WEBVTT 1 00:00:48.240 --> 00:00:51.359 Okay. 2 00:00:52.770 --> 00:01:05.010 Hello. 3 00:01:22.140 --> 00:01:29.189 Oh, hey, good afternoon class. 4 00:01:29.189 --> 00:01:37.620 Probability and I think I'm recording this and for the people who are not here. 5 00:01:37.620 --> 00:01:42.329 Just do a quick check over here, but it will. 6 00:01:37.650 --> 00:01:42.329 Just do a quick check over here, but it will. 7 00:01:44.159 --> 00:01:47.430 Some feedback probably. 8 00:01:47.430 --> 00:01:52.920 But I'll just do it for. 9 00:01:57.144 --> 00:02:23.724 Okay. 10 00:02:35.729 --> 00:02:42.180 Okay. 11 00:02:42.180 --> 00:02:45.599 Let's see. 12 00:02:45.599 --> 00:02:59.039 Hello. 13 00:03:00.294 --> 00:03:01.014 Screen 14 00:03:01.944 --> 00:03:17.034 okay. 15 00:03:17.310 --> 00:03:22.080 Good. 16 00:03:23.969 --> 00:03:34.620 Is actually working so. 17 00:03:44.580 --> 00:03:50.250 So, what is happening today, is we're continuing on in chapter 5. 18 00:03:50.250 --> 00:03:53.430 And I'm teaching you by means of examples. 19 00:03:53.430 --> 00:03:59.849 From the book, and so on 1st exam, the 2nd exam will be coming up in a couple of weeks. Like. 20 00:03:59.849 --> 00:04:05.520 3 weeks, 2 or 3 weeks on a Thursday same rules as the 1st exam. 21 00:04:05.189 --> 00:04:09.960 Except you have 2 credit sheets, they won't base it on great scope multiple choice questions. 22 00:04:05.520 --> 00:04:09.960 Except you have 2 credit sheets at 1 base it on great Scott, multiple choice questions. 23 00:04:13.289 --> 00:04:16.439 To be equipped for people, um. 24 00:04:20.759 --> 00:04:34.978 Spring equinox. Um, so day and night are the same. 25 00:04:36.418 --> 00:04:42.088 Both 12 hours however, if I pull up the weather app on my phone. 26 00:04:42.088 --> 00:04:49.348 I find out the day is in fact, day, like, 12 hours and 12 minutes long not 12 hours. So. 27 00:04:49.348 --> 00:04:53.908 You can watch the why question? 28 00:04:55.254 --> 00:05:08.723 2nd thing, I mean, this is unrelated, of course, but I like gadgets. I don't know if you guys like gadgets all. So so my latest gadget is a USB cable that has a little hour display built into the cable. 29 00:05:09.269 --> 00:05:15.059 So, I can look at that, and I can see, like, right now, my iPad is drawing 28 watts. 30 00:05:15.059 --> 00:05:21.149 So little slow little device here, I can see how things are like. 31 00:05:22.588 --> 00:05:27.959 And I have a little gadget at home, which not only shows power, but shows voltage and current. 32 00:05:27.959 --> 00:05:33.809 For the USB cable plugs in line with USB cable. So I can I can see how things are going. 33 00:05:31.678 --> 00:05:36.269 I can see how things are going. I like yeah, it just like that. 34 00:05:33.809 --> 00:05:39.389 I like just like that. Okay, so. 35 00:05:37.588 --> 00:05:42.209 Okay, so I'm going to pick, um. 36 00:05:40.588 --> 00:05:47.848 I'm gonna pick I'm gonna pick some examples from this actually. Um. 37 00:05:44.519 --> 00:05:47.848 Kind of pick some examples from this actually um. 38 00:05:47.848 --> 00:05:55.108 Not precisely from that, but there was 1 I started last time, but I didn't finish. It would be. 39 00:05:55.019 --> 00:05:59.218 Okay. 40 00:05:55.108 --> 00:05:59.218 Okay. 41 00:05:59.218 --> 00:06:03.629 So, it'll chance to review it, and it will be a chance to finish it. I started it. 42 00:06:03.629 --> 00:06:08.369 And this was example, 530 on page 263. 43 00:06:03.658 --> 00:06:08.369 And this was example, 530 on page 263. 44 00:06:09.478 --> 00:06:19.858 Come on here continued, so. 45 00:06:19.858 --> 00:06:28.379 And again, um, what we have is that, um. 46 00:06:29.999 --> 00:06:34.649 Here's a chip with defects. 47 00:06:37.738 --> 00:06:41.608 And, um, X is the random variable. 48 00:06:41.608 --> 00:06:50.519 Are the number of defects okay. And, um, X is, um. 49 00:06:51.869 --> 00:07:00.418 Plus, on with me equals alpha. Okay. Um, so here's a region are. 50 00:07:03.059 --> 00:07:10.528 And, um, so each, the FAQ. 51 00:07:14.249 --> 00:07:21.298 On the chip, it falls into our. 52 00:07:21.298 --> 00:07:25.798 With probability P. W. P. I'll write it out me in the West. 53 00:07:25.798 --> 00:07:29.728 Overbill okay. Okay. 54 00:07:29.728 --> 00:07:33.449 Um. 55 00:07:35.038 --> 00:07:40.199 So, why is the random variable for the number of defects? 56 00:07:42.329 --> 00:07:47.249 And R, and basically, um. 57 00:07:53.848 --> 00:07:57.928 Tell me why. 58 00:07:57.928 --> 00:08:05.218 Okay, um, so what I want is something like, um. 59 00:07:57.959 --> 00:08:05.218 Okay, um, so what I want is something like, um. 60 00:08:07.288 --> 00:08:17.189 I so what is probability of why for some particular value, Jay and it's discrete it's the probability mass function. Okay. 61 00:08:17.189 --> 00:08:21.538 Um, how do I do that? 62 00:08:21.538 --> 00:08:26.278 Okay, um. 63 00:08:31.288 --> 00:08:39.328 So, now for the region, why so, why is, um. 64 00:08:43.408 --> 00:08:49.259 So, basically, the number of defects that fall into ours, so binomial. 65 00:08:49.259 --> 00:08:53.308 To buy random variable. So, um. 66 00:08:53.278 --> 00:08:57.239 Basically. 67 00:08:53.308 --> 00:08:57.239 Basically. 68 00:09:00.749 --> 00:09:06.538 So there are K, defects in the whole chip, then the. 69 00:09:06.538 --> 00:09:11.609 Of J, defects in the region why given K in the whole chip. 70 00:09:12.688 --> 00:09:16.048 This is going to be a meal. 71 00:09:16.048 --> 00:09:24.119 And that's going to be oh, 0, if J, greater than K you know. Okay. 72 00:09:19.438 --> 00:09:22.859 0, if J, greater than K, you know. 73 00:09:22.859 --> 00:09:27.719 Okay, and otherwise it's, um, K, Tuesday. 74 00:09:24.119 --> 00:09:27.719 And otherwise it's, um, K, Tuesday. 75 00:09:33.778 --> 00:09:38.639 Okay, these are all the same Jay. Let's see. Okay, so it's just each. 76 00:09:38.639 --> 00:09:44.038 Defect in the whole chip independently falls into the region are with probability. 77 00:09:44.038 --> 00:09:47.038 P, they're all independent of the number of defects. 78 00:09:47.038 --> 00:09:55.889 That hit the region, it's this binomial rent here. Okay. The P is a problem if anyone defect landing. Okay. Um. 79 00:09:58.769 --> 00:10:04.288 And the probability for the total number of defects being K. 80 00:10:06.448 --> 00:10:11.129 As I said was paused and that's going to be, um. 81 00:10:16.769 --> 00:10:20.038 Okay, what's on again? 82 00:10:20.038 --> 00:10:23.489 Part of the problem statement. 83 00:10:23.458 --> 00:10:28.198 So now, what's the probability mass function for the number of defects in. 84 00:10:23.489 --> 00:10:28.198 So now, what's the probability math function for the number defects in. 85 00:10:31.349 --> 00:10:38.369 And I had the conditional 1 up here, given that there were K total. So the unconditional probability. 86 00:10:38.369 --> 00:10:44.339 For J, defects hitting the chip, it's going to be the sum of all the conditional probabilities. 