WEBVTT 1 00:00:46.679 --> 00:00:51.090 Okay, good afternoon. Everyone. 2 00:00:51.090 --> 00:00:56.429 1st question is, and you hear me. 3 00:01:01.859 --> 00:01:07.469 Care necklace so this is probability. Class 17 I guess. 4 00:01:07.469 --> 00:01:11.280 March 29th and also. 5 00:01:11.280 --> 00:01:16.769 If you have trouble seeing my shared screen, tell me and then. 6 00:01:18.000 --> 00:01:22.230 Oh, okay. 7 00:01:22.230 --> 00:01:27.810 So, what's happening today is. 8 00:01:27.810 --> 00:01:35.250 Or to variable stuff, and chapter 5, I'll do the expected value of them. 9 00:01:35.250 --> 00:01:41.010 Of a function and see, now, thank you. 10 00:01:44.700 --> 00:01:47.879 It it does work. 11 00:01:47.879 --> 00:01:52.439 Okay, so. 12 00:01:52.439 --> 00:01:55.709 Start off by doing this thing that. 13 00:01:57.870 --> 00:02:02.730 Caused a bit of a hassle last time. 14 00:02:04.379 --> 00:02:09.509 Silence. 15 00:02:09.509 --> 00:02:15.750 Okay. 16 00:02:15.750 --> 00:02:19.889 So. 17 00:02:19.889 --> 00:02:23.520 Just to click. 18 00:02:23.520 --> 00:02:28.740 Quick reminder about some things, just this actually working. 19 00:02:30.060 --> 00:02:37.740 Silence. 20 00:02:37.740 --> 00:02:42.569 On here. 21 00:02:44.729 --> 00:02:55.560 2nd screen bearing was working 10 minutes ago. 22 00:02:55.560 --> 00:02:58.650 Working again. 23 00:03:01.919 --> 00:03:09.330 Silence. 24 00:03:09.330 --> 00:03:13.979 Silence. 25 00:03:13.979 --> 00:03:18.419 Silence. 26 00:03:19.560 --> 00:03:23.280 Silence. 27 00:03:23.280 --> 00:03:27.150 Silence. 28 00:03:31.469 --> 00:03:36.000 Silence. 29 00:03:36.000 --> 00:03:41.639 Silence. 30 00:03:41.639 --> 00:03:47.219 Silence. 31 00:03:51.210 --> 00:04:01.199 Testing testing. 32 00:04:01.199 --> 00:04:05.789 For class, and it was working before class. 33 00:04:08.669 --> 00:04:12.300 Right now, of course. 34 00:04:12.300 --> 00:04:17.519 Some more time. 35 00:04:20.250 --> 00:04:23.699 Silence. 36 00:04:23.699 --> 00:04:28.259 Silence. 37 00:04:32.129 --> 00:04:36.478 Silence. 38 00:04:36.478 --> 00:04:44.848 Silence. 39 00:04:50.428 --> 00:04:55.048 Silence. 40 00:04:55.048 --> 00:04:59.009 Silence. 41 00:05:00.569 --> 00:05:08.129 Silence. 42 00:05:09.598 --> 00:05:16.738 Good. Okay. Wow. 43 00:05:18.598 --> 00:05:26.069 Finally, and let's see. 44 00:05:28.769 --> 00:05:33.509 Good. Okay. Now. 45 00:05:36.778 --> 00:05:40.288 Silence. 46 00:05:45.749 --> 00:05:49.168 Silence. 47 00:05:49.168 --> 00:05:54.389 So just a quick review if we have something like. 48 00:05:55.829 --> 00:06:03.059 F x X. okay. So that's a probability density function. 49 00:06:03.059 --> 00:06:08.728 And if we integrate after we get 1. 50 00:06:08.728 --> 00:06:13.678 Putting the I'm putting the subscript to big X. 51 00:06:13.678 --> 00:06:22.408 The thing that's right here to indicate that this is that's the name of the distribution big X. 52 00:06:22.408 --> 00:06:28.139 Okay, and now we have the expected value just as a reminder is the undergo X. 53 00:06:30.088 --> 00:06:36.538 Okay, if the article exists integrating minus infinity, if you need to whatever and give an example. 54 00:06:36.538 --> 00:06:40.858 Uniform then. 55 00:06:42.209 --> 00:06:45.538 F, X equals 1 to. 56 00:06:45.538 --> 00:06:50.728 Okay, and interval 0 to 1 X. 57 00:06:50.728 --> 00:06:56.009 I got a little 11 and then the main is. 58 00:06:57.178 --> 00:07:02.009 X x x equals 0. 0 2 1. 59 00:07:02.009 --> 00:07:06.988 Of X, the cost. 60 00:07:06.988 --> 00:07:10.288 Okay, so far. 61 00:07:10.288 --> 00:07:15.478 So far, so good now, if we define a new random variable. 62 00:07:15.478 --> 00:07:23.399 Let's just say we to find a new random variable. Let's say why equals to X or something. 63 00:07:23.399 --> 00:07:26.968 So now. 64 00:07:26.968 --> 00:07:36.028 So, now, the question is, I noticed I got a super secret, why they're. 65 00:07:38.069 --> 00:07:44.218 Well, we can do it with probabilities and. 66 00:07:47.309 --> 00:07:57.418 Probability say again, lower cases of particular value. Upper cases is the name of their random variable. 67 00:08:00.059 --> 00:08:06.358 That is going to be. 68 00:08:06.358 --> 00:08:13.079 F of Y. D. why now? Since why is why is 2 effects. 69 00:08:13.079 --> 00:08:21.178 Then it's going to you that some value of X. 70 00:08:34.918 --> 00:08:40.469 So, if why is some value, why X is going to be why over 2? Because why it was 2 x so. 71 00:08:43.379 --> 00:08:50.129 And also whatever we put here for DX, it doesn't actually matter. 72 00:08:50.129 --> 00:08:55.619 Because that's going to be f of. 73 00:08:55.619 --> 00:09:04.318 Times X, and those things are, are. 74 00:09:04.318 --> 00:09:07.438 I actually can say X here. 75 00:09:10.288 --> 00:09:22.649 Now, and whatever this happens, we could get whatever we have for the interval. It doesn't matter. And any case what it notes out to. 76 00:09:22.649 --> 00:09:25.708 Waving my hand, so. 77 00:09:25.708 --> 00:09:28.979 Nets out to f. 78 00:09:30.839 --> 00:09:34.048 Is going to be. 79 00:09:36.178 --> 00:09:41.369 I mean, I mentioned this before, but it's it's a little confusing to know if I motivated. 80 00:09:41.369 --> 00:09:45.119 A little, okay, let's suppose a uniform thing here is this. 81 00:09:46.979 --> 00:09:53.158 The uniform act, so this is so this will be. 82 00:09:53.158 --> 00:10:00.658 F effects and from 0 to 1, it's going to be 1. otherwise it's 0. that's what makes the integral work. 83 00:10:00.658 --> 00:10:08.129 Um, now what we have, why X? So basically f, a. why. 84 00:10:09.208 --> 00:10:13.739 Going to be greater than 0 for 0. listen to. 85 00:10:13.739 --> 00:10:23.278 X axis paused it from 0 to 1. so why it's going to be 0 It took his wife was 2 effects. So now if we have integral f of why. 86 00:10:25.948 --> 00:10:31.048 What this implies that half of Y equals 1 half. 87 00:10:31.048 --> 00:10:36.719 And more generally. 88 00:10:38.188 --> 00:10:44.668 Is what the scale factor is going to be if we eFax. 89 00:10:44.668 --> 00:10:48.208 It's going to be D. Y, over D X. 90 00:10:49.408 --> 00:10:55.318 Um, in a sense, and I'm being a little sloppy here because. 91 00:10:58.229 --> 00:11:02.578 So this is the probability of the random variables and little interval. 92 00:11:02.578 --> 00:11:07.469 Of what's the why? And that's this has got to be equal to. 93 00:11:11.278 --> 00:11:16.889 And that's what we get to thing up there, because he's being a little sloppy basically that. 94 00:11:16.889 --> 00:11:21.359 Now, if I use this in a more general thing, just wife was 2 X. 95 00:11:21.359 --> 00:11:26.428 What, if no, let's try. 96 00:11:26.428 --> 00:11:30.418 Y, equals X squared so now. 97 00:11:31.589 --> 00:11:36.839 Why would he X equals 2 acts so what this means now, is that f of Y. 98 00:11:36.839 --> 00:11:42.448 Equals. 99 00:11:42.448 --> 00:11:46.438 Over 2 X and. 100 00:11:50.068 --> 00:11:53.099 So, what this means is, I actually. 101 00:11:54.239 --> 00:11:57.298 That's going to be bigger so. 102 00:12:00.359 --> 00:12:06.058 So, it's actually going to look something like that. Actually. 