WEBVTT 1 00:00:28.379 --> 00:00:37.679 Hello. 2 00:00:51.390 --> 00:00:59.579 Hello. 3 00:01:06.810 --> 00:01:18.359 Okay, so. 4 00:01:18.359 --> 00:01:21.480 Good afternoon. People hope you had a nice break. 5 00:01:22.920 --> 00:01:26.459 I am attempting to project straight from my iPad. 6 00:01:26.459 --> 00:01:30.780 Today so showing the slides to people that are remote. 7 00:01:32.730 --> 00:01:38.700 And the theory is also that I'm recording it. 8 00:01:38.700 --> 00:01:42.030 Um. 9 00:01:44.250 --> 00:01:50.159 Trouble is if I watch it, try watching it remotely and see what happens. 10 00:01:50.159 --> 00:01:54.150 Problem becomes that then leads to. 11 00:01:59.819 --> 00:02:01.194 This is working. 12 00:02:22.139 --> 00:02:26.460 Okay. 13 00:02:29.370 --> 00:02:35.520 Yeah, I get okay I can't watch what I'm doing. I just think it's coming out that I can't tell for sure. 14 00:02:35.520 --> 00:02:43.469 Okay, so where are we, we're, we're in chapter 5 we're talking about, um. 15 00:02:43.469 --> 00:02:48.689 2 variables, and just this to remind you. 16 00:02:54.240 --> 00:02:59.460 Oh, Mark sorry about that. 17 00:02:59.460 --> 00:03:02.639 Tomorrow. 18 00:03:02.639 --> 00:03:07.319 In chapter 5, we're talking about 2 variables so I'm glad I want to do today. 19 00:03:07.319 --> 00:03:13.139 Is. 20 00:03:15.180 --> 00:03:18.719 Many examples and. 21 00:03:22.050 --> 00:03:26.789 Real quick here. 22 00:03:30.449 --> 00:03:35.340 Problem is that. 23 00:03:35.340 --> 00:03:40.169 I'm also watching what I'm doing, then it tends to cause. 24 00:03:40.169 --> 00:03:44.729 Audio feedback and I'm not completely certain. 25 00:03:46.705 --> 00:03:57.205 Okay. 26 00:04:15.539 --> 00:04:18.930 Some weird low. Okay. 27 00:04:32.785 --> 00:04:45.774 What. 28 00:05:01.259 --> 00:05:08.369 Hello. 29 00:05:37.709 --> 00:05:42.569 Oh, you can see from the TV. Well, thank you then. Okay so. 30 00:06:02.038 --> 00:06:05.548 Well, that's not my fault. Oh, okay. Okay. 31 00:06:05.548 --> 00:06:11.819 Okay, um. 32 00:06:12.928 --> 00:06:19.678 Hopefully that I write on the blackboard, then it's hard to record it. So, um. 33 00:06:20.699 --> 00:06:23.999 Okay, all I can suggest is that. 34 00:06:25.978 --> 00:06:29.639 If you are there to move there sorry. 35 00:06:29.639 --> 00:06:40.379 Um, 100 years ago, the classrooms at Blackboards on all the walls, and the prompt just rolled in Europe. Um. 36 00:06:40.379 --> 00:06:44.939 And see, the problem is the students go up to the blackboard and solve problems. 37 00:06:44.939 --> 00:06:48.238 Which is probably pedagogically better than today, but. 38 00:06:48.238 --> 00:06:56.218 Um, and as always, if you have other classes that make this technology work better than me. 39 00:06:56.218 --> 00:06:59.788 Tell me how they do it, so. 40 00:07:03.449 --> 00:07:09.449 Okay um, and so I can't see what's projecting if it doesn't work then. 41 00:07:09.449 --> 00:07:13.949 Okay. 42 00:07:17.608 --> 00:07:23.038 I'll call fix so chapter 5, it's, um, basically 2 variables. 43 00:07:26.519 --> 00:07:29.668 And I've got a pile of stuff on the blog. 44 00:07:29.668 --> 00:07:33.598 Um, if I just naturally go up. 45 00:07:33.598 --> 00:07:44.038 Who, um. 46 00:07:44.038 --> 00:07:51.869 Okay, um, this year actually. 47 00:07:58.889 --> 00:08:10.738 Okay, so, and other word technology things is, I'm using tech math on the blog type equations. 48 00:08:10.738 --> 00:08:13.918 And on my iPad, the equations. 49 00:08:13.918 --> 00:08:18.778 Look nice on my think, uh, the equations. 50 00:08:18.778 --> 00:08:21.988 The mathematics does not get rendered, so. 51 00:08:21.988 --> 00:08:32.849 That's, um, how Firefox is implemented on 2 different platforms yeah. Okay. So I want to play with and, um. 52 00:08:32.849 --> 00:08:38.788 Got 3, Sacramento, and then work on some examples that we've got here. 53 00:08:38.788 --> 00:08:46.859 And whatever, so maybe black is easier to see. 54 00:08:46.859 --> 00:08:52.918 Okay, um, 1st, just to show I mean, so the Gaussian, um. 55 00:08:56.849 --> 00:09:02.729 So, I just I know it's going to do mean 0, Sigma equals 1. 56 00:09:08.489 --> 00:09:15.839 Now, I'm gonna show you try to show you now, in theory what we should have integral ActiveX. 57 00:09:15.839 --> 00:09:19.469 D, X minus Anthony to and they should be 1. 58 00:09:19.469 --> 00:09:25.979 Let me say, let's try to prove it. Um. 59 00:09:25.979 --> 00:09:30.448 Well, I could let a equals, um. 60 00:09:36.568 --> 00:09:43.798 Okay, and a also equals, I could do. 61 00:09:46.259 --> 00:09:51.328 That's why I squared over to you. Why. 62 00:09:52.828 --> 00:09:59.399 Okay, I mean, next slide to change the variable to the same thing. But now what I can do is that. 63 00:09:59.399 --> 00:10:03.658 I can multiply I can say that, um. 64 00:10:03.658 --> 00:10:17.339 Squared is going to be 1 over 2 Pi integral. He is minus X squared over 2-Y squared over 2. okay. And all that all that sort of stuff. 65 00:10:18.839 --> 00:10:23.729 Okay, um, oops. 66 00:10:25.198 --> 00:10:29.879 So again, so, hey, squared. 67 00:10:31.889 --> 00:10:46.739 Um, so InterCall minus infinity to infinity minus affinity entity. Um. 68 00:10:58.229 --> 00:11:06.058 Okay, um, now what I'm going to do is I'm going to change variables. Um. 69 00:11:12.958 --> 00:11:21.239 And I'm going to let, um, our equals, um. 70 00:11:24.178 --> 00:11:28.708 I mean, this is and not changed the folder and, um. 71 00:11:29.908 --> 00:11:33.928 And, like, close it or the other 1. 72 00:11:36.509 --> 00:11:44.129 Our science data, um, so we can get our determinant matrix. Um. 73 00:11:49.408 --> 00:11:54.178 So, we're going to get his and get all our partial derivatives. So. 74 00:11:54.178 --> 00:11:58.048 You know, was. 75 00:11:58.048 --> 00:12:01.229 Okay. 76 00:12:01.229 --> 00:12:06.058 Dx data is, um. 77 00:12:07.259 --> 00:12:16.859 When it's assigned data by, um, BR equals, um. 78 00:12:16.859 --> 00:12:23.428 All right. Okay g, Y, by D is, um. 79 00:12:23.428 --> 00:12:29.249 Or close data so then our Nicole is, um. 80 00:12:29.249 --> 00:12:35.399 You know, we put our matrix of DX by. 81 00:12:35.399 --> 00:12:38.578 Did you say to. 82 00:12:38.578 --> 00:12:44.879 Equals, um. 83 00:12:47.849 --> 00:12:51.719 Data science data. 84 00:12:51.719 --> 00:12:54.808 Or. 85 00:12:54.808 --> 00:13:02.788 Post data, and the determinant is our coast is, um. 86 00:13:02.788 --> 00:13:07.739 Okay, um. 87 00:13:07.739 --> 00:13:11.308 Let's go square plus science 1. 88 00:13:11.308 --> 00:13:14.698 Okay, so now, um. 89 00:13:18.119 --> 00:13:23.489 So, again, we're, we're doing this under girl here. We want to find the integral, so. 90 00:13:26.458 --> 00:13:30.808 I am I change things around. Sorry? 91 00:13:32.339 --> 00:13:36.178 Hey, squared is okay what I was trying to do. 92 00:13:36.178 --> 00:13:42.149 Was I was trying to show that if we integrate the density function for the galaxy and I'm going to get 1. 93 00:13:42.149 --> 00:13:46.438 So, I got a equal the inner girl here for the in here. 94 00:13:46.438 --> 00:13:52.528 Um, actually write up here. 95 00:13:54.568 --> 00:13:58.078 Okay, and I want to show that equals 1. 96 00:13:59.129 --> 00:14:06.389 So, what I haven't got there yet I'm halfway I'm part way to it. 97 00:14:06.389 --> 00:14:13.048 So, I want to show that, let me write that down explicitly then, um. 98 00:14:13.048 --> 00:14:16.918 To show it goes 1. 