87 00:10:44.339 --> 00:10:51.089 Um, okay and probability. Okay. 88 00:10:51.089 --> 00:10:56.999 So, exit in here and send over, um. 89 00:10:58.408 --> 00:11:04.828 To in Canada, so John, conditional probability is that being J defects on the sub region? 90 00:11:04.828 --> 00:11:16.048 Conditional probability given the Katie text total times are probably okay. Defects. Total summed over. Okay. Okay. So now I just plug in the formulas and simplify. 91 00:11:16.048 --> 00:11:28.198 So, um, now the 1st thing here is that, um. 92 00:11:28.198 --> 00:11:33.869 K has to be at least as big as Jay K. cannot be less than Jay. That is probability. 0. 93 00:11:33.869 --> 00:11:36.899 Okay, so, um. 94 00:11:36.899 --> 00:11:42.869 So, the 1st thing I can do is, I can say, um. 95 00:11:48.269 --> 00:11:52.259 Since Jason, I say, okay. Okay. Um. 96 00:11:52.259 --> 00:11:55.379 So, in any case I can now. 97 00:11:56.489 --> 00:12:08.249 Simplify the whole thing um, okay, so that 1st thing is there okay. Choose Jay. 98 00:12:10.078 --> 00:12:16.469 It is a J1-P K minus J times um. 99 00:12:14.578 --> 00:12:22.739 Times, um, elephant is a K over K Victoria alpha. 100 00:12:16.469 --> 00:12:22.739 If it is a K over K Victoria either my alpha. 101 00:12:22.739 --> 00:12:26.099 Okay, now, um. 102 00:12:50.339 --> 00:12:55.798 Okay, now what can we start doing with this? 103 00:12:55.798 --> 00:13:02.788 Um, well, 1st, of course, uh, let me see. 104 00:13:06.808 --> 00:13:14.788 This goes out, um, we can start pulling out constant things. Um. 105 00:13:14.788 --> 00:13:24.538 Oh, pile of these things actually can Co in front of the summation. So I got to go to the next page and it probably is as well. 106 00:13:25.589 --> 00:13:32.188 Right what we're finding is a probability of some J. 107 00:13:33.538 --> 00:13:37.109 I'll scroll up in in a minute. Um, so. 108 00:13:40.889 --> 00:13:46.739 Uh, what can we pull out here? Um. 109 00:13:48.178 --> 00:13:53.788 Let's see, that would be enough for the 1st step. 110 00:14:06.869 --> 00:14:14.219 Okay, and if I scroll too fast for any 1, let me know, um. 111 00:14:12.448 --> 00:14:17.698 You know, um, what else can I pull out here? Um. 112 00:14:14.219 --> 00:14:17.698 What else can I pull out here? Um. 113 00:14:22.379 --> 00:14:25.798 So, I've got 1 more piece here, um. 114 00:14:29.038 --> 00:14:37.288 Okay, okay. 115 00:14:39.899 --> 00:14:47.428 Okay, um, let's see what we can pull out here. Um. 116 00:14:54.389 --> 00:14:57.688 Let me go. 117 00:15:12.749 --> 00:15:19.769 I'm gonna use the fact that alpha is a k8 goes out is a J time zone, which is a K minus K. 118 00:15:19.769 --> 00:15:25.379 Um, and now this thing equals. 119 00:15:29.308 --> 00:15:34.438 You know, minus alpha alpha to the J over J factorial. 120 00:15:34.438 --> 00:15:47.969 Some of all the, um, hey, minus J. A. J. 121 00:15:49.198 --> 00:15:54.479 Let me pull out, um, over. 122 00:15:53.639 --> 00:15:59.698 Over okay. 123 00:15:58.198 --> 00:16:08.219 Okay, okay. Um. 124 00:16:05.609 --> 00:16:09.389 Okay, um, let's see. 125 00:16:08.219 --> 00:16:12.928 See, now this summation here. 126 00:16:09.389 --> 00:16:12.839 Now, this summation here. 127 00:16:12.839 --> 00:16:21.688 I can do a slide K down by J. actually, um. 128 00:16:12.928 --> 00:16:21.688 I can do a slide K down by J. actually, um. 129 00:16:27.418 --> 00:16:31.558 And this will then be the same as, um. 130 00:16:39.448 --> 00:16:43.168 K. over K. factorial. 131 00:16:44.489 --> 00:16:47.519 And the thing on the left, I can simplify a little. 132 00:16:47.489 --> 00:16:50.489 As alpha P to the J. 133 00:16:47.519 --> 00:16:53.158 As alpha to the J, um. 134 00:16:52.078 --> 00:16:57.208 Um, either myself over J factorial. 135 00:16:54.448 --> 00:17:04.138 Either mine itself over J. factorial. Okay. So the thing you may want to think a few seconds about is a summation here. 136 00:16:57.208 --> 00:17:04.138 Okay, so the thing you may want to think a few seconds about is estimation here. I. 137 00:17:04.138 --> 00:17:11.638 Instead of J. to infinity all slide reduced Kay. Bye de, replace K minus J. 138 00:17:11.638 --> 00:17:23.909 I'll reduce by J. O. everywhere basically to replace K by K. plus J. and so it says K starting a J. we'll start at 0 and wherever we had a K minus J. 139 00:17:23.909 --> 00:17:29.398 It will be a K. okay, so if that looks reasonable. 140 00:17:31.949 --> 00:17:37.318 For this summation here, it's a it's an exponential sum. 141 00:17:37.318 --> 00:17:40.409 And this equals, um. 142 00:17:40.409 --> 00:17:44.878 It is a -1-P. 143 00:17:46.378 --> 00:17:50.969 Um, and. 144 00:17:56.398 --> 00:18:00.088 The, the plus alpha 1-P 1 of my thinking, um. 145 00:18:01.378 --> 00:18:08.429 So, what we have is alpha P to the J. you didn't mind itself, but he did the alpha. 146 00:18:08.429 --> 00:18:12.479 1-P over J factorial. 147 00:18:08.578 --> 00:18:12.479 1-P over J factorial. 148 00:18:12.479 --> 00:18:15.778 These things simplified nets out to. 149 00:18:15.778 --> 00:18:18.989 The alpha P. J. 150 00:18:18.989 --> 00:18:25.318 Um, let's do it on the next line here. Um. 151 00:18:31.679 --> 00:18:36.419 P to the J. E. the minus, um. 152 00:18:36.419 --> 00:18:43.679 Alpha P. over J. factorial. So this is parcel with mean, alpha. 153 00:18:45.179 --> 00:18:58.259 The corner here, so what I just showed you. 154 00:18:58.259 --> 00:19:02.939 Was that the number of defects in the sub region? 155 00:19:02.939 --> 00:19:09.689 It's pass on with me, and I'll put in the whole region is the probability P of being in the sub regions. So. 156 00:19:09.689 --> 00:19:15.989 So this makes sense somewhat if you think about it, but it's showing you some techniques was working with these things. 157 00:19:17.068 --> 00:19:24.209 Okay, any questions about. 158 00:19:24.209 --> 00:19:29.009 About that so. 159 00:19:36.148 --> 00:19:45.898 Oh, okay. Next 1 I want to work on this example 531 on page 264. 160 00:19:45.898 --> 00:19:49.979 Let's do it in black because it's easier to read. Um. 161 00:20:00.269 --> 00:20:05.308 Okay, binary communication system, we're going to see this example. 162 00:20:05.308 --> 00:20:10.858 My mind is going binary communicate, you know, quite a lot. 163 00:20:17.699 --> 00:20:29.669 Okay, um, for each time I show it to you, I'll make it a little more complicated, which means more a little more realistic. So. 164 00:20:31.048 --> 00:20:36.419 There I got some notes on this. Um, so basically, what we do is that. 165 00:20:38.338 --> 00:20:46.528 Transmit signal X and X is going to be, um, +1 or -1. 166 00:20:46.528 --> 00:20:50.098 But they're not equally probable so, +1. 167 00:20:50.098 --> 00:20:54.028 Has probability 4th. 168 00:20:54.028 --> 00:20:58.019 And -1 is probability 2 thirds. They're not even. 169 00:20:59.308 --> 00:21:02.999 Now, what's the real example where you'd have uneven equal. 170 00:21:02.999 --> 00:21:12.358 Probabilities imagine that I'd take a, a scan of this page is an image P. G, whatever your favorite image format is. 171 00:21:12.328 --> 00:21:18.269 Then the black pixels, and the White pixels are not equal probability on that page. There. 