WEBVTT 1 00:00:30.359 --> 00:00:33.780 Okay, good afternoon. I think. 2 00:00:33.780 --> 00:00:38.759 So, I can share. 3 00:00:45.929 --> 00:00:53.609 Good. Okay. Um, so the subset of stuff today, so continuing on. 4 00:00:53.609 --> 00:00:57.630 Chapter 6 vector random variables from the on Garcia. 5 00:00:57.630 --> 00:01:00.630 But 1st, I thought, um. 6 00:01:00.630 --> 00:01:07.469 A couple of light things, and how you can make money from probability. These are actual stories. 7 00:01:07.469 --> 00:01:14.129 Um, so Wired magazine a few years ago. 8 00:01:14.129 --> 00:01:18.269 I mentioned this brief before. I don't think I gave you the link. Um. 9 00:01:18.269 --> 00:01:22.769 Your local Geological statistician found out that. 10 00:01:22.769 --> 00:01:28.769 Scratch off game and Ontario use the same math. This is day job. So. 11 00:01:28.769 --> 00:01:34.319 I can read that and. 12 00:01:35.819 --> 00:01:39.930 There's a number of cases of people making money on. 13 00:01:39.930 --> 00:01:47.969 Various state lotteries in the past, and this is 1 person. John kinnser who made a number of, um. 14 00:01:47.969 --> 00:01:50.969 Big wins. She didn't Tom. 15 00:01:50.969 --> 00:01:57.000 Say how she's done it, but people are trying to guess how she did it and, um. 16 00:01:59.129 --> 00:02:03.959 My assumption is, she did it completely legally so there are patterns or something. 17 00:02:03.959 --> 00:02:09.300 Okay, so kind of looking at that, um. 18 00:02:11.969 --> 00:02:15.840 Some things that she possibly use this was, um. 19 00:02:15.840 --> 00:02:20.099 Again, a few years ago, 1, um. 20 00:02:20.099 --> 00:02:24.449 1 idea. 21 00:02:24.449 --> 00:02:30.479 Certain, I'm actually Webex incorrectly. Um. 22 00:02:31.860 --> 00:02:36.840 Start video here. Okay. 23 00:02:40.469 --> 00:02:50.909 To make sure that things are working here, sharing the screen we're recording. Okay. 24 00:02:50.909 --> 00:03:00.000 So, yeah, 1st, which she might have been doing was using the fact that. 25 00:03:00.000 --> 00:03:06.389 They, um, they didn't want it to be a big winner at the start of a lottery. So the tickets. 26 00:03:07.194 --> 00:03:19.944 So as a store sold out of its tickets, at that time, the state library commission would send more packets of tickets to stores that needed them and the later packets of tickets and more winners. And, um, because they did not want there to be winter, started to start at the game. 27 00:03:20.335 --> 00:03:24.955 And she was exploiting that by buying tens of thousands of tickets from particular stores. 28 00:03:25.199 --> 00:03:28.319 Okay, any case so. 29 00:03:33.985 --> 00:03:44.905 Going back to the course. So I want to get on teach from giving you several examples and I just listed some of them here. Some examples and sections I might work on. Um. 30 00:03:45.449 --> 00:03:57.569 So, let me see here, give you some examples. Um. 31 00:03:59.550 --> 00:04:03.629 Okay. 32 00:04:05.550 --> 00:04:11.550 Okay, so. 33 00:04:16.410 --> 00:04:22.829 So, we've got, um, that's working. Okay. So we've got 3 variables here. 34 00:04:22.829 --> 00:04:26.399 And X is uniform and 0 and 1. 35 00:04:26.399 --> 00:04:32.069 Is uniform. 36 00:04:32.069 --> 00:04:35.999 And 0, 2, X1. 37 00:04:35.999 --> 00:04:43.738 An X X3 is uniform in 2. 38 00:04:43.738 --> 00:04:48.238 And what we want to know is, um. 39 00:04:50.158 --> 00:04:55.108 Basically join. 40 00:04:55.108 --> 00:05:02.009 Pdf of X1 X2 X3 and then just the PDF. 41 00:05:02.009 --> 00:05:05.879 Okay 3. okay. Um. 42 00:05:07.889 --> 00:05:12.749 It's a chance to show some conditional probabilities and so on. So, um. 43 00:05:13.949 --> 00:05:23.639 So, um, so that's uniform in 021. so 1 equals 1 for. 44 00:05:24.869 --> 00:05:33.119 Okay, put a little just to be explicit. It's 2, it's uniform and 0 to X1. 45 00:05:34.649 --> 00:05:37.949 So, to. 46 00:05:37.949 --> 00:05:44.848 That's going to be scaled. Um, so it's going to be enough of X to give an. 47 00:05:45.899 --> 00:05:57.149 Um, given some next 1 and that will be, um. 48 00:06:04.259 --> 00:06:07.408 Otherwise, okay. 49 00:06:08.639 --> 00:06:11.999 So equals 1 half. 50 00:06:11.999 --> 00:06:17.009 F, X2 given equals 1 half equals 2. 51 00:06:17.009 --> 00:06:22.168 X2 that's it for 1. half. Okay. 52 00:06:22.168 --> 00:06:26.129 0, okay, so it, it makes it scale up. 53 00:06:29.009 --> 00:06:34.588 And now, X3 is uniform from 0, 2. 54 00:06:34.588 --> 00:06:43.559 Then f, X3 given some X2 is going to be 1 over X2 0. 55 00:06:45.869 --> 00:06:49.649 Okay, otherwise it's 0. okay. 56 00:06:53.459 --> 00:06:58.678 So half of X1 X2 X3. 57 00:06:58.678 --> 00:07:04.048 F3 given X2 times 2. 58 00:07:04.048 --> 00:07:12.658 It goes off of given X2 times. F X2 give an X1 X1. 59 00:07:12.658 --> 00:07:17.999 Okay, so. 60 00:07:17.999 --> 00:07:24.149 This 1, over why it was the 1 of our X200 racks 1. so this is, um. 61 00:07:24.149 --> 00:07:32.098 On the RX if, um. 62 00:07:35.579 --> 00:07:40.678 To a, um, everything in here. 63 00:07:43.259 --> 00:07:52.738 Okay. 64 00:07:52.738 --> 00:07:57.298 So that's why okay. 65 00:08:00.149 --> 00:08:05.579 And then that's a joint thing now. Um. 66 00:08:10.139 --> 00:08:15.358 You'll get for the top, so this is. 67 00:08:15.358 --> 00:08:23.639 Okay, now, for just half of X3 we'd integrate out the other is, let's say. 68 00:08:24.233 --> 00:08:24.624 So, 69 00:08:24.653 --> 00:08:44.274 okay. 70 00:08:47.489 --> 00:08:48.058 Well, 71 00:08:48.264 --> 00:08:48.923 um, 72 00:09:04.793 --> 00:09:05.333 now, 73 00:09:05.333 --> 00:09:06.114 um. 74 00:09:08.519 --> 00:09:13.798 So 1 goes from 0 to 1, goes from 0 to 1. 75 00:09:13.798 --> 00:09:25.078 So, and we'll do the inner 1 will be X2. Let's say. 76 00:09:26.999 --> 00:09:37.259 Okay. 77 00:09:38.668 --> 00:09:42.928 A little more cleanly and or grow. 78 00:09:50.219 --> 00:09:54.568 Okay. 79 00:09:55.889 --> 00:09:58.979 Um, this thing here. 80 00:10:02.969 --> 00:10:08.999 Um, sorry calls, um. 81 00:10:11.729 --> 00:10:12.774 Basically log 82 00:10:26.663 --> 00:10:27.533 something like. 83 00:10:30.509 --> 00:10:36.089 Logo next, um, there's 2 actually. 84 00:10:36.089 --> 00:10:41.879 Um, and, uh. 85 00:10:55.619 --> 00:11:06.239 Should it would be if they log 2 or something like that and then we, um, 1, 1, 1, 1. 