 119 00:14:57.778 --> 00:15:05.369 Okay, you can get it better. So the, um, for 1 question is. 120 00:15:06.568 --> 00:15:14.158 And so, okay, let me check. 121 00:15:14.158 --> 00:15:24.119 I mean, if it's going to be a probability density function, this has to be true. 122 00:15:25.139 --> 00:15:29.969 Well, how would we do that? So we're going to have, let's call it. 123 00:15:32.068 --> 00:15:40.769 Silence. 124 00:15:40.769 --> 00:15:46.109 Silence. 125 00:15:46.109 --> 00:15:50.219 Well, 1 thing I can do, I mean. 126 00:15:50.219 --> 00:15:53.548 I could have a different I, it's also equals 1. 127 00:15:55.469 --> 00:16:05.908 You know, I couldn't use why is my dummy variable instead of X. 128 00:16:05.908 --> 00:16:09.839 You know, silly little syntactic change. 129 00:16:09.839 --> 00:16:12.899 But if I do that. 130 00:16:12.899 --> 00:16:21.328 I could say that a squared equals and a, what I want is that a equals 1 okay. 131 00:16:25.109 --> 00:16:32.938 So, what I'm trying to prove. Okay. Okay. Okay. So this is the definition of the. 132 00:16:32.938 --> 00:16:42.629 Here definition of the normal distribution, and I could say X, I could say why so so, but now I can multiply these 2 versions and I'm going to get. 133 00:16:50.668 --> 00:16:59.009 Silence. 134 00:17:01.948 --> 00:17:05.338 It's a trick, but that will work. I'll multiply like that. 135 00:17:05.338 --> 00:17:09.088 Okay, and. 136 00:17:10.528 --> 00:17:16.558 Okay, so when I. 137 00:17:16.558 --> 00:17:24.239 Scroll to the next page, I'm going to lose that but so now what I'm going to do is I'm going to rework this and and do a change of variables. So. 138 00:17:27.659 --> 00:17:31.259 So, what's going to happen is that a square is. 139 00:17:36.749 --> 00:17:41.368 Silence. 140 00:17:44.278 --> 00:17:48.419 Okay. 141 00:17:48.419 --> 00:17:53.249 Now, what I'm going to do is change variables. 142 00:17:53.249 --> 00:17:56.999 Photo change on Cartesian to polar coordinates. 143 00:17:58.858 --> 00:18:04.169 Silence. 144 00:18:04.169 --> 00:18:07.348 Silence. 145 00:18:09.058 --> 00:18:14.098 Okay, and so. 146 00:18:17.548 --> 00:18:21.959 Okay, Eva. Okay. 147 00:18:21.959 --> 00:18:27.239 And Andy J Columbian. 148 00:18:44.909 --> 00:18:52.979 Well, DX by D R's coast say to do you. 149 00:18:52.979 --> 00:18:57.689 The X by the data is minus. 150 00:18:57.689 --> 00:19:03.388 Our sign sayda D. Y, ID are assigned. 151 00:19:03.388 --> 00:19:08.368 Is our. 152 00:19:11.038 --> 00:19:14.969 The contract and so, Jay calls, um. 153 00:19:16.739 --> 00:19:22.108 R, okay. It's our signed squared. Let's close. 154 00:19:22.108 --> 00:19:27.269 Squared equals. R. okay. So what this means now. 155 00:19:27.269 --> 00:19:32.398 Is that D x D. Y. will be our. 156 00:19:32.398 --> 00:19:41.638 The data. Okay so I have the thing up there for a I'm going to change. 157 00:19:41.638 --> 00:19:46.528 From Cartesian to Polar. 158 00:19:46.528 --> 00:19:51.778 Coordinates, and what we get is a square is equals 1. 159 00:19:55.528 --> 00:20:00.479 So. 160 00:20:03.509 --> 00:20:08.909 Our, and that's our there. Okay. 161 00:20:08.909 --> 00:20:11.939 Now, in this case, um. 162 00:20:11.939 --> 00:20:19.769 So are going from 0 to infinity and. 163 00:20:19.769 --> 00:20:24.118 Data going from there are the 2. 164 00:20:24.118 --> 00:20:35.068 Okay now, so say it is a constant in there so I integrate over data that just means a 2 pie. So. 165 00:20:37.858 --> 00:20:51.473 To pie. Okay. Good. Yeah, there's no data in that thing that we're integrating. 166 00:20:51.473 --> 00:20:53.124 So to the costs and this goes out to 1. 167 00:20:53.398 --> 00:21:04.199 Okay, so now we have to integrate that. 168 00:21:10.378 --> 00:21:15.959 Okay, but so the integral of. 169 00:21:15.959 --> 00:21:19.798 Are either the life squared over. 170 00:21:19.798 --> 00:21:23.098 Is going to be a. 171 00:21:31.439 --> 00:21:39.388 And let's see how everything works out are squared over 2 that's to our over 2. that's and are there. 172 00:21:39.388 --> 00:21:42.868 And we have a. 173 00:21:44.669 --> 00:21:48.929 Minus sign, we've got to put in there. Okay. 174 00:21:48.929 --> 00:21:53.489 Okay, so the integral from. 175 00:21:55.108 --> 00:21:59.999 0, to infinity close that arrow to infinity. 176 00:21:59.999 --> 00:22:03.719 And he was. 177 00:22:03.719 --> 00:22:08.098 1, so. 178 00:22:10.288 --> 00:22:14.848 Squared equals 1 equals 1 and the undergo. 179 00:22:20.338 --> 00:22:25.588 Okay. 180 00:22:25.588 --> 00:22:36.719 So that's a basic thing you can do with the galaxy and so this definition of the Gaussian it is, in fact, legal probably on the density density function. So, on. 181 00:22:44.909 --> 00:22:52.798 That's going to go. Okay. That's an example of the sort of thing that you can do with it. 182 00:22:52.798 --> 00:22:57.179 That's about all you can do with it. You can't. 183 00:22:58.409 --> 00:23:04.019 Do many other things as closed enter girls. Okay. 184 00:23:04.019 --> 00:23:15.118 The things we might want to do if. 185 00:23:15.118 --> 00:23:18.808 This and bring up the text book. 186 00:23:24.628 --> 00:23:31.798 Let me hit another continuous before I go back to disagree would be another example, a continuous thing you can do. 187 00:23:43.769 --> 00:23:49.798 Do some examples to remind you of means and so on. 188 00:23:49.798 --> 00:24:01.499 Play with exponential level, just to show you what you can do with that. 189 00:24:02.759 --> 00:24:08.638 Yeah, I started with page. 190 00:24:11.759 --> 00:24:24.179 Well, let me show you, um. 191 00:24:24.179 --> 00:24:27.239 Okay, so. 192 00:24:31.378 --> 00:24:35.459 Okay, so again, the basic definition. 