103 00:12:06.058 --> 00:12:12.208 Now, we can just do a check on this. What about the integral. 104 00:12:17.818 --> 00:12:22.589 Um, so. 105 00:12:22.589 --> 00:12:32.188 Let's say access from 0 to 1 let's say so, it's also going to be 0 and whitelisted equal. 1. let's do a test on this and a girl. 106 00:12:35.188 --> 00:12:40.859 That's going to be. 107 00:12:40.859 --> 00:12:44.219 1, over 2. 108 00:12:47.308 --> 00:12:51.269 And that is, if we work it out right. Is going to be something like lawn. 109 00:12:52.469 --> 00:12:59.759 0 line upon, and so on okay if we try this on expected values so. 110 00:13:01.558 --> 00:13:08.999 Again it all X so again, I was integral effects. 111 00:13:08.999 --> 00:13:18.599 Equals X squared close 1 half if we try that on why we would get the same thing. Now. 112 00:13:18.599 --> 00:13:23.038 Now, let's let's try another case here. Let's try. 113 00:13:23.038 --> 00:13:26.548 Y, equals a X plus B. 114 00:13:26.548 --> 00:13:30.239 So, I'm scaling X and I'm also adding something to it. 115 00:13:30.239 --> 00:13:36.719 So, half of why do why is going to be. 116 00:13:38.788 --> 00:13:41.969 After the next steps. 117 00:13:41.969 --> 00:13:51.418 Over, but the, why over DX is going to be equal to a here. So if I am. 118 00:13:52.528 --> 00:13:56.249 Okay, so what we have is. 119 00:13:58.739 --> 00:14:02.188 Okay. 120 00:14:02.188 --> 00:14:05.879 So, if we try a few things, let's say. 121 00:14:07.619 --> 00:14:15.328 It's trying to find the expected value a quasi integral of why. 122 00:14:18.568 --> 00:14:21.899 Ok, and that now is, um. 123 00:14:25.589 --> 00:14:36.359 Silence. 124 00:14:36.359 --> 00:14:41.548 No, for. 125 00:14:41.548 --> 00:14:45.958 Oops, here x X. 126 00:14:45.958 --> 00:14:49.528 Now, do, why is going to be. 127 00:14:49.528 --> 00:14:57.568 Because D, Y DX, Jose. 128 00:14:58.889 --> 00:15:05.489 Okay, so so our ace cancel out. 129 00:15:05.489 --> 00:15:16.798 And now we can pull this apart into pieces and this will be a undergo X. 130 00:15:19.889 --> 00:15:24.778 Silence. 131 00:15:24.778 --> 00:15:29.609 He acts. 132 00:15:29.609 --> 00:15:36.028 Okay, now so that 1st thing there is expected value of X. 133 00:15:36.028 --> 00:15:39.239 And then the 2nd, thing is just speak is there going to. 134 00:15:39.239 --> 00:15:45.208 Okay, so the linear thing happens with the expected values. 135 00:15:45.208 --> 00:15:50.129 Okay, which is what I was getting messed up back. 136 00:15:50.129 --> 00:15:55.048 A week ago, so now we can have it out. 137 00:15:55.048 --> 00:16:00.119 Yeah, quest away. So you see how it says why goes B. 138 00:16:00.119 --> 00:16:04.229 Yes, yes. Okay. And then what is like that. 139 00:16:04.229 --> 00:16:07.438 I'm sorry, I'm just confused by writing, but. 140 00:16:07.438 --> 00:16:21.149 Can you just read out to me what the next line says you're accusing me of writing allegedly the correct but still. Okay. Yeah, this is f of lower 2nd here. 141 00:16:21.149 --> 00:16:28.259 And that's a capital Y, parentheses Lowercase Y equals f subscript capital X. 142 00:16:28.259 --> 00:16:35.698 Argument is lower case X divided by by DX, which is equal a equal to a. 143 00:16:36.234 --> 00:16:47.514 So that is where do you get the over DX? I get that equals. Oh, well, we do a derivative here. Why? 144 00:16:47.514 --> 00:16:54.144 Because they X plus B then, I mean, that's random random virus. So specific things are. 145 00:16:54.418 --> 00:16:57.448 So, in future values, why say X. 146 00:16:57.448 --> 00:17:02.158 Us being able to do to dramatize. So T Y, axis equal say. 147 00:17:03.208 --> 00:17:06.239 Okay, got it Thank you. 148 00:17:06.239 --> 00:17:10.739 It gets confusing. I like to look at this example. 149 00:17:12.419 --> 00:17:18.538 That I had before is if we're converting say between. 150 00:17:18.538 --> 00:17:23.459 Feet and yards, so. 151 00:17:25.318 --> 00:17:29.669 So, X is a distance. 152 00:17:29.669 --> 00:17:35.759 And yards why it was distance. 153 00:17:37.798 --> 00:17:45.269 And feed, so why it's going to be 3 X and then the probability. 154 00:17:45.269 --> 00:17:53.219 So so, so we want the probability that X is from 1 to 2 yards or something. 155 00:17:55.318 --> 00:18:07.469 2, and so this is going to be the probability that why if axis and 1 to 2 yards, and why it will be less than or equal to we will be 3 to 6 feet. 156 00:18:07.469 --> 00:18:14.338 And and now we can work our way down doing playing with stuff like that. So. 157 00:18:15.989 --> 00:18:20.308 So, this here is effectively f of X. 158 00:18:20.308 --> 00:18:23.429 Time so here, 1. 159 00:18:23.429 --> 00:18:29.909 And and this is going to be equal to. 160 00:18:29.909 --> 00:18:34.318 Silence. 161 00:18:34.318 --> 00:18:37.499 And here, do you, why is going to be 3. 162 00:18:37.499 --> 00:18:42.868 3, to 61 to 2 and so what this shows is f of X. 163 00:18:45.148 --> 00:18:49.558 It was 3 times, so. 164 00:18:49.558 --> 00:18:53.219 And then the 3, why is 3 X then? 165 00:18:54.598 --> 00:19:05.159 3, so way of looking at is f of X DX equals f of why. 166 00:19:05.159 --> 00:19:09.328 Why, and half hopes and why. 167 00:19:09.328 --> 00:19:12.509 Yeah. 168 00:19:12.509 --> 00:19:17.189 In this case. 169 00:19:18.239 --> 00:19:23.548 Over 3, does this make sense? Or am I confusing and more. 170 00:19:28.019 --> 00:19:40.348 So, if we have a random variable X, and we from it, we derive or directly we defined a new random variable. Why effects? Wise? 3 X suffixes 5 points 15. 171 00:19:40.348 --> 00:19:46.919 Then the density function for why is a density function of X divided by 3. 172 00:19:50.669 --> 00:19:56.308 And this is something you actually want to get nailed down. 173 00:19:56.308 --> 00:19:59.848 Things like this and. 174 00:20:01.769 --> 00:20:05.219 Same here. 175 00:20:05.219 --> 00:20:12.719 You want to sort of get things like this nailed down because they are. 176 00:20:12.719 --> 00:20:15.959 That's sort of a key to. 177 00:20:15.959 --> 00:20:22.378 Understanding some deeper stuff, so. 178 00:20:29.368 --> 00:20:44.068 Well, if you have questions later on, then you can ask them. Let me showing an extension. So I'm talking about expected value. Let me show another thing. This is I was going to do later today, but it's sort of falls along the theme that's suppose. 179 00:20:48.148 --> 00:20:52.229 You start with 2 random variables X and Y. 180 00:20:52.229 --> 00:20:57.689 And we define a new random, a variable. 181 00:20:57.689 --> 00:21:03.898 Z this some of the 2 random variables. 182 00:21:06.479 --> 00:21:10.469 Now, um. 183 00:21:10.469 --> 00:21:13.679 So, the question is. 184 00:21:18.659 --> 00:21:26.729 Is that okay so how can we do that? Let's see if this is in the book, but let's try to. 185 00:21:26.729 --> 00:21:29.878 Work it out? No. 186 00:21:29.878 --> 00:21:33.598 Silence. 187 00:21:33.598 --> 00:21:39.628 Silence. 188 00:21:39.628 --> 00:21:44.098 Now. 189 00:21:54.239 --> 00:22:00.209 Question is what is f. F. C. 190 00:22:00.209 --> 00:22:08.759 If we think about this, um. 191 00:22:10.679 --> 00:22:15.659 Silence. 