172 00:21:12.358 --> 00:21:18.269 Then the black pixels, and the White pixels are not equal probability on that page. There. 173 00:21:18.269 --> 00:21:21.538 You know, the black pixels are. 174 00:21:21.538 --> 00:21:27.328 Maybe 3%, 5% or something in the White pixels are maybe 95%. 175 00:21:27.328 --> 00:21:35.249 So, it's quite common that the black and white P, pixels of the 2 bits were transmitting to values + and -1. 176 00:21:35.249 --> 00:21:40.108 It's quite common that they might not be equal probability. And here I'm just making it. 177 00:21:40.108 --> 00:21:48.689 4th 2 thirds, um, keeping them discreet. Next example, they'll be Kelsey and distributed. Um. 178 00:21:48.689 --> 00:21:52.828 And then we had noise, um. 179 00:21:58.259 --> 00:22:01.709 And and noises go see, and. 180 00:22:02.999 --> 00:22:07.709 Um, just to keep it simple mean equals 0, Sigma equals 1. 181 00:22:07.709 --> 00:22:12.989 And then what's received is Y, equals X plus N. 182 00:22:12.989 --> 00:22:21.868 Okay um, and ultimately, what we would like to know is. 183 00:22:13.019 --> 00:22:21.868 Okay, and ultimately what we would like to know is. 184 00:22:21.868 --> 00:22:26.459 What's the best guess for us when we see a particular Y. 185 00:22:27.749 --> 00:22:32.308 Um, so the, the big question we're going to. 186 00:22:32.308 --> 00:22:38.939 The big question is, um, if we see. 187 00:22:38.939 --> 00:22:47.969 Some Y, what do we get. 188 00:22:47.969 --> 00:22:51.419 That ex. 189 00:22:51.419 --> 00:22:56.519 Okay, and, um. 190 00:22:56.489 --> 00:23:01.769 So, they aren't going to be working my way up to that. Um. 191 00:22:56.519 --> 00:23:01.769 So, they aren't going to be working my way up to that. Um. 192 00:23:01.769 --> 00:23:07.888 Now, and ideally we'd want to say the probability so we'd have to say. 193 00:23:07.888 --> 00:23:17.308 Probability of X equals +1, given some Y versus the probability. The X. 194 00:23:17.308 --> 00:23:22.648 It was -1 given some why? And we'd pick the bigger. Okay. 195 00:23:23.909 --> 00:23:31.709 Okay um, makes it a bigger perhaps. 196 00:23:31.709 --> 00:23:35.098 And you might be taking base, in fact. 197 00:23:35.098 --> 00:23:39.179 And you might be right. Okay. Now, um. 198 00:23:35.189 --> 00:23:39.179 And you might be right. Okay. Now, um. 199 00:23:40.439 --> 00:23:49.588 The real world, um, in fact, I just put some blogs some points here on the right. It's on the blog that I typed in. Um, while back. 200 00:23:51.509 --> 00:23:57.838 That in the real world, there are other things you would do. Of course, the probability would only be part of it. Um. 201 00:23:57.808 --> 00:24:05.459 Because you could imagine an intermediate thing where you might see a, why that's so large. You knew that X was 1. 202 00:23:57.838 --> 00:24:05.489 Because you could imagine an intermediate thing where you might see a, why that's so large. You knew that X was 1. 203 00:24:05.459 --> 00:24:10.919 You might see a why that's so negative. You knew that X was -1 or you might see a Y in the middle. 204 00:24:05.489 --> 00:24:10.919 You might see a why that's so negative. You knew that X was minus water you might see a Y in the middle. 205 00:24:10.919 --> 00:24:14.278 So suppose you receive a signal of 0. 206 00:24:14.278 --> 00:24:20.788 So, it's, you know, you don't know, was that -1 or X was 1 and the real world is different things you could do. 207 00:24:20.788 --> 00:24:23.999 And I mentioned some of them there, you might increase. 208 00:24:23.999 --> 00:24:27.868 You might increase the transmitted signal to increase a transmitted voltage. 209 00:24:27.868 --> 00:24:36.778 Um, if you're able to now, that might have some problems historically, the 1st, trans, Atlantic telegraph cable that went. 210 00:24:36.778 --> 00:24:43.259 Added a Newfoundland and heart's content Bay, which I visited actually um. 211 00:24:43.259 --> 00:24:47.909 So, it's like 1500 or more miles long and so. 212 00:24:47.909 --> 00:24:55.348 The received signal was so weak, they had trouble deciding what it was. The transmission speed was under 1 bit per. 2nd. 213 00:24:47.939 --> 00:24:55.348 The received signal was so weak, they had trouble deciding what it was. The transmission speed was under 1 bit per. 2nd. 214 00:24:55.348 --> 00:24:59.219 So they decide to increase the transmission voltage. 215 00:24:59.219 --> 00:25:03.989 To make the receipts single stronger, uh, amplifiers have not yet been invented. 216 00:25:03.808 --> 00:25:09.358 So, the problem was, they increased the, um. 217 00:25:03.989 --> 00:25:09.358 So, the problem was, they increased the, um. 218 00:25:09.358 --> 00:25:17.038 Transmission voltage so much that they punched through the installation on the cable in the middle of the Atlantic, and they destroyed the cable. 219 00:25:17.038 --> 00:25:21.568 After 3 weeks, that was an operation for 3 weeks and they destroyed it. 220 00:25:21.568 --> 00:25:29.308 And it took sorry field for the vendor several years to round up the money until lay a 2nd cable. 221 00:25:29.308 --> 00:25:40.078 And, of course, he was being accused of fraud because the thing failed after 3 weeks. Maybe it never actually worked but any case increase in signal transmitted signal. Perhaps um. 222 00:25:40.078 --> 00:25:44.608 You might try to reduce the noise somehow better installation or something. So. 223 00:25:44.608 --> 00:25:50.699 On the they take the translated cable is an exit another example. 224 00:25:50.699 --> 00:25:55.108 They wrapped some of their Telegraph cables with a special metal. 225 00:25:55.108 --> 00:26:02.128 That counteracted the, um, the inductance of the cable and sort of reduced the. 226 00:26:02.128 --> 00:26:06.898 The doctor and so they reducing the noise. That'd be step 2 here. 227 00:26:06.868 --> 00:26:12.298 Step 3, again, you might go to some error correcting method. 228 00:26:06.898 --> 00:26:12.269 I mean, step 3, again, you might go to some error correcting method. 229 00:26:12.269 --> 00:26:16.169 If you can, you know, if you can send a negative acknowledgement back. 230 00:26:12.298 --> 00:26:16.169 If you can, you know, if you can send a negative acknowledgement back. 231 00:26:16.169 --> 00:26:24.449 It says, I didn't understand you, then the transmitter would say, transmit the single again and hopefully the 2nd time it might come through better. 232 00:26:20.308 --> 00:26:24.449 Transmit the single again and hopefully the 2nd time it might come through better. 233 00:26:24.449 --> 00:26:34.709 Would be an idea, um, you might put in check psalms and again, it's it it's received with an invalid check. Some. Then you send a Nac negative acknowledgment back and that's what. 234 00:26:34.709 --> 00:26:38.878 Transmission control protocol does, for example, for the Internet. 235 00:26:38.489 --> 00:26:48.689 Is that each packet has a check some if the check summarize wrong then and and negative acknowledgement is sent back. So it's re, transmitted. 