193 00:24:42.148 --> 00:24:45.269 And again, just a. 194 00:24:45.624 --> 00:24:47.064 Notation here. 195 00:25:12.659 --> 00:25:22.229 And and again, you just well, just walking through here you can see that. 196 00:25:22.229 --> 00:25:28.648 Okay, and again, just some notation here. 197 00:25:37.439 --> 00:25:41.759 Okay, that is, um. 198 00:25:43.348 --> 00:25:48.479 But all, it's a cumulative distribution function. 199 00:25:48.479 --> 00:25:51.808 This here is the name. 200 00:25:54.808 --> 00:26:01.949 And here at some value. Okay. 201 00:26:01.949 --> 00:26:09.749 Um, and again, well, no need to work through that so much, but. 202 00:26:09.749 --> 00:26:14.068 But also lower case at. 203 00:26:16.648 --> 00:26:27.659 And again. 204 00:26:27.659 --> 00:26:30.749 Okay. 205 00:26:30.749 --> 00:26:33.778 Okay, and again. 206 00:26:33.778 --> 00:26:37.378 So. 207 00:26:38.848 --> 00:26:44.669 And you might leave that. So then that would be. 208 00:26:44.669 --> 00:26:52.108 Okay, yeah. Okay. That's how we get that. So that's the main thing there. 209 00:26:52.108 --> 00:26:57.148 Okay. 210 00:26:57.148 --> 00:27:02.578 Let me go back and do some examples from I'll do some homework questions from. 211 00:27:04.469 --> 00:27:08.398 Good question. 212 00:27:08.398 --> 00:27:14.038 There are some good questions here. 213 00:27:20.578 --> 00:27:26.278 I got 346 maybe. 214 00:27:29.278 --> 00:27:33.749 Silence. 215 00:27:40.558 --> 00:27:44.098 I don't know 346 page 1. 216 00:27:45.419 --> 00:27:49.558 Hello. 217 00:27:49.558 --> 00:27:57.088 Madness page whatever. 218 00:28:01.588 --> 00:28:05.249 Page 149 or something. I'm not certain about that. 219 00:28:05.249 --> 00:28:14.489 Okay, so we want to solve this. We have to look at what's the important stuff versus. 220 00:28:14.489 --> 00:28:18.118 What says the irrelevant things. 221 00:28:21.118 --> 00:28:25.138 Okay. 222 00:28:29.249 --> 00:28:32.249 So, the important thing here is we've got 8. 223 00:28:32.249 --> 00:28:36.118 A component and they're independently. 224 00:28:36.118 --> 00:28:40.348 Active alive or not. 225 00:28:40.348 --> 00:28:49.798 Equals so it's say, um, so this is a combination question. This will be the, this will be working with the binomial. 226 00:28:49.798 --> 00:28:54.778 Probability distribution. 227 00:28:54.778 --> 00:28:57.868 So, word problem. 228 00:29:08.159 --> 00:29:12.659 Whatever distribution. 229 00:29:12.659 --> 00:29:20.519 Okay, and we've got any equals a p equals. 230 00:29:23.759 --> 00:29:27.269 1 quarter. Okay. 231 00:29:28.858 --> 00:29:38.159 And let's let that gives systems. 232 00:29:39.659 --> 00:29:45.388 So, what we remember, so we're going to have of any K is going to be. 233 00:29:45.388 --> 00:29:50.038 8 K. 234 00:29:52.288 --> 00:29:57.088 Okay. 235 00:29:59.578 --> 00:30:06.479 So oh, 1Billion or active that's 8 2. 0. 236 00:30:08.638 --> 00:30:12.689 Um. 237 00:30:12.689 --> 00:30:17.098 8 0 is 1. 238 00:30:17.098 --> 00:30:25.709 This is 1, and this is the corners to the whatever that happens to be. 239 00:30:25.709 --> 00:30:32.429 Something okay exactly. 1 is active. 240 00:30:33.598 --> 00:30:38.189 Choose 11 quarter times. 241 00:30:38.189 --> 00:30:43.199 3 quarters the 7th and so on more than. 242 00:30:44.338 --> 00:30:50.459 Um, a B and C probability that. 243 00:30:50.459 --> 00:30:57.358 More than for active. 244 00:30:58.618 --> 00:31:02.189 Okay. 245 00:31:09.598 --> 00:31:12.989 And so often. 246 00:31:14.489 --> 00:31:20.338 Okay, so I level with what we had on the test and so on. 247 00:31:20.338 --> 00:31:24.449 Questions about that sort of thing. 248 00:31:42.719 --> 00:31:50.429 I know if I look at 352, let's say. 249 00:31:55.469 --> 00:32:02.788 Okay, so that's going to be geometric. 250 00:32:05.969 --> 00:32:13.288 Silence. 251 00:32:16.469 --> 00:32:24.269 Well, basically, naively, for a, the answer is 100. 252 00:32:25.318 --> 00:32:26.068 Ok, 253 00:32:26.453 --> 00:32:41.183 Eva. 254 00:32:48.659 --> 00:33:00.898 Okay, so the probability of. 255 00:33:05.878 --> 00:33:12.659 Yeah, to 1 minus here and then that is so the probability that there is exactly. 256 00:33:14.489 --> 00:33:21.898 K. error free characters. 257 00:33:26.308 --> 00:33:29.699 Errors. 258 00:33:33.328 --> 00:33:41.068 Is going to be 1 minus P error free and then Pete, the next 1. 259 00:33:41.068 --> 00:33:47.009 It was an error. Okay. And now you can. 260 00:33:51.058 --> 00:33:55.169 So that's going to be. 261 00:33:55.169 --> 00:34:00.179 So that so then this is your probability mass function. So. 262 00:34:00.179 --> 00:34:05.699 Okay, they're having it as and so the PM. 263 00:34:05.699 --> 00:34:10.438 Is okay for, and is 1 minus P. 264 00:34:10.438 --> 00:34:13.949 Okay. 265 00:34:15.418 --> 00:34:22.318 And the expected value of that, is that. 266 00:34:22.318 --> 00:34:25.889 Silence. 267 00:34:25.889 --> 00:34:30.478 Silence. 268 00:34:30.478 --> 00:34:34.108 Silence. 269 00:34:34.108 --> 00:34:39.268 Okay. 270 00:34:39.268 --> 00:34:43.528 And that was worked out in the book before I won't work it out again. 271 00:34:43.528 --> 00:34:47.429 It's like. 272 00:34:49.469 --> 00:34:58.048 Now, if you want to be 99% sure that at least a 1000 characters are received correctly. 273 00:35:01.469 --> 00:35:11.849 Well. 274 00:35:22.739 --> 00:35:30.688 Well, the probability that K received correctly. 275 00:35:40.378 --> 00:35:44.608 Okay. 276 00:35:44.608 --> 00:35:50.338 And now basically. 