192 00:22:26.814 --> 00:22:30.324 Well, how would we do that if we go back to definitions? Oops here. 193 00:22:31.199 --> 00:22:39.148 It's getting away from me. Well, the probability that some random variables Z is between. 194 00:22:44.788 --> 00:22:50.578 Z Z. 195 00:22:50.578 --> 00:22:57.239 Well, this file here. 196 00:22:57.239 --> 00:23:02.699 Equals X plus Y, so. 197 00:23:02.699 --> 00:23:10.348 So, what's the probability. 198 00:23:12.328 --> 00:23:16.679 Can we do something like that? 199 00:23:22.828 --> 00:23:31.798 Well, that's actually going to be some sort of joint probability distribution. So this is a chance to talk about some joint probability distribution. 200 00:23:36.628 --> 00:23:39.628 What can I just do it here? 201 00:23:40.709 --> 00:23:51.179 I read my poem on 1 wrong point on the screen. 202 00:23:55.888 --> 00:24:07.138 Well, yeah, we think what we're going to have to do here and. 203 00:24:11.548 --> 00:24:17.398 And it could be correlated also. So what we have is. 204 00:24:22.078 --> 00:24:25.888 Yeah, we think about now, what would we do here? 205 00:24:25.888 --> 00:24:32.969 On X and Y. 206 00:24:32.969 --> 00:24:40.769 And C as X plus Y, so you've got some sort of band here. 207 00:24:40.769 --> 00:24:45.298 Um. 208 00:24:45.298 --> 00:24:50.159 So. 209 00:24:53.459 --> 00:24:59.249 Yeah, we think about it, let us see so so this thing up here. 210 00:25:04.078 --> 00:25:09.269 I didn't. 211 00:25:11.759 --> 00:25:15.868 Well, let me see, how can we do it here? So the probability that. 212 00:25:18.449 --> 00:25:21.929 Silence. 213 00:25:21.929 --> 00:25:28.439 A close. 214 00:25:28.439 --> 00:25:33.778 The called X could be anything naturally. 215 00:25:45.749 --> 00:25:50.009 And and then why. 216 00:25:50.009 --> 00:25:59.249 Is any value so why is going to be minus X and. 217 00:26:01.078 --> 00:26:04.469 Yeah, you might a sex less and. 218 00:26:07.858 --> 00:26:11.398 Silence. 219 00:26:13.769 --> 00:26:19.798 And we integrate this overall values of X actually. So. 220 00:26:23.969 --> 00:26:29.969 So, Z is in a certain value effects is in. 221 00:26:31.259 --> 00:26:40.618 Some X could be anything and why is between C minus X and see minus X capacity? Why. 222 00:26:42.479 --> 00:26:46.528 I actually going to continue that later. I'm going to go back to what I was planning to do. 223 00:26:46.528 --> 00:26:53.939 Which is. 224 00:26:53.939 --> 00:27:07.979 Which is, I'll work my way up to the joint thing, but, let me look at some examples I'll work. So, then I'll come back to that, starting to get complicated to 42 example, 59 and 252 example 515. 225 00:27:07.979 --> 00:27:13.528 Silence. 226 00:27:24.148 --> 00:27:27.959 For the. 227 00:27:30.598 --> 00:27:33.749 This will help us work our way up to here. 228 00:27:35.368 --> 00:27:42.568 I mentioned this before I wanted to see if I can work this thing out in more detail, more detail. Now. 229 00:27:43.588 --> 00:27:48.328 We're transmitting messages and bytes long a geometric distribution. 230 00:27:48.328 --> 00:27:52.348 And it's getting split. 231 00:27:52.348 --> 00:27:59.308 And to and let me make it. Let me make it easy. And let's. 232 00:27:59.308 --> 00:28:03.028 Say, the parameter is. 233 00:28:03.028 --> 00:28:06.449 1, half, let's say. 234 00:28:08.249 --> 00:28:12.148 So this is an example. 235 00:28:12.148 --> 00:28:15.538 5.9 page. 236 00:28:15.538 --> 00:28:19.229 242 I'm going to let. 237 00:28:19.229 --> 00:28:22.739 It was 1. 238 00:28:22.739 --> 00:28:27.719 So so the probability here. 239 00:28:27.719 --> 00:28:39.449 Of and the ranges, and it's going 0 up. So the probability it's a message says 90 equals 1 half Providence length 1 equals 1 quarter. 240 00:28:39.449 --> 00:28:42.509 And so on Providence lanes K. 241 00:28:42.509 --> 00:28:48.179 8 walls to to the minus. 242 00:28:48.179 --> 00:28:51.628 K plus 1. okay. 243 00:28:51.628 --> 00:29:00.358 Now, so we're cutting this, so we're. 244 00:29:01.528 --> 00:29:06.719 So, this could be a potentially a very long message. Now what we're doing is that was. 245 00:29:06.719 --> 00:29:10.348 Were breaking the thing up into blocks, so. 246 00:29:13.558 --> 00:29:16.739 And to blocks. 247 00:29:17.878 --> 00:29:23.038 And let's suppose I'm going ahead the blocks of length. 248 00:29:24.148 --> 00:29:30.419 Who, I don't know, box of lines 2. 249 00:29:31.648 --> 00:29:37.138 Plus plus a bit plus a partial block. Okay. 250 00:29:38.608 --> 00:29:42.778 Let me come back up to. 251 00:29:42.778 --> 00:29:46.108 Here come on. 252 00:29:46.108 --> 00:29:50.278 Okay, blocks of length to plush a partial block. 253 00:29:51.598 --> 00:30:01.199 So so, in other words, and we want to do probabilities on this. 254 00:30:01.199 --> 00:30:05.159 And so if I use a notation from the book. 255 00:30:05.159 --> 00:30:10.499 Go and bite. 256 00:30:10.499 --> 00:30:14.519 Okay, so here that that's to him. 257 00:30:14.519 --> 00:30:24.989 Okay, now trying to put both of these things up together. So I won't be able to see the chat window. So speak up. 258 00:30:26.308 --> 00:30:29.638 I've seen it, so. 259 00:30:30.989 --> 00:30:34.888 Silence. 260 00:30:34.888 --> 00:30:39.419 Silence. 261 00:30:41.999 --> 00:30:48.598 Okay, so now we want to start playing games with this. 262 00:30:55.709 --> 00:31:03.088 So then invite message goes to. 263 00:31:05.219 --> 00:31:09.538 And over 2 blocks. 264 00:31:11.098 --> 00:31:16.739 Bus wherever you do the quotient, plus perhaps. 265 00:31:18.509 --> 00:31:23.278 I'll send over to piece. 266 00:31:23.278 --> 00:31:26.818 Saying that percent is modern or something. 267 00:31:26.818 --> 00:31:29.878 Okay, so in other words. 268 00:31:31.499 --> 00:31:34.949 7 bites goes to 3 blocks. 269 00:31:36.358 --> 00:31:45.239 Us 1 bite. Okay. So now we want to do the probability that of whatever. 270 00:31:50.638 --> 00:32:00.148 So, how do we start doing this? 271 00:32:00.148 --> 00:32:15.118 Well, so the probability that some particular, so Q, is the quotient, that's the number of full blocks and are is the remainder that's the length of the fractional block. And that's a probability. 272 00:32:16.288 --> 00:32:22.888 Of some, and so and is now going to be 2? Q plus R. 273 00:32:22.888 --> 00:32:26.909 Okay. 274 00:32:28.888 --> 00:32:32.848 And so for and. 275 00:32:35.278 --> 00:32:39.598 I should have a lower case and there actually, so this will be. 276 00:32:39.598 --> 00:32:44.969 2 to them to it as a minus. 277 00:32:46.469 --> 00:32:50.098 And that is going to be. 278 00:32:50.098 --> 00:32:53.548 Down here is a bigger. 279 00:32:55.199 --> 00:33:02.548 Silence. 280 00:33:02.548 --> 00:33:06.269 It's a problem if you're saying, and now. 281 00:33:07.469 --> 00:33:10.949 So that's a value of. 282 00:33:12.239 --> 00:33:16.259 Combo of Q and art now, if we want to. 283 00:33:25.528 --> 00:33:28.769 Suppose we want to integrate out. 284 00:33:38.368 --> 00:33:43.979 I want to integrate our so we've got that thing up here. 285 00:33:45.778 --> 00:33:52.979 And that's going to be equal to the probability. 