WEBVTT 1 00:00:20.550 --> 00:00:26.129 Silence. 2 00:00:27.989 --> 00:00:38.159 Silence. 3 00:00:38.159 --> 00:01:18.750 Silence. 4 00:01:21.450 --> 00:01:31.530 Silence. 5 00:01:31.530 --> 00:01:41.189 Silence. 6 00:01:41.189 --> 00:01:55.319 Silence. 7 00:02:05.670 --> 00:02:09.120 Silence. 8 00:02:10.319 --> 00:02:19.500 Silence. 9 00:02:22.650 --> 00:02:40.800 Silence. 10 00:02:43.199 --> 00:02:49.860 Silence. 11 00:02:54.030 --> 00:03:02.009 Silence. 12 00:03:04.319 --> 00:03:08.189 Silence. 13 00:03:08.189 --> 00:03:13.289 Silence. 14 00:03:14.310 --> 00:03:25.469 Silence. 15 00:03:33.300 --> 00:03:42.030 Okay, so good afternoon class. 16 00:03:42.030 --> 00:03:47.849 As usual, I cannot tell if anyone can hear me. 17 00:03:47.849 --> 00:03:52.650 So. 18 00:03:54.780 --> 00:03:59.250 Thank you Nicholas. 19 00:04:00.895 --> 00:04:15.685 Again, the reason I cannot tell is if I put on audio feedback on another device, then the delay, it would be half a 2nd, delay give or take. And, of course, you cannot speak when you're hearing yourself half a 2nd later it would drive you. 20 00:04:15.685 --> 00:04:20.004 Insane. So, I can tolerate the delay when I'm. 21 00:04:21.959 --> 00:04:28.769 You know, for the video, but so actually, what I have in front of me, I have my main laptop that I'm. 22 00:04:28.769 --> 00:04:38.459 Doing the display from I've got a 2nd older laptop where I can see what I think you can see. Yeah thanks Thomas also. And so that helps me. 23 00:04:38.459 --> 00:04:48.059 Get a sent to it and that shows the chat window and then I've got an iPad here that I'm writing on. So too much technology, but I like technology. That's why. 24 00:04:48.059 --> 00:04:53.608 I'm at and just for fun unrelated to the course, but I got the. 25 00:04:54.204 --> 00:05:07.074 Sold ourselves on my roof are working better than I thought this is 1st week in March. And as I write this, the test starts the week on Monday, and it's positive for the week. 26 00:05:07.103 --> 00:05:11.303 And so this is not even spring. So I think in. 27 00:05:11.579 --> 00:05:25.619 It will really get positive. Well, it would really get positive in the summer, except that I'm looking at buying a Tesla car and that will use all the extra electricity up. So. 28 00:05:25.619 --> 00:05:30.088 Okay, real world electrical engineering. 29 00:05:30.088 --> 00:05:35.999 So, getting back to engineering probability, we are. 30 00:05:35.999 --> 00:05:48.869 In chapter 3, and so my teaching style is to walk you through the textbook, actually giving my opinions on stuff. Skipping things. I don't think are interesting and going rather fast. 31 00:05:48.869 --> 00:06:00.358 But so we're in chapter 3 at the moment with discrete random variables and their random variables that you can assign a probability to each outcome. That could be finite. Utah. The coin. 32 00:06:23.338 --> 00:06:33.418 Such as you measure the temperature today or something, and so you can measure this infinitely. Precisely. And the thing with the. 33 00:06:33.418 --> 00:06:46.283 Continuous random variables is any specific. There's an uncountable number of infinite values and what that means is, you cannot take all the real numbers and put them in a 1 to 1 correspondence with the natural numbers. 34 00:06:46.944 --> 00:06:58.403 And so any specific temperature will have a probability 0T. So we have to talk about the probabilities of intervals of ranges of the probability of the temperature is between 40 and 50. let's say. 35 00:06:58.673 --> 00:07:06.954 So, that's where how continuous has fundamentally different now. Now, a particular random variable could be. It could be a mixed, random variable. 36 00:07:06.954 --> 00:07:17.334 It could be both like, if I arrive at the airport and I need an Uber, then maybe there's a number waiting their probabilities in over waiting there. Let's say, maybe I dunno. 37 00:07:17.639 --> 00:07:23.459 50% haven't flown in so long. I can't remember but 50% and then. 38 00:07:24.658 --> 00:07:38.608 So that's that precise number. Wait time. 0T has a probability. That's that part of the distribution. That's great. But then, if it's not an over there, I may have to wait some sort of might be geometric probability distribution of some sort. 39 00:07:38.608 --> 00:07:44.158 So it'd be or sorry, some sort of continuous thing. Geometric is discreet. And so it'd be a mix. 40 00:07:44.158 --> 00:07:49.468 That's in the future, so we're talking about comm, discreet. 41 00:07:49.468 --> 00:08:01.528 Before I start just question for the class. So how is is going remote affecting people I mean, we're already remote in probability, but. 42 00:08:01.528 --> 00:08:09.778 I'm just curious how how, how is it affecting you that courses that were in person or how online you can. 43 00:08:09.778 --> 00:08:13.408 I mean, Mike can, you know, talk if you want come. 44 00:08:14.639 --> 00:08:19.978 And, yes, I should on mute something to hear. 45 00:08:19.978 --> 00:08:23.488 And. 46 00:08:23.488 --> 00:08:27.358 No, 1 has. 47 00:08:27.358 --> 00:08:30.538 Just checking here on sound. 48 00:08:35.788 --> 00:08:39.958 Yeah, so I have I have some sound like I can hear you if you. 49 00:08:39.958 --> 00:08:45.448 Chime in, but see. 50 00:08:48.389 --> 00:08:53.369 Not particularly good, but, um. 51 00:08:54.928 --> 00:08:58.769 Yeah, well. 52 00:08:58.769 --> 00:09:04.948 Yeah, so I can hear you also, if you on mute your Mike and speak, but. 53 00:09:04.948 --> 00:09:10.528 I just don't get good feedback. I can't tell what you're hearing. Okay. 54 00:09:10.528 --> 00:09:15.479 So, nothing there. Okay, so let's go to the park and then and. 55 00:09:17.129 --> 00:09:24.239 I put the book. Okay, so we're talking about variants here and so it captures. 56 00:09:24.239 --> 00:09:27.869 It captures the spread, so. 57 00:09:31.798 --> 00:09:40.828 So, if I have some distribution, which looks something like this or whatever, so, sweet V, X and some sort of probability. 58 00:09:42.028 --> 00:09:55.828 And so the variants would be something like that would be small if I had if I had a distribution like this, which is something like that out of a smaller variance if I got something like that, that would be a wider variance. So, this is. 59 00:09:57.958 --> 00:10:01.229 Bigger variants and the blue here. 60 00:10:01.229 --> 00:10:08.068 Smaller variance. Okay. It's how it's how spread out things are. 61 00:10:08.068 --> 00:10:19.379 Okay, and we have the definition. We have 2 different definitions here. I'll show you both just remind you, there's the 1 or variants. 62 00:10:20.639 --> 00:10:29.009 It calls the expected value of X squared, minus, expected value of X. 63 00:10:29.009 --> 00:10:33.418 Squared and that thing there. 64 00:10:34.948 --> 00:10:45.089 Can we mix that's, um, 1 definition? Another definition is expected value of X minus expected value of X. 65 00:10:46.288 --> 00:10:49.708 Squared, um. 66 00:10:50.729 --> 00:10:55.019 This is give a simple example here. 67 00:10:55.019 --> 00:10:59.639 I don't know that's toss. Let's do binomial. 68 00:10:59.639 --> 00:11:08.788 Equals 1 half and then equals 2 or something so it means we tossed the coin twice. Okay. 69 00:11:10.198 --> 00:11:15.269 So so the various probability to the various outcomes are. 70 00:11:17.668 --> 00:11:22.349 On the random variable X equals the number of heads. Okay. 71 00:11:23.609 --> 00:11:28.948 So the outcomes 0, 1 and 2 in the probability. 72 00:11:28.948 --> 00:11:33.568 Outcomes is. 73 00:11:34.859 --> 00:11:39.149 Do choose K, basically, um. 74 00:11:39.149 --> 00:11:43.048 After the K1 half to the 2, the minus K. 75 00:11:43.048 --> 00:11:46.889 And I'm just writing a general saying, so basically. 76 00:11:48.538 --> 00:11:57.058 That's 1 corridor to choose K and now to choose K again happens to be. 77 00:11:57.058 --> 00:12:04.379 1, 2 or 1, so probably 0T equals 1 quarter probability of 1 is 1 half. 78 00:12:04.379 --> 00:12:15.778 Probability to the close 1 quarter. Okay. You see, it's total review now. So the expected value is. 79 00:12:18.359 --> 00:12:27.869 Equals 1 quarter to 0, 0T plus a half times 1 plus 1, quarter times 2, which is. 80 00:12:27.869 --> 00:12:33.328 1, which makes sense. Okay. Now. 81 00:12:37.649 --> 00:12:43.288 No, so the variance, um, come on. 82 00:12:45.089 --> 00:12:49.168 Well, if I do method 1, um. 83 00:12:51.058 --> 00:12:54.719 Expected value of X minus the mean of X. 84 00:12:55.739 --> 00:13:03.989 Squared well, the mean of Axis 1, so effects so effects as 0, 1 or 2. 