277 00:36:07.048 --> 00:36:11.909 Well, what I would do is I would graph it, I would use. 278 00:36:13.378 --> 00:36:17.429 Some favorite my favorite calculation program and just draft it. 279 00:36:23.188 --> 00:36:27.179 Silence. 280 00:36:28.318 --> 00:36:34.228 Silence. 281 00:36:37.409 --> 00:36:50.009 Okay, I mean, I don't think there's necessarily some super powerful solution that gives you an exact answer just as a formula. 282 00:36:51.418 --> 00:36:56.668 Okay, see how that works question. 283 00:37:01.018 --> 00:37:04.409 Oh, question a while back. 284 00:37:07.108 --> 00:37:19.829 Eva, yeah. 285 00:37:22.139 --> 00:37:30.509 You're right. 286 00:37:34.588 --> 00:37:40.438 That was it on trivial error Nicolas. So, send me an email. 287 00:37:41.518 --> 00:37:45.898 Other corrections and I'll give you a point. 288 00:37:47.188 --> 00:37:51.358 How could you grab this? Find it like that? I can't hear you. 289 00:37:51.358 --> 00:37:56.579 Sorry for them next the last 1 you just did I could do that to get an answer. 290 00:37:56.579 --> 00:38:02.039 Here could I do what to use a computer to graph and look. 291 00:38:02.039 --> 00:38:12.269 Yeah, what I would do is I would grab the sum as a function of P. P. C. only pretty variable there. 292 00:38:13.320 --> 00:38:25.860 And and I would just look at the graph and see at what point it becomes greater than, you know, I would have the computer calculate to some for any given P. 293 00:38:25.860 --> 00:38:32.909 And then just look at that and see where it becomes greater than point 99. so. 294 00:38:34.380 --> 00:38:44.070 And you could use Matlab, connect this. Yeah, I'd have to think I don't want to take the whole class this time right now to work it out but yeah, I'll work it out with them. 295 00:38:44.070 --> 00:38:48.449 Yeah, go on more detail on Monday, so. 296 00:38:49.650 --> 00:38:55.349 Next. 297 00:39:00.175 --> 00:39:13.824 For things like summing Kelsey and some croissant and so on there's no closed formula says, no simple. Exact formula for, like, partial sums partial into girl's of these things. 298 00:39:14.099 --> 00:39:17.760 So, you use approximations or nowadays, you just. 299 00:39:17.760 --> 00:39:24.630 The 4 computers got cheap enough. People worked with a table of numbers. Now you just. 300 00:39:24.630 --> 00:39:39.150 Run it on the computer, so here is 365. let's work on that. Let's say, chance to use plus on. 301 00:39:43.050 --> 00:39:47.429 Okay, so. 302 00:39:49.230 --> 00:39:52.679 Um, so there's lots of, um. 303 00:39:53.820 --> 00:39:58.710 So, and equals 750,000. 304 00:39:58.710 --> 00:40:01.949 Probability. 305 00:40:03.449 --> 00:40:11.159 It calls, um, so the expected number of so, let's say. 306 00:40:11.159 --> 00:40:14.159 X equals the number of bad fix. 307 00:40:17.099 --> 00:40:24.480 Expected value of X equals 750,000 times. 308 00:40:24.480 --> 00:40:28.110 Divided by 100,000 equals 75. 309 00:40:30.989 --> 00:40:34.469 No, 7.5 that see. 310 00:40:36.239 --> 00:40:48.210 Yeah, so pulse on is appropriate. 311 00:40:48.210 --> 00:40:52.380 Um, binomial. 312 00:40:55.800 --> 00:41:00.809 Would work. 313 00:41:02.099 --> 00:41:07.409 You know, you could compute. 314 00:41:07.409 --> 00:41:11.039 And factorial, et cetera, but, you know, but. 315 00:41:11.039 --> 00:41:18.179 It's suboptimal. 316 00:41:18.179 --> 00:41:22.619 Gaussian would work here also. 317 00:41:22.619 --> 00:41:34.559 The reason is so the mean. 318 00:41:34.559 --> 00:41:43.920 8 of 7.5, the Sigma for plus on the Sigma here is square root of 7.5. 319 00:41:45.090 --> 00:41:49.860 I chose, I don't know, 2.3 or something. 320 00:41:49.860 --> 00:41:56.369 So, it's not too close to 0 so. 321 00:41:58.349 --> 00:42:01.500 Of the curve. 322 00:42:01.500 --> 00:42:05.909 It has a left tail. 323 00:42:05.909 --> 00:42:12.210 So so the normal would work, but it would work in this case. 324 00:42:12.210 --> 00:42:19.199 On so. 325 00:42:19.199 --> 00:42:23.219 Well, I'll show you both. I'll, I'll show you go see, and actually here. 326 00:42:23.219 --> 00:42:28.980 So, okay, a little ahead for a review of. 327 00:42:31.110 --> 00:42:37.829 Okay, so the parameter is a say. 328 00:42:37.829 --> 00:42:46.800 Orlando other people outside Leon Garcia use when I'm done, I'll use a okay, so the thing was the. 329 00:42:49.289 --> 00:43:01.139 If I remember it okay, it's an integer use K for integers expert, real numbers often. So. 330 00:43:03.030 --> 00:43:07.860 Or some, a parameter. 331 00:43:07.860 --> 00:43:12.000 Sometimes use the name of underground. Sometimes you use a parameter. 332 00:43:12.000 --> 00:43:23.039 Whichever okay, so that is. 333 00:43:31.380 --> 00:43:35.940 Me okay. 334 00:43:35.940 --> 00:43:41.730 Okay, now the mean is the mean. 335 00:43:41.730 --> 00:43:46.590 Equals a, and a standard deviation equals a also. 336 00:43:46.590 --> 00:43:51.719 So, equal square root. Okay. 337 00:43:51.719 --> 00:43:58.320 Now, so what this means is if we plot it, so a, is 7.5. 338 00:43:59.789 --> 00:44:03.480 Kind of look like something like this. Um. 339 00:44:05.909 --> 00:44:12.869 That mean, it's pretty close to the mode and this distance here is about. 340 00:44:15.000 --> 00:44:23.670 2 and 2 or something, so it's, it's, it's all not all pushed up at the, at the Y, axis which things normal would work. 341 00:44:23.670 --> 00:44:28.110 Okay, so we want the probability. 342 00:44:28.110 --> 00:44:35.070 A 15 or few or faulty pixels. Okay. 343 00:44:39.510 --> 00:44:47.519 15 or few bad pixels. 