86 00:11:06.239 --> 00:11:12.448 And then we do the whole thing, and we will get something like, um. 87 00:11:26.188 --> 00:11:31.469 And this will be the marginal okay. 88 00:11:32.609 --> 00:11:41.038 And it's concentrated down around 0. so, um, so what I showed is I took a vector value X, um, 1, X2 X3. 89 00:11:41.038 --> 00:11:44.759 And actually the conditional on each other and then I, um. 90 00:11:46.318 --> 00:11:50.219 Found the marginal okay. 91 00:11:50.219 --> 00:11:56.879 Let's see what else I said I would do is excited and maybe page. 92 00:11:57.989 --> 00:12:03.269 10. 93 00:12:09.719 --> 00:12:21.298 Okay, next 1 I want to do is some min and Max, I've done it before. That'll just this will take it to a next level of detail. So this will be. 94 00:12:21.298 --> 00:12:25.318 Yeah, let's do black, maybe calls it easier to see. 95 00:12:29.969 --> 00:12:35.129 69, and this will be page 310. 96 00:12:35.129 --> 00:12:41.938 Okay, so I got random variables X1 up to X and. 97 00:12:42.989 --> 00:12:48.839 Um, I've got W, is the minimum of them all is the maximum. 98 00:12:50.969 --> 00:13:00.359 And Z equals the minimum. Oh, when you say the, you're independent. 99 00:13:04.048 --> 00:13:11.129 And they're, they're also the same it's like independent, identically distributed I'd say. Okay. Um. 100 00:13:15.418 --> 00:13:18.658 So, we have the f. 101 00:13:18.658 --> 00:13:22.229 And we want. 102 00:13:22.229 --> 00:13:26.339 Of, um. 103 00:13:26.339 --> 00:13:38.759 The man. Okay. Um, so how do we do this? We can look at the definition of the cumulative and big the distribution function. 104 00:13:38.759 --> 00:13:44.818 Well, um, for some. 105 00:13:47.668 --> 00:13:53.308 It was the probability that random pair of listening to that particular value. 106 00:13:59.158 --> 00:14:05.548 Well, this is the, it's the minimum of, um. 107 00:14:06.869 --> 00:14:10.619 X1 X2, X3 and so on. 108 00:14:13.979 --> 00:14:19.859 Well, this is the probability less than or equal to. 109 00:14:19.859 --> 00:14:28.109 X1 and W, less than or equal to X2 and and so on. 110 00:14:28.109 --> 00:14:32.818 Okay, if it's less than the minimum, it's less than all of them. 111 00:14:33.989 --> 00:14:39.538 Well, this is, um, now they're independent the P, for probability. 112 00:14:39.538 --> 00:14:46.078 Um, W, X1 kinds of 2. 113 00:14:50.129 --> 00:14:57.359 So, the problem is, the joint thing, it's a sensor independent, I can split it up into the separate pieces and multiply. And what is this. 114 00:15:04.589 --> 00:15:10.769 And this is, um, to the end. 115 00:15:13.078 --> 00:15:19.288 So, if I and identical random variables, the Kimble distribution function for their minimum. 116 00:15:19.288 --> 00:15:25.589 Is just the fraud product, or the individual cable distribution function to the end. 117 00:15:27.058 --> 00:15:32.698 So, are you on an example I showed you. 118 00:15:32.698 --> 00:15:39.208 Back if, um, let's say, let's say uniform 0 to 1. 119 00:15:39.208 --> 00:15:44.068 Slowly yeah, um. 120 00:15:44.068 --> 00:15:48.899 Equals equals equals. 121 00:15:50.609 --> 00:15:57.239 And so basic W, say, it's a minimum. I. F. W. 122 00:15:58.499 --> 00:16:02.668 Will be to the eye to the end. 123 00:16:03.808 --> 00:16:09.658 So the 2 random variables, half of Tom view. 124 00:16:09.658 --> 00:16:18.058 And minus is an f. W. W squared. Let me clean this up. So. 125 00:16:21.658 --> 00:16:29.278 Random variables and so on and the density function here. 126 00:16:29.278 --> 00:16:36.208 To view where you W squared so the minimum so the. 127 00:16:36.208 --> 00:16:39.778 Density function for the minimum, um. 128 00:16:43.678 --> 00:16:53.818 Let's see if I got this right entity function for the minimum. 129 00:16:56.938 --> 00:17:03.989 Yeah, whatever okay. That's the minimum is the maximum. 130 00:17:05.999 --> 00:17:09.929 Sorry, I've been switching minimum and maximum along here. Um. 131 00:17:12.239 --> 00:17:15.989 Adobe is the Max, so. 132 00:17:15.989 --> 00:17:23.729 Yeah, so W is less than this is the sorry about that. Let me correct? 133 00:17:25.078 --> 00:17:31.528 So, maximum I was wondering, and so is less than is the maximum. So the. 134 00:17:31.528 --> 00:17:38.159 So, the killer function for dog has to be less than each of them. So it'd be right okay. In this here is Max. 135 00:17:39.778 --> 00:17:44.969 Right, right? So. 136 00:17:44.969 --> 00:17:50.519 If we had the maximum several random variables, the density function has shifted. It's. 137 00:17:50.519 --> 00:17:54.778 It's greater for larger values, so the. 138 00:18:03.358 --> 00:18:09.058 For larger Dolby use okay makes sense. 139 00:18:09.058 --> 00:18:13.378 Um, for W, the max. 140 00:18:13.378 --> 00:18:19.108 Um, for the minimum. 141 00:18:24.419 --> 00:18:27.689 We use, we can use the compliments, so. 142 00:18:27.689 --> 00:18:31.108 So, 1-f Z. 143 00:18:32.189 --> 00:18:36.778 Because of probability that, um. 144 00:18:39.628 --> 00:18:46.979 Random a variable it's greater than or equal to some particular value. 145 00:18:46.979 --> 00:18:52.949 But that particular value is the, uh, minimum. 146 00:18:58.439 --> 00:19:03.269 See, it's it's, it's the, um, the minimum of the. 147 00:19:03.269 --> 00:19:14.848 So so, uh, basically the Gary go to the minimum. 148 00:19:14.848 --> 00:19:20.038 Of the means, it has to be greater Nicole, um, all of them, um. 149 00:19:21.479 --> 00:19:30.328 Basically, um, sorry, my mind is going today to be crosstalk. 150 00:19:36.929 --> 00:19:45.808 Right. That one's correct. And, um, right so. 151 00:19:48.778 --> 00:19:54.388 Yeah, and then there is a minimum of the. 152 00:19:54.388 --> 00:20:02.068 Hi, so property 0, just some particular value. That's a probability that the. 153 00:20:03.689 --> 00:20:10.259 Minimum the soccer and include that value and That'll be the probability that. 154 00:20:10.259 --> 00:20:16.618 It's 1, Greg ZZ has probabilty. It's 2 grand equals Z and so on. 155 00:20:16.618 --> 00:20:22.318 And this is 1-the density function so I'll leave this up for a minute. 156 00:20:23.338 --> 00:20:30.538 You won't get so oh. 157 00:20:31.769 --> 00:20:41.278 Basically, um, so it will get a negative. Okay. 158 00:20:41.278 --> 00:20:49.108 Um, all right, so if it's uniform, um, so it's or uniform 0 and 1. 