99 00:14:16.918 --> 00:14:28.349 Okay, so okay, so I can say cause that I don't want to find what it is. Well, I can write this also as a wide set of an X. it's the same thing. It's just, you know. 100 00:14:28.349 --> 00:14:37.229 Bulk variable in the equation, but then I can multiply to an, a squared is this 1st, version of 8 times the 2nd, version of a, which is here. 101 00:14:38.458 --> 00:14:43.979 And I can take this and I can rework it and get what we have down here in the middle. 102 00:14:43.979 --> 00:14:48.328 So, now I'm gonna change variables and, um. 103 00:14:49.438 --> 00:14:57.298 I mean, X and Y is Cartesian notation and I want to change the polar notation. The reason I'm gonna do that is it? We'll make the article. 104 00:14:57.298 --> 00:15:10.558 Yes. Okay, well, just go here. 105 00:15:10.558 --> 00:15:13.889 Basic, no, the minus was sorry. 106 00:15:13.889 --> 00:15:17.908 To applying to the whole thing so okay. 107 00:15:17.908 --> 00:15:25.229 So, I got this, I'm going to say, select squared plus Y, squared. 108 00:15:25.229 --> 00:15:29.099 It's our squared and so on and I've got this. 109 00:15:29.099 --> 00:15:32.609 Doctor here are okay, so, um. 110 00:15:37.528 --> 00:15:41.099 You know, what I've got is so now I've got this thing up here. 111 00:15:41.099 --> 00:15:44.698 And I'm going to rework it. 112 00:15:44.698 --> 00:15:52.589 Um, so, let me write that down again. 113 00:15:52.589 --> 00:15:56.788 So, we have a square vehicles. 114 00:16:02.578 --> 00:16:05.698 Okay. 115 00:16:05.698 --> 00:16:12.958 Um, so that is then going to be I rework it. 116 00:16:14.099 --> 00:16:19.139 Then, um, instead of this is going to be either the, the minus R squared over to. 117 00:16:19.139 --> 00:16:25.019 Um, you are, you. 118 00:16:25.019 --> 00:16:28.318 And, um. 119 00:16:29.399 --> 00:16:35.278 So our will be 0 to infinity and data will be 0 to 2. hi. 120 00:16:35.278 --> 00:16:38.609 And we have to bring in a. 121 00:16:38.609 --> 00:16:46.558 Nicole, BN, um, which will be in with our there. I will put a, a, um. 122 00:16:46.558 --> 00:16:52.558 I went over our here, I guess, let's see. 123 00:16:52.558 --> 00:16:55.859 Um. 124 00:17:00.418 --> 00:17:06.088 All right, I'm going over our her, whichever okay now, um. 125 00:17:06.088 --> 00:17:15.838 Something like, that always opened the corrections now. Um, now I could split it apart again since our data don't affect each other. 126 00:17:17.638 --> 00:17:24.719 Integral of to pie and to grow for. 127 00:17:29.638 --> 00:17:36.868 Okay um, okay um. 128 00:17:39.538 --> 00:17:44.249 This thing here goes to pie, um, and this thing here. 129 00:17:44.249 --> 00:17:53.729 Actually, I got the R. 130 00:17:53.729 --> 00:18:01.409 I can't just pull our out and it's, um. 131 00:18:05.759 --> 00:18:12.959 I got our backwards probably, but you can correct me. Okay. Now, this inner girl here is, um. 132 00:18:15.388 --> 00:18:16.288 See, 133 00:18:29.544 --> 00:18:30.894 I got an hour backwards, 134 00:18:30.894 --> 00:18:31.463 but you can, 135 00:18:31.493 --> 00:18:32.903 you can correct me on that. 136 00:18:33.354 --> 00:18:34.223 Um. 137 00:18:39.598 --> 00:18:43.888 This thing if I do it, right will integrate out to 1. um. 138 00:18:47.429 --> 00:18:51.749 You know, I'm skipping a step or 2, and it nets out too. 139 00:18:54.689 --> 00:18:58.199 1, so a square equals 1. 140 00:18:59.969 --> 00:19:06.719 Equals 1, so the undergo 1 over squared or 2 pie either minus X squared over 2. 141 00:19:06.719 --> 00:19:14.788 Dx equals 1 so so it is a so for the calcium is a genuine PDF. So. 142 00:19:16.528 --> 00:19:21.929 Okay, got a little hand way you've been here on just mentioned. 143 00:19:24.088 --> 00:19:31.858 And waving, and maybe, and maybe ours backwards. Um. 144 00:19:34.138 --> 00:19:41.969 Ours backwards, but okay, but it shows you how you can do things. 145 00:19:41.969 --> 00:19:47.009 But this is almost the only thing, this is the 1 variable. Now, if we go to 2 variable. 146 00:19:47.009 --> 00:19:50.098 Um, oh, okay. To scroll up. 147 00:19:51.929 --> 00:19:55.048 So, 2 variable Gaussian, um. 148 00:19:58.858 --> 00:20:05.308 That was a warm up the 2 variable girls and, um. 149 00:20:05.308 --> 00:20:08.909 Is on page. 150 00:20:15.719 --> 00:20:23.788 Okay. 151 00:20:23.788 --> 00:20:32.098 Occurred, um, this example 518 on page 253. 152 00:20:32.098 --> 00:20:42.118 Oh, um. 153 00:20:42.118 --> 00:20:46.199 So the way this works is, um. 154 00:20:47.759 --> 00:20:51.419 So, there's basically so the parameters. 155 00:20:54.388 --> 00:20:58.648 I mean, X has a mean and standard deviation. 156 00:20:58.648 --> 00:21:05.519 Why, as a mean and standard deviation and then there's a new variable role. 157 00:21:05.519 --> 00:21:18.838 That's a correlation coefficient and it's absolutely less than 1. 158 00:21:18.838 --> 00:21:22.469 And it's how the, um. 159 00:21:22.469 --> 00:21:26.189 X and Y, track each other linearly. So. 160 00:21:28.199 --> 00:21:34.979 And again, this is, um, and the way this works, if we plot it, and I'm really bad at flooding. 161 00:21:36.028 --> 00:21:43.858 Plotting stuff, and, um, to so say, if we, you know, plot the density. 162 00:21:46.739 --> 00:21:53.699 Um, it's going to be some, uh, I can do a counter plot actually. 163 00:21:57.179 --> 00:22:06.538 That'd be a counter plot here so here we'd have and say SIG, Max. 164 00:22:06.538 --> 00:22:11.128 1 and 0, it's nice and round. 165 00:22:11.128 --> 00:22:17.608 Um, okay, let's say for all equals 1 half or something. 166 00:22:17.608 --> 00:22:28.048 It's going to look something like this. Um, you see, they're tracking each other a large will suggest that why is large package that. 167 00:22:28.048 --> 00:22:34.259 If we do role equal to -1 half, it will be the other way. They'll track each other. 168 00:22:34.259 --> 00:22:37.979 Opposite okay so, um. 169 00:22:37.979 --> 00:22:43.348 So, there's a formula for that it's on, um, I wrote down the page number. It's on page. 170 00:22:43.348 --> 00:22:50.788 253 and, um, oh, I, what I want to do is play with this equation a little. 171 00:22:50.788 --> 00:22:54.419 And again, if I scroll too fast, let me know. 172 00:22:56.578 --> 00:23:09.689 And what it is, is that, um, X, and Y, kinda go a little sub scripts there where those are the names of the random variables. And the arguments are particular values. 173 00:23:09.689 --> 00:23:23.993 Um, so if we stop right there, then you see role has to be absolutely less than 1. 174 00:23:23.993 --> 00:23:25.433 cause I've got a square root 1-row squared here. 175 00:23:27.173 --> 00:23:28.344 Either the minus. 176 00:23:40.913 --> 00:23:50.723 Okay. And that whole thing is, it's up in exponent there so. 177 00:23:51.028 --> 00:23:54.449 Okay, so it's a lot Messier, um. 178 00:23:54.449 --> 00:24:01.409 If I can try some particular value well, if we do roll exactly equal to 1. yes. A question. 179 00:24:05.729 --> 00:24:12.269 Okay, you can't read my handwriting. Uh, let me rewrite it again more clearly then, um. 180 00:24:24.354 --> 00:24:45.473 Okay. 181 00:24:46.919 --> 00:24:52.798 That's why I like the type stuff into my blog so much, because you can read my typing. Okay. 182 00:24:52.798 --> 00:24:56.429 And that is actually from. 183 00:24:57.628 --> 00:25:05.759 Um, that's that's actually equation 5,018. 184 00:25:07.949 --> 00:25:18.419 No, we can't put row exactly. 