236 00:26:38.878 --> 00:26:48.689 Is that each packet has a check some if the check summarize wrong then and and negative acknowledgement is sent back. So it's re, transmitted. 237 00:26:48.689 --> 00:26:54.959 Um, how well, that works, of course, depends on things like latency. 238 00:26:54.959 --> 00:27:05.159 If you're doing computer communications to a robot on Mars and you're round trip is what? 15 minutes or something at best? I can't remember. Then this. 239 00:26:55.019 --> 00:27:05.159 If you're doing computer communications to a robot on Mars and you're round trip is what? 15 minutes or something at best? I can't remember. Then this. 240 00:27:05.159 --> 00:27:09.148 You know, requesting a re, transmit doesn't work quite as well, but, um. 241 00:27:10.378 --> 00:27:16.588 So this is, um, and you could adapt only do re, transmit. The signal is uncertain. Let's say. 242 00:27:16.588 --> 00:27:23.128 In any case, so those are a lot of real world things you might do in this noisy communication. 243 00:27:23.128 --> 00:27:26.608 Thing I, of course, you can do all of them at once. Um. 244 00:27:26.608 --> 00:27:30.989 You know, go for less noise and do air correction and so on. But, um. 245 00:27:32.308 --> 00:27:38.608 Any case so, this is being a probability. Course I'll talk about the probability part of that broader solution. 246 00:27:38.608 --> 00:27:41.638 Oh. 247 00:27:38.638 --> 00:27:41.638 Oh. 248 00:27:41.638 --> 00:27:47.939 But you could also have, or another thing, of course, real world is, you could transmit, um. 249 00:27:47.939 --> 00:27:55.798 You might say, have multiple transmitter and Ken is multiple receiver in tennis. This is the way. 250 00:27:55.798 --> 00:28:01.588 You know, sometimes done with transmitting and receiving both WI, Fi and cell phone to get multiple antennas. 251 00:28:01.588 --> 00:28:07.409 And 1 antenna might receive the signal better than another antenna. That's a few inches away. 252 00:28:07.378 --> 00:28:10.679 So, you know, 1 possibility so. 253 00:28:07.409 --> 00:28:10.679 So, you know, 1 possibility so. 254 00:28:10.679 --> 00:28:17.638 And the 2 tenants don't have to be very far apart if we're transmitting say, at, um. 255 00:28:18.868 --> 00:28:28.709 Well, let's take a gigahertz, for example, if somebody signals a couple of gigahertz, then gigahertz the wavelengths and vacuum is 1 foot. So that. 256 00:28:28.709 --> 00:28:34.138 If you have your 2 antennas say, 6 inches apart, that's half a wavelength. So. 257 00:28:34.138 --> 00:28:40.229 You know, 1 is being subjected to destructive interference. The next antenna might actually work better. That's a few inches away. 258 00:28:40.229 --> 00:28:43.318 Okay, any case, um. 259 00:28:44.368 --> 00:28:50.638 So probability, um, talk about the probability aspects here. 260 00:28:50.638 --> 00:28:59.519 Start a new page, like days here. 261 00:28:59.519 --> 00:29:03.628 I mean, I want the backward, um, probabilities as I call them. 262 00:29:03.628 --> 00:29:09.269 Well, 1st, we'll compute the, um, the forward 1 so let's say it's do some sort of. 263 00:29:09.269 --> 00:29:13.949 Kim, which of things say we're talking about, why. 264 00:29:13.949 --> 00:29:23.219 And, um, it came the probability, um, of some why given that the transmitted X was, um, +1. 265 00:29:24.598 --> 00:29:29.519 Right. +1 well, that's the, um, probability that. 266 00:29:29.519 --> 00:29:34.288 Random variable Y, is less than equal to the specific given value. 267 00:29:34.259 --> 00:29:38.249 Given that X was 1. okay. 268 00:29:34.288 --> 00:29:38.219 Give it in that X was 1. okay. 269 00:29:38.219 --> 00:29:46.499 Now, why, but if we remember him use a different color for contrast here, remember that why it was X? Plus N. 270 00:29:38.249 --> 00:29:46.499 Now, why, but if we remember him, he's a different color for contrast here. Remember that why it was X? Plus N. 271 00:29:48.028 --> 00:29:54.179 So so this thing equals the probability that, um. 272 00:29:56.068 --> 00:29:59.939 Um, so why is N +1? That's very cool to why. 273 00:29:59.939 --> 00:30:05.848 Given X equals 1 equals the probability that N, it's less than or equal to. 274 00:30:07.439 --> 00:30:11.489 By -1 give an X equals. 275 00:30:11.489 --> 00:30:16.949 Well, I've used the fact that X equals 1 here. I don't have that any more. Don't need that any more. Okay. 276 00:30:16.949 --> 00:30:20.519 That is used that I plugged in. Okay. 277 00:30:20.519 --> 00:30:32.788 Um, okay, um, that now, but remember and I'll put comments in a different color. Okay, remember that and is. 278 00:30:35.459 --> 00:30:44.939 Okay, so now this here, this is a, a, well, it's the left tail of the calcium distribution. 279 00:30:44.939 --> 00:30:49.979 And this will be here equally in a girl from minus infinity to. 280 00:30:49.979 --> 00:30:53.788 By -1. 281 00:30:55.679 --> 00:31:01.019 Is the minus square over 2? 282 00:31:01.019 --> 00:31:04.588 Okay, um. 283 00:31:05.909 --> 00:31:10.348 For the ghost yet. Oh, okay. Um. 284 00:31:14.278 --> 00:31:19.288 Now, what I may want is if access, we might be interested that. 285 00:31:19.288 --> 00:31:26.098 Effects was +1. What's the probability the received signal is why it's positive. Let's say. 286 00:31:26.098 --> 00:31:31.919 So, um, so maybe, let's look at something that's probably say why greater than 0. 287 00:31:31.919 --> 00:31:38.578 Given X equals 1, that's going to be there goal of, um. 288 00:31:45.118 --> 00:31:53.969 It's basically, um, it's gonna net out was my. 289 00:31:53.969 --> 00:31:59.729 Well, it's going to net out if I, um, wave my hands a little, um. 290 00:32:06.689 --> 00:32:11.308 Because I have the probability here is the probability that, um. 291 00:32:06.808 --> 00:32:11.308 Because I have the probability here is the probability that, um. 292 00:32:15.838 --> 00:32:24.778 Let's see, now, um, this here up here is the probability that. 293 00:32:26.398 --> 00:32:29.878 That the noise is less than or equal to -1 is that. 294 00:32:30.989 --> 00:32:37.439 And we so the problem is why I greater 0, that's the probability. Um. 295 00:32:37.439 --> 00:32:41.459 That X was was greater than -1. 296 00:32:41.459 --> 00:32:46.798 Okay, well this sequence a problem is, the noise is greater than -1 here. 297 00:32:48.778 --> 00:32:53.098 No, he's greater than equal to -1 which is that okay. 298 00:32:53.098 --> 00:33:02.189 Um, and that we can actually look up into our tables here and that will turn to point 84, um, from from the tables. 299 00:33:06.449 --> 00:33:10.169 And that will be called some 1-queue of 1. um. 300 00:33:10.169 --> 00:33:14.578 Q is just a short engineering shorthand for the. 301 00:33:16.648 --> 00:33:21.749 The right tail of the probability, um, density function here. Um. 302 00:33:26.669 --> 00:33:33.028 I tell the go see, and and you can look it up in tables. Okay. Um. 303 00:33:32.999 --> 00:33:43.949 Yesterday, so we got here, this is, this is the probability that if we transmit it a 1, then we receive a positive signal. 