159 00:20:49.108 --> 00:20:54.509 So, I. 160 00:20:54.509 --> 00:20:58.739 So, for Z, being the minimum of the. 161 00:20:58.739 --> 00:21:04.499 Okay, well, if say for an equals 2, what we will have is, um. 162 00:21:08.338 --> 00:21:11.638 It's still here, so 1-half of X. 163 00:21:11.638 --> 00:21:18.749 This 1, by the 1-C squared and f, and Z will be. 164 00:21:20.519 --> 00:21:26.219 Squared minus Tuesday. 165 00:21:27.929 --> 00:21:32.608 And it will be biased towards the bottom side. Um. 166 00:21:41.398 --> 00:21:46.648 Sorry, um, I'm going slowly today. 167 00:21:49.048 --> 00:21:54.868 So this word, so. 168 00:21:57.148 --> 00:22:03.838 Yeah, so you see minus C squared. 169 00:22:07.318 --> 00:22:12.028 Right so we can what I've showed is that we can find. 170 00:22:12.028 --> 00:22:22.019 Minimum the statistics, the probability distribution function, density function for the minimum of N, random variables in the maximum then random variables. 171 00:22:23.578 --> 00:22:32.909 Okay, and we could also do it for actually combos of min and Max. Yes. 172 00:22:32.909 --> 00:22:44.459 So, on. 173 00:22:47.189 --> 00:22:52.169 Well, this thing here on, I'm defining as the minimum. 174 00:22:52.169 --> 00:22:58.469 Of all the so W. 175 00:22:58.469 --> 00:23:02.729 Um, it's, it's less than or equal to each of them. 176 00:23:02.729 --> 00:23:07.528 Because it's less than it's equal to the minimum so it's less than equal to each of them. So. 177 00:23:09.868 --> 00:23:13.769 Why. 178 00:23:13.769 --> 00:23:20.848 Yeah, they are. Yeah, and then at the end, I got special and said, let's just make them uniform or something. So. 179 00:23:28.648 --> 00:23:32.939 And, yeah, okay. 180 00:23:32.939 --> 00:23:39.179 And so, so, for the minimum, the densities bias to the left so. 181 00:23:41.009 --> 00:23:49.288 Um, you could also do something like this actually. 182 00:23:56.848 --> 00:24:00.239 I could also say, let's say. 183 00:24:00.239 --> 00:24:09.868 Equals say the, um, the middle 1 X1. 184 00:24:09.868 --> 00:24:14.068 X, Ray, and so on and find say, um. 185 00:24:14.068 --> 00:24:17.189 Yeah, for Adobe stuff like that. 186 00:24:17.189 --> 00:24:26.699 Whatever, um, and then even if the X's are uniform, W, will tend to be concentrated in the middle. So. 187 00:24:32.759 --> 00:24:41.219 Say uniform, it'll be concentrated in the middle, so. 188 00:24:48.239 --> 00:24:53.219 Yeah, wait, wait, we could do something like that. If you want it, but you get the idea. So it. 189 00:24:54.298 --> 00:25:05.009 Okay, um, okay, let's look at my little list up here. Um, so. 190 00:25:06.239 --> 00:25:10.169 Right and let me show you 610. let me show you some more examples of this. 191 00:25:12.959 --> 00:25:22.439 Um, you might think is a little artificial so here is a, um. 192 00:25:22.439 --> 00:25:27.479 More apt applied examples of 610 on page. 193 00:25:27.479 --> 00:25:33.449 310, um. 194 00:25:36.719 --> 00:25:39.749 So this is a package switch. 195 00:25:47.848 --> 00:25:51.088 Okay, um, and. 196 00:25:52.558 --> 00:26:01.259 And sources, okay. And, um. 197 00:26:03.298 --> 00:26:12.659 And, and for each source, um, so the here. 198 00:26:16.048 --> 00:26:25.288 Enter arrival time, it's, it's, let's say it's exponential or something. 199 00:26:28.798 --> 00:26:33.118 It's exponential with, um, say, right. 200 00:26:35.278 --> 00:26:39.689 So the mean inter, arrival time is 1 overlap dye. So. 201 00:26:50.729 --> 00:26:55.588 And what we want to know is, um, inter, arrival time. 202 00:26:55.588 --> 00:27:01.288 On the switch for all the sources combined um. 203 00:27:14.519 --> 00:27:18.388 For all the sources combined. Okay. 204 00:27:22.469 --> 00:27:29.939 Um, because it takes to switch a little time to process the packet coming in and. 205 00:27:29.939 --> 00:27:35.038 We'd like to know how much, you know, are we like you to have enough time? Perhaps. 206 00:27:35.038 --> 00:27:38.068 Or, maybe we have to discard a packet, so. 207 00:27:39.538 --> 00:27:43.169 In the real world, depending on the design of the packet switch. 208 00:27:43.169 --> 00:27:50.699 If, like, we get a, we receive a packet, we process it if another packet arise too quickly, we discard it. 209 00:27:50.699 --> 00:27:54.598 Um, so it's like, you know, your telephone. 210 00:27:54.598 --> 00:27:59.999 A company in the line rings busy, because they are still processing the previous call. 211 00:27:59.999 --> 00:28:04.169 So, okay, so let's, um. 212 00:28:06.929 --> 00:28:11.249 Is that, um, okay, so you see what we want to do. 213 00:28:11.249 --> 00:28:17.398 The package, which is an input sources. Each source says inter, arrival, time of, um. 214 00:28:17.398 --> 00:28:21.989 1 overlap I kept packages are coming in Atlanta. I. 215 00:28:21.989 --> 00:28:26.489 And so on, okay, now you might, um. 216 00:28:26.489 --> 00:28:36.209 Think ahead of me of course, each source is providing packets at a rate of lab to I all the sources together provide packets at a rate of, um. 217 00:28:36.209 --> 00:28:45.328 Some of the land I, and you'd be correct, but that case is the rate of all the packets together, but it doesn't give you what the profitability function is for that. 218 00:28:45.328 --> 00:28:49.979 So, what I'm going to do now is calculate the probability. 219 00:28:49.979 --> 00:28:54.179 Density function and distribution function for all the packets together. 220 00:28:54.179 --> 00:29:02.969 Okay, so that's an inter, arrival time. 221 00:29:06.449 --> 00:29:15.058 The, so this is a random variable for the time for the next packet coming from whatever. 222 00:29:15.058 --> 00:29:20.939 Um, so these are random variable. 223 00:29:24.838 --> 00:29:28.019 For the time to the next back end. 224 00:29:29.338 --> 00:29:33.388 But I don't care where it comes from. Okay. 225 00:29:33.388 --> 00:29:46.138 Um, the next 1 is around variable for the time to the next packet from the 1st source exclusive, random variable for the time for the pack next source. So these are random variable. 226 00:29:46.138 --> 00:29:49.679 For the time for the packet coming from anywhere. 227 00:29:49.679 --> 00:29:54.509 Okay, um, if I can scroll up. 228 00:29:55.798 --> 00:30:00.358 And, um, yeah, some screen. 229 00:30:03.028 --> 00:30:11.038 2nd, here. 230 00:30:17.308 --> 00:30:21.808 Share screen start broadcast. 