1 here, because we'll get a device. We'll get a 0 in the denominator and we'd also get to 0 and then we can try something is, let's say. 185 00:25:18.419 --> 00:25:21.808 That's suppose, let's say row equal. 0. 186 00:25:22.949 --> 00:25:26.429 An f X and Y is, um. 187 00:25:26.429 --> 00:25:31.439 1, over 2, Pi, exponential. 188 00:25:34.199 --> 00:25:42.269 Uh, oh, and by the way the minus sign here, um, goes around everything. So. 189 00:25:42.269 --> 00:25:47.429 Um, squared row. 190 00:25:47.429 --> 00:25:51.479 Plus Y, squared over to. 191 00:25:51.479 --> 00:25:57.808 Okay, and what we can do here is then it will split apart. It will be 1 over. 192 00:25:57.808 --> 00:26:07.048 Root of 2, Pi, exponential minus X squared over 2 times 1 over. 193 00:26:12.989 --> 00:26:22.499 So, if we're always Gerald and the thing to splits the part in single variable for the calcium correct times, a single variable calcium for why. 194 00:26:22.499 --> 00:26:29.278 Okay, that works out nicely, um. 195 00:26:33.298 --> 00:26:46.858 Now, now we can compute the, um, things like the marginal, um. 196 00:26:48.598 --> 00:26:54.719 So, you can get to say the marginal PDF, correct? 197 00:26:55.888 --> 00:26:59.278 And this is, this is for the general are. 198 00:26:59.278 --> 00:27:06.269 General role, and that would be just that we integrate out. 199 00:27:06.269 --> 00:27:10.858 Over Y, um, that whole big expression so. 200 00:27:17.848 --> 00:27:21.808 All right and so on, um. 201 00:27:25.679 --> 00:27:33.028 I'm over and I'm not going in and D, why. 202 00:27:33.028 --> 00:27:38.548 And I'm not going to do it by hand they talk about how to do it in the book, you've completed square and so on. 203 00:27:38.548 --> 00:27:44.999 And, um, and what we'll do is that. 204 00:27:48.269 --> 00:27:53.699 Um, we're going to net out has, um, this Equifax. 205 00:27:53.699 --> 00:28:03.088 It'll just be I E, the marginal for ACS is just a Gaussian in 1. 206 00:28:03.088 --> 00:28:06.808 Variable, regardless of the correlation, so. 207 00:28:06.808 --> 00:28:10.019 So base it. 208 00:28:10.019 --> 00:28:20.638 For this joint marginal PDF. 209 00:28:20.638 --> 00:28:23.969 There's a single girl soon. 210 00:28:23.969 --> 00:28:30.298 So, just some fun stuff like that. 211 00:28:30.298 --> 00:28:40.618 So, um, what I would like to do some more examples from the book, and I've numbered some of them in the blog. Um. 212 00:28:41.939 --> 00:28:46.288 The block here and. 213 00:28:46.288 --> 00:28:50.759 So that was on my blog, that's just a review of things. Um. 214 00:29:00.568 --> 00:29:05.519 I dunno example. 215 00:29:20.939 --> 00:29:28.888 Huh. 216 00:29:34.439 --> 00:29:41.068 Here. 217 00:29:43.858 --> 00:29:47.699 Okay, so so you help you understand. 218 00:29:47.699 --> 00:29:54.868 Okay, you start this on another page or something. Okay. So if I do say. 219 00:29:59.368 --> 00:30:02.608 5.5 on page 238. 220 00:30:02.608 --> 00:30:07.138 Well, we have a we have a packet switch here. Um. 221 00:30:09.358 --> 00:30:15.358 And what we have is, um. 222 00:30:17.578 --> 00:30:21.088 We got in 1 and in 2. 223 00:30:21.088 --> 00:30:24.598 1, 2. 224 00:30:27.419 --> 00:30:33.298 So, it's got 2 input lines into output lines and it does it. And so a packet comes in. 225 00:30:33.298 --> 00:30:37.558 Um, and it may go out on either output. Um. 226 00:30:39.479 --> 00:30:45.929 And, um, and I'm looking at what happens, say in this microsecond or something. 227 00:30:45.929 --> 00:30:49.648 So, and the various probabilities are so. 228 00:30:51.058 --> 00:30:55.378 Each input each and foot port. 229 00:30:59.009 --> 00:31:03.058 May have a packet arrive. 230 00:31:05.189 --> 00:31:13.108 Probably a good 1 half and they're independent. Okay. So. 231 00:31:13.108 --> 00:31:19.259 So now a quarter of the time, both ports get a pack at a quarter of the time. Either part gets a packet. 232 00:31:19.259 --> 00:31:24.298 In a quarter of the time, only in Port 1, a quarter of time only part 2. 233 00:31:24.298 --> 00:31:34.259 And so on, so, um. 234 00:31:34.259 --> 00:31:37.888 That's the 1 thing and the next thing is. 235 00:31:37.888 --> 00:31:44.909 Um. 236 00:31:46.259 --> 00:31:51.028 A packet goes out. 237 00:31:52.078 --> 00:31:56.009 On either output port. 238 00:31:57.719 --> 00:32:01.858 It probably was 1 half. Okay. 239 00:32:03.628 --> 00:32:09.419 So that and again oh, correlate Israel independent. So now we want to look at, um. 240 00:32:09.419 --> 00:32:12.838 I want to look at what happens. Let's say, um. 241 00:32:23.068 --> 00:32:35.483 So, basically, um, it comes for input. 242 00:32:35.483 --> 00:32:38.273 Port 1 are, um. 243 00:32:40.259 --> 00:32:45.298 And which meant no input packet input. 244 00:32:45.298 --> 00:32:49.409 Make this 1 half or a 1. 245 00:32:50.608 --> 00:32:55.739 Is input going. 246 00:32:55.739 --> 00:32:59.608 21 probably was 1 port or. 247 00:32:59.608 --> 00:33:05.038 And a 2 input going to port 2, and probability equals 1 quarter. 248 00:33:05.038 --> 00:33:08.999 And so on, okay, um. 249 00:33:08.999 --> 00:33:21.298 And digital input, or too. Okay. Okay. Same thing. Okay. So now, um. 250 00:33:21.298 --> 00:33:28.469 Well, we can talk about is and we can now look start looking at combos and stuff. So. 251 00:33:30.538 --> 00:33:37.558 On to, um, sorry. 252 00:33:39.239 --> 00:33:47.189 Oh, I haven't flipped it yet. So the bottom line you should be you haven't you don't see data input for too. 253 00:33:48.929 --> 00:33:53.429 Okay, good. Okay. 254 00:33:53.429 --> 00:34:00.959 What I need is a mirror or something, you know, at the back wall reflecting back to me what you guys see. 255 00:34:00.959 --> 00:34:04.828 Or camera. Okay. 256 00:34:07.709 --> 00:34:14.068 And again, the reason, I'm not saying having my phone watch this meeting. 257 00:34:14.068 --> 00:34:26.938 It's for some reason, I, I can't fully muted for some reason. It will start a feedback loop. Okay. Even if I think I'm muting the sound. Okay. So now, what can we do with this? Um. 258 00:34:30.898 --> 00:34:43.409 And, um, what was that? 1. 259 00:34:43.409 --> 00:34:48.539 I don't know. No, I'm flipping. Okay. Um. 260 00:34:51.179 --> 00:34:56.039 So, now we can start doing combos. Let's say X is a number of packets. 261 00:34:59.398 --> 00:35:05.909 Out on or 1 why is the number of packets of it? On part? 2. 262 00:35:05.909 --> 00:35:09.179 And now we can start to, um. 263 00:35:11.518 --> 00:35:15.688 Says that it stays that whatever the size, the different inputs. 264 00:35:16.918 --> 00:35:23.909 And we can start getting things like, um, so we could have something like. 265 00:35:25.949 --> 00:35:30.389 We might have no inputs on either pack on either input ports. So. 266 00:35:30.389 --> 00:35:42.208 For 1 gets nothing, I'll put 4 to nothing and so on, or we might have no inputs on Port 1. and the input on port 2 goes to output 1 and then we'll get. 267 00:35:42.208 --> 00:35:49.199 10 and stuff like that and it goes through all the examples. Um, we might have. 268 00:35:49.199 --> 00:35:53.818 And input on each input port, and both go to, uh, put 4, 2. 269 00:35:53.818 --> 00:36:01.079 In which case I'll put 421 gets nothing up. 4.2 gets both and so on. 270 00:36:01.079 --> 00:36:06.179 And now what we can do is we can start getting probabilities of everything. So. 271 00:36:06.179 --> 00:36:14.489 Um, you know, probability for X and Y, both being 0. 