85 00:13:03.989 --> 00:13:08.578 X minus the mean of X is minus 1 0T or 1. 86 00:13:09.629 --> 00:13:12.808 Um, X, minus mean of X. 87 00:13:12.808 --> 00:13:18.359 Squared is 1 0T or 1 and the probability of each of these things. 88 00:13:18.359 --> 00:13:23.849 Is 1 quarter 1 half 1 quarter and we sell it all up. 89 00:13:25.048 --> 00:13:30.749 We're going to get a quarter times 1, Sarah plus this, and we are going to get. 90 00:13:30.749 --> 00:13:34.109 If the variance we're going to get 1 half. 91 00:13:34.109 --> 00:13:38.759 Sum everything up that's the 1 method. If I take the other method. 92 00:13:39.958 --> 00:13:44.399 Expected value of X squared minus X. 93 00:13:44.399 --> 00:13:48.599 Squared the expected value of X squared. 94 00:13:48.599 --> 00:13:51.658 So, X squared is 0. 95 00:13:51.658 --> 00:13:57.869 1, or 4 is probability 1 quarter 1, half 1 quarter. 96 00:13:57.869 --> 00:14:01.318 So expected value is. 97 00:14:05.129 --> 00:14:09.869 This is a half plus some. 98 00:14:09.869 --> 00:14:15.359 Expected value of X equals 1. 99 00:14:15.359 --> 00:14:18.778 So we get 3 has minus 1 equals 1 half. 100 00:14:18.778 --> 00:14:22.979 So, the very inside their way is a half so. 101 00:14:24.538 --> 00:14:28.109 Okay. 102 00:14:28.109 --> 00:14:39.658 Whichever method to use, as I mentioned before the 2nd method here, you can lose significant digits when you're evaluating it on a computer. 103 00:14:39.658 --> 00:14:46.349 Okay, so I give that example here for the. 104 00:14:46.349 --> 00:14:50.009 Geometric random variable. There's some tricks you can use. 105 00:14:50.009 --> 00:14:56.609 I started doing a little last time I want to go through to to solve these things. So. 106 00:14:56.609 --> 00:15:00.328 Okay. 107 00:15:00.328 --> 00:15:06.208 And we get something like that. Okay. 108 00:15:06.208 --> 00:15:09.389 I was somewhat of a review, so. 109 00:15:10.918 --> 00:15:19.318 So, we may have a conditional probability mass function. We I mean, we have conditional probabilities and. 110 00:15:20.578 --> 00:15:25.349 I'll just to remind you review additional probabilities for you. 111 00:15:27.778 --> 00:15:31.708 You might have the probably I'll do it a, and B. 112 00:15:31.708 --> 00:15:37.078 Sorry probability it will be. 113 00:15:37.078 --> 00:15:41.068 What we say. 114 00:15:41.068 --> 00:15:47.219 Provide by probably be perhaps. So if I have just to review this, so if I have a. 115 00:15:50.668 --> 00:15:55.859 A, and B. 116 00:15:55.859 --> 00:16:03.208 And they have the outside here, Ben diagram. 117 00:16:03.208 --> 00:16:06.688 And if I throw some numbers in here. 118 00:16:06.688 --> 00:16:13.589 So, I don't know. 119 00:16:15.989 --> 00:16:22.408 Okay, so what we have here, a total number of items is 10. 120 00:16:22.408 --> 00:16:28.109 So, I'll forget this is a, this would be be. 121 00:16:30.058 --> 00:16:33.658 So, probably a, is. 122 00:16:35.668 --> 00:16:39.958 We can't. 123 00:16:41.129 --> 00:16:44.278 Silence. 124 00:16:44.278 --> 00:16:48.208 2 tabs. 125 00:16:48.208 --> 00:16:53.158 It goes to tents, divided by. 126 00:16:53.158 --> 00:16:57.178 I can't equals to this. 127 00:16:58.259 --> 00:17:02.009 And, you know, that makes sense because we look and be here. 128 00:17:02.009 --> 00:17:06.719 There's 5 things and B2 of them are a okay. 129 00:17:06.719 --> 00:17:12.598 Review we have to it on the exam. Of course. No. So the conditional probability mass function. 130 00:17:12.598 --> 00:17:19.709 Nothing new here. 131 00:17:19.709 --> 00:17:25.409 It's basically trying to show you that little thing here. 132 00:17:25.409 --> 00:17:29.669 Um. 133 00:17:29.669 --> 00:17:36.028 But it's best shown with some examples and. 134 00:17:39.568 --> 00:17:44.939 So, we say, take a random clock, we, we spend the clock and. 135 00:17:47.969 --> 00:17:53.338 Whatever spin it, and we look at the probably mass function of the hour and. 136 00:17:57.028 --> 00:18:00.989 So, it starts getting getting interesting here. 137 00:18:00.989 --> 00:18:07.259 Um. 138 00:18:07.259 --> 00:18:12.028 So, for any given hour, of course, it's 112 that, that it's in any 1 hour. 139 00:18:12.028 --> 00:18:15.028 And the probability that. 140 00:18:15.028 --> 00:18:25.199 It's in the 1st, basically, what's a conditional function for what the hour hand is? Assuming it's in 1 of the 1st, 4 hours. 141 00:18:25.199 --> 00:18:39.233 That's what that's saying. So, B is the event that the hand is in 1 of the 1st, 4 hours and chase which it is. Well, if it's not 1 of the 1st, 4, if it's in 1 of the 1st, 4 hours, it's in 1 of the 1st, 4 hours. 142 00:18:39.534 --> 00:18:41.604 So that's why we have down here. 143 00:18:42.298 --> 00:18:50.699 0T otherwise, and it's a quarter that it's in each hour. Each that's 1 of the 1st. 4. this example is actually, it's a little simple. It's. 144 00:18:50.699 --> 00:18:54.898 It's get silly, but. 145 00:18:54.898 --> 00:18:59.009 Residual waiting time starts getting more interesting here. 146 00:18:59.009 --> 00:19:11.669 So, we're transmitting a message and we know that it's within it will take 1 to L. 147 00:19:11.669 --> 00:19:16.618 Seconds let's say, let's say seconds to transmit and. 148 00:19:16.618 --> 00:19:21.538 Will not take more than now so, maybe I'll is 10. it won't take more than 10 seconds. 149 00:19:21.538 --> 00:19:31.259 And so now what let's say, it's been already transmitting for 3 seconds then what's the distribution on the remaining transmission time? 150 00:19:31.259 --> 00:19:36.989 Now, if we know it's more than 3 so. 151 00:19:38.159 --> 00:19:42.628 If it's already, I'll do this with an example here. Okay. 152 00:19:42.628 --> 00:19:46.229 Oh, this says example. 153 00:19:47.398 --> 00:19:55.528 24, but that's going to laptop say, let al equals 10. let's say 10 seconds perhaps. So, the thing is that. 154 00:19:57.419 --> 00:20:04.949 And so exits the transmission time and basically. 155 00:20:04.949 --> 00:20:11.939 The probability the probability, mass function for any value of X. a K. 156 00:20:11.939 --> 00:20:15.328 It is going to be on 10th. 157 00:20:16.858 --> 00:20:24.898 If K is from okay and 0T otherwise. 158 00:20:25.919 --> 00:20:29.278 Okay, so that is our. 159 00:20:29.278 --> 00:20:39.568 That's our probability mass function here. Little execute. I mean, it's a variable X we're talking about if we had more than 1, random variable. Now. 160 00:20:39.568 --> 00:20:43.499 Before it so now, but suppose that. 161 00:20:43.499 --> 00:20:49.138 Um. 162 00:20:49.138 --> 00:20:53.608 But now, here, the problem is, let's say. 163 00:20:53.608 --> 00:20:58.019 Seconds have passed. 164 00:21:01.138 --> 00:21:04.378 And message. 165 00:21:04.378 --> 00:21:07.949 Hasn't finished. 166 00:21:09.659 --> 00:21:15.388 Okay, so, let's suppose say, um. 167 00:21:15.388 --> 00:21:20.368 And equal 6 so now. 168 00:21:22.469 --> 00:21:27.328 What's happening? So, I mean, before I scroll up again, I'm the trouble with the. 169 00:21:27.328 --> 00:21:41.398 Tool I'm using I can't show you half a page so so the message we know it's going to take from 1 to 1010 seconds. Max. I said, all equals 10 and say 6 seconds of past and the message hasn't. 170 00:21:41.398 --> 00:21:45.239 Gone through yet, so it's going to take so the message. 171 00:21:49.229 --> 00:21:54.568 It will take 7 to 10 seconds. 172 00:21:54.568 --> 00:21:57.719 The probability of 7 will be a quarter. 173 00:21:57.719 --> 00:22:04.078 Age 1, corridor and sell and 9 and 10 on quarter. Each. Okay. So how do we do that in. 174 00:22:05.219 --> 00:22:11.128 In math, we use our, we use our conditional thing so we want to say. 175 00:22:11.128 --> 00:22:17.818 So, the probability that X equals some value and. 176 00:22:21.088 --> 00:22:24.328 Oh, they're saying in plus J. 177 00:22:24.328 --> 00:22:30.058 Given that X is greater than M. 178 00:22:30.058 --> 00:22:33.568 Okay, well that's going to be. 179 00:22:35.159 --> 00:22:41.249 I am plus Jay and X greater than that is basically always true. So. 180 00:22:42.449 --> 00:22:48.058 So, how do we do this now? 181 00:22:48.058 --> 00:22:53.699 Silence. 182 00:22:57.028 --> 00:23:06.088 Okay, okay the thing at the top here, the numerator probably if X equals. 