231 00:30:28.348 --> 00:30:32.969 Okay, good enough. 232 00:30:37.169 --> 00:30:41.398 Okay, now I don't know why it does that. 233 00:30:41.398 --> 00:30:50.219 Um, so we'll do is being minimum we'll do the, we're so 1. 234 00:30:53.759 --> 00:31:02.699 Um, so this is the probability that, um. 235 00:31:05.759 --> 00:31:09.838 The random variables Z is greater and equal to Z. 236 00:31:09.838 --> 00:31:14.969 Oh, greater. Okay. Um, just because. 237 00:31:14.969 --> 00:31:23.308 Is the probability that the less than the random variable? Um, greater than equal doesn't matter. Okay. So now, um. 238 00:31:24.419 --> 00:31:30.269 The 1-f Z is a probability that the minimum. 239 00:31:32.489 --> 00:31:40.318 Of the, um, oops, minimum square nickel to Z greater than to see. 240 00:31:41.489 --> 00:31:53.489 And so this is that there is that they're all graded equal to Z. 241 00:31:58.499 --> 00:32:02.548 Et cetera and since they're independent. 242 00:32:05.128 --> 00:32:10.499 As a program X2 K and Z and so on. 243 00:32:11.759 --> 00:32:15.598 And this will be 1-half of X1. 244 00:32:15.598 --> 00:32:22.138 Um, X1, 1-2. 245 00:32:22.138 --> 00:32:28.199 2, and so on, and now these being exponentially distributed. 246 00:32:28.199 --> 00:32:36.719 Um, um. 247 00:32:36.719 --> 00:32:39.929 Since the exponential. 248 00:32:43.739 --> 00:32:56.338 Um, yeah, it will be 1-. 249 00:33:03.148 --> 00:33:09.509 Sorry, these are all Z here actually so you can see and so on, um. 250 00:33:14.429 --> 00:33:18.419 And so half of that. 251 00:33:18.419 --> 00:33:25.558 Z, remind mind is slammed. 252 00:33:26.638 --> 00:33:33.388 I see. So 1-half of Z. 253 00:33:35.009 --> 00:33:39.959 It is. 254 00:33:41.278 --> 00:33:48.868 Whatever. 255 00:33:51.179 --> 00:33:59.788 So, what I showed was that if I have these and independent random variables, and each of them is exponential. 256 00:33:59.788 --> 00:34:05.759 And they've got different parameters and the minimum the minimum of them is also exponential. 257 00:34:05.759 --> 00:34:09.148 With a parameter of the some of those separate parameters. 258 00:34:09.148 --> 00:34:12.898 So, they are. 259 00:34:12.898 --> 00:34:21.358 Random variables it's exponential with some sort of right? 260 00:34:21.358 --> 00:34:24.688 Hold on. 261 00:34:24.688 --> 00:34:31.768 The Capitol letter is the random variable as a random variable. The lowercase letters of particular values. 262 00:34:31.768 --> 00:34:39.298 So something like half of Z put a subscript for naming random variables Z. but the argument's particular value. 263 00:34:39.298 --> 00:34:50.878 So, what we showed is that we had the, if we have these separate sources and each sources exponential. 264 00:34:50.878 --> 00:34:54.719 With its own rate Lambda I, that. 265 00:34:54.719 --> 00:35:07.648 And I have all these sources piling up on this 1 package switch, then the distribution of inter arrival times is exponential with the right to, to some of them. It makes sense at the rate. 266 00:35:07.648 --> 00:35:15.418 We add up the rates, you know, package per 2nd, on all the separate sources, but we didn't know that the probability. 267 00:35:15.418 --> 00:35:21.659 Function for the minimum was was an exponential itself. It didn't, you know, there's no reason to think it would be. 268 00:35:21.659 --> 00:35:34.349 But what I just did was show that, so I have an exponential probability functions sources all coming together with the package switch all of them together. That looks like 1 source. 269 00:35:34.349 --> 00:35:37.858 Who's in her arrival times are exponentially distributed. 270 00:35:37.858 --> 00:35:43.409 Oh. 271 00:35:46.498 --> 00:35:50.878 Okay, so here was a an application. 272 00:35:50.878 --> 00:35:57.418 We're interested in the minimum cause we want to know the next arrival from any of the end sources. 273 00:35:57.418 --> 00:36:02.608 So, it was the minimum of the arrival times from the end sources. Um. 274 00:36:02.608 --> 00:36:06.389 The next example will work with maximum, so. 275 00:36:06.389 --> 00:36:14.608 This is, um, on page. 276 00:36:14.608 --> 00:36:19.739 So this will this will use maximum. 277 00:36:21.208 --> 00:36:24.358 When he was Max of end. 278 00:36:24.358 --> 00:36:27.989 Then random variables. Okay. Um. 279 00:36:29.068 --> 00:36:41.639 This will be a redundant system here. Okay. And the way that's going to work is that there are. 280 00:36:41.639 --> 00:36:44.938 The system has and components. 281 00:36:49.528 --> 00:36:54.418 And the system works as long as 1 component is still working. 282 00:36:58.079 --> 00:37:04.498 1 component it's working. 283 00:37:07.168 --> 00:37:10.168 You know, 1 or more. Okay. 284 00:37:11.699 --> 00:37:16.349 So let's say the components are exponential. Um. 285 00:37:21.418 --> 00:37:24.568 So each component, it's lifetime. 286 00:37:29.699 --> 00:37:35.518 Is exponential it's exponential with some lamb. I. 287 00:37:35.518 --> 00:37:39.599 Okay, um. 288 00:37:39.599 --> 00:37:42.958 So, what that means is. 289 00:37:42.958 --> 00:37:51.418 It's its lifetime probability that, um. 290 00:37:51.418 --> 00:37:58.259 It's greater than some value K or whatever will be either the minus. Um. 291 00:37:59.759 --> 00:38:02.818 Okay, so. 292 00:38:02.818 --> 00:38:07.438 Going down and so the, um. 293 00:38:08.878 --> 00:38:14.820 To function again. Okay. Is 1. 294 00:38:16.619 --> 00:38:23.639 Okay, now so the system has the end component. 295 00:38:25.590 --> 00:38:29.639 So, let's say W, it's the random variable for the life. 296 00:38:30.960 --> 00:38:37.260 Of the system, and that's going to be the maximum of all of the. 297 00:38:37.260 --> 00:38:45.750 So, the system, these are the random barriers to the lifetimes of each separate component. The system is alive. 298 00:38:45.750 --> 00:38:50.460 If at least 1 component is still alive so. 299 00:38:52.710 --> 00:39:03.630 Um, and so what we've got is going back. 300 00:39:03.630 --> 00:39:10.260 Half an hour or so um. 301 00:39:15.599 --> 00:39:22.230 Well, they're all the same, let's say, um. 302 00:39:22.230 --> 00:39:36.719 And that, um, and this will be 1-either the - and in this case dokie, till the end. 303 00:39:36.719 --> 00:39:40.050 I certainly know the components of the same, let's say. 304 00:39:40.050 --> 00:39:44.400 And we can expand that as a, as a series of course. 