272 00:36:14.489 --> 00:36:19.168 Well, that requires that that will be 1 quarter. 273 00:36:19.168 --> 00:36:26.398 Because it means the only way that could happen is input 1, got nothing in and put 2 got nothing. For example. 274 00:36:26.398 --> 00:36:35.579 Um, it's a little more complicated we might have. 275 00:36:35.579 --> 00:36:43.588 Say, output port 1, got nothing output 42 got a packet. Well, that could be a case of. 276 00:36:44.938 --> 00:36:53.668 Input Port 1 got a packet going out somewhere and input. 42 got nothing which would turn out. Actually will be a quarter. 277 00:36:53.668 --> 00:36:57.418 Um, okay. 278 00:36:57.418 --> 00:37:03.059 There'll be an age plus an 8, because this, this 1st thing here. 279 00:37:03.059 --> 00:37:10.318 Would be the case of packet in on. 280 00:37:10.318 --> 00:37:15.599 Fort 1 going out. 281 00:37:15.599 --> 00:37:20.458 242 and no packet. 282 00:37:22.228 --> 00:37:30.239 And on for 2, and this pack it in on, and this will packet it on the 4th 1 is a half. 283 00:37:30.239 --> 00:37:34.409 It's going out to 4 2 is a half. 284 00:37:34.409 --> 00:37:40.438 Is that half of a half? This will be a quarter. It also has to be no packet in on 42 and all that out. 208. 285 00:37:40.438 --> 00:37:48.659 Yes, number of packets going out on output port 1. 286 00:37:51.353 --> 00:37:52.434 Let me write that down 287 00:38:06.233 --> 00:38:06.983 for 2. 288 00:38:09.478 --> 00:38:15.960 Hmm. 289 00:38:19.139 --> 00:38:29.369 And this is, I want to complete the probability that there's no output packet on Port 1 and, and also 1 output packet on 42 and that's. 290 00:38:29.369 --> 00:38:37.949 And so it's an 88 the 1st dates, assume packet comes in on input port 1 that goes out to port 2. nothing comes in and input port 2. 291 00:38:37.949 --> 00:38:51.630 And that case is, is innate to the other 8 tier would be nothing coming in on Port 1, but a packet coming in port 2 that chooses to go up to 42. That'll be another 8 and they seem to give a quarter. So, we can start doing these calculations. 292 00:38:51.630 --> 00:38:56.550 Okay, okay. 293 00:38:56.550 --> 00:39:01.920 And the book does it in more detail and so on if you want, but that. 294 00:39:01.920 --> 00:39:05.730 Hey, good idea. 295 00:39:10.349 --> 00:39:13.710 Okay. 296 00:39:16.440 --> 00:39:21.090 And, okay, so you get the idea of what you can do here. Um. 297 00:39:27.269 --> 00:39:30.510 I want to go 1. it's a little more complicated now. Um. 298 00:39:31.014 --> 00:39:45.804 Where do we have that? 1 here? Weirder. 299 00:39:45.835 --> 00:39:46.764 Let's see. 300 00:39:49.260 --> 00:39:54.480 Jumping around a little, um. 301 00:40:08.460 --> 00:40:13.679 Go okay. 302 00:40:21.210 --> 00:40:27.539 Okay, let me do example, say. 303 00:40:30.090 --> 00:40:31.375 Page, um, 242 around. Okay. 304 00:40:31.375 --> 00:40:32.005 So now, 305 00:40:33.355 --> 00:40:33.625 here, 306 00:40:34.525 --> 00:40:47.155 what we do is we're transmitting messages and we're going to have to split the messages up into packets or restoring a pile on a disk drive and we split the file up into blocks. 307 00:40:48.929 --> 00:41:01.710 Okay, um, so, um, so, as an earlier example in the book, which does this somewhat, also. 308 00:41:01.710 --> 00:41:09.090 Okay, so and transmit a message. 309 00:41:10.920 --> 00:41:17.309 And equals the length. Okay. Um. 310 00:41:17.309 --> 00:41:23.309 So, what we're doing is, um, you must split the message. 311 00:41:27.599 --> 00:41:32.130 Blocks of length. 312 00:41:33.510 --> 00:41:39.630 M, plus a partial block. Okay. 313 00:41:42.690 --> 00:41:48.059 Also say for this is also for say. 314 00:41:48.059 --> 00:41:57.510 Um, to files on the desk. Okay. 315 00:41:59.909 --> 00:42:06.900 Okay, it's a file on a disk modern discs. There's a block size 4 K bytes for example. 316 00:42:07.980 --> 00:42:16.230 Um, it was 4 K. 317 00:42:16.230 --> 00:42:24.719 Okay, so, um, so am, is the block line? Okay I say, I always set it up here. Okay. 318 00:42:26.130 --> 00:42:30.929 Okay, so, um, so the number of whole blocks. 319 00:42:35.190 --> 00:42:38.699 That's Q, that's going to be, um. 320 00:42:38.699 --> 00:42:44.130 And divided by, em, and take a floor to get down to the next. 321 00:42:44.130 --> 00:42:47.219 And the, um, the link. 322 00:42:49.769 --> 00:42:56.400 Of the fragment oh, there are for the remainder. 323 00:42:56.400 --> 00:43:01.800 That, um. 324 00:43:01.800 --> 00:43:05.849 So, for example. 325 00:43:05.849 --> 00:43:13.949 Let's suppose suppose 10,000. 326 00:43:13.949 --> 00:43:17.309 And that may calls. 327 00:43:17.309 --> 00:43:21.360 Say, 256, um. 328 00:43:21.360 --> 00:43:27.960 Well, let me make it, uh, make it to say 496 then that will be. 329 00:43:29.280 --> 00:43:33.630 2 full blogs and the remainder is. 330 00:43:33.630 --> 00:43:40.139 I think something like that. Okay. 331 00:43:41.849 --> 00:43:47.969 And now we want all I want to do statistics on Q and R. okay. Now. 332 00:43:50.579 --> 00:44:02.159 Okay. 333 00:44:07.679 --> 00:44:13.800 Okay, so let's say so and let's say end is a geometric distribution. 334 00:44:13.800 --> 00:44:27.264 Has geometric distribution. Okay. 335 00:44:27.565 --> 00:44:29.905 And let's say that, um. 336 00:44:32.519 --> 00:44:37.559 Then so, man. 337 00:44:38.639 --> 00:44:42.150 Is going to be, let's say, um. 338 00:44:47.579 --> 00:44:51.389 At the end, so we have a parameter. 339 00:44:51.389 --> 00:44:56.010 Parameter P and and is, um. 340 00:44:59.820 --> 00:45:05.880 Okay, if I did this right now, um. 341 00:45:09.000 --> 00:45:23.250 Okay, okay. 342 00:45:25.980 --> 00:45:30.929 I'm looking like, I've got a slight Erin. Some of all the PS is, um. 343 00:45:49.860 --> 00:45:53.730 Sorry 1 mine yeah. 344 00:45:53.730 --> 00:45:59.760 Courses in the numerator. Okay. Good. So. 345 00:45:59.760 --> 00:46:08.940 Okay, now the question is, what are the, what's the probably distribution for Q and R. 346 00:46:11.519 --> 00:46:18.480 And, um, and by that, just as a quick test, um, some of all the P, then. 347 00:46:18.480 --> 00:46:22.110 Thorough to infinity should be 1. 348 00:46:22.110 --> 00:46:25.349 Okay, now, um. 349 00:46:37.199 --> 00:46:45.389 The question is what's the probability? Distribution? 350 00:46:47.760 --> 00:46:51.119 For Q and R. 351 00:46:52.469 --> 00:46:57.900 Um, so the original message, the length is geometric distribution. 352 00:46:57.900 --> 00:47:06.539 And, okay, what can we do here? Um. 353 00:47:08.280 --> 00:47:12.780 So, again, the probability of then for some and. 354 00:47:18.030 --> 00:47:27.840 Probability to the end. I'm sorry did that right before. Okay. 355 00:47:27.840 --> 00:47:31.289 Well, and equals, um. 356 00:47:32.940 --> 00:47:36.150 It was some Q times and plus are. 357 00:47:39.630 --> 00:47:53.460 So the probability that, um, with some value, and our has an R, has some value that's going to be, um. 358 00:47:59.820 --> 00:48:08.730 For, um, 0, less than equal to less than infinity doesn't matter and are less than. 359 00:48:09.840 --> 00:48:18.389 Okay um, just a 2nd here. Yeah. M is the block size okay. 360 00:48:18.389 --> 00:48:21.420 Um. 361 00:48:24.329 --> 00:48:28.320 So, what are what, what are we going to have to do here? Um. 362 00:48:29.579 --> 00:48:35.159 Well, we want to get a distribution for the number of whole blocks that are used. So. 363 00:48:38.429 --> 00:48:44.789 Is going to be the sum of the probability of some queue. It goes some value. 