84% of the time. 304 00:33:33.028 --> 00:33:43.949 Yesterday, so we got here, this is, this is the probability that if we transmit it a 1, then we receive a positive signal. 84% of the time. 305 00:33:43.949 --> 00:33:56.068 Um, okay, go to the next page and we can also ask what's the probability that we receive a positive signal? 306 00:33:56.068 --> 00:34:02.459 If, um, if we transmit at a negative signal. 307 00:34:02.459 --> 00:34:07.499 And it'll, it'll net out to the similar thing 1, to infinity. 308 00:34:11.429 --> 00:34:23.009 Point 16. okay. So if we try and spend a negative signal, 16% of the time, -116ofthe time will receive a positive signal. 309 00:34:24.298 --> 00:34:28.889 And, um, so the probability that we receive a positive signal. 310 00:34:28.889 --> 00:34:33.838 It's the combo cause they receive a positive signal given. 311 00:34:35.009 --> 00:34:40.559 We transmit it to pause those signal times. It probability transmitting a positive signal. 312 00:34:40.559 --> 00:34:46.528 Plus the probability receiving a positive signal given we transmit in a negative signal terms of probability. 313 00:34:40.619 --> 00:34:46.528 Plus the probability receiving a positive signal given we transmit in a negative signal terms of probability. 314 00:34:47.579 --> 00:34:51.958 Receiving a negative signal, um. 315 00:34:54.539 --> 00:35:06.719 And, um, and I said the problem and this 1, add out to, um, point 84 times 4+point, 16 times 2 thirds. And that will net out too. 316 00:35:06.719 --> 00:35:14.518 20 39, so the 40% chance that we will receive a positive signal. 317 00:35:14.518 --> 00:35:17.878 Um, so, no, um, basically. 318 00:35:17.878 --> 00:35:25.708 So, just just observe, um, the probability of transmitting a positive signal was. 319 00:35:25.708 --> 00:35:31.559 33, but the probability of receiving a positive signal. 320 00:35:31.559 --> 00:35:36.059 39, so. 321 00:35:36.059 --> 00:35:43.829 What happened and this is worth, you know, I think you can think about this here actually. Um. 322 00:35:45.449 --> 00:35:54.148 Think about it, because the thing is, the signals were blurred by the noise. 323 00:35:54.148 --> 00:35:57.748 So, the things that I got. 324 00:35:57.748 --> 00:36:00.748 Closer to 50 here so. 325 00:36:00.748 --> 00:36:10.378 Okay um, but what we wanted is to go backwards. 326 00:36:12.958 --> 00:36:17.699 Then the problem can scroll again. If I ever scroll up to query. Oops. 327 00:36:17.699 --> 00:36:27.509 Let me know um, again, so I go back into black here. Um. 328 00:36:29.728 --> 00:36:33.509 To the global thing we wanted was the probability that we transmitted. 329 00:36:33.509 --> 00:36:39.119 A 1, if we received a positive signal. Okay. 330 00:36:40.889 --> 00:36:46.228 Um, and that, um. 331 00:36:46.228 --> 00:36:50.068 You can add up to, um. 332 00:36:55.168 --> 00:37:03.809 Probability we transmitted a 1 and we received a positive signal divided by the probability that, um. 333 00:37:03.809 --> 00:37:07.528 We received a positive signal, um. 334 00:37:09.898 --> 00:37:15.989 So, we travel bill, we transmitted a 1 and receive the positive signal. So, this thing here. 335 00:37:18.719 --> 00:37:22.108 Um, that equals, um. 336 00:37:24.148 --> 00:37:35.458 Work it down here probability we received the pause, so, you know, given we transmitted a 1 kinds of probability, we transmitted a 1. 337 00:37:35.458 --> 00:37:44.429 Okay, well pause this thing and transmit it to 1 probability. 4th received a positive signal given we transmit it a 1. 338 00:37:35.489 --> 00:37:44.429 Okay, well pause this thing, transmit it to 1 probability. 4th received a positive signal given we transmit it a 1. 339 00:37:44.429 --> 00:37:50.938 Um, was 84%. 340 00:37:57.503 --> 00:39:53.603 Okay, 341 00:39:53.634 --> 00:39:54.083 getting a little. 342 00:39:54.088 --> 00:40:02.070 For example, because the thing is the transmitted signals, they're not split 50, 50, it's plus a 3rd of the. 343 00:39:59.340 --> 00:40:04.769 50 50, it's plus a 3rd of the time and . 344 00:40:03.570 --> 00:40:13.530 Time and -, and if you take account of that tag, there's in fact a better cut off than 0 for deciding between the 2 possible transmitted values. 345 00:40:04.769 --> 00:40:13.530 And if you take account of that tag, there's in fact, a better cut off then 0, for deciding between the 2 possible transmitted values. 346 00:40:13.530 --> 00:40:18.389 Um, but we can actually, um. 347 00:40:18.389 --> 00:40:22.980 Worry about that later so okay. 348 00:40:24.659 --> 00:40:30.239 No questions about this example so it's, um, it shows you a couple of different things here. 349 00:40:30.210 --> 00:40:37.679 But ultimately, what we want to do with a lot of our things here is make a make a policy decision. 350 00:40:30.239 --> 00:40:37.679 But ultimately, what we want to do with a lot of our things here is make make a policy decision. 351 00:40:37.679 --> 00:40:45.360 But what should be our rule for referring the transmitted signal okay when it's been distorted by noise and we just have the received signal. 352 00:40:45.360 --> 00:40:49.170 Okay, what happened here. 353 00:40:49.170 --> 00:40:54.659 Um, there okay, so. 354 00:41:00.449 --> 00:41:07.079 Okay, um, spend some time on this example, but it's worth going through fairly well. Um. 355 00:41:10.980 --> 00:41:14.429 We have conditional density functions here. 356 00:41:14.429 --> 00:41:18.539 Let me demonstrate it with an example. 357 00:41:23.789 --> 00:41:30.449 Okay, this is, um. 358 00:41:37.170 --> 00:41:42.659 That's the conditional PDFs and we'll have an example. 359 00:41:44.519 --> 00:41:47.760 On page 267. 360 00:41:47.760 --> 00:41:51.210 And it refers to, um. 361 00:41:52.559 --> 00:41:57.659 It refers to example, point 8. 362 00:41:58.889 --> 00:42:08.280 Let's go back to wherever that says, um. 363 00:42:27.144 --> 00:42:47.335 Okay, 364 00:42:47.574 --> 00:42:48.445 go here. 365 00:43:16.380 --> 00:43:20.460 They actually mean 516 actually, but, um. 366 00:43:26.909 --> 00:43:34.559 516. 367 00:43:34.559 --> 00:43:37.829 On page 250 to. 368 00:43:39.659 --> 00:43:45.090 Um, we have an X and Y. 369 00:43:45.090 --> 00:43:54.300 Is, um, um. 370 00:43:57.420 --> 00:44:05.309 Okay, um, and then if we do again, if we do marginal. 371 00:44:05.309 --> 00:44:08.969 So this will be to see an overlap of. 372 00:44:08.969 --> 00:44:16.320 X and Y, Y, Y, being from X to infinity. 373 00:44:17.519 --> 00:44:28.559 And this will net out to, um, f, of Y. 374 00:44:28.559 --> 00:44:35.820 Which is the of X and Y, integrate out X and X has to be. 375 00:44:39.929 --> 00:44:46.199 From 0, to X, I think, um, on that out to to, you know, minus Y. 376 00:44:48.179 --> 00:44:52.380 For anything okay now if I go back to page 267. 