183 00:23:06.088 --> 00:23:09.719 Jay is now negative than. 184 00:23:11.128 --> 00:23:18.328 Then, basically, this here, I mean, this is going to be true. 185 00:23:19.798 --> 00:23:26.788 True for say January 0T. Okay so what this thing goes down to is probability. 186 00:23:28.709 --> 00:23:36.358 Black, see. 187 00:23:36.358 --> 00:23:41.909 Because the 2nd condition didn't add anything over problem. Okay. 188 00:23:44.038 --> 00:23:54.509 Okay, well the probably exit specific value on the top that we want over L1 over and and then the bottom ex, greater than M. is that fraction there. 189 00:23:55.648 --> 00:24:01.409 And that is the 1. okay. Oh, my Sam. Okay, so it's showing you a conditional thing. 190 00:24:02.818 --> 00:24:12.989 These sorts of conditional things real world thing is device lifetime. 191 00:24:14.999 --> 00:24:21.959 So, what's happening here is we got 2 types of devices and 1 is more reliable than the other. 192 00:24:21.959 --> 00:24:30.778 And we're assuming a geometric distribution for their lifetime, such like, radioactive decay under this model. 193 00:24:30.778 --> 00:24:34.048 The life the remaining lifetime. 194 00:24:34.048 --> 00:24:38.429 Of the wage, it does not depend on how old it is. 195 00:24:38.429 --> 00:24:42.358 So, is this the case for real world objects? 196 00:24:42.358 --> 00:24:47.338 Probably not, but it makes a simple example is the case it radioactive Adams see. 197 00:24:48.084 --> 00:24:59.903 The past history does not affect the Adam, or you might say that if the device will live until a cosmic re headset again, the probability of a cosmic re hitting it in the next. 198 00:24:59.903 --> 00:25:06.413 2nd does not depend on how all the devices under those cases. Geometric distribution. Would be. 199 00:25:06.989 --> 00:25:13.199 Would be reasonable any case, you know, we have to make the examples simple enough that we can actually work with them. 200 00:25:13.199 --> 00:25:17.368 So, what's happening here is you got the 2 types of devices. 201 00:25:17.368 --> 00:25:22.378 The, the both has a geometric distribution. 202 00:25:22.378 --> 00:25:27.358 For their lifetime and a parameter. 203 00:25:27.358 --> 00:25:31.019 Well, 1st, we got 2 mixes so the type 1. 204 00:25:31.403 --> 00:25:45.894 Probability alpha of the whole batch are type 1 and 1 minus alpha. The whole batch are type 2 and then the 2 types have different lifetime parameters about your decision as a parameter. And these 2 types of widgets are 2 different parameters. 205 00:25:46.169 --> 00:25:58.858 Our and s, so we'd like to find the probability mass function of a random device in the batch, or we don't know what it is. 206 00:25:58.858 --> 00:26:04.439 And the way it does it, so. 207 00:26:04.439 --> 00:26:16.828 Again, those 2 different, slightly different versions of the geometric distribution, but doing this version here. 208 00:26:17.939 --> 00:26:27.868 so the lifetime k the parameter is the lifetime or does it die in the first second second second third second or something and . 209 00:26:27.868 --> 00:26:32.429 Are is the probability of dying in any given? 2nd. 210 00:26:32.429 --> 00:26:38.249 So, if it dies exactly in the case, 2nd, this means that. 211 00:26:38.249 --> 00:26:45.868 It did not die in in the K minus set 1 seconds before, and then died in the case. 2nd, so this is the geometric distribution here. 212 00:26:45.868 --> 00:26:49.888 For the 1st device. 213 00:26:49.888 --> 00:26:56.278 And so what this is saying, this is the probability mass function given that it's the 1st type of wedge at. 214 00:26:56.278 --> 00:27:02.578 And then here, we have below it for the 2nd type of wedge yet the only differences as for our. 215 00:27:03.838 --> 00:27:07.318 So the way you get the, you might say the combo. 216 00:27:07.584 --> 00:27:19.314 Probability mass function is we use the total probability thing that was in the previous chapter we take the 2 probability mass function, and we just wait them with the probability of the device being type 1 or type 2. 217 00:27:19.314 --> 00:27:30.923 so, we get the math for the device for type 1 devices, have the probability of being a type 1 device plus the for type 2 devices times, that probability of being a type 2 device. 218 00:27:31.169 --> 00:27:35.459 So, we can do this combo thing for probably building mass functions. 219 00:27:38.068 --> 00:27:41.909 Okay, now. 220 00:27:41.909 --> 00:27:45.358 So, we saw we've got these conditional. 221 00:27:45.358 --> 00:27:53.249 So, in combo, probably, as long as we saw, expectation, expected value also called the mean also called the arithmetic mean. 222 00:27:53.249 --> 00:27:58.169 Now, you may want the. 223 00:27:58.169 --> 00:28:03.298 The conditional mean, the conditional expected value. 224 00:28:03.298 --> 00:28:14.098 And that's this here, the expected value of equity, random variable, given that condition B, and we take all we do is to take all the conditional probabilities. 225 00:28:14.098 --> 00:28:17.368 And we wait them. 226 00:28:17.368 --> 00:28:24.989 And, oh, take each possible value for the output and weighted by the conditional probability. And that gives us. 227 00:28:24.989 --> 00:28:28.259 The conditional mean. 228 00:28:28.259 --> 00:28:31.439 And if you're thinking ahead, we are going to have a conditional variance. 229 00:28:33.058 --> 00:28:39.778 So, okay, um, this. 230 00:28:39.778 --> 00:28:46.199 Right. Just right here it's exactly the same thing. We have the formula for the parents, the expected value of the. 231 00:28:46.199 --> 00:28:50.128 Divergence from the main squared. 232 00:28:51.659 --> 00:28:55.469 Times. 233 00:28:55.469 --> 00:29:00.959 Wait it. Okay. There's a question here. I didn't see sorry yet to do. 234 00:29:02.999 --> 00:29:06.028 Oh, I'm online. Okay. 235 00:29:07.288 --> 00:29:14.999 Would the some, not be 1 for the expected value of X, minus the mean X? 236 00:29:16.318 --> 00:29:20.848 So, I didn't see that question a few minutes ago wasn't looking over there. 237 00:29:22.108 --> 00:29:29.818 Um. 238 00:29:29.818 --> 00:29:33.808 Sexual expected value of X minus mean X. 239 00:29:33.808 --> 00:29:39.509 That would be 0T at Charlie, but it's the expected value of X minus mean X squared. 240 00:29:39.509 --> 00:29:46.348 So, it depends where the square is. 241 00:29:46.348 --> 00:29:50.669 So, I'm going back to answer this question. I didn't seem before. 242 00:29:50.669 --> 00:29:57.209 I see here. 243 00:29:58.769 --> 00:30:02.489 We went back here. 244 00:30:08.699 --> 00:30:13.138 Well, let me do it again. Let me do a really simple 1 here. Let's take a look at that. 245 00:30:15.689 --> 00:30:19.169 So this is about the question. Okay. 246 00:30:21.689 --> 00:30:25.138 About a chat question. 247 00:30:25.138 --> 00:30:30.598 Um. 248 00:30:30.598 --> 00:30:34.409 You can even like, let's say. 249 00:30:37.199 --> 00:30:40.828 I'd say, binomial, I'm not do the same thing. 250 00:30:43.019 --> 00:30:47.788 That's by knowing. Okay. Equals 1. so we're tossed the coin once. Okay. 251 00:30:49.318 --> 00:30:57.088 But, let me make it not fair. So probably the equals. I don't know. 252 00:30:57.088 --> 00:31:01.798 3rd. 253 00:31:01.798 --> 00:31:06.449 Okay, so probably 0T was 2 thirds. 254 00:31:06.449 --> 00:31:12.989 1 equals 4th, expected value of X is. 255 00:31:15.509 --> 00:31:22.048 Equals. 256 00:31:23.548 --> 00:31:27.388 Now, for the variance. 257 00:31:29.669 --> 00:31:34.558 Well, if we do if it is the expected value of X, minus. 258 00:31:34.558 --> 00:31:38.128 Mean of X squared. 259 00:31:41.939 --> 00:31:48.509 So so 0T or 1 X, minus the mean of X minus 4th. 260 00:31:48.509 --> 00:31:52.229 So this is minus 4th or plus 2 thirds. 261 00:31:54.209 --> 00:31:59.068 Squared is 1 nights. 262 00:31:59.068 --> 00:32:03.239 Or 4 night, and then. 263 00:32:03.239 --> 00:32:09.628 The probability for this is 2 thirds and 4th. 264 00:32:09.628 --> 00:32:13.648 So, we multiply, we are going to get. 265 00:32:13.648 --> 00:32:20.909 27th and 427, so we saw. 266 00:32:20.909 --> 00:32:25.409 We get 627 equals 2 nights. 267 00:32:25.409 --> 00:32:31.949 For the variants here, if I do it the other way, the variance. 268 00:32:31.949 --> 00:32:37.199 Equals say, expected value of X squared, minus effective value X. 269 00:32:38.249 --> 00:32:45.058 Square it's expected value of X squared. So x0T or 1 x squared is 0T or 1. 270 00:32:46.288 --> 00:32:49.648 The probability is. 271 00:32:49.648 --> 00:32:54.628 2 thirds 4th multiply you. 0. 