364 00:48:44.789 --> 00:48:49.440 And are some value hard. 365 00:48:49.440 --> 00:48:56.099 -1. 366 00:49:04.739 --> 00:49:10.829 Okay, um. 367 00:49:10.829 --> 00:49:14.130 Okay, um. 368 00:49:14.130 --> 00:49:19.050 Well, we can pull stuff out and, um. 369 00:49:33.719 --> 00:49:43.679 Okay, and this thing here. 370 00:49:48.119 --> 00:49:58.440 Calls, um, let's see. 371 00:50:00.570 --> 00:50:06.150 Okay, so the whole thing equals, um. 372 00:50:18.929 --> 00:50:24.510 Yeah okay. Um, I did it right? 373 00:50:26.190 --> 00:50:30.210 And this is a geometric distribution. Yes. 374 00:50:35.429 --> 00:50:39.539 Because R, is the length of that partial less block. 375 00:50:40.650 --> 00:50:47.340 Um, so we, this message is end bytes long and the file is then by. 376 00:50:47.340 --> 00:50:52.199 We use, um, a number a queue full blocks. 377 00:50:52.199 --> 00:50:58.019 And each block is laying them +1, partial block of lights 0 to a -1. 378 00:51:00.119 --> 00:51:04.139 So our for remainder queue for quotient. 379 00:51:04.139 --> 00:51:09.179 Okay. 380 00:51:10.349 --> 00:51:18.239 So this is the probability of the number of full blocks and this is geometric. Okay just with a, um. 381 00:51:18.239 --> 00:51:23.639 Different parameters of, um, 80 PM. So this is geometric. 382 00:51:26.219 --> 00:51:35.309 Okay, so, in other words, a number of original, uh, the number, the length of a. 383 00:51:36.510 --> 00:51:49.800 A file message, whatever is geometric then, and we cut it into full blocks. Plus a piece. The probably the distribution for the number of full blocks is also geometric. 384 00:51:49.800 --> 00:51:55.320 So, okay. 385 00:51:59.639 --> 00:52:03.659 Um. 386 00:52:03.659 --> 00:52:08.099 We could also look at the distribution for that last piece here. 387 00:52:08.099 --> 00:52:13.409 What we do, what we would need to do is some over. 388 00:52:13.409 --> 00:52:16.829 2, instead of summing over are. 389 00:52:25.260 --> 00:52:29.610 Okay, so again, probabilty some queue. 390 00:52:29.610 --> 00:52:34.590 They said that are equal some value again. That is, um. 391 00:52:34.590 --> 00:52:38.610 Um. 392 00:52:42.599 --> 00:52:52.739 Plus it's plus here I'm sorry, I take a put a minus somewhere. Well, there's a minus in here I put in, I should take out oh, P of, um. 393 00:52:54.389 --> 00:53:06.960 Here are okay, so probably, we'll do some that are, is something that is we sum over queue. 394 00:53:08.190 --> 00:53:16.320 Okay, okay so we pull out. 395 00:53:16.320 --> 00:53:19.949 1-P and we pull out, um. 396 00:53:22.440 --> 00:53:26.730 You are, and then it's the sum of all the. 397 00:53:27.960 --> 00:53:38.639 M, R. and just saying. 398 00:53:41.070 --> 00:53:45.869 Is going to be, um. 399 00:53:51.840 --> 00:54:02.849 The whole thing equals, um, sorry that it is. 400 00:54:08.309 --> 00:54:17.489 And the whole thing then equals. All right. 401 00:54:19.559 --> 00:54:22.650 Geometric all so, it just a different term. 402 00:54:22.650 --> 00:54:26.699 Scale factor to make it work and this is, um. 403 00:54:30.119 --> 00:54:33.510 So little light, um. 404 00:54:33.510 --> 00:54:42.900 No, real world question is, why would you care about the distribution of the size of these fragments? 405 00:54:42.900 --> 00:54:46.500 Well, let's suppose you designing a file system. 406 00:54:46.500 --> 00:54:54.900 Um, and you may want to, you have to decide how do you handle these, these fractional pieces of the file. 407 00:54:54.900 --> 00:55:06.389 That are less than a full block and some file systems pack these little pieces together in a block. It saves space on the disk, but it makes the file system more complicated. 408 00:55:06.389 --> 00:55:14.250 So, with mathematics like this, you could determine, you know, whether it's worth it and so on. 409 00:55:14.250 --> 00:55:20.639 So, in any case, it's going back to the start here. So this was an example where we had a message. 410 00:55:20.639 --> 00:55:25.980 Of length and bites, or could be a file on a disk. It's the math. It's the same. 411 00:55:25.980 --> 00:55:30.210 And we to transmit the message or to store the file. 412 00:55:30.210 --> 00:55:45.150 We had to chop it up into chunks that were M, bytes long on a just a typical chunk called a block on the disk would be 4,004 K bytes. So your long Messenger file gets cut into chunks of M bytes. Plus a piece left over. 413 00:55:45.150 --> 00:55:50.460 The number of complete blocks is queued the quotient the lengths of the. 414 00:55:50.460 --> 00:55:56.789 Piece left over is our for remainder and I just computed the probability distribution for Q and R. 415 00:55:56.789 --> 00:56:02.219 If the probability distribution of the original link to the original file, was it geometric. 416 00:56:04.260 --> 00:56:16.139 So questions about that, um, and then what I can do is that I can ask is. 417 00:56:16.139 --> 00:56:19.889 Are they independent of each other. 418 00:56:19.889 --> 00:56:25.440 Um. 419 00:56:36.059 --> 00:56:43.079 Well, yes, it's relevant because if we look up here all these little Q. M. plus, are you see. 420 00:56:43.079 --> 00:56:48.510 That equals in, so that's relevant for computing. Um. 421 00:56:50.280 --> 00:56:54.659 Um, let me rehashed that here. Um. 422 00:56:54.659 --> 00:56:59.670 A different color for. Okay. Um, yeah, so. 423 00:57:03.000 --> 00:57:10.800 And that means floor. 424 00:57:10.800 --> 00:57:17.880 And our equals BOD. 425 00:57:17.880 --> 00:57:25.199 And let's say module, um. 426 00:57:25.199 --> 00:57:28.199 Hello. 427 00:57:28.199 --> 00:57:33.929 That means floor. 428 00:57:33.929 --> 00:57:44.219 So, for example, let, let's say, um, an equals, um, 20 and amicable 7. 429 00:57:44.219 --> 00:57:47.579 The 2 will be to. 430 00:57:47.579 --> 00:57:54.989 And, um, just go here. Okay, good. 431 00:57:58.110 --> 00:58:05.250 Okay, um. 432 00:58:08.429 --> 00:58:12.570 Goes to and articles 6. okay. 433 00:58:20.250 --> 00:58:28.980 Okay, so next question is, um, our, um. 434 00:58:30.750 --> 00:58:37.079 Easier to read. 435 00:58:40.380 --> 00:58:41.340 Whatever, 436 00:58:53.184 --> 00:58:54.985 and so the question is, 437 00:58:54.985 --> 00:58:55.434 is. 438 00:58:56.190 --> 00:58:59.250 The probability that queue has some value. 439 00:58:59.250 --> 00:59:03.690 And R, has some value equal the probability that. 440 00:59:03.690 --> 00:59:08.039 Queue of some guy times the probability that our heads that. 441 00:59:08.039 --> 00:59:20.730 2nd, here times a probability that are has that value. 442 00:59:20.730 --> 00:59:27.599 And it turns out if we multiply, then they will be the same. So. 443 00:59:27.599 --> 00:59:35.760 Um, and the answer will be. 444 00:59:35.760 --> 00:59:40.170 I may skip it on cause we're going slowly, but. 445 00:59:42.510 --> 00:59:57.329 Okay um, let me see. 446 01:00:09.570 --> 01:00:13.260 Okay. 447 01:00:20.070 --> 01:00:31.559 Okay, let's try to find another good 1 other questions about that. 448 01:00:53.485 --> 01:00:56.005 Try to pick up another nice 1 here. 449 01:00:56.309 --> 01:01:05.760 I do. 450 01:01:40.650 --> 01:01:46.260 Okay, do another 1 here, um, working with joints. 451 01:01:48.090 --> 01:01:55.469 Joint good example. 