305 00:40:00.505 --> 00:40:03.954 + and square it over to the -2. 306 00:40:05.940 --> 00:40:09.659 Okay, um. 307 00:40:15.090 --> 00:40:22.590 Sure, how far hey, when. 308 00:40:38.489 --> 00:40:47.070 Okay, so these are applications where you'd want min and Max let's say, um. 309 00:40:48.179 --> 00:40:53.429 So, when you got redundant systems, and at least 1 of them has to be working. So, yeah. 310 00:40:53.429 --> 00:40:57.840 Okay, okay. 311 00:40:57.840 --> 00:41:04.949 So so, on my next example, I'll show you, let's see. 312 00:41:06.360 --> 00:41:10.260 Let's move on to page 318 section. 6, 3. 313 00:41:11.730 --> 00:41:15.840 Okay. 314 00:41:20.730 --> 00:41:24.900 Giving some stuff I might do it Thursday or maybe not. Um. 315 00:41:26.670 --> 00:41:37.679 Hello. 316 00:41:47.159 --> 00:41:55.440 Section 631. okay. Um. 317 00:41:57.239 --> 00:42:01.469 And, um, we can get matrices so. 318 00:42:07.559 --> 00:42:11.369 Co variances and so on. 319 00:42:14.969 --> 00:42:20.849 Um, so we've, we've got our ex, um. 320 00:42:20.849 --> 00:42:24.989 Brandon on exercise. 321 00:42:24.989 --> 00:42:34.349 Um, we can take all the means and put all the means in a vector or something. So he can get something called a mean vector. 322 00:42:34.349 --> 00:42:38.639 And it'd be expected value of all the access and match just. 323 00:42:38.994 --> 00:42:53.485 Um, no, nothing deep there take. We've got end components in the random back. 324 00:42:53.485 --> 00:42:58.824 It would take the mean of each 1. we put it in a vector and we call that the main factor. Let's say. 325 00:42:59.070 --> 00:43:04.710 Um, no, it's easy enough. Um. 326 00:43:12.510 --> 00:43:16.559 Um, we can get a correlation matrix, um. 327 00:43:27.449 --> 00:43:35.489 And this will be, and we just have all of the squares expected value of X1 squared. 328 00:43:35.489 --> 00:43:40.710 Expected value of Exelon next 2 expected value of X1 X3. 329 00:43:40.710 --> 00:43:48.929 Sector value of X1 X2 2 detected value X2. 330 00:43:48.929 --> 00:43:54.389 Squared and so on big matrix here. 331 00:43:56.159 --> 00:44:02.880 So, it has the, um, expected value of all the 2nd order. 2nd degree things here. 332 00:44:02.880 --> 00:44:12.690 Um, so basically our, this correlation matrix. 333 00:44:14.280 --> 00:44:17.670 J, that's the expected value. 334 00:44:17.670 --> 00:44:22.139 X times X. J. okay. 335 00:44:23.460 --> 00:44:35.159 And the problem is here, if we translate the random variables and this changes, so we'd like, is a so these, these are 2nd order moments. Okay. 336 00:44:43.260 --> 00:44:47.760 The thing is central moments are more useful actually. So. 337 00:44:58.769 --> 00:45:06.389 Are more useful cause we don't much care if we translate something. 338 00:45:06.389 --> 00:45:11.309 So, let me know when I can. 339 00:45:11.309 --> 00:45:17.280 So, what we have is like, a variance matrix. 340 00:45:23.400 --> 00:45:30.630 And this what we call it a K X, and that will be expected value. I'll say. 341 00:45:35.099 --> 00:45:43.469 Expected value it's 2-M 2, et cetera. 342 00:45:45.690 --> 00:45:49.440 So, okay, the CO variance matrix. 343 00:45:49.440 --> 00:45:52.739 jay's expected value. 344 00:45:52.739 --> 00:45:58.739 Minus M. J. - and J. 345 00:45:59.789 --> 00:46:08.820 And these are some metric and so on. And the idea is that. 346 00:46:11.039 --> 00:46:16.110 Hi, next Jay are on correlated to this term is 0, so. 347 00:46:16.614 --> 00:46:17.635 Very independent. 348 00:46:35.005 --> 00:46:35.514 0. 349 00:46:35.849 --> 00:46:40.769 Are you in on it? Actually. 350 00:46:45.659 --> 00:46:54.420 Call enter this is sufficient, so. 351 00:47:04.980 --> 00:47:10.289 And you can no examples with this. The trouble is they're getting harder. 352 00:47:10.289 --> 00:47:15.239 It's if I start writing them down, as I talk about them. 353 00:47:15.239 --> 00:47:23.400 The examples get messy enough, it's you can't you won't be able to read my handwriting so I won't walk you through some examples. Maybe on a. 354 00:47:23.400 --> 00:47:29.820 Maybe later, but okay. Um. 355 00:47:34.380 --> 00:47:41.280 Questions okay. Um. 356 00:47:43.110 --> 00:47:52.050 Now, big thing you do with variables and engineering you to transform them music, rotate them and so on. So. 357 00:47:52.050 --> 00:47:55.860 Do something simple say linear transformation? Um. 358 00:47:55.860 --> 00:47:59.039 Let's say. 359 00:48:09.389 --> 00:48:12.659 Your transformation and that's page. 360 00:48:15.329 --> 00:48:23.730 Um. 361 00:48:25.920 --> 00:48:30.269 What would be an example? Let's say. 362 00:48:32.940 --> 00:48:44.099 Say your processing color video so, um. 363 00:48:44.099 --> 00:48:48.420 So, pixel color. 364 00:48:49.860 --> 00:48:55.019 Say the vector R. G. 365 00:48:55.019 --> 00:49:00.000 For red, green, blue, and then. 366 00:49:01.559 --> 00:49:05.880 What you actually do is, um, in in the process thing. 367 00:49:07.440 --> 00:49:13.739 So, what the processing does, um, they rotates the factor. 368 00:49:18.179 --> 00:49:21.179 Do something like Y IQ. 369 00:49:23.639 --> 00:49:29.789 Um, so we've got the caller. 370 00:49:29.789 --> 00:49:34.769 Is the color vector? R. J. V. 371 00:49:36.449 --> 00:49:44.730 Transform it to Y, que why for example would be point 6 R plus point. 372 00:49:44.730 --> 00:49:48.539 Point 1 flu, um. 373 00:49:48.539 --> 00:49:55.050 And why it could be the same. So why I. 374 00:49:55.050 --> 00:49:58.889 There'll be some matrix a times. R. J. 375 00:49:58.889 --> 00:50:05.250 Um, and we'd like to say, RGB, they're random. Um. 376 00:50:07.019 --> 00:50:13.500 There's some random things, so you might call this vector here X and the speaker vector why I have to say. 377 00:50:13.500 --> 00:50:19.739 Now, why 1st of all, why they want to do this transformation. 378 00:50:20.335 --> 00:50:31.434 Is, um, it's because of characteristics wall of electronics and of people and of images. So, why is the total brightness of a pixel? 379 00:50:31.764 --> 00:50:36.625 So, 61st so you take the red time 60% the green time 30% and the blue times 10%. 