272 00:32:54.628 --> 00:32:58.048 4th. 273 00:33:01.439 --> 00:33:05.368 So expected value of X squared equals 4th. 274 00:33:05.368 --> 00:33:08.848 Expected value X. 275 00:33:08.848 --> 00:33:12.868 2nd value of X was 4th. 276 00:33:12.868 --> 00:33:16.378 Is 1, 9, and then the variance. 277 00:33:16.378 --> 00:33:20.699 Equals a 3rd minus a night. 278 00:33:21.959 --> 00:33:25.769 And we get the same thing each way. 279 00:33:25.769 --> 00:33:32.548 Does that sort of make things differently confusing? 280 00:33:37.378 --> 00:33:40.679 The thing is note the, um. 281 00:33:43.229 --> 00:33:46.409 You got to watch. 282 00:33:47.459 --> 00:33:52.769 Expected value of X squared versus suspected value of X. 283 00:33:52.769 --> 00:33:56.489 Squared okay. 284 00:34:04.078 --> 00:34:08.849 Okay, uh, so. 285 00:34:09.898 --> 00:34:12.989 Okay. 286 00:34:12.989 --> 00:34:16.588 How do you have a question? Sure. Question away. 287 00:34:16.588 --> 00:34:19.949 So, how do you do the. 288 00:34:19.949 --> 00:34:24.208 Like, what do you look for for the 1st, the 1st 1 there E that. 289 00:34:24.208 --> 00:34:33.028 Expected value of X squared. Well, X is a random variable X squared. It's just a square of X. 290 00:34:33.028 --> 00:34:37.259 So, let me take this case here. 291 00:34:37.259 --> 00:34:42.568 So this again, so this is the, this is the unfair coined. 292 00:34:42.568 --> 00:34:48.628 Silence. 293 00:34:50.998 --> 00:34:56.159 So, basically people's 4th, so, X is 0T or 1. 294 00:34:56.159 --> 00:35:00.449 The probability of X being is. 295 00:35:00.449 --> 00:35:07.798 In the series and 4th, this is just part of the definition. This is just the definition of the problem. 296 00:35:07.798 --> 00:35:18.509 But now we can talk about X squared it just and then we have the probability of X squared. 297 00:35:18.509 --> 00:35:21.568 Is also the thirds 4th. 298 00:35:21.568 --> 00:35:25.498 And so we're just looking at. 299 00:35:25.498 --> 00:35:36.418 So, it just happens to be that it's 0 1 and that. It's the same. Yeah, I just hit me as I was doing this. Let me do 1, which is, doesn't have that distraction. 300 00:35:36.418 --> 00:35:45.418 Well, let me go back to the 1st, starting to page here, because I can't. 301 00:35:45.418 --> 00:35:51.898 Um, yeah. 302 00:35:51.898 --> 00:35:56.938 Are so fair coin twice? 303 00:35:56.938 --> 00:36:10.289 Um, very good. Thrice make a little harder. Okay so this means and equals 3 and we're going to make it fair because life is complicated enough. Okay. So, values of X. 304 00:36:10.289 --> 00:36:19.648 Is there 1, 2 or 3 heads and it's your binomial distribution? 305 00:36:19.648 --> 00:36:26.248 Creates 3, 8 and 100. okay. And so the expected value. 306 00:36:27.173 --> 00:36:36.233 It's this week, add this 0T plus 3, AIDS, plus 6, 8, plus 3 eights and it sums to 12 8. 307 00:36:36.233 --> 00:36:44.003 so which is 3 has X squared is 0, 1, 4 and 9 has the same probabilities. 308 00:36:44.309 --> 00:36:52.349 And so he got the accepted value on the X squared. We're going to solve their problem be 0. 309 00:36:52.349 --> 00:37:05.818 3, 8, 3, H, times 4, just 12, 8, 1, 8 times 9, which is 9 8 and we, some, we get 15 and 924 8, which is 3. 310 00:37:05.818 --> 00:37:10.108 So, I just a good thing. It's the expected value of X. 311 00:37:10.108 --> 00:37:14.099 Was 3 have expected value X squared. 312 00:37:14.099 --> 00:37:21.239 Was 3 in this case of tossing the fair coin 3 times. So the variance. 313 00:37:22.588 --> 00:37:26.518 Will be accepted value that squared. 314 00:37:26.518 --> 00:37:31.378 Minus expected value of X and notice the parentheses squared. 315 00:37:31.378 --> 00:37:35.309 Which is 3 minus 9 quarters. 316 00:37:35.309 --> 00:37:43.798 Which is not 3 quarters if I didn't screw up too badly. 317 00:37:46.228 --> 00:37:49.619 But I could also go back here and now, look at. 318 00:37:49.619 --> 00:37:52.768 If I look at also. 319 00:37:52.768 --> 00:37:58.858 Yeah, I got started another call. Okay before I scroll to a new page to just sort of makes sense here. 320 00:38:00.329 --> 00:38:12.269 Okay, so if I do this again. Okay I mean, I continuing the same thing. Let me do the other formula for variance. So, again, X was 0, 1, 2 or 3. 321 00:38:12.269 --> 00:38:17.760 X minus the mean, the main was 3 halves. 322 00:38:17.760 --> 00:38:22.079 Minus 3 halves minus 1, half us 1 half. 323 00:38:22.079 --> 00:38:27.360 Of 3 halves, X, minus the mean squared. 324 00:38:28.409 --> 00:38:38.760 Was 9 corridors, quarter, 1, quarter 9 quarters the probabilities again were 1. 325 00:38:38.760 --> 00:38:46.380 On 8, 3, 8, 3, eighths and 1 8 you multiply them. You get. 326 00:38:48.960 --> 00:38:52.409 930 seconds. 327 00:38:52.409 --> 00:38:56.550 330 seconds 330 seconds and. 328 00:38:56.550 --> 00:39:06.449 930 seconds and we sum them we get 122430 seconds, which is 2 thirds. So we get the variance. 329 00:39:06.449 --> 00:39:09.510 What did I get last time? 330 00:39:09.510 --> 00:39:13.860 Recorders okay. 331 00:39:13.860 --> 00:39:24.960 I didn't I do wrong here. I added wrong probably 9 plus 3 that's 122430 seconds. 332 00:39:24.960 --> 00:39:30.929 Silence. 333 00:39:30.929 --> 00:39:37.019 It was 3 quarters and they said vulgar fractions are obsolete. 334 00:39:38.190 --> 00:39:41.519 Okay. 335 00:39:42.539 --> 00:39:46.710 Does that make more sense now? Yeah, in that last column there. 336 00:39:46.710 --> 00:40:00.594 That's E, I was just yeah, I wanted to find the mean of this here. The mean of X minus means squared. So these are the, it's possible values. This is the probability of each value. It's a probability of the original value of X here. Okay. 337 00:40:00.594 --> 00:40:03.804 So, I just see what I. 338 00:40:04.619 --> 00:40:08.610 Times. 339 00:40:08.610 --> 00:40:12.809 Okay, and that's the sum down here. 340 00:40:16.920 --> 00:40:21.389 Make more sense? Yes. Okay. Cool. 341 00:40:21.389 --> 00:40:30.539 So, they got variance conditional variance actually. Nothing interesting. The same thing applies. 342 00:40:32.130 --> 00:40:36.000 Here so. 343 00:40:36.000 --> 00:40:40.619 Has to do with an example actually, I think. 344 00:40:40.619 --> 00:40:48.780 And, okay, so we have our box of widgets of the 2 types 1, last longer than the other. 345 00:40:48.780 --> 00:40:52.469 And maybe. 346 00:40:53.639 --> 00:40:59.579 To remind you how you can do the mean and so on and. 347 00:40:59.579 --> 00:41:04.380 I scroll back up to remind you. 348 00:41:04.380 --> 00:41:07.860 Sorry, I'm sorry about that. 1 come back. 349 00:41:09.059 --> 00:41:13.650 Yeah, let's do it. We're doing example 326 here. 350 00:41:13.650 --> 00:41:18.599 Okay, if I scroll back up and so. 351 00:41:18.599 --> 00:41:22.650 So, okay. 352 00:41:27.900 --> 00:41:31.320 Silence. 353 00:41:31.320 --> 00:41:35.820 Okay, minus 1 or it has some. 354 00:41:37.320 --> 00:41:40.619 Or, yes. Okay. Um. 355 00:41:40.619 --> 00:41:46.769 Okay, so again, just to remind you the way, you would get to mean here. 356 00:41:46.769 --> 00:41:50.309 But something retinue where you would get. 357 00:41:50.309 --> 00:41:55.980 Just some of all the probability. 358 00:41:57.630 --> 00:42:02.250 Times K, but you'd be the sum of all the. 359 00:42:04.469 --> 00:42:13.739 Moves come on just a 2nd here. K. 360 00:42:13.739 --> 00:42:18.329 And that would they say would be going over our. 361 00:42:18.329 --> 00:42:25.199 Well, I don't know if I actually worked that out in detail. I could derive that if you want. 362 00:42:25.199 --> 00:42:29.159 I drive part of it so but any case. 363 00:42:29.159 --> 00:42:36.449 So the conditional mean, this is just the main given its device type 1 or divisive type 2 and. 364 00:42:36.449 --> 00:42:42.570 And the unconditional mean you multiply the conditional means by the. 365 00:42:42.570 --> 00:42:49.949 By their weights, and that's what they have there and to working out the variants. 366 00:42:49.949 --> 00:42:54.090 We the same thing you take the conditional variance as. 367 00:42:56.880 --> 00:43:00.420 You know, you'd have the mean here. 368 00:43:03.150 --> 00:43:16.920 The unconditional mean of X squared, or the conditional means of X squared times their weights and you take the unconditional mean of X squared minus the unconditional Square. The unconditional mean and you get the, you get the unconditional variance. 