286 00:34:01.679 --> 00:34:06.689 It's the 2 different cases for our I'm doing this a little simpler than we have in the PoC. 287 00:34:06.689 --> 00:34:09.929 And that will be 2 to the minus. 288 00:34:11.969 --> 00:34:15.478 Silence. 289 00:34:15.478 --> 00:34:22.289 Silence. 290 00:34:22.289 --> 00:34:27.298 And this is going to net out to. 291 00:34:45.329 --> 00:34:51.449 Silence. 292 00:34:55.259 --> 00:35:03.449 If I have this, right? So, in other words. 293 00:35:07.619 --> 00:35:10.739 And I could do a check on that. 294 00:35:10.739 --> 00:35:17.278 So this is the probability distribution on the number of complete blocks. 295 00:35:17.278 --> 00:35:20.608 So, it's also geometric distribution, so. 296 00:35:23.579 --> 00:35:29.608 Silence. 297 00:35:29.608 --> 00:35:37.739 Silence. 298 00:35:48.179 --> 00:35:51.179 And if I'm being. 299 00:35:53.728 --> 00:35:58.679 Particularly daring, I could see if that sums up to 1, but. 300 00:35:58.679 --> 00:36:02.338 Okay, now. 301 00:36:03.389 --> 00:36:08.458 Now, what's the probability distribution on the size of the remainder. 302 00:36:08.458 --> 00:36:16.619 Well, um. 303 00:36:21.389 --> 00:36:27.898 Well, what we can do is we can take the probably distribution. 304 00:36:27.898 --> 00:36:34.079 For a particular value van and our being setups and just sum up, we could get something like. 305 00:36:44.039 --> 00:36:48.628 Okay. 306 00:36:48.628 --> 00:36:52.619 And now if I scroll up. 307 00:36:54.568 --> 00:37:07.679 This is going to be see here. 308 00:37:17.338 --> 00:37:22.679 Silence. 309 00:37:34.708 --> 00:37:37.739 If I got this, right. 310 00:37:40.139 --> 00:37:45.329 And how we do that is that we can pull out. 311 00:37:51.958 --> 00:37:56.938 Silence. 312 00:37:58.469 --> 00:38:03.148 Silence. 313 00:38:03.148 --> 00:38:06.958 Do something like that? 314 00:38:08.128 --> 00:38:15.659 Okay, and. 315 00:38:17.489 --> 00:38:20.940 Which I think is sort of similar to what's in the. 316 00:38:38.159 --> 00:38:42.780 Something like that, I think. Okay so, um. 317 00:38:43.860 --> 00:38:48.869 That's how example, 5, and I simplified 5Million people to a half and so on. 318 00:38:48.869 --> 00:38:52.590 In any case the. 319 00:38:52.590 --> 00:39:03.360 Well, this is a truncated geometric distribution, so if we have the packets and bytes, and it's a geometric distribution. 320 00:39:03.360 --> 00:39:07.440 Then the random variable to number complete packets. 321 00:39:07.440 --> 00:39:16.019 It's also geometric and the random variable for the size of the remainder leftover, it's truncated geometric because the size goes up from. 322 00:39:16.019 --> 00:39:20.460 0, to M, minus 1 and being the size of the complete box. 323 00:39:20.460 --> 00:39:25.260 So, that's what they have now. I knew that most discrete. Okay. 324 00:39:27.300 --> 00:39:31.139 Now, see what I have in the. 325 00:39:33.000 --> 00:39:40.710 Okay, that's sorry about 240. now 516 on 252. 326 00:39:48.690 --> 00:39:52.559 Silence. 327 00:39:52.559 --> 00:39:59.579 Well, actually, let me go up to the related 1, which is 517. 328 00:40:00.840 --> 00:40:04.530 Come back to this, because type 17 is the same example here. 329 00:40:05.760 --> 00:40:10.079 No, so I did I thought things. 330 00:40:10.079 --> 00:40:14.699 Like, 20 sorry. 331 00:40:14.699 --> 00:40:18.389 Page. 332 00:40:18.389 --> 00:40:28.590 Okay, so the question is, okay, so again, so, 520 I want to do it now because that's continuing on. 333 00:40:29.730 --> 00:40:38.309 So our Q and R independent. Well, let's see, I could live dangerously here and. 334 00:40:40.320 --> 00:40:51.599 See, see what happens. So, probably the combo probability of human hours is further up in the page. Let me. 335 00:40:53.280 --> 00:40:56.820 That asset would be. 336 00:40:56.820 --> 00:41:03.750 Here okay and, um. 337 00:41:03.750 --> 00:41:08.579 And the thing is, they're independent are okay. 338 00:41:08.579 --> 00:41:12.780 So now. 339 00:41:12.780 --> 00:41:18.030 So example, 520. 340 00:41:18.030 --> 00:41:22.800 5, so questions are. 341 00:41:22.800 --> 00:41:26.429 The 1 are independent. 342 00:41:26.429 --> 00:41:30.809 Well, yes. 343 00:41:30.809 --> 00:41:35.579 Yes. 344 00:41:35.579 --> 00:41:38.579 Silence. 345 00:41:38.579 --> 00:41:42.210 Silence. 346 00:41:42.210 --> 00:41:51.750 I'm going to work this through if, and only have oh work just to part way and see. 347 00:41:51.750 --> 00:41:58.230 What happened so probability that was Q is if I go back a little. 348 00:42:00.000 --> 00:42:03.389 3 times 2 to the minus 2, 2 plus 2. 349 00:42:03.389 --> 00:42:07.530 Silence. 350 00:42:07.530 --> 00:42:11.159 I got time, right and our. 351 00:42:12.780 --> 00:42:23.309 Is okay, we're going to have to sum this thing. 352 00:42:26.579 --> 00:42:31.829 And, oh, that's equal to. 353 00:42:34.710 --> 00:42:42.659 And that's going to be equal to 4 thirds. 354 00:42:42.659 --> 00:42:46.050 So this will be. 355 00:42:48.090 --> 00:42:53.219 Silence. 356 00:42:54.840 --> 00:43:01.110 And let's give this a test, I probably made a mistake, but let us see. 357 00:43:02.820 --> 00:43:07.440 Probably article is 0, is going to be. 358 00:43:07.440 --> 00:43:10.829 5, 0, there is. 359 00:43:10.829 --> 00:43:15.929 Terrence articles 1 is. 360 00:43:15.929 --> 00:43:19.440 Wow, it's consistent. 361 00:43:19.440 --> 00:43:24.809 Pull that up to I can only be 01. hey. Wow. Okay. 362 00:43:24.809 --> 00:43:28.440 So, the problem is the probability. 363 00:43:28.440 --> 00:43:32.099 That are equals R. is. 364 00:43:32.099 --> 00:43:40.320 For search 2 to them. Well, okay, well, this simplifies here. Okay if I go back up here, come on. 365 00:43:41.369 --> 00:43:50.280 Thank you this here simplifies to. 366 00:43:51.300 --> 00:43:55.380 Okay. 367 00:43:55.380 --> 00:44:00.630 Concerts to the minus R. 368 00:44:00.630 --> 00:44:05.369 Have clean that up. 369 00:44:08.130 --> 00:44:12.269 Okay, so go back down here. 370 00:44:15.389 --> 00:44:18.690 So, if we combine these 2 things, probably. 371 00:44:36.780 --> 00:44:42.000 Modify those 2 things together and we're going to get 2. 372 00:44:43.079 --> 00:44:48.389 Go to the minus 2 2. 373 00:44:48.389 --> 00:44:52.739 2 minus R. 374 00:44:52.739 --> 00:44:55.949 Um. 375 00:44:55.949 --> 00:45:01.619 Silence. 376 00:45:06.239 --> 00:45:09.869 And what did we want it to have. 377 00:45:09.869 --> 00:45:20.340 Turn to the minus 22 plus R plus 1. 378 00:45:22.829 --> 00:45:26.670 Yeah, it actually works out because the 2 it goes into. 379 00:45:31.110 --> 00:45:37.440 If I did it, right? 380 00:45:37.440 --> 00:45:41.909 So, they're independent. 381 00:45:47.760 --> 00:45:52.769 Okay, so example on 520. 382 00:45:52.769 --> 00:45:58.019 So worked out. Okay. 383 00:45:59.760 --> 00:46:04.380 So, I was so I was continuing on here, the example of splitting. 384 00:46:04.380 --> 00:46:12.869 A long message up into a whole packet so fraction the distribution for the whole number of whole package is independent from the size of the. 385 00:46:12.869 --> 00:46:17.190 Of the fraction at the end. Okay. 386 00:46:17.190 --> 00:46:27.900 Back up here, so I was going to be okay, so, let me go back to 560 and so on phase 2. 