380 00:50:38.460 --> 00:50:43.619 That gives the brightness of the pixel, um, Y, Y, axis. 381 00:50:43.619 --> 00:50:47.789 And I don't remember the numbers for I and Q. 382 00:50:47.789 --> 00:50:56.579 But I, um, basically looks like a red component and queue looks like a. 383 00:50:56.579 --> 00:51:00.269 Blue component of the pixel and. 384 00:51:00.269 --> 00:51:07.889 I actually stands for in phase component and Q stands for Quadrant for component and. 385 00:51:07.889 --> 00:51:13.710 You transformed in a couple of things they do is, is that you can see brightness. 386 00:51:13.710 --> 00:51:25.255 Better than more accurately than you can see color than you can see here. So if they do the rotation, they can now code these 3 new vector components. Separately. 387 00:51:25.255 --> 00:51:30.655 The Mo use, for example, more bits per 2nd for Y, than for I in queue. 388 00:51:30.929 --> 00:51:36.389 The 2nd thing they can do after they've done the rotation. 389 00:51:36.389 --> 00:51:40.380 Is that the Y, I, and Q are now less correlated with each other. 390 00:51:40.380 --> 00:51:50.579 So, the initial RGB, if our is very high g's very high also, probably, because it's a bright pixel you do this rotation it correlates the, um. 391 00:51:50.579 --> 00:51:53.699 Components which can be useful when there. 392 00:51:53.699 --> 00:52:07.590 When they're encoding them, so, and they also since you, so they can represent why it was more bits per 2nd that I and Q and then the decorrelated. So they can do things. So what I'm giving you is engineering reasons. 393 00:52:07.590 --> 00:52:10.679 Why they want to do this rotation. 394 00:52:10.679 --> 00:52:14.250 Us getting beyond this course somewhat, but, um. 395 00:52:14.250 --> 00:52:22.260 It's an interesting thing as they can do that rotation with discrete components like resisters and so on. 396 00:52:22.260 --> 00:52:26.190 And they don't need fancy electronics. They were 1st doing this. 397 00:52:26.190 --> 00:52:31.019 Rotation and electrical hardware 60 years ago, or something um. 398 00:52:32.670 --> 00:52:36.809 Of course, the problem with using discrete components was that. 399 00:52:36.809 --> 00:52:41.280 If the component value's drifted, then the matrix would drift. 400 00:52:41.280 --> 00:52:47.309 And then, um, the colors would be wrong. So well, another advantage of the rotation. 401 00:52:47.309 --> 00:52:52.829 Is they were creating the standard at a time when. 402 00:52:52.829 --> 00:52:56.070 Black and white T. V. was what people had. 403 00:52:56.070 --> 00:53:02.460 And they're creating a future standard for when they did color TV and they needed it to be upward. 404 00:53:02.460 --> 00:53:08.429 And downward compatible, so they did the rotation for the color signal transmitted. Why. 405 00:53:08.429 --> 00:53:19.829 Which is like, a brightness, black and white component as they transmitted any black and white TV signal. And so there was a color transmitter. The black and white TV should receive the why signal? 406 00:53:19.829 --> 00:53:23.099 And all black and white image, ignore the IQ. 407 00:53:23.099 --> 00:53:27.750 So, it was all sort of cool. Okay. 408 00:53:27.750 --> 00:53:32.940 So, in any case, um, so what you've got is. 409 00:53:34.829 --> 00:53:39.329 You have a vector of random variables that gets a linear transformation. 410 00:53:39.329 --> 00:53:45.179 2 of another vector, Y, doesn't have to be a rotation for the color things. It's actually a rotation. 411 00:53:45.179 --> 00:53:51.300 But, um, just as an aside. 412 00:53:51.300 --> 00:53:57.300 Because it's crazy if you look in the history of how they were trying to invent color. 413 00:53:57.300 --> 00:54:03.030 There are a lot of ideas that were not to say, not the most successful. 414 00:54:03.030 --> 00:54:09.179 1 of them is that you had a black and white with a rotating color wheel in front of it. 415 00:54:09.179 --> 00:54:12.480 So the had a diameter of a foot. 416 00:54:12.480 --> 00:54:17.219 The color wheel would have a radius of noticeably more than a foot to the color wheel. 417 00:54:17.219 --> 00:54:23.880 But happy, maybe 3 feet in diameter, and it would be spinning at something like, you know, at least 30 Hertz. 418 00:54:23.880 --> 00:54:33.059 And so the color wheel had red, green and blue filter color filters on it spinning in front of the black and white. See again, if you look at it, and it's spinning fast enough on the. 419 00:54:33.059 --> 00:54:40.860 The black and white transmitting the red green blue components has to be synchronized with the rotating color wheel. 420 00:54:40.860 --> 00:54:48.869 Potassium we got at least 30 Hertz and preferably say 60, because your eye is accused of these separate images. 421 00:54:48.869 --> 00:54:55.739 Let's just say that was not a big success. It was an idea. Oh, okay. 422 00:54:56.880 --> 00:55:00.659 So, back to here probability, so you got Y, equals. 423 00:55:00.659 --> 00:55:07.199 And then, um, you want to do things. 424 00:55:08.309 --> 00:55:11.969 And then, so if you know, um. 425 00:55:14.099 --> 00:55:18.090 If, you know, means and covariances, et cetera. 426 00:55:21.059 --> 00:55:26.969 Et cetera for X, you can find them for why so. 427 00:55:30.750 --> 00:55:35.070 You can find them or why. 428 00:55:35.070 --> 00:55:38.250 It's linear algebra, um. 429 00:55:39.900 --> 00:55:48.690 So, basically, um, demeans for the Y, it's just, uh, any times the means for the axis. 430 00:55:48.690 --> 00:55:54.809 And I won't work it out. You can take it and work it out yourself. Um, Co, variants for the wise. 431 00:55:54.809 --> 00:56:07.139 Is, um, little fancy or 8 K so you can find, um, you can do find stuff like that and so on. 432 00:56:07.139 --> 00:56:12.630 Um, okay, um. 433 00:56:15.659 --> 00:56:18.840 And is a new thing. Um. 434 00:56:20.039 --> 00:56:29.039 And then, like, there's a cross cross Co variance. 435 00:56:30.780 --> 00:56:38.550 Between X and Y, and you can find that. So, um. 436 00:56:49.079 --> 00:56:57.030 So, it will be how components of X relate to components of why so, again, if I use my color video, as an example. 437 00:56:57.030 --> 00:57:00.750 This would say how the, why of the brightness. 438 00:57:00.750 --> 00:57:05.550 Say correlated with the green or something of the. 439 00:57:05.550 --> 00:57:10.500 Input thing, so that can be useful. So okay. 440 00:57:14.130 --> 00:57:25.469 To do, um. 441 00:57:30.750 --> 00:57:40.409 And 1 thing you can work with the covariant says that you can plug it in for the definition of a joint. So this could be this could be interesting. 