344 00:44:47.519 --> 00:44:52.139 It close this. 345 00:44:55.079 --> 00:45:01.079 Okay, it was here 115. okay. 346 00:45:01.079 --> 00:45:06.059 And you've got all the numbers, you could work it out, but don't do it by hand. 347 00:45:06.059 --> 00:45:10.289 That's using what's on. 348 00:45:11.699 --> 00:45:18.360 Now, using normal, not this is an excuse for me to teach you some normal stuff. 349 00:45:19.739 --> 00:45:26.880 Again, so, so, in this case, the mean equals 7.5. 350 00:45:26.880 --> 00:45:32.460 Sigma around 2.3, whatever. 351 00:45:32.460 --> 00:45:35.880 Square root of 7.5. 352 00:45:35.880 --> 00:45:39.750 I'm doing, like, 1, significant digit basically. Now. 353 00:45:42.840 --> 00:45:46.800 So, 15. 354 00:45:46.800 --> 00:45:56.820 That's going to be, um. 355 00:45:59.760 --> 00:46:10.650 Whatever 3 times, 2.3 I'm being really, really rough here. Okay. Um, so that's me plus 3 segment about 3 and a half stigma. 356 00:46:10.650 --> 00:46:17.190 Um, so. 357 00:46:20.880 --> 00:46:24.599 So, what we want here is in a normal distribution. 358 00:46:27.329 --> 00:46:32.309 No. 359 00:46:33.929 --> 00:46:40.800 Again, so 7.5 and this is 15. you want all of this so okay. 360 00:46:42.719 --> 00:46:46.800 And this is the probability that this normal. 361 00:46:49.530 --> 00:46:53.460 What the probability that this normal variable. 362 00:46:53.460 --> 00:46:58.710 It's less than or equal to. 363 00:46:58.710 --> 00:47:05.340 You plus 3 segment and that is something like. 364 00:47:06.570 --> 00:47:11.460 Silence. 365 00:47:14.489 --> 00:47:23.039 Really small so the way you do that is that there are cables, you can use a computer or. 366 00:47:23.039 --> 00:47:26.789 There are tables engine in the book, so. 367 00:47:37.349 --> 00:47:42.329 So, we'll see that more thorough of us. That would be a way. You could do 365. so. 368 00:47:46.260 --> 00:47:51.809 Okay. 369 00:47:53.880 --> 00:48:06.449 Silence. 370 00:48:09.090 --> 00:48:15.420 369 for fun. 371 00:48:16.679 --> 00:48:22.440 Well. 372 00:48:25.590 --> 00:48:28.949 Okay, minus 3 to 4. 373 00:48:30.329 --> 00:48:36.929 Nate was saying our sample space says minus 3 minus 2. 374 00:48:36.929 --> 00:48:40.559 4, okay. 375 00:48:45.269 --> 00:48:52.199 So, to me, is this a sum of all the xa over 8 and that's going to be whatever. 376 00:48:52.199 --> 00:48:58.679 And then the variants use different formulas. 377 00:49:00.059 --> 00:49:04.469 1 would be. 378 00:49:04.469 --> 00:49:07.650 Would be 1 message, for example. 379 00:49:10.739 --> 00:49:16.679 Now, for that would be for a. 380 00:49:16.679 --> 00:49:20.130 Let me, let me do it for a simpler example here. 381 00:49:21.360 --> 00:49:24.989 Or simplify it. 382 00:49:24.989 --> 00:49:29.190 Simplify let's let the sample space be, um. 383 00:49:31.260 --> 00:49:36.360 Equals 4. 384 00:49:36.360 --> 00:49:41.280 Okay, I mean, is. 385 00:49:42.840 --> 00:49:52.800 1, half. Okay the expected value of X squared. 386 00:49:52.800 --> 00:49:58.860 8 calls. 387 00:50:01.710 --> 00:50:05.190 We have variants. 388 00:50:06.750 --> 00:50:10.980 Hey. 389 00:50:15.659 --> 00:50:20.219 Silence. 390 00:50:22.409 --> 00:50:26.159 Silence. 391 00:50:28.380 --> 00:50:33.719 Okay, so now for 369 B. 392 00:50:37.440 --> 00:50:42.300 Um, there are 2 ways you could do this. 393 00:50:53.849 --> 00:50:56.849 At least okay. 394 00:50:56.849 --> 00:51:00.480 Way 1 is you just. 395 00:51:00.480 --> 00:51:12.539 Compute the wise, so the wise are minus 2 x squared plus 3 and again, my axis again, minus 1. 0 1. 396 00:51:12.539 --> 00:51:15.869 And 2, so the wise are going to be. 397 00:51:15.869 --> 00:51:21.809 2 squared plus 3 is, um, X squared is 1 just 2 plus C. 398 00:51:22.829 --> 00:51:27.989 5, 3 5. 399 00:51:29.039 --> 00:51:33.659 Um, X squared is 4 drinks 8 plus 3 is 11. 400 00:51:33.659 --> 00:51:36.809 And then, meanwhile, close. 401 00:51:36.809 --> 00:51:40.920 5, plus 10, sir chain. 402 00:51:40.920 --> 00:51:44.070 24. 403 00:51:44.070 --> 00:51:47.849 In 6 and so on. 404 00:51:49.289 --> 00:51:55.079 And cetera, or way 2 is that we use transformation rules. 405 00:51:56.940 --> 00:52:04.739 And just. 406 00:52:04.739 --> 00:52:07.860 You could do some things, but. 407 00:52:12.150 --> 00:52:16.710 Well, you could use it on the 2 and the 3. so. 408 00:52:18.000 --> 00:52:23.489 The expected value of X squared and so on. 409 00:52:24.719 --> 00:52:28.079 So the expected value of X squared I saw. 410 00:52:28.079 --> 00:52:34.019 Was 3 halves so if why. 411 00:52:34.019 --> 00:52:39.840 Equals a minus in there. I forgot. 412 00:52:43.980 --> 00:52:51.630 Well, this assumed why plus. 413 00:52:51.630 --> 00:52:56.400 X squared plus 3. okay so. 414 00:52:59.280 --> 00:53:03.269 Expected value why it was twice. 415 00:53:03.269 --> 00:53:06.900 Expected value of X squared 3. 416 00:53:06.900 --> 00:53:13.590 Expected value of X squared was 3 has that'd be 3 plus 3 equals X. 417 00:53:13.590 --> 00:53:18.000 Hey, and what do you know it agreed. 418 00:53:21.119 --> 00:53:26.340 Well, and for variance. 419 00:53:27.630 --> 00:53:30.630 There's no shortcuts on the X squared thing, but okay. 420 00:53:35.190 --> 00:53:41.610 And for CO sign, you basically have to work it out, find a transformed variable. W. so. 421 00:53:50.550 --> 00:53:56.519 Okay, let's look at 308 for fun. 422 00:54:03.780 --> 00:54:09.300 Okay, okay so we got defective variables. 423 00:54:14.489 --> 00:54:17.760 Okay, so I have to invent. 424 00:54:18.989 --> 00:54:24.570 Location here or events. 425 00:54:29.039 --> 00:54:32.039 Huh. 426 00:54:33.539 --> 00:54:38.699 Silence. 427 00:54:38.699 --> 00:54:42.150 Silence. 428 00:54:42.150 --> 00:54:47.070 Silence. 