369 00:43:16.920 --> 00:43:21.090 So, okay. 370 00:43:21.090 --> 00:43:24.570 Now. 371 00:43:24.570 --> 00:43:29.369 What we have here page 115. 372 00:43:29.369 --> 00:43:35.789 Might want to note it because this is worth remembering it's a summary of different. 373 00:43:35.789 --> 00:43:46.650 Random variables so I'll put it on the blog. Remember. 374 00:43:47.730 --> 00:43:55.889 I don't remember I said, I'm not forcing you to memorize so much stuff in the scores because. 375 00:43:55.889 --> 00:43:59.219 You know, in the real world, you can look stuff up, but know how to find it. 376 00:43:59.219 --> 00:44:02.760 So just a quick review here we've seen. 377 00:44:03.144 --> 00:44:16.585 For newly calling once binomial tossed the coin and times and random variable is the number of heads. We don't care what order they occurred in geometric again 2, slightly different versions. 378 00:44:19.349 --> 00:44:31.679 And you dig for gold until you find it the number of years you have to dig before you succeed, or the number of years, including the success here or something and. 379 00:44:31.679 --> 00:44:43.860 So, it's slightly different. It just mess things up because you never know which version you are, you generating function. Here. You see, maybe you've seen generating functions and I know there. 380 00:44:43.860 --> 00:44:55.980 Course just ways to derive moments and so I go a little light on them. I can cover them if people are interested in any case. So, these here are 4 random variables. So far we're going to see more. 381 00:44:57.809 --> 00:45:03.659 Okay, here are some new ones uniform down at the bottom. You see. 382 00:45:03.659 --> 00:45:07.139 Here are some, some important new ones. 383 00:45:09.869 --> 00:45:12.869 So, negative binomial. 384 00:45:12.869 --> 00:45:18.420 Okay, so the geometric was, you keep trying until you succeed. 385 00:45:18.420 --> 00:45:25.050 Negative binomial is to keep trying until you got our successes. 386 00:45:25.050 --> 00:45:28.530 So. 387 00:45:28.530 --> 00:45:42.869 Trying to think of a real, real world example. Actually, if anyone can think of 1, you can chime in, but I'll be thinking, as I'm talking the croissant random variable is very important distribution. 388 00:45:42.869 --> 00:45:49.019 And let me give you some real examples of it. 389 00:45:49.019 --> 00:45:53.250 Talk to you about cosmic grace let's say a radioactive decay. 390 00:45:53.250 --> 00:46:00.059 Let's suppose I got a gram of some Radium or something and. 391 00:46:00.059 --> 00:46:09.030 So well, let's assume that on the average, it has 10 decays a 2nd. 392 00:46:09.030 --> 00:46:20.429 And I'm counting, I wouldn't be Graham, but I've got some radioactive substance here and that's suppose on the average there are 10 decays a 2nd. 393 00:46:20.429 --> 00:46:35.010 Now, the atoms all decay independently of each other, they were not in an atomic bomb situation and not stimulating each other. Okay if what Adam decays. It doesn't cause another Adam to D. K. 394 00:46:35.010 --> 00:46:38.940 Okay, a, that's a lesson for another day. Um. 395 00:46:38.940 --> 00:46:43.469 How to build telecoms or something that's outside of course. Okay. 396 00:46:44.849 --> 00:46:51.570 Got to give you a story though effect of errors. 397 00:46:51.570 --> 00:46:55.409 You know, if everyone makes team, see errors, you know. 398 00:46:55.409 --> 00:47:01.650 Mathematicians engineers, physicists making errors. 399 00:47:01.650 --> 00:47:11.579 We're talking 1954 give or take us the physicists where designing an atomic bomb that was going to be tested. 400 00:47:11.579 --> 00:47:16.679 Out in the West Pacific, and it was intended to be 5 Mega tons. 401 00:47:17.820 --> 00:47:24.900 The physicist made any teensy analysis error, they got the physics wrong. 402 00:47:24.900 --> 00:47:32.610 Seeing who hasn't made a narrow designing an atomic bomb. Okay. And the bomb that was intended to be 5 Mega tons. 403 00:47:32.610 --> 00:47:35.909 In fact was 15, Mega tons. 404 00:47:35.909 --> 00:47:39.059 Factor of 3 error. 405 00:47:39.059 --> 00:47:48.059 That turned out to be the biggest tomic test the US ever did. Actually, it's called castle gravel and. 406 00:47:48.059 --> 00:47:55.769 Any errors from not related to the course, but real world engineering I'll show you and. 407 00:47:55.769 --> 00:48:02.369 Just show you for 5 for I come back to. 408 00:48:03.840 --> 00:48:11.070 Yeah. 409 00:48:11.070 --> 00:48:19.500 So, yeah. 410 00:48:19.500 --> 00:48:23.099 2 and a half times the predicted value. 411 00:48:23.099 --> 00:48:27.090 Hey, like I said, errors occur, hey. 412 00:48:28.260 --> 00:48:36.119 That motivated Godzilla by that's the barb that inspired Godzilla. Okay. Back to profitability. 413 00:48:36.119 --> 00:48:49.380 So, crosstalk random variable. Got me is radioactive decay. Let's say. So, let's suppose you got some substance you got a chunk of it and you have an average of 10 decays a 2nd. 414 00:48:49.380 --> 00:48:59.880 Now, so, but the number of decays in each 2nd, is a random variable on 3rd it might be 8 and another 2nd, it might be 13 and another. 2nd, it might be 11, whatever. 415 00:48:59.880 --> 00:49:04.170 So, what's the probability distribution of that? Random variable. 416 00:49:04.170 --> 00:49:10.559 That's possible and it is when you have a very large number of atoms, like. 417 00:49:10.559 --> 00:49:16.139 Perhaps in this chunk, you have a mole of Adams that'd be 6 times 10 to the 23rd. 418 00:49:16.139 --> 00:49:19.590 And maybe the probability of any given 1. 419 00:49:19.590 --> 00:49:25.769 Decaying in the next 2nd might be 6 times 10 minus. 420 00:49:28.704 --> 00:49:41.815 Whatever 1020, whatever so it's a very large number of things might happen but each particular thing is very unlikely so that the expected mean number of them happening is a small. 421 00:49:42.599 --> 00:49:50.489 Natural number where do i50 K you got a lot of Adams and each Adam's unlikely but. 422 00:49:50.489 --> 00:50:03.570 You multiply the number of atoms probably if any 1 atom you get 10 per. 2nd, let's say so that's a brand. That's a croissant random variable. So, in some 2nd, you might have 0T you might have won. You might have to whatever. 423 00:50:03.570 --> 00:50:09.119 And right there on the page, we have the formula for it right there. 424 00:50:09.119 --> 00:50:18.389 The mean is a, it's alpha, the probability of K occurring in the 2nd is alpha to the K over K factorial time seated the minus alpha. 425 00:50:18.389 --> 00:50:22.110 I'll give you other examples for it. 426 00:50:22.110 --> 00:50:31.170 Let's suppose you take the 86 field behind the engineering center if you're ever allowed to be on campus again and. 427 00:50:31.170 --> 00:50:39.449 You divided into 100 square, say, 10 by 10 squares then you all go up on to the roof of the engineering center and tossed paper airplanes. 428 00:50:39.449 --> 00:50:47.070 On to the field so let's suppose that you toss a 1000 paper airplanes on to the field that got divided into 100 squares. 429 00:50:47.070 --> 00:50:51.059 Okay, let me draw that here. What's on? 430 00:50:55.230 --> 00:50:58.409 So. 431 00:50:58.409 --> 00:51:02.940 That's 11st of all value of 10. 432 00:51:02.940 --> 00:51:16.679 Okay, if airplanes, okay, they're all independent of each other. Each paper airplane has an equal probability of landing in any Square. 433 00:51:16.679 --> 00:51:19.679 So, what's the probability that a given Square. 434 00:51:19.679 --> 00:51:27.780 As a paper airplane, so let me write that out a little for you sounds and important distribution. You got to understand it. 435 00:51:27.780 --> 00:51:34.469 Okay, come on go. That's better. Okay, so what we have, um. 436 00:51:38.670 --> 00:51:42.000 Silence. 437 00:51:53.010 --> 00:51:57.300 So alpha equals, say. 438 00:51:57.300 --> 00:52:00.480 10, and that's the number. 439 00:52:01.949 --> 00:52:05.309 So, you can see the airplanes question. 440 00:52:05.309 --> 00:52:11.820 I can't, I mean, I can't see that. I know it's just my screen sorry there. Okay. There we go. Yeah. 441 00:52:14.099 --> 00:52:17.820 So redraw the figure here. 442 00:52:19.050 --> 00:52:22.349 Paper. 443 00:52:22.349 --> 00:52:33.449 Okay, so we talked a 1000 airplanes out of the field had 100 squares. The alpha is the main number of paper airplanes per square and our random variable. 