452 01:01:55.469 --> 01:01:59.250 5.011 on page. 453 01:02:00.750 --> 01:02:05.880 45, okay um. 454 01:02:10.949 --> 01:02:16.380 So, the random variables X and Y, so what we have is a. 455 01:02:18.510 --> 01:02:28.769 Point, um, and what I'm going to do is I'm going to plot it. The, the book is the equation. 456 01:02:28.769 --> 01:02:39.059 But, um, basically. 457 01:02:39.059 --> 01:02:43.380 And if we are. 458 01:02:43.380 --> 01:02:52.260 Anymore put in another color, just to be different um, anywhere out here to there on 0. 0. um. 459 01:02:54.239 --> 01:02:58.889 And here it 6 times why up here is here. 460 01:02:58.889 --> 01:03:02.639 Here it's why and over here, it's 1. 461 01:03:02.639 --> 01:03:07.289 Okay. 462 01:03:07.289 --> 01:03:10.679 And again, um, just the definition. 463 01:03:10.679 --> 01:03:18.840 X and Y, it was a probability. 464 01:03:18.840 --> 01:03:23.340 That the random variable excellent X and also. 465 01:03:23.340 --> 01:03:29.340 Less than or equal to Y. okay. 466 01:03:29.340 --> 01:03:32.820 So, we have that thing up here so if I pick a point up here. 467 01:03:32.820 --> 01:03:35.849 I'll give you some examples. Um. 468 01:03:38.099 --> 01:03:41.760 I suppose I pick a point here that is, um. 469 01:03:41.760 --> 01:03:49.170 Point 2.1 is going to be point 0.2. 470 01:03:49.170 --> 01:03:56.880 And that's the probability that X is less than point to. And also why is less than that? If I pick another 1? Um. 471 01:03:58.110 --> 01:04:03.570 Here, let's say 2 and point 5. 472 01:04:06.750 --> 01:04:14.099 That is going to be point 5 so probability the exit last week will to 2 will actually be up here. So. 473 01:04:14.099 --> 01:04:17.159 Okay, so now, um. 474 01:04:19.739 --> 01:04:24.750 We can ask, okay, that's the cable distribution function for the 2 variables. 475 01:04:24.750 --> 01:04:28.800 And so if I want to joint 1 for 1 particular. 476 01:04:28.800 --> 01:04:39.659 Variable, um, well, the 1st thing I can do, that's the cumulus, the density function. Um. 477 01:04:43.619 --> 01:04:47.760 Put it in black, actually, the density, um. 478 01:04:52.650 --> 01:04:55.860 Eventually for continuous, which is what this is. 479 01:04:59.369 --> 01:05:02.849 Maybe this do you by. 480 01:05:02.849 --> 01:05:08.610 Why okay, if it's continuous. 481 01:05:08.610 --> 01:05:17.429 So, in in the case of and 0, less wireless into 1, the cumulative thing. 482 01:05:20.070 --> 01:05:24.929 X Y, so the, uh, density. 483 01:05:26.159 --> 01:05:30.869 Was 1. okay. So. 484 01:05:32.820 --> 01:05:35.820 That's in that region. 0, 1, 2 1. 485 01:05:35.820 --> 01:05:39.300 Outside and it's, um, 0, so. 486 01:05:46.079 --> 01:05:52.260 Okay, now, if you want the marginal, um. 487 01:05:52.260 --> 01:06:02.909 See, what it will be, is that. 488 01:06:10.079 --> 01:06:15.030 For the particular X, and for why it being infinity actually. 489 01:06:16.139 --> 01:06:21.659 So, in this case, it will be, um. 490 01:06:24.570 --> 01:06:32.940 Turns out, it will be X if, um, 0 X I think the 1 so. 491 01:06:32.940 --> 01:06:44.909 Um, and dental for why, um, why. 492 01:06:44.909 --> 01:06:48.659 Will be for. 493 01:06:50.760 --> 01:06:57.269 So, we can get, um, if I can scroll back without making a for a moment. 494 01:06:59.610 --> 01:07:03.780 So, here, I just drew the, um, came up the function so. 495 01:07:05.369 --> 01:07:09.329 So, for why being very large, an X being finite. 496 01:07:09.329 --> 01:07:14.699 It's up at the top here. It's, um, the CDF is X. um. 497 01:07:16.019 --> 01:07:23.099 So, that's the, the single fraction for why it'd be a very large infinity and then be why. 498 01:07:23.099 --> 01:07:28.769 So. 499 01:07:31.320 --> 01:07:36.179 Okay. 500 01:07:56.815 --> 01:08:00.985 Now, we could also, since he's the kingdom of those things can also find the density. 501 01:08:01.260 --> 01:08:04.260 Webex will be 1 again there. 502 01:08:06.239 --> 01:08:10.530 And for why will also be 1. 503 01:08:12.570 --> 01:08:17.340 So, now we can ask our X and Y, independent. Um. 504 01:08:27.689 --> 01:08:35.340 And again, let's just do the interesting region. Interesting region is, um. 505 01:08:39.510 --> 01:08:43.739 What we have here, is that the, um. 506 01:08:46.500 --> 01:08:51.510 1, and a half of X equals 1 and half of why. 507 01:08:51.510 --> 01:08:55.109 Equals 1, so this equals. 508 01:08:55.109 --> 01:08:59.039 Times for why it says yes. 509 01:08:59.039 --> 01:09:07.470 We are independent, so okay. 510 01:09:10.350 --> 01:09:19.199 Um. 511 01:09:19.199 --> 01:09:30.359 So, let me do 1 more example, maybe if we got time, we might have were random variables of different types. Um. 512 01:09:31.590 --> 01:09:35.069 This could be 531 page 247. 513 01:09:50.725 --> 01:09:53.125 So, we might have, um. 514 01:09:53.399 --> 01:09:56.399 Noisy transmission channel let's say. 515 01:10:01.229 --> 01:10:05.460 Transmission, um. 516 01:10:09.090 --> 01:10:14.670 X is the input and it's -1 or +1. 517 01:10:16.560 --> 01:10:22.409 Okay, um, we had a noise. 518 01:10:22.409 --> 01:10:26.729 And it's due in a form for -2 to 2. 519 01:10:26.729 --> 01:10:31.260 And then the output is why he goes. 520 01:10:31.260 --> 01:10:45.960 Okay, and so the input is discreet to possible values -1 or 1 but the output is continuous. It gets this continuous noise added to it. So it's smeared out. 521 01:10:45.960 --> 01:10:53.130 Um, but now what we're very interested in is, um. 522 01:10:54.300 --> 01:11:02.640 Okay to guess what was transmitted. 523 01:11:07.619 --> 01:11:15.779 And instead of yes, you could say in for or determine, guess is a nice 1 still word but you could, you know, if you're. 524 01:11:15.779 --> 01:11:21.569 You know, you could expand it. Okay. So what we want is something like this um. 525 01:11:24.149 --> 01:11:29.430 Let's say, um. 526 01:11:32.220 --> 01:11:43.890 You know, things like given if I received output signal was less than greater than a certain cut off. 527 01:11:43.890 --> 01:11:48.000 What's the probability? The transmitted signal was +1. 528 01:11:48.000 --> 01:11:51.569 Or -1 or something. Okay, so. 529 01:11:51.569 --> 01:12:00.810 And we, and if this is more than 50% or what? Well, depending on input to prior probabilities, we use this to have a rule for decoding the noisy output. 530 01:12:00.810 --> 01:12:09.149 Okay, so so we can do things like that. Um. 531 01:12:12.060 --> 01:12:19.560 And we could, how could we do it? Um. 532 01:12:25.289 --> 01:12:31.529 A modifying example. Well, if we have the probability say that, um. 533 01:12:39.270 --> 01:12:45.960 Let's do this direction, um, you know, the, the, you know, bring base rolling and all. 534 01:12:45.960 --> 01:12:51.539 So, the probability that why is less than a certain value, given this well. 535 01:12:51.539 --> 01:12:59.310 Why is X plus noise. Okay so this is the probability of X plus noise. Let's say a certain value given. 536 01:12:59.310 --> 01:13:02.789 Equals 1, which is the probability that the. 537 01:13:02.789 --> 01:13:06.300 Noise is less than or equal to Y -1. 538 01:13:06.300 --> 01:13:13.050 Okay, and, um. 539 01:13:14.340 --> 01:13:19.649 And the noise is uniform from -2 to 2. I think the books yes. 540 01:13:19.649 --> 01:13:29.010 So, and so now we can start calculating this. I'll just give you some here. Let's say, um. 541 01:13:31.140 --> 01:13:38.909 If let's say why it was 0, then the problem with the noise is last year for -1. 