387 00:46:27.900 --> 00:46:33.900 52. 388 00:46:33.900 --> 00:46:39.090 Silence. 389 00:46:39.090 --> 00:46:43.889 Silence. 390 00:46:43.889 --> 00:46:53.429 Okay, so this is a chance to show a review, the marginal probability distributions and. 391 00:46:54.630 --> 00:47:04.260 I won't write everything down by hand, but will give you a sense. 392 00:47:04.260 --> 00:47:07.559 A joint distribution and. 393 00:47:08.610 --> 00:47:13.110 1st question is what is the constant C then. 394 00:47:13.110 --> 00:47:27.119 We integrate this over X and Y, so they're only from 0 to infinity. Okay. Now, this gets interesting here. I may start writing it down notice excess of 0 to infinity. But why is from 0 to X? 395 00:47:27.119 --> 00:47:35.579 So, in fact, let me write this down here I started a page come on. 396 00:47:35.579 --> 00:47:38.610 Okay. 397 00:47:38.610 --> 00:47:42.030 So this is example, 516. 398 00:47:43.170 --> 00:47:49.079 So, excess 0 to infinity. 399 00:47:49.079 --> 00:47:53.159 But why it's only in here so. 400 00:47:55.050 --> 00:48:05.309 On several things. Okay. That's what makes it interesting because you've got a question also later R, X and Y. 401 00:48:05.309 --> 00:48:12.929 Dependent on each other, so say, as a joint thing up there and now to get. 402 00:48:12.929 --> 00:48:19.829 The distribution function, the density function on X or Y, on X you integrate out why. 403 00:48:19.829 --> 00:48:23.550 On why integrate out X so. 404 00:48:24.659 --> 00:48:31.139 There's the whole thing you integrate integrate to do to do, and you're going to. 405 00:48:31.139 --> 00:48:34.440 See, over to the contents to oversee. 406 00:48:34.440 --> 00:48:37.739 Okay, constant is to. 407 00:48:37.739 --> 00:48:43.619 Okay, so if of X integrate out, why why disclose. 408 00:48:43.619 --> 00:48:50.340 It's just integrating otherwise the density is 0 to do to do, and we get. 409 00:48:50.340 --> 00:48:55.889 Eventually function on X over here. Now why. 410 00:48:57.630 --> 00:49:04.739 A little trickier we want the density function on why? So integrating 0 to X and Sarah to infinity. 411 00:49:04.739 --> 00:49:11.010 But this is the density function is positive only if Y, is some 0 to X. 412 00:49:11.010 --> 00:49:16.769 So, it's pause that it's the same thing as saying X is bigger than why. 413 00:49:17.969 --> 00:49:24.840 If it's exactly equals probably. So you say is greater than just to be say quicker. Okay. So. 414 00:49:24.840 --> 00:49:28.079 Density function is non 0 when. 415 00:49:28.079 --> 00:49:42.840 X is bigger than Y. so, and just integrate over from, for why? To infinity and we're going to get that. So we have the density function on X. so either the minus X times 1, minus 16 and that's the function on Y2 either the minus 2. why. 416 00:49:42.840 --> 00:49:52.440 Now, the question is, is our X and Y, independent of each other they're independent of each other. If, and only if. 417 00:49:52.440 --> 00:49:55.679 These 2 things small separate. 418 00:49:55.679 --> 00:50:01.739 Single value density, if I'm just modify out to the joint 2 value density function up there. 419 00:50:04.710 --> 00:50:11.159 So, and I'm looking at them without work it out thinking that probably is not the case. 420 00:50:11.159 --> 00:50:16.949 Okay, so what did I do wrong here? 421 00:50:16.949 --> 00:50:21.659 I should give myself take a point off from myself or something. 422 00:50:23.610 --> 00:50:27.269 Silence. 423 00:50:27.269 --> 00:50:34.619 Okay, really? What it should be. Yeah. I don't know. And no 1 caught it. 424 00:50:37.079 --> 00:50:42.809 Why is bigger than X so for any value of HAX here, why it's out here. 425 00:50:43.829 --> 00:50:47.969 Okay oh, okay. 426 00:50:47.969 --> 00:50:51.719 Okay, so that was 516. 427 00:50:52.949 --> 00:50:56.130 Was the next 1 I was talking about handling. 428 00:50:56.130 --> 00:51:03.929 Is in 517 so the probability that. 429 00:51:03.929 --> 00:51:11.610 X plus Y squared is less than 1. okay. So, here, if I have my example here, then come on. 430 00:51:13.829 --> 00:51:17.280 We want to so what we want this is the line. 431 00:51:19.260 --> 00:51:26.909 What's 1 then what we want. 432 00:51:26.909 --> 00:51:31.199 Is basically the integral here? 433 00:51:32.340 --> 00:51:33.210 Okay, 434 00:51:33.204 --> 00:51:54.445 silence. 435 00:51:55.800 --> 00:52:01.829 Is. 436 00:52:01.829 --> 00:52:06.869 Eyes less than X and X plus wise less than 1. 437 00:52:06.869 --> 00:52:16.500 Okay, and they're giving an example here specifically. 438 00:52:16.500 --> 00:52:23.789 Save her access a halfs and why financing from a half up to. 439 00:52:26.340 --> 00:52:30.630 Okay, basically have so. Okay. 440 00:52:42.329 --> 00:52:50.190 Skipping some details and so next 1 is so. 441 00:52:50.190 --> 00:53:01.320 And we saw okay, getting up stuff. We saw jointly calcium before, and they get stuff like this just quick review. 442 00:53:01.320 --> 00:53:10.679 And and this, if X and Y are not related to each other, if they're independent, this will just be a circular. 443 00:53:10.679 --> 00:53:20.250 At type function, but when it's this long, thin Ridge, it's because they're related to each other if you know something about X and you know something about why. 444 00:53:20.250 --> 00:53:23.400 Okay, we saw the definition of independence. 445 00:53:23.400 --> 00:53:27.750 Okay, joint. 446 00:53:28.920 --> 00:53:37.440 Okay here so we had the expected value in the variance of a single random variable. 447 00:53:38.670 --> 00:53:45.750 At a function of a single 1, like white with some function of X, for example. Well, here is the. 448 00:53:47.820 --> 00:53:51.719 Also be another way to do it. Let me say, okay. 449 00:53:53.280 --> 00:54:04.079 Come on. 450 00:54:08.010 --> 00:54:12.690 Silence. 451 00:54:20.369 --> 00:54:24.989 It's just expected value of a function, so. 452 00:54:26.699 --> 00:54:38.639 So, why is the function of X here? 453 00:54:42.510 --> 00:54:46.079 Okay, so then expected value of Y. 454 00:54:46.079 --> 00:54:51.030 Is going to be. 455 00:54:51.030 --> 00:54:54.150 Silence. 456 00:54:54.150 --> 00:54:57.389 Silence. 457 00:54:57.389 --> 00:55:05.639 Okay, very, very, very quick way to do it. I saw way before. Okay. 458 00:55:06.780 --> 00:55:16.019 So now coming back to the thing, I did. 459 00:55:16.019 --> 00:55:25.019 Somewhat earlier i2nd value the sum of 2, random variables. So we have to random variables X and Y, we to find a new 1 Z. 460 00:55:26.940 --> 00:55:34.079 So then just use this definition here, it's expected value of a function of 1 or 2, random variables. 461 00:55:37.380 --> 00:55:47.400 And I'll walk you through this, I can write it down and comment as I write it come on here. 462 00:55:47.400 --> 00:55:51.989 Just a 2nd, to. 463 00:56:06.269 --> 00:56:16.440 Okay, what happened is Apollo, in my hand, touched the iPad at the wrong time. 464 00:56:16.440 --> 00:56:19.500 Okay, well I'll walk you through this. 465 00:56:19.500 --> 00:56:25.920 Said I can write it down by hand if you want. So if I don't need the iPad, I can also bring up the. 466 00:56:28.710 --> 00:56:32.820 Bring up the chat window. Okay. So. 467 00:56:32.820 --> 00:56:42.630 So using the thing up here about the expected value for functional 2, grand and variables. So the function is X plus Y, so. 468 00:56:42.630 --> 00:56:46.650 So the expect, so just the integral to the function. 