377 00:45:09.420 --> 00:45:15.960 Okay, um, so that. 378 00:45:21.150 --> 00:45:25.949 So the book goes through the proof, I'll just give you the answer here so. 379 00:45:25.920 --> 00:45:32.699 The conditional and then I'll go back to the exam, the conditional PDF. Um. 380 00:45:25.949 --> 00:45:32.699 Can the conditional and then I'll go back to the exam, the conditional PDF um. 381 00:45:34.679 --> 00:45:42.510 Basically here, um. 382 00:45:44.130 --> 00:45:56.730 To go here so why given a value. 383 00:45:56.730 --> 00:46:00.210 Equals the, um. 384 00:46:00.210 --> 00:46:04.289 The joint divided by. 385 00:46:07.199 --> 00:46:11.280 Um, the same thing we saw before, um. 386 00:46:11.280 --> 00:46:15.300 And did over half of ex given why? Okay so. 387 00:46:18.150 --> 00:46:22.110 So, in this particular case here, we would find that. 388 00:46:24.389 --> 00:46:28.170 And if I scroll to pass and did a f, uh. 389 00:46:28.170 --> 00:46:33.659 Ex, given why. 390 00:46:33.599 --> 00:46:38.159 Will be put a little X there the joined. 391 00:46:33.659 --> 00:46:38.190 Will be a little X there the joined. 392 00:46:39.539 --> 00:46:45.059 Divided by 5 by why. 393 00:46:45.059 --> 00:46:52.530 Okay, so, in our example, if I'm going back to this example, the 516, um. 394 00:46:45.090 --> 00:46:52.530 Okay, so, in our example, if I'm going back to this example, the 516, um. 395 00:46:56.519 --> 00:47:02.219 If I want say, why give an X. 396 00:47:02.219 --> 00:47:08.219 Or the joint was, um, to eat the minus X minus Y. 397 00:47:10.739 --> 00:47:14.489 Divided by, and if I scroll back up. 398 00:47:14.489 --> 00:47:17.820 That was to either the minus, um. 399 00:47:23.010 --> 00:47:26.369 I got it, right? Yeah. 400 00:47:26.369 --> 00:47:31.050 Okay, and then we would simplify out and so on, um. 401 00:47:32.880 --> 00:47:43.769 Yeah, um, okay for. 402 00:47:45.840 --> 00:47:52.260 Figure out why so you can get the conditional. So, in this case, if we, um. 403 00:47:52.260 --> 00:48:00.929 If we have a given value of access would be the density function for why given that particular value impacts. So we can do stuff like that. 404 00:48:00.929 --> 00:48:08.579 Oh, okay. Um. 405 00:48:19.440 --> 00:48:29.099 Okay um, let me go back to my communications thing again. Let me start a new page paper shape. Um. 406 00:48:33.059 --> 00:48:39.150 168, so the previous example, the noisy. 407 00:48:39.150 --> 00:48:46.619 Channel we computed the probability that a 1 was transmitted. If the received signal was positive, but I just. 408 00:48:46.619 --> 00:48:53.940 Hinted that maybe making the cut off at 0 was not the best idea. Maybe you could do a little better. 409 00:48:53.940 --> 00:48:59.159 This example is going to look at that. Um, basically. 410 00:49:01.050 --> 00:49:04.769 Um, so again. 411 00:49:04.769 --> 00:49:08.429 Plus -1 or +1 probability. 412 00:49:08.429 --> 00:49:17.010 2 thirds probably equals 4th noises. Calcium. 01, that's how we represented. Y, equals X + and. 413 00:49:18.030 --> 00:49:24.150 And now, um, what we want is. 414 00:49:38.099 --> 00:49:43.050 We want to have for a general why. 415 00:49:45.090 --> 00:49:48.780 What is the probability that, um. 416 00:49:50.130 --> 00:49:54.960 X equals day 1, given that value of why. 417 00:49:57.000 --> 00:50:03.510 Okay, um, and then the next stage after that will be, um. 418 00:50:05.699 --> 00:50:17.219 Are there some values of the cutoff, but for any value fly? So we see a particular value of why what's the probability that the, what was transmitted was positive? 419 00:50:17.219 --> 00:50:28.650 Okay um, but here's the problem that why is continuous because the noises continuous. 420 00:50:17.250 --> 00:50:28.650 Okay um, but here's the problem that why is continuous, because the noise is continuous. 421 00:50:28.650 --> 00:50:32.519 So, if we try something like this, um. 422 00:50:35.880 --> 00:50:46.679 So, what about this idea the probability of X equals 1 given Y, equals some value Y, equals um. 423 00:50:48.659 --> 00:50:55.920 And why was that why divided by the probability that why was that? Why um. 424 00:51:00.000 --> 00:51:06.570 Um, the probability is since why is continuous for the probability, the wise particular value. 425 00:51:06.570 --> 00:51:13.829 It's going to be 0, um, and actually we'll, I mean. 426 00:51:13.829 --> 00:51:16.889 Right the top thing also, um. 427 00:51:18.599 --> 00:51:33.295 Probabilty is 428 00:51:33.295 --> 00:51:33.894 amazing. 429 00:51:34.344 --> 00:51:37.465 Here's the thing that these 2 things here. 430 00:51:41.519 --> 00:51:49.679 They're both 0 sense. Why is continuous? Okay. 431 00:51:52.019 --> 00:51:57.780 So this formula would be 0 divided by 0 um. 432 00:51:57.780 --> 00:52:01.409 So, what do we do about that? Um. 433 00:52:03.420 --> 00:52:08.610 It's using something which sort of looks like role and calculus. 434 00:52:08.610 --> 00:52:15.329 We're going to take we're going to sort of approach this as a limit. Um. 435 00:52:15.329 --> 00:52:24.960 Will use the derivative, um, we're going to take a very small interval and then, so it won't be a 0 divided by a 0 here. Um. 436 00:52:27.030 --> 00:52:37.559 So so instead of, um. 437 00:52:41.280 --> 00:52:46.860 Call bill, why he goes why which is 0. 438 00:52:48.690 --> 00:52:53.039 Let's use, um, probabilty that, um. 439 00:52:56.369 --> 00:53:04.710 Um, and then take the limit as Delta goes to 0. 440 00:53:08.969 --> 00:53:16.170 Okay. Um, okay and. 441 00:53:17.849 --> 00:53:21.510 So, the thing is that the probability that, um. 442 00:53:25.619 --> 00:53:30.900 That's going to be the density of that particular value. Why times? Delta. 443 00:53:32.130 --> 00:53:37.469 Okay um, okay um. 444 00:53:37.440 --> 00:53:45.449 Okay. 445 00:53:45.449 --> 00:53:49.409 Okay, it says my screen broadcast to stopped. 446 00:54:08.849 --> 00:54:13.409 Okay, sharing broadcast days. 447 00:54:13.409 --> 00:54:18.000 Let me just go over here for a quick check and see if it's working. 448 00:54:22.920 --> 00:54:26.159 And this again. 449 00:54:32.460 --> 00:54:37.170 Okay, it appears to be working. 450 00:54:40.650 --> 00:54:43.889 Okay, okay so just remind you. 451 00:54:43.889 --> 00:54:54.599 We wanted, so we had a general received voltage Y, and for this general Y, um, we wanted to say, what's the probability that was transmitted? Was +1. 452 00:54:54.599 --> 00:54:57.929 And so. 453 00:54:59.400 --> 00:55:05.670 So, if I may scroll back, I got this form because it probably the exit was 1 given why it's why. 454 00:55:05.639 --> 00:55:09.690 It says here, but, um. 455 00:55:05.670 --> 00:55:09.690 This is here, but, um. 456 00:55:15.929 --> 00:55:21.840 Europe is wise continuous so I'm going to say, use wide being in a small. 457 00:55:26.880 --> 00:55:33.239 Again, and we're, we're going to end up. 458 00:55:36.690 --> 00:55:40.559 Probability that 1 given. 459 00:55:55.409 --> 00:56:04.019 Again, on the top, the probability of why being a particular value, given X equals 1, because the probability of X equals 1. 460 00:56:08.099 --> 00:56:12.630 Um, which is. 461 00:56:16.769 --> 00:56:20.550 Times because we're not doing a little interval times. 