444 00:52:33.449 --> 00:52:37.889 X equals the number airplanes. 445 00:52:37.889 --> 00:52:43.769 And say this square here. 446 00:52:50.969 --> 00:52:57.420 Okay, okay so X is distributed. 447 00:52:59.940 --> 00:53:03.750 Okay. 448 00:53:05.280 --> 00:53:08.519 Give you a 3rd example. 449 00:53:08.519 --> 00:53:13.710 Is service like, let's suppose. 450 00:53:13.710 --> 00:53:22.679 So, like I said, this, all these assumptions that assumes the paper airplanes are independent of each other, et cetera and saturated uniform. 451 00:53:22.679 --> 00:53:30.119 Your service you've got a server and you've got customers coming in randomly. 452 00:53:30.119 --> 00:53:34.170 So, maybe they are calls for service. 453 00:53:34.170 --> 00:53:43.829 You know, connecting to a switch people trying to make a call on their cell phone or something. And let's suppose that an average of 10 people a 2nd. 454 00:53:43.829 --> 00:53:47.340 Are trying to place a phone call and. 455 00:53:47.815 --> 00:53:55.375 But it's a very large. Maybe there's Troy has 100000 people and of them 10 per 2nd are trying to place a phone call. 456 00:53:55.375 --> 00:54:08.755 So that probably, if any 1 person placing a phone call with 1 out of 10000, I mean, Troy say, register county Troy is like, 20000 people who are 50. okay. So that's pulse on distributed. So, if the potential customers are independent of each other. 457 00:54:11.039 --> 00:54:19.710 And everything is uniform, then the number of customers trying to place a call and the next 2nd is a croissant variable. 458 00:54:19.710 --> 00:54:24.300 So, the distribution pops up all the time. 459 00:54:24.300 --> 00:54:28.619 So, let me type to type this other ones. So. 460 00:54:36.869 --> 00:54:42.360 Customers trying to call was on. 461 00:54:43.710 --> 00:54:49.949 If that they're independent. 462 00:54:52.139 --> 00:54:55.530 And you in a form. 463 00:54:55.530 --> 00:55:08.070 Uniform means a number of customers trying to call is like the same and independent isn't not affecting each other. If that's the case. Then the number of customers calling in a 2nd. 464 00:55:08.070 --> 00:55:13.380 Is on, or like, the number of airplanes hitting a square on the field. 465 00:55:13.380 --> 00:55:25.019 Is possible, or the number of at radioactive Adams that became the next 2nd, is croissant the number of cosmic rays that are going to hit your chip in the next. 2nd. That's croissant. 466 00:55:25.019 --> 00:55:31.619 So, whenever you have a large number of independent things that might happen. 467 00:55:31.619 --> 00:55:39.869 The probability of any 1 of them happening is small, but the but the product the expected number happening is some small natural number. 468 00:55:39.869 --> 00:55:43.679 It's not not solid and then it's a distribution. 469 00:55:43.679 --> 00:55:47.460 So, and so this is an important random variable. 470 00:55:48.719 --> 00:55:51.989 And the formula there in the book. 471 00:55:51.989 --> 00:55:56.550 I'm not going to attempt to derive it, but you can find the. 472 00:55:57.630 --> 00:56:05.369 You can find the expected derives the expected value for fun. Oops. Sorry. 473 00:56:05.369 --> 00:56:10.559 Go I'd say, so. 474 00:56:10.559 --> 00:56:13.739 So so plus on is on. 475 00:56:13.739 --> 00:56:18.300 Notation is not standard in the book drives me off the wall. 476 00:56:18.300 --> 00:56:24.599 Sometimes a subscript is the name of the random variable. Sometimes it's the parameter alpha to the K. 477 00:56:24.599 --> 00:56:35.099 hey factory well the first thing if you sum up pete is a k k equals zero to infinity we can't see that yes . 478 00:56:38.005 --> 00:56:50.364 There okay. Yeah the issue is that when I scroll it part way on the iPad for summaries and the screen mirroring program doesn't scroll the screen mirror program doesn't mirror the screen. 479 00:56:50.699 --> 00:56:56.219 All the time. Exactly. That's the okay so I had this is the. 480 00:56:56.219 --> 00:57:04.019 Probably mass function for the facade here and then if you sum it all up, you can either minus alpha. 481 00:57:04.019 --> 00:57:08.429 Some of all the help to the camera, in fact, Charles here. 482 00:57:08.429 --> 00:57:12.090 And just thing here, of course, is. 483 00:57:14.730 --> 00:57:18.210 I say equals 1. okay. 484 00:57:18.210 --> 00:57:22.829 And if you wanted to get say, the mean. 485 00:57:24.570 --> 00:57:30.119 To 2nd here so the expected value of X. 486 00:57:30.119 --> 00:57:38.039 Being a name and the random variable. So that would also that would be the sum of all the K time. Speed is a K. 487 00:57:38.039 --> 00:57:42.239 And that you could. 488 00:57:46.739 --> 00:57:50.460 Silence. 489 00:57:50.460 --> 00:58:01.440 So, how could we, how could we do that? Well, you pull the minus alpha. It's a constant of course. 490 00:58:02.730 --> 00:58:05.969 Whoops. Times K. okay. 491 00:58:07.139 --> 00:58:13.650 Well, what can we do to that? The minus alpha it's just a distraction. 492 00:58:17.699 --> 00:58:21.989 Okay, so we got K over K factorial. 493 00:58:21.989 --> 00:58:30.719 And, of course, you know, that's 1 over K minus 1. factorial. Okay. So this equals. 494 00:58:32.369 --> 00:58:36.539 Some of all the alpha to the K or. 495 00:58:36.539 --> 00:58:50.280 This 1 factorial. Okay so I got to go to the next page because again, because I can't show you half and half your pages. So I'll leave this up for a sec. 496 00:58:53.429 --> 00:58:57.269 Okay, so this is going to be. 497 00:58:57.269 --> 00:59:05.010 I'll rewrite it, so that was the minus alpha. Some of all the alpha to the K minus 1. factorial. 498 00:59:05.010 --> 00:59:11.099 Well, actually what I can do, if I scroll back. 499 00:59:14.699 --> 00:59:21.750 Let me scroll back a 2nd, go back, come on. Go. 500 00:59:24.900 --> 00:59:38.699 That has better. Okay what I wanted to do here when cake was 0T. This whole thing is 0T up here. So, what I can actually do is change that to K equals 1 because the cake was 0 0T. Okay. 501 00:59:38.699 --> 00:59:41.849 Does that make that cables? 1? 502 00:59:43.050 --> 00:59:50.309 And that would be 1. okay. And so now, what I do is that I can. 503 00:59:53.579 --> 01:00:01.739 on an alpha alpha ear the minus alpha some of all k equals one to infinity off it is a k minus one over . 504 01:00:03.630 --> 01:00:07.889 Now, I will change my variables instead of. 505 01:00:07.889 --> 01:00:14.219 For 1, I can just change, I can offset K by 1 or what I can do it here is say, let. 506 01:00:14.219 --> 01:00:24.420 say i don't know l equals k minus one so now that thing equals alpha either minus alpha some of the l equals zero to infinity . 507 01:00:24.420 --> 01:00:28.110 Alpha to the L over factorial. 508 01:00:29.880 --> 01:00:35.460 And just saying is, um. 509 01:00:37.079 --> 01:00:42.389 You to Z saying this, this thing here is you to the alpha actually. 510 01:00:42.389 --> 01:00:48.929 And it goes to alpha, so that was the mean here. 511 01:00:48.929 --> 01:00:57.750 That's how you can find the meat of the posts all random variable and variance be similar thing. Okay. 512 01:00:57.750 --> 01:01:02.820 Uniform is the obvious thing went up to L. 513 01:01:02.820 --> 01:01:10.050 Mean and variance work it out if anyone's interested. Okay. Zip. 514 01:01:13.800 --> 01:01:23.489 Zip f*** in cases, like, you're looking at cities in a country and I know the distribution of words, let's say. 515 01:01:23.489 --> 01:01:33.059 An English, and maybe the frequency of the Kate's most common word or something might be as if distribution. 516 01:01:33.059 --> 01:01:43.079 I personally don't like, does if distribution I think it's southern overall. I think it sort of coincidental that it's not so significant, but. 517 01:01:44.369 --> 01:01:54.269 You see it from time to time. Okay so what we had here on this page was lots of useful, random variables. There's more but this is a SAP. 518 01:01:54.269 --> 01:02:02.280 Okay, so this is a rehashing, but normally that starts the coin. Once we already worked out. 519 01:02:02.280 --> 01:02:07.860 The mean, the, and they can compute the variance. 520 01:02:07.860 --> 01:02:15.480 Now, if we plot, so the variances the spread. Okay. 521 01:02:15.480 --> 01:02:21.480 In the random variables. So let's suppose a coin is really unfair. 522 01:02:21.480 --> 01:02:25.800 99 times its heads in 1 time out of a 100 its tails. 