442 00:57:42.000 --> 00:57:49.469 Well, just send them a side cause I just personally thinking who's talking about history of video. 443 00:57:49.469 --> 00:57:54.360 Um, so again, real world engineering you're designing with constraints now. 444 00:57:54.360 --> 00:58:02.429 And so they had black. Oh, 1st, black and white TV again, had some big historical design. 445 00:58:02.429 --> 00:58:06.960 Issue is like the way it worked was. 446 00:58:06.960 --> 00:58:11.280 The, um, there was an electron gun would scan its way. 447 00:58:11.280 --> 00:58:15.210 Scan its way across the, um. 448 00:58:15.210 --> 00:58:23.340 And rose like this, and the pattern of scanning was just the same as if a farmer was plowing his field. 449 00:58:24.719 --> 00:58:35.369 Which was not a coincidence, because it was invented by a guy called vital key parts with the grew up on a farm around the Idaho. 450 00:58:35.369 --> 00:58:41.219 Utah border, and when he was a kid, he was helping to plow his family farm. 451 00:58:41.219 --> 00:58:44.489 And when he, um, grew up. 452 00:58:44.489 --> 00:58:50.670 And was became the inventor of what's called electronic television, because it wasn't using moving parts. 453 00:58:50.670 --> 00:58:58.829 But it's a coincidence or not, but he might have been remembering how he plowed a whole field when he designed how an electron beam would scan. 454 00:58:58.829 --> 00:59:11.880 The back of the, and there's a statue to him actually, in the base of the United States capital, and in the basement I saw it once years ago. So. 455 00:59:11.880 --> 00:59:16.349 Each state got to nominate a couple of famous people and the staches would be. 456 00:59:16.349 --> 00:59:23.820 Play somewhere on the Capitol and I think Idaho, Utah, I forget which 1 cause both claim nominated file. Key far. It's worth. 457 00:59:25.199 --> 00:59:29.340 A competing thing was a mechanical television. 458 00:59:29.340 --> 00:59:34.199 Where they had this rotating wheel with had. 459 00:59:34.199 --> 00:59:41.340 Pinned holes in it in a spiral fashion and the wheel would rotate in front of just a light source and a light source would change in density. 460 00:59:41.340 --> 00:59:55.349 As this wheel with a rotate with a moving hole located in front of it, and you would see an image and this was manufactured in the 920 s tells us it didn't scale up the good resolutions you can do a 30 by 30 pixels or something. But that's not a very good image. 461 00:59:55.349 --> 01:00:01.079 So, but they actually had experimental broadcasts that way, but then electronic television came over. 462 01:00:01.079 --> 01:00:04.800 But then the next issue was so the head black and white T. V. 463 01:00:04.800 --> 01:00:12.809 And now they wanted to make it color and they wanted to make it uploaded backward compatible and use the same bandwidth as black and white. 464 01:00:13.980 --> 01:00:20.579 So, the constraints and manufacturable was 1950 technology no integrated circuits. 465 01:00:20.579 --> 01:00:24.449 No transistors for that matter. Vacuum tubes. 466 01:00:24.449 --> 01:00:34.739 So, they look at characteristics of images, and they looked at characteristics of people they looked at how you saw a color image and they realized you didn't see colors as well. 467 01:00:34.739 --> 01:00:43.260 As you saw brightness, for example, and they realize they look at images and as you go from fix little pixels, you scan across an image the. 468 01:00:43.260 --> 01:00:46.349 The brightness changes faster than the caller does. 469 01:00:46.349 --> 01:00:51.030 And they work these things into a design for compressing. 470 01:00:51.030 --> 01:00:55.800 The image it worked tolerably well, for black for color T. V. so. 471 01:00:55.800 --> 01:00:59.820 Quite sophisticated again, not using any. 472 01:00:59.820 --> 01:01:03.389 Discreet component not using any integrated components. 473 01:01:03.389 --> 01:01:07.139 The problem was it had some issues. 474 01:01:07.139 --> 01:01:10.710 So, this is done for the national television standards committee. 475 01:01:10.710 --> 01:01:15.690 And the, and the job goes, so it would never twice the same color. 476 01:01:16.710 --> 01:01:20.909 But the United States that at 1st, and in other countries got to. 477 01:01:20.909 --> 01:01:28.860 Other countries got to look at the United States, the American experience when they designed their own system. 478 01:01:28.860 --> 01:01:35.519 So, they designed their system for having to work around some of the problems that have been observed here. Um. 479 01:01:35.519 --> 01:01:44.639 But they had other issues also, so the French designed a system, which had actually memory, it could actually store 1 line of the image in memory. 480 01:01:44.639 --> 01:01:50.039 Um, there's some sort of acoustic register or something. 481 01:01:50.039 --> 01:01:56.340 And so, in French, their system was sequential color with memory. Um. 482 01:01:56.340 --> 01:01:59.940 Sequential yeah. 483 01:01:59.940 --> 01:02:04.590 But, of course, the job was installed for system essentially contrary to the American method. 484 01:02:04.590 --> 01:02:12.659 And back when this was being done, Germany was in 2 parts, West Germany and East Germany, the Federal Republic, the Democratic Republic. 485 01:02:12.659 --> 01:02:18.539 So, West Germany did, um, a certain color standard than East. Germany did a color standard later. 486 01:02:18.539 --> 01:02:21.539 That was not compatible with the West German standard. 487 01:02:21.539 --> 01:02:26.639 And every 1 assumed it was so his shermans could not watch West German color broadcasts. 488 01:02:26.639 --> 01:02:31.800 They could see him black and white, but not in color. Politics is coming into standards. 489 01:02:31.800 --> 01:02:35.130 Any case what has happened to here. 490 01:02:35.130 --> 01:02:38.130 You don't see, I'm seeing a message that. 491 01:02:38.130 --> 01:02:46.559 Green broadcasting stopped again, so sorry about that people watching. Let's started up again. 492 01:02:51.929 --> 01:02:57.329 Okay, I don't see anyone here. 493 01:02:59.250 --> 01:03:04.920 Several people. Okay let us, um, we're still recording. 494 01:03:07.829 --> 01:03:11.909 So, in any case, we'll work with an engineering electrical engineering you've got factor. 