542 01:13:41.130 --> 01:13:45.989 Um, this will be 1 quarter, for example um. 543 01:13:47.250 --> 01:13:54.300 Sense the noise is uniform from -2 the 2 so you can start doing these calculations. Yes. 544 01:13:56.729 --> 01:14:02.550 The book does it more general but you got the idea. So. 545 01:14:02.550 --> 01:14:08.100 Okay. 546 01:14:11.039 --> 01:14:14.279 And you can say, you can see book. 547 01:14:17.699 --> 01:14:21.779 Um. 548 01:14:24.689 --> 01:14:29.369 On the book works out the probability that X equals 1. 549 01:14:29.369 --> 01:14:33.449 And also why let's say close their own. So. 550 01:14:35.069 --> 01:14:42.420 Okay, I do it Thursday. Okay, so just let me refresh a little what I was doing today. 551 01:14:42.420 --> 01:14:47.880 As I showed you that for the calcium, the normal distribution that they. 552 01:14:47.880 --> 01:14:51.539 Claimed density function, in fact, works. 553 01:14:51.539 --> 01:14:56.010 That if we integrate it from minus infinity to infinity, we, in fact, do get 1. 554 01:14:56.010 --> 01:15:06.840 So, it's legal, and then I showed you the Gaussian function for 2 variable texts and Y, which are correlation coefficient. Um. 555 01:15:06.840 --> 01:15:12.000 Roll roll from -1 to 1 so it's a lot more complicated. 556 01:15:12.000 --> 01:15:21.000 But if you take it, um, and, you know, the marginal on X, and you get a single variable Gaussian, which is good. 557 01:15:22.920 --> 01:15:27.510 And then I felt, um, some examples like a packet switch. 558 01:15:27.510 --> 01:15:31.229 Um, were to input ports and to output ports. 559 01:15:31.229 --> 01:15:38.880 And each input port independently has an input arriving with probability 1, half and the next microsecond. Maybe. 560 01:15:38.880 --> 01:15:44.640 And each input packet independently goes to either of the 2 output ports. 561 01:15:44.640 --> 01:15:49.770 So, we can compute probabilities for the output. 562 01:15:49.770 --> 01:15:54.479 You know, we're different combos of the output ports, getting different numbers of packets and so on. 563 01:15:54.479 --> 01:15:59.250 And then I spent more time on a more complicated 1 with the. 564 01:15:59.250 --> 01:16:05.640 Long message or file that to be transmitted or stored has to be cut up into blocks. 565 01:16:05.640 --> 01:16:13.229 A whole number of blocks each of lane them and might be a 5,000 whatever plus a, um. 566 01:16:13.229 --> 01:16:19.229 Fractional block and we could do probability distributions. If the input message length is. 567 01:16:19.229 --> 01:16:25.350 Geometric then the number of whole blocks, there's also geometric and is the length of the, um. 568 01:16:25.350 --> 01:16:30.300 Fractional last packet is also geometric. 569 01:16:30.300 --> 01:16:38.039 And I, and then I started to show you that the queue. The number of whole blocks is actually independent of our. 570 01:16:38.039 --> 01:16:43.680 The fractional last block I worked with 2 partway. I showed you how to compute it. If I didn't draw it all through. 571 01:16:44.699 --> 01:16:48.300 And then was showing you here a mixed, um. 572 01:16:48.300 --> 01:16:51.810 The 2 random variables are. 573 01:16:51.810 --> 01:17:02.340 That 1 might be discreet, and the other might be continuous that could easily happen when you're transmitting a discrete bit 0 or 1-1 or +1. And it gets a continuous noise. 574 01:17:02.340 --> 01:17:09.539 Um, add it to it, and this is a relevant problem to work out because then you're receiving a continuous. 575 01:17:10.590 --> 01:17:15.210 Transmitted variable and from it, you want to. 576 01:17:15.210 --> 01:17:20.609 Do something like base role and infer what the discrete transmitted signal was? 577 01:17:21.989 --> 01:17:25.199 So, we're in chapter 5 to, um. 578 01:17:25.199 --> 01:17:32.039 And variables that are related to each other, probably and will continue this on Thursday. 579 01:17:33.090 --> 01:17:38.760 If there's any questions okay. 580 01:17:38.760 --> 01:17:44.220 With luck, I was recording this, although I never trust these things. 581 01:17:44.220 --> 01:17:48.989 So, um. 582 01:18:02.579 --> 01:18:06.539 Okay. 583 01:18:12.210 --> 01:18:17.550 Testing. 584 01:18:19.020 --> 01:18:23.039 Hello. 585 01:18:23.039 --> 01:18:27.840 Hello. 586 01:18:27.840 --> 01:18:31.409 Hello. 587 01:18:31.409 --> 01:18:34.470 Okay. 588 01:18:36.239 --> 01:18:42.210 Hello. 589 01:18:42.210 --> 01:18:45.270 But the variable this before I forget. 590 01:19:00.270 --> 01:19:07.350 Hello. 591 01:19:09.270 --> 01:19:19.140 Okay, so I got on here, but I'm not sure how they went from here to here. 592 01:19:20.399 --> 01:19:29.130 Um, it's a geometric series and the, some of the geometric series well, you can pull out, um. 593 01:19:30.899 --> 01:19:36.210 Yeah, well I'll give it to you. Um, here the other time. 594 01:19:37.529 --> 01:19:43.770 I'm guessing they also have a formula for this 1. yeah. Um, I took this in high school. 595 01:19:45.720 --> 01:19:50.159 The sum of all the ages again, and it was 0 to infinity. 596 01:19:50.159 --> 01:19:55.770 Is I think 1 over 1 minor say you can check me on this okay. I could have gotten it wrong. 597 01:19:55.770 --> 01:20:02.760 And then the sum of all the, a, to Z, and equals 0 to, um. 598 01:20:02.760 --> 01:20:07.649 I don't know be -1, let's say. 599 01:20:10.619 --> 01:20:14.069 Minus ages the B1 might say. 600 01:20:14.069 --> 01:20:19.590 You can check that. Okay. I may have it wrong, but something like that. 601 01:20:19.590 --> 01:20:25.529 And, um, so that so, in here, um. 602 01:20:26.880 --> 01:20:37.350 So, the 1st thing, they pull up the 1-feed up as a factor. Okay. And now what the summing over is K. so, P to the Q. M. 603 01:20:37.350 --> 01:20:43.140 That's a factor. So they pull that up right now there's the summing P to the K. 604 01:20:43.140 --> 01:20:48.329 And that is 1-and minus B foreign, like. 605 01:20:48.329 --> 01:20:52.020 Okay, so, um. 606 01:20:53.430 --> 01:20:59.520 With the, uh, so they say 2 in the division of them. 607 01:20:59.520 --> 01:21:11.430 Q equals divisional and, um, correct floor, take the floor to the next next floor. It means that it's no, it's not a square bracket. It's a thing. 608 01:21:11.430 --> 01:21:14.699 Um, it's like that. 609 01:21:14.699 --> 01:21:19.229 Oh, yeah sort of like a now and then the backwards though. 610 01:21:20.340 --> 01:21:25.170 And it means floor, it means just take off the decimal points. Exactly. 611 01:21:25.170 --> 01:21:30.869 Um, does that mean when they do that? It's actually like, westing to. 612 01:21:30.869 --> 01:21:38.670 Um, less than 2, like, but I get an Apple by the way. Um. 613 01:21:38.670 --> 01:21:45.689 Hm. 614 01:21:47.220 --> 01:21:51.239 Okay. 615 01:21:51.239 --> 01:22:03.210 Um, well, there's a little example right here, I'd say. 616 01:22:07.470 --> 01:22:12.239 And equals 20 M equals. 7 is 2 and 6. 617 01:22:27.060 --> 01:22:30.659 You know, I'm going to go to the union access budget. 618 01:22:30.659 --> 01:22:37.350 Oh, I gotta pick up, uh, yeah, we're gonna go in a 2nd, but I had to pick up the. 619 01:22:37.350 --> 01:22:42.600 Oh, okay, okay. And remember you got your ratio here so. 620 01:22:42.600 --> 01:22:56.875 Yeah, I have a question about what are the top, I guess 1 of the former problems. So I know how to find the, um, by finding the joint PDF. 621 01:22:57.625 --> 01:22:59.725 However, I'm not sure what the, um. 622 01:23:00.000 --> 01:23:11.189 Like, the 1, I should use for the, um, intervals of integration. It'd be better to ask the, because they created the question. Okay. I see it. Yeah. Um, I did it. 623 01:23:11.189 --> 01:23:16.409 You, what I did is i1st, pull out see. 624 01:23:16.409 --> 01:23:21.029 Right. And then I integrated from excellent. 