469 00:56:52.769 --> 00:57:03.900 Okay um, and so we just split it out X and Y, then here on the left, you see. 470 00:57:03.900 --> 00:57:07.949 And this thing on the left is a constant, we're integrating over x X. 471 00:57:11.219 --> 00:57:15.780 And so that is. 472 00:57:18.719 --> 00:57:23.610 Well, oh, this becomes a marginal thing here. We're integrating. 473 00:57:23.610 --> 00:57:27.960 On the left for integrating over, why is that? Just ends up. 474 00:57:27.960 --> 00:57:31.199 X Y. X Y. 475 00:57:31.199 --> 00:57:36.659 A good time see why to discuss the marginal f effects, which come down here. 476 00:57:36.659 --> 00:57:48.000 Access comes down here, this comes out to executive value and same thing on the right. Expected value of X times expected value of Y, so expected values will some. 477 00:57:48.000 --> 00:58:02.670 Now, the interesting thing is that this does not depend on any relation between X and Y, X and Y, can be anything they can be correlated independent, whatever the expected values will. This is the only case. 478 00:58:02.670 --> 00:58:06.869 And something this obvious happens. It's very nice. 479 00:58:06.869 --> 00:58:15.869 And very unusual, the product thing works. 480 00:58:15.869 --> 00:58:20.369 Only, if they're independent independent, so. 481 00:58:20.369 --> 00:58:23.699 Here, the function is the product of the 2. 482 00:58:23.699 --> 00:58:27.030 And we're assuming they're independent. 483 00:58:29.130 --> 00:58:36.539 So g is X times Y, so it splits out if they're independent, we've got this and can work that thing out. 484 00:58:40.769 --> 00:58:54.000 Okay, yeah, this is a stuff I've had before with some new stuff. We've seen this before more fruitful. I get things a couple of times have greater and greater detail. 485 00:58:54.000 --> 00:59:00.119 You can have you, you can have moments of any order so. 486 00:59:01.260 --> 00:59:06.809 Means variances and so on so sexuality back to the J terrified of the K. 487 00:59:09.630 --> 00:59:18.539 The 11 moment is the correlation of them, the central correlation to be the variable. So okay. 488 00:59:19.769 --> 00:59:27.269 So, the, which we've seen before is like, the 11 moment, the expectation of X minus either the EXT. 489 00:59:27.269 --> 00:59:35.429 That's why I'm going to see why. So now you can take this and this is a variance. 490 00:59:35.429 --> 00:59:40.320 And you can all apply it out and get this. 491 00:59:41.969 --> 00:59:49.349 So, you see this simplifies here and it will simplify to. 492 00:59:49.349 --> 00:59:53.159 Right here so the variance. 493 00:59:55.349 --> 01:00:00.360 This this form here is. 494 01:00:00.360 --> 01:00:06.090 Easier to look at and remember how are the previous form up here? 495 01:00:06.090 --> 01:00:10.139 Is actually better for computation sometimes. 496 01:00:10.139 --> 01:00:18.449 If you're worried about that sort of thing, the problem is with this thing here, let's suppose action wise means are not years here on very large. 497 01:00:18.449 --> 01:00:25.349 Sex times wise, we're sending a lot of large numbers and then subtracting out numbers, which are only a little bit smaller. 498 01:00:25.349 --> 01:00:28.650 And so you lose a lot of significant Patriots. 499 01:00:29.699 --> 01:00:34.139 Car sick shorthand way to solve that. Is you stumble precision? 500 01:00:38.309 --> 01:00:45.570 It would affect you only like I said, you got numbers, which are very large. 501 01:00:45.570 --> 01:00:59.519 It's a case it might affect me. Let's suppose I'm doing navigation. I do a lot of hiking. I've got I bought many different over the public positioning system, handheld things over the years. I've got mapping apps and so on. 502 01:01:01.050 --> 01:01:05.610 So, let's suppose I want to do some statistics on calculating my. 503 01:01:05.610 --> 01:01:08.849 So, taking on Saturday. 504 01:01:08.849 --> 01:01:12.000 On my path, which is logging on my TPS. 505 01:01:12.000 --> 01:01:15.630 So, the latitude we're about. 506 01:01:17.489 --> 01:01:25.829 For mail, we're about 4 and a half 1Million meters to the equator at the moment. So for using universal transfers indicator, then. 507 01:01:25.829 --> 01:01:34.889 X is going to be around 4 and a half 1Million. So, you know, I was hiking only a couple of miles. I'm getting lazy if I wanted to find my main location. 508 01:01:34.889 --> 01:01:41.639 So, my XS are, there are a lot of numbers that are all about 4 and a half 1Million that are varying only by. 509 01:01:41.639 --> 01:01:45.090 About a 1000 or so because, you know, kilometer. 510 01:01:46.349 --> 01:01:54.840 So so, if I'm trying to find say variance, I'm summing up numbers that are 4 and a half 1Million squared. I'm going to be getting numbers that are. 511 01:01:54.840 --> 01:02:02.610 And a range of 2 times, 10 of the 13th, unless I dropped to 0, but they're varying only by. 512 01:02:02.610 --> 01:02:05.880 A much smaller number, so you can see that. 513 01:02:05.880 --> 01:02:10.050 You can lose a lot of significant digits with this method here. 514 01:02:12.030 --> 01:02:17.130 Any case if they're independent, then Nicole variances 0. 515 01:02:18.900 --> 01:02:27.059 And then I mentioned before, reviewing the correlation coefficient to CO variant says units of distance squared of length squared. 516 01:02:27.059 --> 01:02:37.739 So, we divide through by the 2 standard deviations. It's now a dimensionalize. The correlation coefficient is now a dimension number from minus 1 to 1. 517 01:02:37.739 --> 01:02:40.949 Correlation coefficient. 518 01:02:40.949 --> 01:02:49.949 And realize the correlation coefficient has to be fairly close to 1 to be actually meaningful. So. 519 01:02:50.215 --> 01:03:02.155 There is a sensitive statistics where you can talk about how much of the variance of 1 variable is explained by another variable that it's correlated with the amount that's explained to square the correlation coefficient. 520 01:03:02.155 --> 01:03:10.945 So, it's correlation coefficient is point 5 then only a quarter of the variance and the 1 variable is explained by knowing the other variable. Okay. 521 01:03:12.840 --> 01:03:24.449 And mentioned this before, that variables can be dependent, but not linearly card because the correlations, a linear concept and cell case, or. 522 01:03:24.449 --> 01:03:34.710 They're both coasts and sign of an angle. They're quite clearly dependent on each other. If you know the coast, there's only 2 possible values for the sign plus and minus 1 minus go squared. 