495 01:03:11.909 --> 01:03:15.929 Variables and so on. 496 01:03:15.929 --> 01:03:22.469 Okay, um, so we get crosscore variance and so on between different things here. 497 01:03:24.420 --> 01:03:29.010 Examples start getting a little messy to do by hand, but, um. 498 01:03:31.829 --> 01:03:35.219 Okay, and, um. 499 01:03:37.260 --> 01:03:42.119 1 thing, so we have the transformation. Um, okay, so we have. 500 01:03:42.119 --> 01:03:45.630 Say X. 501 01:03:45.630 --> 01:03:54.090 Are correlated and maybe, um. 502 01:03:57.900 --> 01:04:03.449 Today, so the why I. 503 01:04:05.699 --> 01:04:08.849 Or less correlated, so. 504 01:04:12.780 --> 01:04:26.844 It may help with compression, perhaps, for example, so we can start getting stuff like that. 505 01:04:27.355 --> 01:04:28.494 So, pick a why? 506 01:04:29.130 --> 01:04:33.150 Okay um, other big things here. 507 01:04:37.349 --> 01:04:51.864 Um, okay, um, okay, I may work on that detail, but I hit you with the big things and anything. I write down. We sold little type. 508 01:04:51.864 --> 01:04:56.425 You want to understand it so our next big topic in chapter 6. 509 01:04:56.699 --> 01:05:03.809 Is, um, section 6.5. 510 01:05:03.809 --> 01:05:10.739 Estimation okay and it's page. 511 01:05:10.739 --> 01:05:16.980 332 okay. Um. 512 01:05:18.210 --> 01:05:30.480 We've seen a little of this before. Um, you know, we're we have a noisy channel, you transmit X, you receive why? And you want to guess what X is given why. 513 01:05:30.480 --> 01:05:35.280 Okay, so now this is well, now we're doing on, um. 514 01:05:36.900 --> 01:05:46.440 So, basically, you know, well, let's say, give an example. 515 01:05:49.289 --> 01:05:55.739 Sex. 516 01:05:55.739 --> 01:06:02.849 Receiver, why? And, um. 517 01:06:05.670 --> 01:06:12.690 So, I don't know what X, um. 518 01:06:12.690 --> 01:06:19.289 Maximizes, um. 519 01:06:27.690 --> 01:06:35.429 Mac, so, um, so it takes a little thinking there. Um. 520 01:06:38.130 --> 01:06:43.500 So, we, we so we see some why then what X. 521 01:06:43.500 --> 01:06:48.929 Has the highest probability given that we're, we're using base and so on. So, um. 522 01:06:50.429 --> 01:06:54.750 So the terminology might be maximize X. 523 01:06:54.750 --> 01:07:05.010 Um, and this is called a maximum a. 524 01:07:07.920 --> 01:07:13.920 Um, estimator. 525 01:07:16.679 --> 01:07:22.409 Which is M. A. P. okay. Um. 526 01:07:24.059 --> 01:07:27.539 And we've seen that before, so this is a bit of a review here. 527 01:07:27.539 --> 01:07:32.429 Um, so the probability that goes X given. 528 01:07:44.010 --> 01:07:50.760 Okay, and then the thing on the top, we, we do that again. Um, using conditionals. 529 01:07:55.019 --> 01:08:06.480 Probability and again what this notation means it probably the random variables, the value X. okay. 530 01:08:06.480 --> 01:08:15.809 Good. Um, so what we have to do to find this map Estimator probabilty then. 531 01:08:15.809 --> 01:08:23.010 Well, we well, this requires knowing the power probability of X. okay. 532 01:08:24.060 --> 01:08:28.170 Yes, the review. Okay. So, um. 533 01:08:32.369 --> 01:08:36.689 To review using base and so on, put a little note on days down here. 534 01:08:37.404 --> 01:08:51.864 Right. Okay. Um, so if we know we obviously have to have the code. The conditional is probably why give an X. we have to know the prior profitability events. 535 01:08:52.109 --> 01:09:00.989 And the problem, we see everything that we can get this, the conditional, and we can find the X, which is most likely. Okay. Um. 536 01:09:03.659 --> 01:09:09.539 So, if I can scroll. 537 01:09:20.729 --> 01:09:27.630 Sometimes, we don't know the probability has, he's given access. 538 01:09:29.550 --> 01:09:35.159 Okay um, but we still want to do something. Okay. 539 01:09:41.279 --> 01:09:48.029 I want to do something. Okay. Um. 540 01:09:50.369 --> 01:09:56.250 So, and this is called an estimator, um. 541 01:09:57.840 --> 01:10:01.770 So, we use a maximum likelihood estimator. 542 01:10:05.220 --> 01:10:16.170 You can't read that see spell right? 543 01:10:18.750 --> 01:10:23.939 It's called an L estimator. 544 01:10:23.939 --> 01:10:30.659 And what it does is, it basically assumes all the X's are equally probable. So, um. 545 01:10:36.569 --> 01:10:40.560 And that we don't do a max, find the exit, maximize this. 546 01:10:46.710 --> 01:10:51.960 So we find the X, which is most likely to create this. Y, and, um. 547 01:10:58.739 --> 01:11:01.859 It sort of assumes, um. 548 01:11:01.859 --> 01:11:09.869 Half of X is uniform. Um. 549 01:11:11.340 --> 01:11:14.640 Even if the domain is infinite. Um, so. 550 01:11:17.069 --> 01:11:20.729 You can't do that directly. Would you do something like that? So. 551 01:11:22.739 --> 01:11:29.340 So, in any case, the maximum Estimator is the thing that. 552 01:11:29.340 --> 01:11:32.699 We talked about before using base and so on. 553 01:11:32.699 --> 01:11:36.689 And the maximum likelihood Estimator is, we don't know. 554 01:11:36.689 --> 01:11:41.069 What the source 2 probabilities of the different source. 555 01:11:41.069 --> 01:11:45.659 Sorry, sorry the bar so we do what we can and we have. 556 01:11:45.659 --> 01:11:49.560 We assume it's uniform and. 557 01:11:49.560 --> 01:11:53.609 Hope it gives us something. So, in any case, um. 558 01:11:57.029 --> 01:12:01.890 Reasonable point to stop now. Um. 559 01:12:04.079 --> 01:12:10.619 So, I'll continue more on this Thursday so what I'll show you today it was more chapter 6 stuff on. 560 01:12:10.619 --> 01:12:15.060 Sector random variables. We can find conditional. 561 01:12:15.060 --> 01:12:19.020 You know, the joint probabilities, conditional probabilities for example. 562 01:12:19.020 --> 01:12:25.800 Um, working with stuff like that, but means there's several examples. 563 01:12:25.800 --> 01:12:30.960 Um, or basically, excuse me, running down chapter 6 so. 564 01:12:30.960 --> 01:12:34.560 1, more day, and then the following Friday, of course, this. 565 01:12:34.560 --> 01:12:41.369 Exam so open for questions now, other than that, have a good week. So. 566 01:12:47.130 --> 01:12:51.479 Okay, see you Thursday. 567 01:12:52.529 --> 01:12:56.069 Okay. 568 01:13:17.460 --> 01:13:25.859 The 2nd. 569 01:13:33.569 --> 01:13:36.779 We stopped.