4th. 462 00:56:22.079 --> 00:56:25.500 Because probably the 1 being a 3rd, and in the bottom. 463 00:56:25.500 --> 00:56:29.400 The total probability that lies in a certain interval will turn out to be. 464 00:56:30.659 --> 00:56:36.150 Probably wiping a certain God given X equals 1 possibility. Why being a given X equals? -1. 465 00:56:36.150 --> 00:56:48.750 And that out too, um, Delta times 4+the, Pro. 466 00:56:48.750 --> 00:56:55.019 Give and -1 time sales at times 2 thirds probably. 467 00:56:55.019 --> 00:56:58.650 Of X being -1 is 2 thirds and, um. 468 00:57:02.460 --> 00:57:09.750 For the delta is immediately cancel out. Okay. 469 00:57:11.460 --> 00:57:18.989 And now, um, and now we plug in because, um. 470 00:57:23.070 --> 00:57:26.610 So, let me just mention 1 of these as an example here. 471 00:57:31.289 --> 00:57:36.150 It's a guy I was going on, um. 472 00:57:37.170 --> 00:57:44.099 -1 over 2 um. 473 00:57:45.449 --> 00:57:49.980 And we do et cetera, so we're going to plug something like that into all 3 of them here. 474 00:57:49.980 --> 00:57:53.579 And it will net out to. 475 00:58:01.079 --> 00:58:07.860 What I did is each of these conditional wise looks something like what I've written in red here and multiply by 32 thirds of that and divide. 476 00:58:07.860 --> 00:58:13.199 Um, and it will net out to what I've got down at the bottom here. So again. 477 00:58:15.329 --> 00:58:21.989 This was and again, just remind you this was the probability that X equals 1, given some particular value. 478 00:58:21.989 --> 00:58:27.599 Why okay, what am I doing? Yeah, 2. 479 00:58:22.019 --> 00:58:27.599 Why okay, what am I doing? Yeah, 2. 480 00:58:27.599 --> 00:58:33.210 The 2nd here right? It again more neatly. 481 00:58:39.239 --> 00:58:43.829 Okay, okay. Um. 482 00:58:46.380 --> 00:58:54.360 So, we look at this and what does this look like? Um, if why is incredibly small, like, say, -100orsomething. 483 00:58:55.590 --> 00:59:02.400 The eat of the -200is0 and that gets out to 1 so effects, um. 484 00:59:04.530 --> 00:59:15.000 The 2nd, um, I do here, um. 485 00:59:17.699 --> 00:59:20.880 Sorry uh, if Y is very small. 486 00:59:20.880 --> 00:59:26.369 Say, -100thenthis will be will be really so the V of the mind either the +200. 487 00:59:26.369 --> 00:59:31.559 Almost an affinity, and this will be a 0, so why is there a small. 488 00:59:26.579 --> 00:59:31.559 Be almost infinity and this will be a 0, so why is there a small. 489 00:59:31.559 --> 00:59:39.630 The probability that the transmitted signal was 1 is like 0, if Y is really big. 490 00:59:39.599 --> 00:59:45.659 Then either minus big number 0 and this that's out to 1. 491 00:59:39.630 --> 00:59:45.659 Then either the minus big number 0 and this, that's out to 1. 492 00:59:47.460 --> 00:59:54.210 So, it does the intuitively right thing. Okay. So, let me write this down this year. Um. 493 00:59:56.820 --> 00:59:59.940 That's suppose why it was -100. 494 00:59:59.940 --> 01:00:08.940 Probably X equals 1 given why it was -100equals1 over 1+um. 495 01:00:08.940 --> 01:00:14.789 Oops, come on on number 1. 496 01:00:18.150 --> 01:00:22.230 This is about 0, which makes sense of why it was +100. 497 01:00:23.909 --> 01:00:32.610 Probably the X equals 1 given why it was +100is1 over. 1+he is a -200. 498 01:00:28.949 --> 01:00:32.610 Is 1 over 1+here is a -200. 499 01:00:32.610 --> 01:00:37.889 Why do we, um, which nets out to, um. 500 01:00:37.889 --> 01:00:41.550 1, okay, that makes sense. Okay. 501 01:00:41.550 --> 01:00:48.059 Um, so next question is, um, and what value of why is it? 50, 50. 502 01:00:48.059 --> 01:00:55.800 Um, if why it was 35. 503 01:00:57.539 --> 01:01:04.829 Then probably the X equals 1 given why it was point 35 equals 1 half. 504 01:01:06.329 --> 01:01:12.090 Um, so maybe use point 35 as you cut off. So. 505 01:01:16.230 --> 01:01:23.639 Has cut off perhaps before I greater than point 35 guess. 506 01:01:25.590 --> 01:01:30.570 X equals +1, let's say. 507 01:01:30.570 --> 01:01:34.170 If Y, lessens point 35. 508 01:01:36.389 --> 01:01:45.150 Yes, it was -1 perhaps. So it's better to use the cut off and then we can compute what the error is for this thing. 509 01:01:45.150 --> 01:01:50.730 Um, but the panel, why is but this would. 510 01:01:45.179 --> 01:01:50.730 Um, but the panel, why is but this would. 511 01:01:50.730 --> 01:01:58.619 Give us, um, a smaller, total error if we use point 35 as a cut off, instead of using 0 as a cut off. 512 01:01:58.619 --> 01:02:03.750 Oh, okay. Um. 513 01:02:05.010 --> 01:02:08.909 So, we took this example, the noisy communication channel, and we just. 514 01:02:08.909 --> 01:02:12.599 You know, made it a touch base here. Oh. 515 01:02:14.429 --> 01:02:18.510 Okay, questions about that. 516 01:02:26.010 --> 01:02:30.210 Now, you can do conditional expectations and so on I did an average. 517 01:02:32.610 --> 01:02:36.179 You know, talk about something, um, for a few minutes um. 518 01:02:39.210 --> 01:02:45.389 Hello. 519 01:02:55.679 --> 01:03:00.000 And that would be page 268. 520 01:03:01.500 --> 01:03:05.250 3rd section 572 so. 521 01:03:11.880 --> 01:03:19.650 Equals it's in here, it's like, it's, um, it's your expectation thing, but we had staff of the conditional probability. 522 01:03:20.789 --> 01:03:24.630 X of D. Y. 523 01:03:25.980 --> 01:03:32.489 Go for a while. So, um, is there any value of X we can find the conditional. 524 01:03:32.489 --> 01:03:40.260 Thanks expected so we could find the expected receive signal effects was 1 effects was -1 and so on. 525 01:03:41.429 --> 01:03:56.369 Okay, okay. 526 01:04:01.409 --> 01:04:04.739 Sure, let me go back to my defect example here. 527 01:04:08.519 --> 01:04:15.599 Age number is 269 and again, just remind you that we have a chip. 528 01:04:17.699 --> 01:04:23.610 Um, defects request on. 529 01:04:23.610 --> 01:04:27.300 I also have a region here. 530 01:04:30.210 --> 01:04:33.449 Are in the probability P. so, um. 531 01:04:35.400 --> 01:04:40.769 The question is what's the expected number of defects in R. okay. 532 01:04:48.210 --> 01:04:55.889 So, maybe your chip manufacturer and you're bidding your output. So, like, you're in video perhaps and maybe, you. 533 01:04:55.889 --> 01:05:02.849 Make a chip, which has perhaps maybe 16 symmetric, multi processing units on the chip. 534 01:04:55.949 --> 01:05:02.849 Um, make a chip which has perhaps maybe 16 symmetric, multi processing units on the chip. 535 01:05:02.849 --> 01:05:12.480 You manufacture the chip and you test the 16 units. If they all 16 work, you sell it at a high price. 536 01:05:12.480 --> 01:05:17.429 If only 15 work you sold at a lower price of only 14 or. 537 01:05:17.429 --> 01:05:20.820 Do you sell it to the game or community or something? I'm joking. 538 01:05:20.820 --> 01:05:30.210 But not, um, so you'd want to know what's expected number defects or what's the problem, you know, in this region and what's the probability? Okay. 539 01:05:30.210 --> 01:05:35.130 So, um, so. 540 01:05:35.130 --> 01:05:38.969 Why was the random variable for the expected number of defects? 