523 01:02:26.574 --> 01:02:37.614 So that's not much of a spread. So the variance is quite small. If the coin is, if we pretty certain what the client's going to be, the variance is greater than we don't know anything when it's a fair coin. So that's what this is doing. 524 01:02:38.545 --> 01:02:41.635 Binomial again toss the coin and times. 525 01:02:41.880 --> 01:02:47.400 Out the number of ads, we don't care what order that's our distribution down here. 526 01:02:47.400 --> 01:02:56.190 And this is what the thing looks like for different values of PE, let's say, 20%. 527 01:02:56.190 --> 01:03:01.440 Empire has a few. Okay. And you can. 528 01:03:01.440 --> 01:03:07.920 Okay, so there's a question here. Let me go back to what the formula is what you've seen before. 529 01:03:07.920 --> 01:03:18.719 You got this and choose K and minus K. Victoria. So how do you evaluate that? Because factorial scroll really fast. 530 01:03:18.719 --> 01:03:23.820 What you do is you call a function in your favorite programming language and evaluated for you. 531 01:03:23.820 --> 01:03:30.719 Mail light solution is other ways to evaluate it. They talk about. 532 01:03:31.769 --> 01:03:40.014 Overflow problems are, because there's limits to the biggest possible flow to single precision floats on a machine. 533 01:03:40.014 --> 01:03:49.614 The maximum flow size is 10 to the 38 double precision floats are better, but still, you're doing factorial for a large and they grow so fast. 534 01:03:50.760 --> 01:04:03.630 What they talk about various ways, you can do it interactively and so on the right we just actually did that. I think meat of a random binomial, random variable and. 535 01:04:05.159 --> 01:04:10.829 Me know, then you can get very insulted by me in the X squared. 536 01:04:10.829 --> 01:04:14.820 Okay. 537 01:04:16.260 --> 01:04:22.800 Look at this example. 538 01:04:22.800 --> 01:04:26.250 Here, um. 539 01:04:26.250 --> 01:04:32.099 Oh, maybe I should work out. I'm parents of I know me a random variable. 540 01:04:35.159 --> 01:04:39.480 And. 541 01:04:39.480 --> 01:04:43.110 Sure, let's see how we can work it out. 542 01:04:44.489 --> 01:04:50.190 Silence. 543 01:04:50.190 --> 01:04:53.969 Silence. 544 01:04:53.969 --> 01:04:58.980 Okay. 545 01:04:58.980 --> 01:05:02.039 Well, I'll do the mean 1st, actually. 546 01:05:03.119 --> 01:05:06.599 Okay, so again, so. 547 01:05:06.599 --> 01:05:10.409 Called K. K. and choose K. 548 01:05:16.500 --> 01:05:21.150 Okay, so the mean of acts also written as that. 549 01:05:22.530 --> 01:05:26.070 Some of all the K times and choose K. 550 01:05:29.820 --> 01:05:33.659 Okay, so how do we do that? 551 01:05:35.789 --> 01:05:41.940 Well, we can take this thing here and that is K. 552 01:05:41.940 --> 01:05:52.710 In factorial over, which is we cancel out the case and pull out and then. 553 01:06:03.900 --> 01:06:07.289 Okay. 554 01:06:07.289 --> 01:06:11.429 So that then is and minus 1 choose. 555 01:06:12.869 --> 01:06:18.360 So, that thing there is. 556 01:06:20.670 --> 01:06:24.809 Then times and minus 1 choose K minus 1. 557 01:06:27.000 --> 01:06:32.969 Okay, so, um, so the mean is, um. 558 01:06:35.489 --> 01:06:38.519 So, the sum of all the. 559 01:06:39.869 --> 01:06:46.889 Silence. 560 01:06:48.630 --> 01:06:52.800 Okay, what can we do here? 561 01:06:52.800 --> 01:06:58.079 We're going to take that. 562 01:06:58.079 --> 01:07:01.349 It out, we're going to take 1 of the keys. 563 01:07:01.349 --> 01:07:09.630 And pull it out and we're going to net out as end times P, times the sum of all the and minus 1 K minus 1. 564 01:07:09.630 --> 01:07:13.320 8 K minus 1. 565 01:07:15.989 --> 01:07:20.010 And, um. 566 01:07:21.960 --> 01:07:28.739 Sort of thing here it goes 1 side equals in. 567 01:07:28.739 --> 01:07:35.369 I'm I dropped a detail or 2 in the but that's the general idea. 568 01:07:35.369 --> 01:07:47.010 Okay, that's how we find the expected value. The variance is the same sort of concept. 569 01:07:47.010 --> 01:07:54.570 So, yeah, maybe I'll just leave it at that. If people want. 570 01:07:54.570 --> 01:07:58.559 I'll do the variance, so maybe just means and off. 571 01:07:58.559 --> 01:08:07.380 Okay, okay so the example of redundant systems. 572 01:08:07.380 --> 01:08:16.500 Um, so we're assuming here here is. 573 01:08:18.810 --> 01:08:30.324 Actually, we're seeing a new random variable, continuous 1 called an exponential distribution. Sorry? It doesn't much matter, but it could be an integer. He could be real number. It actually doesn't matter. 574 01:08:30.654 --> 01:08:34.135 Probably the microprocessor is still alive after 2 seconds. Is. 575 01:08:34.380 --> 01:08:42.270 Either a minus lamb to T, for some constant lamb depth and the bigger Lambda is the faster the device dies. 576 01:08:42.270 --> 01:08:48.239 So, it shows interesting technique here. 577 01:08:48.239 --> 01:08:54.510 It's we've got triple redundancy and our. 578 01:08:54.510 --> 01:09:00.810 Machine runs as long as, at least 1 of the 3 processors is running. 579 01:09:00.810 --> 01:09:13.829 So, and we want the probability there. Well, the way we can do is we can flip it. So, the probability that, at least 1 processor is operating is 1 minus the probability that they're all dead. 580 01:09:13.829 --> 01:09:25.829 This just makes it a touchy easier this way, because at least 1, we'd have to find the probability of exactly. 1 process probably. Exactly to probably exactly. 3 we could do that. But in this case, it's easier just to find the problem is 0T and. 581 01:09:25.829 --> 01:09:31.319 Subtracted from 1 Central, just showing you here an example. See, 29. so. 582 01:09:31.319 --> 01:09:37.890 So, we have here is the probability that. 583 01:09:37.890 --> 01:09:47.909 That a device that 1 processor is still alive at time. T, this is the probability that that specific processor is dead. 584 01:09:47.909 --> 01:09:53.760 At time T, this cube is the probability of all 3. 585 01:09:53.760 --> 01:09:57.420 Are dead at time T, assuming they're independent. 586 01:09:57.420 --> 01:10:01.949 Oh, it's in the real world you got to wonder about, but. 587 01:10:01.949 --> 01:10:08.729 And why would they be dependent that they die? Well, maybe they're killed by a voltage spike. 588 01:10:08.729 --> 01:10:12.329 Okay, so you get a, you get a lightning hits the power. 589 01:10:12.329 --> 01:10:19.020 And all 3 of them get sapped. Okay. So this is a case where they could be dependent, but ignoring that. 590 01:10:19.555 --> 01:10:20.604 So again, 591 01:10:20.664 --> 01:10:30.055 either the minus time to teach the probability that a specific processor is still alive at time T1 minus is probably that specific processor is dead at, 592 01:10:30.145 --> 01:10:37.255 by time T1 minus that cube is the probability that all 3 processors are dead at time. 593 01:10:37.255 --> 01:10:45.774 T. and 1 minus, that is a probability that it's not true that all 3 are data time, which is probably that at least 1. 594 01:10:46.979 --> 01:10:51.840 1 or more are alive. So, this is here is the probability that. 595 01:10:51.840 --> 01:10:58.800 Your your device is still operating, but you said, that's the reason for showing this is we. 596 01:10:58.800 --> 01:11:01.800 Look at the reverse thing. Okay. 597 01:11:01.800 --> 01:11:07.350 Geometric, I've talked about before probability of. 598 01:11:07.350 --> 01:11:11.279 A coin until the 1st head. Okay. 599 01:11:12.479 --> 01:11:21.090 So, the 1st head occurs exactly on the case toss which mean you had K minus 1 failures plus a success. 600 01:11:21.090 --> 01:11:24.300 Okay, assuming they're independent. 601 01:11:25.409 --> 01:11:30.720 Why do I keep saying assuming it's independent until you're sick of it because. 602 01:11:30.720 --> 01:11:37.439 Let's take this course into the real world era at here. You guys are smart and hard working. 603 01:11:37.439 --> 01:11:44.819 You'll get the math, you'll learn the math. I'll give you a sample going up in this. Course you can pick up other math in the future. 604 01:11:44.819 --> 01:11:47.819 You can learn to mass, you need to solve the problem. 605 01:11:47.819 --> 01:12:02.460 Given that you got the math, the hard part is, which math do you use to solve the problem and you get disasters and stuff in society financial failures. 