429 00:54:47.070 --> 00:54:50.460 Silence. 430 00:54:50.460 --> 00:54:55.170 Silence. 431 00:54:55.170 --> 00:54:59.969 Okay, and. 432 00:54:59.969 --> 00:55:06.900 The probability that it's defective, it calls. 433 00:55:06.900 --> 00:55:18.480 P, okay. And if it's defective the probability it's identified correctly so probably is identified correctly given its defective. 434 00:55:18.480 --> 00:55:23.699 It was a. 435 00:55:23.699 --> 00:55:36.150 Okay, now we're assuming that non defective items are identified correctly all the time. So the problem has identified correctly given it's not effective. 436 00:55:36.150 --> 00:55:36.804 Is 1, 437 00:55:44.304 --> 00:55:45.114 okay, 438 00:55:45.114 --> 00:55:52.255 now we're asking the probability of K items or test until the defective item is identified so that's. 439 00:55:52.320 --> 00:55:56.909 It's gonna be geometric so what we want. 440 00:55:57.960 --> 00:56:03.840 Is a probability that we identify a. 441 00:56:03.840 --> 00:56:07.260 Effective item. Okay. So. 442 00:56:07.260 --> 00:56:16.590 So, what we want. 443 00:56:19.050 --> 00:56:24.119 The probability that it's effective and identified correctly. 444 00:56:24.119 --> 00:56:28.050 Okay, and then later geometric is going to be relevant. 445 00:56:30.239 --> 00:56:33.780 That'd be relevant. Okay. I'm later. 446 00:56:33.780 --> 00:56:38.880 Yeah, okay. What's the probability that is defective and identified. 447 00:56:38.880 --> 00:56:45.119 Correctly, how can we do that? 448 00:56:56.699 --> 00:57:02.340 Well, that's going to be equal to the probability. 449 00:57:02.340 --> 00:57:06.300 And it's effective times of probability. 450 00:57:07.320 --> 00:57:11.400 That it was identified correctly, given that it's effective. 451 00:57:11.400 --> 00:57:16.679 Unless I get it wrong long day. 452 00:57:16.679 --> 00:57:23.760 Okay, so. 453 00:57:26.219 --> 00:57:29.909 So, probably the K items are defective. Um. 454 00:57:31.590 --> 00:57:35.099 So this is the probability that each test. 455 00:57:35.099 --> 00:57:40.650 We find a defective item. It was the fact of and we found it. Okay. 456 00:57:40.650 --> 00:57:47.579 And this makes sense a problem, but it's the fact of times the probability that we identified it correctly a. 457 00:57:48.295 --> 00:57:49.405 So, um. 458 00:58:17.429 --> 00:58:22.050 It's going to be. 459 00:58:28.469 --> 00:58:31.829 Okay. 460 00:58:32.940 --> 00:58:42.329 Answer a here leave that up for people can't think about for a 2nd. 461 00:58:53.940 --> 00:58:59.070 And to be if we remove the identified defectives. 462 00:58:59.070 --> 00:59:03.090 What proportion of the remaining items. 463 00:59:03.090 --> 00:59:11.190 Are defective well, we can work that out just by thinking actually. 464 00:59:11.190 --> 00:59:23.489 Silence. 465 00:59:26.670 --> 00:59:29.730 Silence. 466 00:59:34.829 --> 00:59:39.809 Silence. 467 00:59:47.579 --> 00:59:51.659 So, the fraction. 468 00:59:52.980 --> 00:59:57.269 So. 469 00:59:58.320 --> 01:00:02.429 Silence. 470 01:00:06.329 --> 01:00:15.239 Okay, okay. 471 01:00:31.590 --> 01:00:34.739 Now, B, you could do different ways. 472 01:00:34.739 --> 01:00:39.059 I'll do it. I'll do it just with intuitively almost. 473 01:00:39.059 --> 01:00:44.460 Or see, I mean, so no. 474 01:00:44.460 --> 01:00:54.960 Questions okay. 475 01:00:54.960 --> 01:00:58.800 Well, he's a way to do it, so all the items. 476 01:01:02.309 --> 01:01:06.809 The factor. 477 01:01:08.730 --> 01:01:13.079 Good and now we got Tom. 478 01:01:14.309 --> 01:01:22.679 Identified. 479 01:01:22.679 --> 01:01:26.070 Here, okay now. 480 01:01:27.420 --> 01:01:33.059 Here we say some good items, so be. 481 01:01:34.469 --> 01:01:38.639 They're called defective. 482 01:01:38.639 --> 01:01:42.059 And 1 minus, be. 483 01:01:44.130 --> 01:01:48.659 Is still good? Good, good. 484 01:01:48.659 --> 01:01:58.019 Okay, so so this nets out, there's a p are removed. 485 01:01:59.699 --> 01:02:02.849 Um, 1 minus a p. 486 01:02:02.849 --> 01:02:06.389 They remain kept. 487 01:02:07.739 --> 01:02:14.099 Okay, here what the good ones is, we're going to have B1 minus P removed. 488 01:02:15.690 --> 01:02:20.610 Wrongly and. 489 01:02:20.610 --> 01:02:24.750 1, minus be on minus PE, kept. 490 01:02:24.750 --> 01:02:29.039 Okay, so. 491 01:02:34.710 --> 01:02:38.340 Um, so kept. 492 01:02:39.539 --> 01:02:43.769 Is 1 minus a E plus. 493 01:02:43.769 --> 01:02:47.670 1, minus 1 minus P. 494 01:02:48.869 --> 01:02:54.059 Bad are the bad ones, kept our. 495 01:02:56.639 --> 01:03:01.739 Okay, so the fraction. 496 01:03:01.739 --> 01:03:04.739 Bad in the kept items. 497 01:03:06.150 --> 01:03:09.150 Is 1 minute to say. 498 01:03:10.500 --> 01:03:14.159 Silence. 499 01:03:14.159 --> 01:03:17.550 Okay, so. 500 01:03:26.639 --> 01:03:34.230 And we hope that we've improved the number of a fraction of good items. So, but if their error rates are bad enough, maybe not. 501 01:03:36.150 --> 01:03:42.840 I'll leave this up for a 2nd. 502 01:04:25.315 --> 01:04:27.204 Good questions. Okay. 503 01:04:27.750 --> 01:04:35.639 The girls here, I'll do more of these Monday, but you're getting an idea. 504 01:04:38.190 --> 01:04:43.139 Okay, it's enough questions. Let me do a little stuff from the park and then. 505 01:04:43.139 --> 01:04:51.780 We'll do more of these Monday. 506 01:04:51.780 --> 01:04:57.389 I want to go on to. 507 01:05:04.079 --> 01:05:08.099 A, 2nd, here. 508 01:05:08.099 --> 01:05:19.710 Silence. 509 01:05:19.710 --> 01:05:23.039 Silence. 