625 01:23:21.029 --> 01:23:25.050 Both above 0 and Beth and pile for 2. so I just integrated. 626 01:23:25.050 --> 01:23:30.630 The for both of them, I went from 0 to 5 over to. 627 01:23:30.630 --> 01:23:35.850 Oh, and I found the error to okay. And I, and then I found the area then. 628 01:23:35.850 --> 01:23:40.470 Then, you know, so the 1 divided by. 629 01:23:40.470 --> 01:23:47.550 Too, that's what I got to see, because 3 times the all the area, and needs to be convert to. 630 01:23:47.550 --> 01:23:54.090 You know, 1 okay, so then so you're saying you did X for both of them from 0 to priority. So. 631 01:23:54.090 --> 01:23:58.199 I just. 632 01:23:58.199 --> 01:24:03.899 2, why would it be. 633 01:24:03.899 --> 01:24:09.239 0 priority about, because it only goes up from here to. 634 01:24:09.239 --> 01:24:12.359 Okay, okay. Then thanks. 635 01:24:12.359 --> 01:24:18.119 Okay, thanks. Mm. Hmm. And the other ones was module. 636 01:24:18.119 --> 01:24:22.319 Almost like a model. 637 01:24:22.319 --> 01:24:27.359 Yeah, it's, um, and it's the remainder. 638 01:24:28.439 --> 01:24:32.369 So, if it's 2720 years, 2 times 7+6. 639 01:24:32.369 --> 01:24:38.939 But I'm confused is that it's top of that, but I wasn't sure where it used it in the of the question or. 640 01:24:38.939 --> 01:24:44.970 We just it's a, something we gotta know where you gotta know where. 641 01:24:51.359 --> 01:24:54.630 For here, um, we don't use the, uh. 642 01:24:54.630 --> 01:25:03.539 You know, floor or module anywhere. Well, we use them to compute to or is that is that how we come up with this? 643 01:25:03.539 --> 01:25:08.189 Take a note that formula right there cause that is N. timestamp. Plus are. 644 01:25:08.994 --> 01:25:22.885 And well, for example, when you're summing are from 0, to -1 K for -1, that's because the legal values for our from 0 to -1. and so so it's implicit in, Ah. 645 01:25:24.899 --> 01:25:29.189 Oh, so I thought. 646 01:25:29.189 --> 01:25:38.310 So, I didn't really use this to find the I just went back to 5.3. mm. Hmm. And then. 647 01:25:41.130 --> 01:25:56.100 It's quite a bit up here. Yeah, I skipped 5.0. it was too simple but, uh, 5.3 here we are just now. Yeah, it just explained it. So I didn't really use it to find. So I was a little bit confused on why I gotta use it. 648 01:25:56.100 --> 01:26:04.170 So, um, what I'm getting is like, the, uh, what is the floor and the module is used to find the. 649 01:26:04.170 --> 01:26:08.430 Boundaries, well he is to find, um. 650 01:26:08.430 --> 01:26:14.220 In this case, it's used to find the oh, okay. Go back. We'll bring that thing back up again. Um. 651 01:26:14.220 --> 01:26:17.340 Not to 5.3 the bigger 1, but. 652 01:26:17.340 --> 01:26:20.520 Um, it's what makes. 653 01:26:20.520 --> 01:26:23.699 These things like this valid. 654 01:26:23.699 --> 01:26:30.510 Cause and equals 2 times M plus K or 2 times 10+are. And the reason that's. 655 01:26:31.979 --> 01:26:35.579 You know, and then that works because. 656 01:26:36.930 --> 01:26:40.170 Do an M, or, you know. 657 01:26:40.170 --> 01:26:44.609 That's that's a computer that way. Otherwise this thing wouldn't work there. 658 01:26:45.810 --> 01:26:51.930 We just gotta memorize it or well, that's any, you know, you're dividing a file into. 659 01:26:51.930 --> 01:26:55.350 Blocks plus a fragment that, um. 660 01:26:55.350 --> 01:27:00.750 That's the form of the number of blocks, plus the size of the fragment. Yeah. Um. 661 01:27:00.750 --> 01:27:05.760 Um, if you think about it, it it does make sense, but, um. 662 01:27:10.590 --> 01:27:14.159 So, you know, run through some examples of, like. 663 01:27:14.159 --> 01:27:17.250 Hello. 664 01:27:18.359 --> 01:27:23.819 Um, that might be the best way. I could. 665 01:27:23.819 --> 01:27:26.880 I can think about it, but. 666 01:27:26.880 --> 01:27:31.260 Hello. 667 01:27:31.260 --> 01:27:35.760 Thank you all for, uh, I've, it's been. 668 01:27:35.760 --> 01:27:43.050 It's only been away for me. Yeah, so well, that's actually go on vacation there. No, sir. 669 01:27:43.050 --> 01:27:47.069 So, I'll look into. 670 01:27:48.390 --> 01:27:52.289 No. 671 01:27:52.289 --> 01:27:59.970 Yeah, you think about it work them out and may start making sense eventually. So. 672 01:27:59.970 --> 01:28:05.460 Eventually the other parts, otherwise, everything that seems like, you know. 673 01:28:05.460 --> 01:28:10.020 But, I mean, if the course is too easy, you won't think you were getting your money's worth. 674 01:28:11.430 --> 01:28:15.329 Hello. 675 01:28:27.840 --> 01:28:33.119 Yeah. 676 01:28:33.119 --> 01:28:39.659 Too many different types of electronics here. I said 2 different types of. 677 01:28:41.699 --> 01:28:44.939 Hello. 678 01:28:47.939 --> 01:28:51.090 Huh. 679 01:29:11.069 --> 01:29:14.130 All right. Okay. 680 01:29:16.170 --> 01:29:23.609 It'd be worth if someone leaves the laptop here months ago class. 681 01:29:23.609 --> 01:29:37.439 Okay. 682 01:29:40.409 --> 01:29:47.819 Hello. 683 01:29:50.005 --> 01:30:47.604 Hello. 684 01:30:56.850 --> 01:31:07.500 Okay. 685 01:31:10.859 --> 01:31:19.560 Hello. 686 01:31:21.750 --> 01:31:24.869 Okay. 687 01:32:59.248 --> 01:33:04.288 Hello. 688 01:33:40.109 --> 01:33:45.809 Hello. 689 01:34:05.578 --> 01:34:12.569 Hello. 690 01:34:17.368 --> 01:34:31.828 Hello. 691 01:34:54.234 --> 01:35:03.623 Okay. 692 01:35:13.073 --> 01:35:57.533 Hello. 693 01:37:25.918 --> 01:37:29.939 Okay. 694 01:37:37.439 --> 01:37:42.208 Hello. 695 01:37:59.458 --> 01:38:03.088 Hello. 696 01:38:44.904 --> 01:39:17.394 Hello. 697 01:39:22.679 --> 01:39:35.099 Hello. 698 01:39:42.868 --> 01:39:49.378 Okay. 699 01:39:52.314 --> 01:40:22.344 Hello. 700 01:41:41.668 --> 01:41:45.418 Hello. 701 01:41:50.819 --> 01:41:58.918 Hello. 702 01:42:02.399 --> 01:42:07.319 Hello. 703 01:42:13.618 --> 01:42:24.088 Okay. 704 01:42:42.269 --> 01:42:46.948 Hello. 705 01:42:58.559 --> 01:43:06.868 Hello. 706 01:44:03.384 --> 01:44:27.444 Okay. 707 01:44:47.248 --> 01:44:55.918 Okay. 708 01:45:34.613 --> 01:45:53.363 Okay. 709 01:46:20.363 --> 01:46:37.493 Hello. 710 01:47:24.054 --> 01:48:02.963 Okay. 711 01:48:08.698 --> 01:48:20.099 Okay. 712 01:48:25.948 --> 01:48:30.088 Hello. 713 01:48:33.444 --> 01:49:29.453 Hello. 714 01:49:40.439 --> 01:49:43.618 All right. 715 01:50:21.809 --> 01:50:26.668 Hello. 716 01:50:46.559 --> 01:50:55.168 Hello. 717 01:50:56.368 --> 01:51:00.748 Hello. 718 01:51:06.564 --> 01:51:15.923 Hello. 719 01:51:35.519 --> 01:51:45.389 Okay. 720 01:51:57.503 --> 01:52:38.394 Hello. 721 01:52:40.373 --> 01:53:19.823 Hello. 722 01:53:29.604 --> 01:53:45.353 Hello. 723 01:53:57.418 --> 01:54:01.168 Hello. 724 01:54:01.168 --> 01:54:05.969 Hello. 725 01:54:19.918 --> 01:54:24.448 Okay. 726 01:54:47.458 --> 01:54:54.149 Hello. 727 01:55:12.118 --> 01:55:20.009 Hello. 728 01:55:45.298 --> 01:55:51.958 Hello. 729 01:55:51.958 --> 01:55:55.139 Hello. 730 01:56:03.658 --> 01:56:13.588 Hmm. 731 01:56:13.588 --> 01:56:17.009 Okay. 732 01:56:17.009 --> 01:56:23.038 Hello. 733 01:56:30.779 --> 01:56:34.859 Okay. 734 01:56:39.779 --> 01:56:45.118 Hello. 735 01:56:47.423 --> 01:57:17.123 Okay.