523 01:03:34.710 --> 01:03:40.230 So that, but they're not linearly correlated, but clearly dependent. Okay. 524 01:03:41.880 --> 01:03:47.400 So, conditional we talked about before also somewhat so. 525 01:03:48.599 --> 01:03:54.360 Example for the Lord device, and I mentioned this a little quickly last time also. So. 526 01:03:54.360 --> 01:04:00.750 And we mentioned this thing also. Okay. 527 01:04:00.750 --> 01:04:05.909 Let me just hit you with some really good. It's good to see this a 2nd time. 528 01:04:05.909 --> 01:04:14.099 What's happening here is you're transmitting a bit this plus or minus 1, but they're not equally probable. 529 01:04:14.099 --> 01:04:17.610 So, the plus 1 is less likely than the minus 1. 530 01:04:17.610 --> 01:04:20.909 And we had calcium noise to them. 531 01:04:22.980 --> 01:04:26.760 So you want to know what's the most likely. 532 01:04:26.760 --> 01:04:35.400 Input and output voltage it's again, it's a mixed distribution thing here because the 1 variable X is discreet. 533 01:04:35.400 --> 01:04:38.670 But the noise is continuous and the. 534 01:04:38.670 --> 01:04:43.230 Why the output is also continuous it just it's a chance to play with that. 535 01:04:44.489 --> 01:04:50.070 And this is using base rule also so we have the problem. 536 01:04:50.070 --> 01:04:53.159 So the probability okay, so. 537 01:04:53.159 --> 01:05:01.170 X minus 1 or 1 with probabilities. 2 thirds 4th and is calcium just unit variance which unit? Standard? Deviation. 538 01:05:01.170 --> 01:05:06.929 So you want to find the probability that the output voltage is something given the input was 1. 539 01:05:06.929 --> 01:05:11.699 And we're just using the dense the key thing here. That's what the big f means. 540 01:05:11.699 --> 01:05:15.960 And this is partially a goal in the calcium and then. 541 01:05:15.960 --> 01:05:24.630 So, the probability that the output is positive or negative is positive, given that the input is positive or negative and. 542 01:05:24.630 --> 01:05:28.469 We get that here. 543 01:05:28.469 --> 01:05:35.039 So, even if the output is net input is negative. This Stella, 16 probability output is positive. So. 544 01:05:35.039 --> 01:05:38.699 And then you can apply bay's roll and go backwards. 545 01:05:39.625 --> 01:05:53.695 And then, I mean, the best rule is that you judge perhaps based on you, look at the output, is it positive or negative who it's positive use of the inputs positive at the outputs negative. We suddenly input his negative. That might not be the best rule. Actually. But to see. 546 01:05:53.940 --> 01:05:59.369 Obvious simple 1 and now you can calculate that the output was positive. Then there's a. 547 01:05:59.369 --> 01:06:02.730 73% chance the input was positive, so. 548 01:06:02.730 --> 01:06:09.989 A 30 37% chance it went wrong so. 549 01:06:12.510 --> 01:06:25.260 And again that if this 73% chance of being transmitted correctly is not good enough for you, then you can start getting the error correction methods. 550 01:06:25.260 --> 01:06:29.940 They really simple 1, as you send it 3 times and a majority fault. 551 01:06:29.940 --> 01:06:35.460 If you had a communications, course you can start using read Solomon and so on. Okay. 552 01:06:37.739 --> 01:06:46.380 Okay, that additional I mentioned a little before I mentioned it some more next time with examples. So. 553 01:06:47.699 --> 01:06:57.630 Okay, now this is interesting here. 554 01:06:57.630 --> 01:07:08.369 I'll walk you through it at a high level basis and then maybe hit a more detailed. Next time. We have that to variable joint thing X and Y, and it's a sort of exponential thing. 555 01:07:08.369 --> 01:07:16.679 And and this is a case for why is again, why is less than or equal to X. 556 01:07:16.679 --> 01:07:30.809 Okay, so there's a correlation and so this is find a conditional density function on why, if, you know, X, X, and Y, is summer between 0 and X. but what's the. 557 01:07:30.809 --> 01:07:36.630 Density, and we can use the stuff in the previous page and calculate it like that. 558 01:07:38.309 --> 01:07:43.050 So, we can find the. 559 01:07:45.385 --> 01:07:55.465 The single variable thing and so that's a typo there. It should be f of access given why right here where I'm circling is a typo. You'd see it up here. 560 01:07:55.704 --> 01:08:05.755 So the marginal condition on X, if we know why so it's got to be bigger than Y, and then the conditional on why if we know X. 561 01:08:06.000 --> 01:08:12.059 So, on okay arrivals. 562 01:08:12.059 --> 01:08:17.670 We talked about that. Let me just rehash some important things here. 563 01:08:17.670 --> 01:08:26.069 We assume there is a lot of customers a very small number of them are coming at a particular. 564 01:08:26.069 --> 01:08:31.109 Instead of whatever your quantum time is. 565 01:08:31.109 --> 01:08:38.130 A minute or something so, the number of customers arriving in a particular minute is a random variable. 566 01:08:38.130 --> 01:08:49.140 Now, what they're making interesting here is once the customer comes, some customers are harder to deal with and other customers. 567 01:08:49.140 --> 01:08:52.800 So each customer takes a different amount of time to service. 568 01:08:52.800 --> 01:09:01.020 And will assume that the time to service the customers at random variable and but this 1 is an exponential distribution with some parameters. 569 01:09:01.020 --> 01:09:06.569 So, now we're interested at that so, while we're serving a customer. 570 01:09:06.569 --> 01:09:14.609 More customers might arrive as the time we're serving the customer at some time, then you throw and you're saying, and you can calculate. 571 01:09:14.609 --> 01:09:25.260 We want to see the number of customers that are new customers that are likely to arrive while we're serving an old customer. 572 01:09:25.260 --> 01:09:32.670 We want, we might even have, you know, want to find it to mean. So these will be the main number of customers that will have to wait. 573 01:09:32.670 --> 01:09:36.869 Because somebody was already being served when they arrived. 574 01:09:36.869 --> 01:09:40.920 And so, and that's what this is working out here. So. 575 01:09:42.869 --> 01:09:46.020 And it's a conditional so it's a 2 step thing. 576 01:09:46.020 --> 01:09:53.069 So, we've got the, uh. 577 01:09:55.890 --> 01:10:01.289 So, we've got that is how many customers arrive and the. 578 01:10:02.760 --> 01:10:07.170 And the time they take that they take okay. 579 01:10:08.760 --> 01:10:14.250 I've, I've skipped over details, but you got the idea. So. 580 01:10:15.989 --> 01:10:22.949 What's happening? Here is again it's a chance to be like the, the earlier 1. 581 01:10:22.949 --> 01:10:25.949 X is uniforms 0 to 1. 582 01:10:25.949 --> 01:10:31.979 But why is uniform from 0 to X? So, why it's always less than X. 583 01:10:33.060 --> 01:10:36.149 And we want to know what's the distribution on why. 584 01:10:36.149 --> 01:10:41.130 Express the CD could be the PDF CDF here it's asking for the CDF. 585 01:10:41.130 --> 01:10:46.229 So obviously why it's going to be clustered more towards 0 than towards 1. 