510 01:05:23.039 --> 01:05:26.489 You know, it means and so on. 511 01:05:26.489 --> 01:05:31.289 Illustrate this with examples here. 512 01:05:39.840 --> 01:05:44.159 Yeah. 513 01:05:46.260 --> 01:05:51.659 Harry and some, so I worked out the, just the integral the. 514 01:05:51.659 --> 01:05:57.030 Density function for the calcium. Now here's the more general Gaussian. 515 01:05:57.030 --> 01:06:08.730 With some mean, it's not 0, so just shift sex and some standard deviation. The standard deviation goes into this in 2 places. It goes up ears a Sigma squared. 516 01:06:08.730 --> 01:06:12.150 Up in this exponent, and it goes to the 1 or 2 Sigma. 517 01:06:12.150 --> 01:06:25.860 Here okay, yeah. And how you can find how you can find the variance in the mean, the more complicated way. So what I was doing here. 518 01:06:25.860 --> 01:06:31.619 Okay, some points about variance and mean, if ice. 519 01:06:31.619 --> 01:06:43.800 Said this Monday, but it's worth saying again, the variance of a constant is 0, if you add a constant to a back to a random variable, the variance doesn't change. If you scale the random variable. 520 01:06:44.969 --> 01:06:50.519 Varying scales by the square, because this a dimensional analysis, which showed has to be a square. 521 01:06:50.519 --> 01:06:54.630 I showed you that. 522 01:06:54.630 --> 01:07:00.690 Yeah okay. This is a cool thing. I'll do this Monday. You can read ahead if you think I'm going to solidly for you. 523 01:07:00.690 --> 01:07:04.769 Speaker here. 524 01:07:04.769 --> 01:07:13.710 Important continuous. This is a table here that it's worth. Um. 525 01:07:14.940 --> 01:07:21.150 Putting on your cheat sheet page 164 the generating functions. I'm. 526 01:07:21.150 --> 01:07:26.309 Not much using. Okay. Yeah. 527 01:07:26.309 --> 01:07:39.840 I mentioned the cow. She's the cool 1. I mean, and variance do not exist. Parade. I'm going to skip data is 1 with a couple of parameters in it that you used to do various things. 528 01:07:39.840 --> 01:07:47.340 Again. 529 01:07:47.340 --> 01:07:54.059 Okay, I'm going to come back to see and again, and I'm going to come back to it until you're sick of it, perhaps, because it is very important. 530 01:07:54.059 --> 01:08:01.829 And again, almost all other random variables. The curves start looking like a Gaussian. 531 01:08:01.829 --> 01:08:05.550 When you add up enough random variables, you take the mean. 532 01:08:05.550 --> 01:08:18.750 Say, Ken whatevers. It doesn't matter what, and probably looks like a Gaussian. It converges surprisingly quick. That's why it's called normal normal and Gaussian are synonymous. 533 01:08:18.750 --> 01:08:25.350 You might say calcium, because you figure that the guy that popularized. 534 01:08:25.350 --> 01:08:28.770 Coffee to cows deserves to be remembered. 535 01:08:28.770 --> 01:08:33.840 So, okay, and again, this is it here. 536 01:08:33.840 --> 01:08:38.220 And. 537 01:08:38.220 --> 01:08:43.140 There is a standard notation here. 538 01:08:44.729 --> 01:08:48.390 This is worth remember it's a big. 539 01:08:48.390 --> 01:08:52.770 Whatever Greek clarify be if you want to say that. 540 01:08:52.770 --> 01:08:59.220 And this is the CDF of a Gaussian of 01, calcium this effects. 541 01:08:59.220 --> 01:09:02.729 Now, why this. 542 01:09:02.994 --> 01:09:16.854 The fee of X5 X is relevant. Is you see what we've done is we've normalized the gals and you might say we scaled and translated to make 0 and Sigma 1. now you get cables of this and you can start like. 543 01:09:18.239 --> 01:09:23.250 For example, 50 is going to be 1 half. 544 01:09:23.250 --> 01:09:29.130 To pie of minus 1 is going to be about 1 6. 545 01:09:29.130 --> 01:09:33.300 5 minus 2 is going to be about point 0, 2. 546 01:09:33.300 --> 01:09:37.050 By minus 3 is going to be point. 547 01:09:37.050 --> 01:09:47.789 Oh, 2 minus 2. so, 2, all 5 or something and minus 3 is point 2, I think and. 548 01:09:49.439 --> 01:09:53.640 So, people remember the width of the tail here so. 549 01:09:53.640 --> 01:09:57.210 So, 50, that's the mid point. 550 01:09:57.210 --> 01:10:02.189 So probably being up to the mid point is a half minus 1. 551 01:10:02.189 --> 01:10:06.479 That's going to be the left tail from up to new minus Sigma. 552 01:10:06.479 --> 01:10:09.960 And so you can start remembering these numbers. 553 01:10:11.340 --> 01:10:18.899 And integrates to 1 this is what I showed, you example, 421 to open up the class here. 554 01:10:18.899 --> 01:10:28.770 Now and electrical engineering, just to be different, they work with a compliment of Fi they call it. Q. and that's the right tale of the normal distribution. 555 01:10:28.770 --> 01:10:32.159 So, and it's 1 minus 5. 556 01:10:32.159 --> 01:10:40.439 Okay, and what they mean by no closed form here is there's no simple. Exact. 557 01:10:40.439 --> 01:10:48.539 Formula computers are ubiquitous, you computer with a computer, or you look it up in a table of numbers. 558 01:10:48.539 --> 01:10:53.760 So, and there's good approximations, but that's a little outside discourse. But. 559 01:10:56.250 --> 01:11:01.140 And it's important everywhere. This is a sort of table here. 560 01:11:04.289 --> 01:11:09.930 Yeah, so from this is the right so from. 561 01:11:09.930 --> 01:11:14.520 So, the probability of the random variable, being more than a Sigma. 562 01:11:14.520 --> 01:11:18.449 Above the mean is 16 roughly. Okay. 563 01:11:18.449 --> 01:11:26.399 1, significant digits, the probability of the random, the calcium variable, being more than 2 Sigma up. 564 01:11:26.399 --> 01:11:32.279 Is 2% 3 Sigma up. 565 01:11:32.279 --> 01:11:38.789 Is 1 in a 1000 so, on this goes to 0 really quickly so. 566 01:11:40.319 --> 01:11:45.630 Well, you can have fun with s. A. t's say they're normally mean 500. 