541 01:05:38.969 --> 01:05:45.150 Why in the chip so the expected number why that's going to be actually. 542 01:05:45.150 --> 01:05:52.079 Um, even some, these things affect your number, why given a particular value of X? 543 01:05:53.369 --> 01:05:57.449 Of probability of that value of X summed over K. 544 01:05:59.369 --> 01:06:03.869 I didn't write that down and skipped the okay. Um. 545 01:06:05.820 --> 01:06:12.090 So, um. 546 01:06:16.440 --> 01:06:22.650 Expected value of why given a particular value OK, this is, um, here. 547 01:06:27.030 --> 01:06:35.070 Is actually K, P. P. is the probability that a general defect is in this region are so. 548 01:06:35.070 --> 01:06:41.429 So, the total number defects is K, the expected value for the number in this region is, um. 549 01:06:41.429 --> 01:06:45.420 So that now we can pull K, we can pull PE out of the. 550 01:06:45.420 --> 01:06:51.269 Um, formula, and it's going to be the sum of all the cake was 0 to infinity. 551 01:06:51.269 --> 01:06:55.469 K time is the probability that X equals K. 552 01:06:55.469 --> 01:07:02.190 And this thing here is expected value of X. 553 01:07:04.440 --> 01:07:08.670 The whole thing, which is alpha the whole thing is P, alpha. 554 01:07:08.670 --> 01:07:13.289 Which makes sense, sometimes the stuff actually does a common sense thing. 555 01:07:13.289 --> 01:07:16.739 You know, stopped clock roll and so on, um. 556 01:07:16.739 --> 01:07:20.760 So the expected number of defects in the whole chip. 557 01:07:20.760 --> 01:07:27.510 His alpha, and the probability of any defect in this region, it's speed expected number of defects in that region. It's P times alpha. 558 01:07:27.510 --> 01:07:30.869 It's the common sense thing. So. 559 01:07:30.869 --> 01:07:34.530 But I just proved it, um. 560 01:07:34.530 --> 01:07:40.619 Because sometimes the common sense thing, in fact, is not true. So. 561 01:07:45.840 --> 01:07:52.230 Okay, um, the questions. 562 01:07:55.440 --> 01:08:05.969 Let me do this condition. Ah, yes here. 563 01:08:09.210 --> 01:08:15.239 3 times alpha, um. 564 01:08:17.909 --> 01:08:21.270 The whole thing here is, um. 565 01:08:22.739 --> 01:08:26.039 Is P, times alpha the denominator this here? 566 01:08:30.960 --> 01:08:35.039 Let me write this whole thing more neatly than how about that. 567 01:08:35.039 --> 01:08:44.279 Um, I'll scroll it up after I write it, because I have to be looking at the old version as I'm writing it. Um. 568 01:08:57.114 --> 01:09:15.534 Okay. 569 01:09:19.140 --> 01:09:24.119 A lot more legible, so. 570 01:09:31.199 --> 01:09:37.439 The book's notation is a little confusing. A lower case. P. is a is a real number afloat. 571 01:09:37.439 --> 01:09:42.899 The upper case P takes an argument and, and a shorthand for probability. 572 01:09:42.899 --> 01:09:46.739 So, um. 573 01:09:49.050 --> 01:09:52.500 Is a function or something. Okay. Um. 574 01:09:55.079 --> 01:10:01.770 Hey, we've got a a minute or 2. let me, um. 575 01:10:03.779 --> 01:10:09.449 Do a simple thing example 537 noisy channel again. 576 01:10:15.989 --> 01:10:21.180 Um, age 70. 577 01:10:22.319 --> 01:10:25.800 What's what's the expected value of? Why. 578 01:10:25.800 --> 01:10:30.420 Given an, let's just talk now, the expected value. 579 01:10:35.789 --> 01:10:44.279 Is that the noisy channels and not just say telegraph lines or telephone lines? It could be of an imaging system. 580 01:10:44.279 --> 01:10:53.789 And, um, just to work current events that you got some video system, you see an object. 581 01:10:53.789 --> 01:10:58.890 Up in the sky, and you're trying to recognize it as a missile versus. 582 01:10:58.890 --> 01:11:03.720 The the rising moon, perhaps real example. Um. 583 01:11:03.720 --> 01:11:09.899 And so it's extremely noisy and you want to compute the problem that probability that you saw a particular object. 584 01:11:09.899 --> 01:11:14.220 Given that you're looking at the noisy signal. So this says. 585 01:11:15.539 --> 01:11:25.199 And so now you also have utility function, you got the cost of your decision of making, you got the cost of if you make the wrong decision 1 way, or the other way. 586 01:11:25.199 --> 01:11:32.579 So, um, and so real world example. 587 01:11:32.579 --> 01:11:35.640 Um, okay. 588 01:11:37.560 --> 01:11:41.789 So expected value why it's picking the value of why. 589 01:11:43.109 --> 01:11:47.430 Given 1 kinds of probability and X equals 1. 590 01:11:47.430 --> 01:11:52.289 Plus the expected value, why give an exit was -1 is the probability. 591 01:11:52.289 --> 01:11:57.600 -1 now. 592 01:11:59.760 --> 01:12:03.270 Expected value of why given X equals 1. 593 01:12:03.270 --> 01:12:07.859 Is the expected value of, um. 594 01:12:09.149 --> 01:12:15.630 +1 and, um. 595 01:12:20.250 --> 01:12:27.539 See, um, it was expected value noise +1, cause expectations add. 596 01:12:27.539 --> 01:12:31.649 Equals 1, 1 and. 597 01:12:31.649 --> 01:12:35.310 And the probability of X equals 1 is 4th. 598 01:12:39.270 --> 01:12:52.710 The expected value, why is expected value? Why given X equals 1, which is 1 times 4+accepted value. Why? Given? X equals -1 with the -1 times 2 thirds. 599 01:12:54.720 --> 01:12:58.050 Um, equals -4th. 600 01:13:01.380 --> 01:13:08.250 Which makes sense because X was -1 twice as often as X. +1. So. 601 01:13:10.050 --> 01:13:13.590 Okay, um, make sense. 602 01:13:13.590 --> 01:13:16.619 So, just. 603 01:13:16.619 --> 01:13:23.310 A refresh what we were doing today was several examples from chapter 5. 604 01:13:23.310 --> 01:13:32.220 With conditional conditional probabilities expectations and. 605 01:13:32.220 --> 01:13:38.430 My 2 unifying examples are the well, the, you know, the chip with defects and you've got the sub region. 606 01:13:38.430 --> 01:13:45.180 Or we have a noisy communications channel so what's going on here? Um. 607 01:13:45.180 --> 01:13:48.659 So, I was sort of following along what I've typed up in my notes. 608 01:13:48.659 --> 01:13:55.170 Dumped around slightly and what will continue on. Thursday is more examples for. 609 01:13:55.170 --> 01:14:00.659 We're getting fairly deep into chapter 5, so we'll, we'll be moving on to chapter 6 at some point. So. 610 01:14:00.659 --> 01:14:06.869 And again, reminder a couple of weeks, 2, and a half weeks exam. 2. 611 01:14:06.869 --> 01:14:14.939 And if you think about my trivia question, why in equinox the day is not in fact, 12 hours long. 612 01:14:14.939 --> 01:14:21.989 It's a, it's a surprisingly sophisticated answer in astronomy, so. 613 01:14:23.760 --> 01:14:28.800 Okay, so just curious. Yeah. 614 01:14:30.750 --> 01:14:43.109 What things were happening here. Okay. 615 01:14:43.109 --> 01:14:47.159 So, see you Thursday. 616 01:14:56.850 --> 01:15:02.069 Security is here. Oh, I actually had 3 people today. Hi guys. 617 01:15:02.069 --> 01:15:03.840 Um.