586 01:10:46.229 --> 01:10:54.329 Okay, and the way they're doing it is that. 587 01:10:56.640 --> 01:11:03.329 Or say the probability that. 588 01:11:06.149 --> 01:11:12.210 Why is less than some value given that access some value? K. 589 01:11:13.319 --> 01:11:20.550 That should be X, their typo type of typo. Why is uniform from 0 to X? 590 01:11:20.550 --> 01:11:23.909 That's us here while for X here um. 591 01:11:26.579 --> 01:11:29.789 So, probably they ran inbound session somewhat says. 592 01:11:29.789 --> 01:11:33.210 This is what this will turn out to be and if. 593 01:11:37.590 --> 01:11:42.840 The 0 there, I would say, see, a lot of typos in this particular example. 594 01:11:42.840 --> 01:11:50.640 In any case, you can take this out and integrate over X. 595 01:11:50.640 --> 01:11:55.710 And and get the single variable. 596 01:11:55.710 --> 01:12:01.020 Function just for why so here they integrated over X. 597 01:12:01.020 --> 01:12:05.760 And got this, so. 598 01:12:08.220 --> 01:12:15.840 Okay, now a little late you'll let me. 599 01:12:18.750 --> 01:12:29.340 Okay, 535 here, it's teaching an important idea. I'll give you the manager's level version of it. 600 01:12:33.689 --> 01:12:45.750 It's our noisy communication system again we're transmitting a bit adding or not random Gaussian noise to it receiving a thing. And then so we observed some output. 601 01:12:47.729 --> 01:12:53.699 Why, and we want to know the probability that the input was. 602 01:12:53.699 --> 01:13:02.729 1, so it's not a simple version of it before here. They're making it. 603 01:13:03.779 --> 01:13:10.770 A little more complicated, so, a more general thing and, um. 604 01:13:14.279 --> 01:13:17.699 I'll tell you what's new I'll tell you what to do here. 605 01:13:17.699 --> 01:13:24.119 The previous version we set. 606 01:13:24.119 --> 01:13:29.939 If the output is positive, then what's the probability that the input was? 607 01:13:29.939 --> 01:13:40.439 Minus 1 or 1, and if the output remember, the previous thing is that if the output was positive, there's a 73% chance that the input was positive. 608 01:13:40.439 --> 01:13:43.560 Well, here. 609 01:13:43.560 --> 01:13:49.020 Instead of saying the, I'll put a specific thing positive. This is the general case of it here. 610 01:13:49.020 --> 01:13:52.500 So, for any output, why. 611 01:13:52.500 --> 01:13:59.310 For arbitrary general, why what's the probability that the input was positive? 612 01:13:59.310 --> 01:14:03.689 So, I'll work through the mass of it later. 613 01:14:03.689 --> 01:14:07.979 But be Thursday, but this is what this would come down to. 614 01:14:07.979 --> 01:14:11.970 So. 615 01:14:11.970 --> 01:14:17.579 And so here's the interesting thing with this. 616 01:14:17.579 --> 01:14:26.609 But I was looking at the previous 1, I said, you know, maybe the cut off should not be why paws attempt to decide if the input is positive. 617 01:14:26.609 --> 01:14:30.329 And what I was doing, I was trying to lead into this slowly. 618 01:14:30.329 --> 01:14:38.670 So, this is saying that the probability that the input is positive. 619 01:14:38.670 --> 01:14:42.689 It becomes a half when the output is 4th. 620 01:14:43.800 --> 01:14:50.399 Not 0, so this 1 you have to sit and think about it a little. Here's what's happening. 621 01:14:52.260 --> 01:14:57.060 If you don't know anything, that's a priority probability. If you don't know anything. 622 01:14:57.060 --> 01:15:02.399 A negative inputs more likely than a positive and that says, that's just a statement. 623 01:15:02.399 --> 01:15:05.430 So, if you don't know anything. 624 01:15:05.430 --> 01:15:12.060 You would guess some so the output was 0, you're going to guess the input was more likely than not negative. 625 01:15:12.060 --> 01:15:21.960 Because it was more likely to be negative anyway, and you added us unbiased noise to it. So, but as the output gets more and more positive, if it's more and more likely that the input was positive. 626 01:15:21.960 --> 01:15:24.989 Me, I suppose the output was a 1,000,000. 627 01:15:25.345 --> 01:15:39.864 And, yeah, the input is positive and your opinion me and Nick sent me unlikely, but at that point, yeah, you think the inputs going to be positive but the question is, at what value for the output, why does this does the probability the input being positive exceed 1, half. 628 01:15:41.399 --> 01:15:45.930 And that's here when the, when the output is point 3 5. 629 01:15:45.930 --> 01:15:56.250 It's a 50 50 chance that the input was positive and why get tired it gets warm. So, this is your this is your optimal cut off here. 630 01:15:56.250 --> 01:16:01.560 For it's called the maximum a receiver, so this is. 631 01:16:01.560 --> 01:16:06.210 But what this is computing is what is the best cut off? 632 01:16:06.210 --> 01:16:11.369 For a decision you're looking at the output that you should decide at the input was paused. 633 01:16:11.369 --> 01:16:20.550 And this is it here at the outputs more than point 35, then it's more than 50 50 at the input was positive. So. 634 01:16:20.550 --> 01:16:24.420 So this is an important thing here. I'll work this out more. 635 01:16:25.920 --> 01:16:29.039 And then we'll get other conditional expectations and. 636 01:16:29.039 --> 01:16:32.880 Things like average summer defects and so on and so. 637 01:16:32.880 --> 01:16:37.260 So this is what I'll continue on remember Monday. 638 01:16:37.260 --> 01:16:42.420 Next week from today, there's an exam and this is. 639 01:16:42.420 --> 01:16:48.899 I'll continue on from here example, 535 and page 268. 640 01:16:48.899 --> 01:16:55.229 And if you'd like to read ahead, if you, you know, some people like to have a reading assignments, so they can prepare for the lecture. 641 01:16:55.229 --> 01:16:58.829 Your reading assignment to start here and read so. 642 01:17:01.829 --> 01:17:06.510 Ok, so if there's any questions. 643 01:17:06.510 --> 01:17:10.500 I'm watching the chat window for a minute or so. 644 01:17:10.500 --> 01:17:14.760 Other than that, have a good week, enjoy the sunshine. 645 01:17:14.760 --> 01:17:18.359 And we'll talk virtually on Thursday. 646 01:17:21.420 --> 01:17:30.869 What red key video are we up to now? 647 01:17:32.699 --> 01:17:35.699 What chapters are on on the exam? 648 01:17:35.699 --> 01:17:40.829 I'll list that in more detail on Thursday, but. 649 01:17:42.600 --> 01:17:46.680 Basically up to what we've had homeworks on, so that will be up through. 650 01:17:46.680 --> 01:17:57.000 Hard way into chapter 5 so, and I'll list the red because professor rag, he's not following the textbook. Precisely. 651 01:17:57.000 --> 01:18:00.600 Which is the issue here so I'll list exactly what. 652 01:18:00.600 --> 01:18:06.569 Are the more interesting? The more relevant videos, but there's a 1 says topics match what we've covered. 653 01:18:06.569 --> 01:18:11.369 So other questions. 654 01:18:12.569 --> 01:18:16.199 Hello. 655 01:18:18.449 --> 01:18:25.859 Where will I post that easiest thing is posted on my blog here. I made duplicate it to Webex. 656 01:18:25.859 --> 01:18:31.260 Using Webex more for temporary stuff and the blog for more permanent stops. So. 657 01:18:35.909 --> 01:18:40.500 Okay anything else okay. 658 01:18:40.500 --> 01:18:43.770 And copy it. 659 01:18:45.960 --> 01:18:51.630 Silence. 660 01:18:51.630 --> 01:18:55.170 Okay. 661 01:18:56.909 --> 01:18:59.939 Bye bye.