567 01:11:45.630 --> 01:11:52.470 Sigma 100, the probability of getting at least 500 is a half the probability of getting at least 600. 568 01:11:52.470 --> 01:11:57.390 Is about 15% probability of getting at least 700. 569 01:11:57.390 --> 01:12:02.909 Is 2% probably of getting 800 would be. 570 01:12:02.909 --> 01:12:09.420 Point 1% roughly these are all very rough numbers, but rough numbers are for said, okay. 571 01:12:10.979 --> 01:12:14.340 Okay, so that's a do do do do do. 572 01:12:14.340 --> 01:12:27.029 Gamma and song, we'll see later, but so I'll leave you thinking about the Gaussian the norm so what we saw today is I just showed you that Matlab exists and gave a hint of how it's our fault. 573 01:12:27.029 --> 01:12:34.170 I'll give you bigger examples on Monday and we did lots of examples from the end of chapter 3. 574 01:12:34.170 --> 01:12:38.130 And just a warm up thing, showing you can integrate the calcium and. 575 01:12:38.130 --> 01:12:44.159 Also there are cables, fear of calcium, random variables you might want to. 576 01:12:44.159 --> 01:12:52.529 Over that table was here page 169 could be worth putting 169 on your cheat sheet also. 577 01:12:52.529 --> 01:12:56.640 Putting a tab in it. Okay. 578 01:12:57.989 --> 01:13:01.079 So enjoy the weekend. 579 01:13:01.079 --> 01:13:05.760 Hope that Blizzard that's supposed to pound Denver doesn't come East or something. 580 01:13:05.760 --> 01:13:09.270 And here welcome and other than that. 581 01:13:10.319 --> 01:13:13.949 I'll stay around for a minute or so and if not. 582 01:13:15.329 --> 01:13:19.409 Morning if I get any skiing in this weekend, I don't know. 583 01:13:20.460 --> 01:13:26.039 Maybe a prospect cross country skiing on you and other than that. 584 01:13:26.039 --> 01:13:31.140 I thought when will the next test be? That's a good point. Thank you. 585 01:13:31.140 --> 01:13:35.069 Let me, let me announce that then, let me just think so. 586 01:13:35.069 --> 01:13:39.329 Send me, send me an email to remind me. 587 01:13:40.829 --> 01:13:46.590 Ask the question about the previous exam certainly, but that question you were talking about. 588 01:13:46.590 --> 01:13:50.399 For 7, 7, I think. 589 01:13:51.569 --> 01:13:55.380 What's, uh, what's a viable great question? 590 01:13:55.380 --> 01:14:01.470 I put the only the discreet random variable has a finite number of outcomes. 591 01:14:01.470 --> 01:14:05.189 That's wrong. Well. 592 01:14:05.189 --> 01:14:09.270 So well, what did you pick as an answer? 593 01:14:11.189 --> 01:14:15.119 And the only the discreet random variable has a find a number of outcomes. 594 01:14:15.119 --> 01:14:24.060 That's true. I mean, that's how you parse the question. Um. 595 01:14:24.060 --> 01:14:36.869 What I was thinking was that continuous does never has a finite variable as a finite number. So only to discreet has the possibility of having a finite number, but it's not required to have a finite number. 596 01:14:36.869 --> 01:14:40.289 But I agree that I could've worded it more clearly. 597 01:14:40.289 --> 01:14:45.989 So just request a re grade and just put a reason class discussion. 598 01:14:45.989 --> 01:14:49.649 Okay or something. 599 01:14:51.119 --> 01:14:55.409 And, um, number 6, what distribution is that? 600 01:14:55.409 --> 01:15:03.630 I always kind of struggle with kind of distribution it is. I mean, pull up the grade scope thing. I can't remember. 601 01:15:33.954 --> 01:15:35.034 Signal those grades. 602 01:15:35.310 --> 01:15:41.699 You said 6 yes geometric. 603 01:15:43.920 --> 01:15:47.100 And that's the only 1. that's correct. Actually. 604 01:15:48.180 --> 01:15:53.039 So, um. 605 01:15:54.420 --> 01:16:03.960 You know, it exists at times 0, probability of a half, it exists still at time, 11 quarter it's still alive at time to it has not decayed to something else. And so on. 606 01:16:05.819 --> 01:16:09.750 Oh, is that what halfway means it has 50% chance of being alive. 607 01:16:09.750 --> 01:16:17.550 Yes, I just sort of assumed that everyone knew that from physics or something or high school physics. 608 01:16:17.550 --> 01:16:20.729 When I did talk about this in a lecture and so. 609 01:16:20.729 --> 01:16:24.329 1 of these things before, so. 610 01:16:24.329 --> 01:16:28.229 I was thinking with newly because I was like a library. 611 01:16:28.229 --> 01:16:32.819 Or, could you talk up please. 612 01:16:32.819 --> 01:16:38.699 I was thinking it was because they're alive or it's not alive. 613 01:16:40.109 --> 01:16:46.649 Well, what this is not just 1 thing. This is a continual probability of decaying. 614 01:16:46.649 --> 01:16:53.909 So, we're newly would be if you looked at it once, like, at time. 615 01:16:53.909 --> 01:17:01.020 37, is it alive or dead? You could say that could be renewal lease, tossing a coin once, but. 616 01:17:01.020 --> 01:17:08.579 The random variable is not. Is it alive at time? 37 or 42? The random variable. Is it. 617 01:17:08.579 --> 01:17:13.260 Is it like, it's how long before it. 618 01:17:13.260 --> 01:17:16.500 Yeah. 619 01:17:16.500 --> 01:17:20.850 Okay, that makes sense. 620 01:17:20.850 --> 01:17:24.539 Welcome Thank you. Sure. Anyone else. 621 01:17:28.140 --> 01:17:37.409 Okay, well, 1 other thing the teaching assistants are available. There's a couple of people are struggling in the course. So the teaching assistants are. 622 01:17:37.409 --> 01:17:50.460 Are available to give help. They've got official well, virtual office hours. If those are not don't work out for you, you can email them. 623 01:17:50.460 --> 01:17:59.159 And set up other times, so, and I think again, not many people actually contact them. 624 01:18:02.460 --> 01:18:10.140 Question 7 also. Well, that's the 1. we talked to we just talked about. 625 01:18:11.369 --> 01:18:17.159 So, if you got a different answer, then just ask for a re grade. So, and. 626 01:18:24.239 --> 01:18:29.069 A different answer that you think is reasonable, and maybe say why. 627 01:18:30.779 --> 01:18:36.300 So, I'm going to also ask that question for. Okay, so 1st, 7. 628 01:18:36.300 --> 01:18:42.359 Um. 629 01:18:43.949 --> 01:18:47.039 Yeah, so 7 we would take either the 1st, 2. 630 01:18:47.039 --> 01:18:56.760 So, for 7, I should have said where those thing is can have, or might have. 631 01:19:17.694 --> 01:19:18.954 Was created by the. 632 01:19:22.800 --> 01:19:28.500 That'd be better to ask them, but 3 engineering books in 3, mass books. Um. 633 01:19:37.289 --> 01:19:45.420 A different way so I might do this. 634 01:19:49.920 --> 01:19:58.170 I could do a tree for example, the 1st book would be math or engineering. If the 1st book is. 635 01:20:01.470 --> 01:20:10.920 His math, then the 2nd book could be master engineering, but if the 1st book is engineering, the 2nd book has to be math. 636 01:20:10.920 --> 01:20:18.420 And then I could work out the tree of them and so, and count go down the tree and counting things. 637 01:20:18.420 --> 01:20:22.199 That would almost be the easiest way. So. 638 01:20:23.970 --> 01:20:34.050 Another way would be to count combinations where there are. 639 01:20:34.050 --> 01:20:37.289 2 engineering books, side by side. 640 01:20:38.430 --> 01:20:42.449 And then the 3rd engineering bucket somewhere else. 641 01:20:44.970 --> 01:20:59.789 But then how you handle the somewhere else, I could say, work out, you got to be subtracting duplications that maybe to engineering book side by side includes 3 side by side also. So you'd have to subtract that 1 out. So. 642 01:21:01.890 --> 01:21:05.279 I would almost find it easier just to do. 643 01:21:05.279 --> 01:21:08.340 At a tree structured thing. 644 01:21:09.479 --> 01:21:17.970 Let me show you. 645 01:21:33.750 --> 01:21:38.609 Silence. 646 01:21:41.520 --> 01:21:54.449 Okay, so. 647 01:22:03.210 --> 01:22:08.729 So this is a start. 648 01:22:10.680 --> 01:22:15.029 As the 1st book is engineering, the 2nd book has to be math. 649 01:22:15.029 --> 01:22:19.229 Is the 1st book is mass? A 2nd book can be engineering or math. 650 01:22:19.229 --> 01:22:22.800 And then after a math, it can be. 651 01:22:24.239 --> 01:22:29.489 That's our engineering after an engineering it has to be math. Mass is. 652 01:22:35.939 --> 01:22:39.600 After mass, it could be either 1. 653 01:22:42.180 --> 01:22:46.170 After engineering, it has to be math. 654 01:22:47.430 --> 01:22:57.659 Aftermath it could be mass or engineering, but now mass, mass math, the next 2 after we engineering. 655 01:22:58.680 --> 01:23:02.310 Mass mass, mass and X3 after be engineering. 656 01:23:02.310 --> 01:23:09.060 Mass mass engineering the next 1 has to be math. 657 01:23:09.060 --> 01:23:12.180 Okay, so here we're going engineering, mass, math, math. 658 01:23:12.180 --> 01:23:16.739 It has to be not possible. 659 01:23:16.739 --> 01:23:29.760 That cannot be, because you'd have to have 2 consecutive engineering doesn't work out engineering, math, math, engineering next 1 has to be math. And the following 1 has to be engineering. 660 01:23:32.039 --> 01:23:36.359 Um, engineering, math, engineering, math. 661 01:23:37.829 --> 01:23:49.199 Could be master engineering and engineering engineering, mass, enjoying mass engineering. 662 01:23:49.199 --> 01:23:54.029 This here. 663 01:23:54.029 --> 01:23:58.590 And math, engineering, math, engineering. 664 01:23:58.590 --> 01:24:04.380 It has to be math that's been 3 mass so it has to be engineering. 665 01:24:05.520 --> 01:24:10.710 Move down here, mass, math, engineering, math, um. 666 01:24:13.050 --> 01:24:18.060 And it's going to not work out so we've gone. 667 01:24:19.800 --> 01:24:25.199 Yes, is this this and this and then this. 668 01:24:25.199 --> 01:24:33.359 123456, but now now we get the permutations. 669 01:24:35.909 --> 01:24:39.029 But now the engineer, the box. 670 01:24:42.449 --> 01:24:46.319 Have 6 permutations. 671 01:24:49.920 --> 01:24:54.090 And the M box also have 6, the total. 672 01:24:54.090 --> 01:25:00.029 Number would be actually, I get a different answer here. 673 01:25:03.569 --> 01:25:06.779 I think put that you included too they couldn't have because. 674 01:25:06.779 --> 01:25:09.960 Engineers can't be together, right? 675 01:25:12.899 --> 01:25:18.630 Oh, thank you. So this is wrong here. 676 01:25:18.630 --> 01:25:21.659 And this is wrong here. 677 01:25:22.890 --> 01:25:26.699 Mass mass engineering, mass engineering. 678 01:25:29.159 --> 01:25:34.439 Because we get a 2nd. Okay, so 1, 2, 3, 4. 679 01:25:43.470 --> 01:25:48.029 Total number is 4 times. 6 times 6. 680 01:25:48.029 --> 01:25:52.680 It calls 144. okay. 681 01:25:54.119 --> 01:26:06.000 That's probably the easiest way to do it so. 682 01:26:11.310 --> 01:26:15.600 Okay. 683 01:26:15.600 --> 01:26:19.199 In that case, see, you. 684 01:26:19.199 --> 01:26:24.960 Monday, and it's not really worth saving the chat window. It's not. 685 01:26:24.960 --> 01:26:32.760 Well, maybe I should say the chat window then talk to, but okay. 686 01:26:58.020 --> 01:27:02.220 Silence. 687 01:27:04.500 --> 01:27:05.880 Okay.