606 01:12:02.460 --> 01:12:07.020 Bridge collapses whatever when the wrong math is use. 607 01:12:07.020 --> 01:12:20.154 So, people assume things are independent when they're not or something I'll give you a real word economic example. So, about a dozen years ago, there was a housing crash across the United States housing prices all over. 608 01:12:20.154 --> 01:12:26.904 The United States went up up up and then down to down down they crashed all over the country and so many. 609 01:12:28.380 --> 01:12:40.375 Housing house prices crashed, because they're all financed on these liars loan mortgages. Elias loan is where the board didn't have to give any documentation he just stated something in the bank accepted it. 610 01:12:40.765 --> 01:12:46.375 So all across the country, then these things and they took banks, it took some good sized banks down with them. 611 01:12:46.680 --> 01:12:53.550 Now, now the mass of mortgages and sony's easy to load. What happen is. 612 01:12:53.550 --> 01:12:56.880 Well, I just caught people by surprise, including some prominent. 613 01:12:57.295 --> 01:13:12.114 Economists is, they had thought that the housing prices housing markets across the country were independent of their assumption is that if house prices crashed in San Francisco, they would not wouldn't mean that they'd crash in Miami or Seattle or Boston. 614 01:13:12.414 --> 01:13:17.904 They assume that these are all independent markets off each other in the house prices in each separate city were independent. 615 01:13:19.350 --> 01:13:24.810 And therefore, when it calculated the risk to the economy is something bad happening, it was very small. 616 01:13:24.810 --> 01:13:29.039 Because they San Francisco prices might crash, but not Miami. 617 01:13:29.039 --> 01:13:35.159 In fact, what happened is they all went up together because of low interest rates. They were in fact. 618 01:13:35.159 --> 01:13:43.289 Correlated with each other, so they all went up together and they all went down together, depending on stuff like availability of money and interest rates. 619 01:13:43.289 --> 01:13:56.579 In fact, so, Alan Greenspan, who is 1 of the lead Congress. He admitted this publicly. He said that he said he didn't foresee this because he had thought that the housing markets in different cities are independent and he admitted he was wrong. 620 01:13:56.579 --> 01:14:01.260 Good takes a big person to admit publicly that you're wrong in such a big thing. 621 01:14:01.260 --> 01:14:10.710 So so this is the case where the math was easy, but it was the assumptions that were wrong and it's cost trillions of dollars, though. 622 01:14:10.710 --> 01:14:13.829 To drag down the economy was it for years so. 623 01:14:13.829 --> 01:14:27.840 Because these random variables house price in different cities were not independent they were dependent on each other. That's why I keep emphasizing the stuff like that. The assumptions underlying the equations any case back to this. 624 01:14:28.920 --> 01:14:34.020 Geometry and you can calculate the mean and the variance and so on. 625 01:14:34.020 --> 01:14:42.779 Okay, now I've mentioned memory listeners before if you're tossing a coin until you. 626 01:14:42.779 --> 01:14:47.399 Get the 1st head how many failures you've had does not affect the future? 627 01:14:47.399 --> 01:14:52.739 So, these people that say, well, you do for a run of success, because you failed for so long. 628 01:14:52.739 --> 01:14:58.949 Well, not with the geometric distribution. That's not true. So. 629 01:14:58.949 --> 01:15:06.899 So, just because you've talked to coin 20 times, and it tells 20 times, does not affect what will happen in the 21st time. 630 01:15:06.899 --> 01:15:10.319 If there is the assumption that. 631 01:15:10.319 --> 01:15:14.880 They're independent, so if it's a. 632 01:15:14.880 --> 01:15:19.170 Take coin maybe it comes up tales. 633 01:15:19.170 --> 01:15:28.289 2 thirds of the time instead of half the time. That's a different matter. Okay. Such geometric facade. I've mentioned. 634 01:15:28.289 --> 01:15:33.420 Hey, it's so important, but so this is the on Garcia. 635 01:15:33.420 --> 01:15:44.520 Event radioactive tell him he mentioned all 3 of the things. I mentioned radioactive decay number of atoms that you can the next 2nd, because the Adams don't trigger each other. 636 01:15:44.520 --> 01:15:48.359 I'm not talking you 235. 637 01:15:49.409 --> 01:15:59.130 And telephone connections, defects, and a semiconductor chip if they're caused by cosmic razor some, that's the formula there. Okay. Now. 638 01:15:59.130 --> 01:16:06.659 the post random variable you can't really do sums and so on by hand except something from zero to infinity so that's . 639 01:16:06.659 --> 01:16:11.850 At the point where I can't be summing for some range of it or something, but. 640 01:16:11.850 --> 01:16:19.409 Okay, this is what it looks like. So, alpha, the expected number of. 641 01:16:19.409 --> 01:16:22.829 It's, it's a real number. It's a fraction so. 642 01:16:22.829 --> 01:16:36.324 So this means alpha, so yeah, and the average takes a paper airplane example, the average square there's 3 quarters of a paper airplane. Okay that doesn't mean that the airplane splits into 2 pieces and mid air and 3 quarters of it lines in the square. 643 01:16:36.444 --> 01:16:39.534 That's the expected value on the expected is There'll be 3 quarters. 644 01:16:39.899 --> 01:16:43.140 Okay or 3 and this is this nice. 645 01:16:43.140 --> 01:16:46.590 You know, balance shaped curve and so on. 646 01:16:49.050 --> 01:16:56.189 Um, for those of you that know about normal distributions as alpha gets big, this starts looking like a normal distribution. 647 01:16:57.210 --> 01:17:04.350 Okay, this is what this is a plus on the call center. Queries thing I mentioned. 648 01:17:05.789 --> 01:17:09.600 Again, so chapel 330. 649 01:17:10.829 --> 01:17:16.800 And so on the average, we have. 650 01:17:16.800 --> 01:17:24.000 They make a little more complicated than they have to bring in T. I would factor out to you if I were writing down the example, but save alpha. 651 01:17:24.000 --> 01:17:28.409 Lamb to tea queries on the average for a 2nd. 652 01:17:28.409 --> 01:17:37.050 And what's the actual number now? Why do you care? Well, it's how much hardware you have to provide so everything can be handled or. 653 01:17:37.050 --> 01:17:42.239 You know, you go to a supermarket to get your week's food. 654 01:17:42.239 --> 01:17:49.739 And how many cashiers or how many of the self serve machines should there be? 655 01:17:49.739 --> 01:17:57.479 And again, because the number of customers wanting to use it as a route is pass on distributed. So now you can calculate the probability that. 656 01:17:57.479 --> 01:18:01.770 You don't have enough machines in this manner, though queries. 657 01:18:01.770 --> 01:18:15.840 Arrivals at a packet multiplex or same sort of thing if the possible packets are again independent of each other, then the number that arrive at the multiplex or in the next 2nd is a random variable. 658 01:18:15.840 --> 01:18:26.460 So, now what they're talking about here is a little slightly it's a compliment to plus on so. 659 01:18:26.460 --> 01:18:30.270 Here, the question is, is how long until. 660 01:18:30.270 --> 01:18:36.810 Something happened, so if I'm thinking of my. 661 01:18:38.340 --> 01:18:44.579 Radioactive decay so maybe an average a. 1 D. K. A. 2nd. So. 662 01:18:44.579 --> 01:18:48.149 The time until the next the case also random variables. So. 663 01:18:48.149 --> 01:19:01.614 What's the random variable for the time to the next decay or if we got the telephone system the time until the next call comes in we have the package system the time until the next packet arrives. That's also a random variable. 664 01:19:01.765 --> 01:19:04.914 That's a continuous random variable. We'll see soon it's called exponential. 665 01:19:05.159 --> 01:19:15.180 As a compliment to the croissant so the is expected number of packets in the next. 2nd, the exponential is the time until the next packet. 666 01:19:15.180 --> 01:19:18.930 They go together, so that's what's happening here. 667 01:19:21.810 --> 01:19:27.510 Also, renewal trials is then gets big starts looking like, crosstalk. 668 01:19:27.510 --> 01:19:31.289 So these things start looking. Okay. 669 01:19:31.289 --> 01:19:34.380 They talk about it here. Um. 670 01:19:34.380 --> 01:19:39.960 Errors same sort of thing if. 671 01:19:39.960 --> 01:19:49.020 We assume errors are happening at touching maybe an average of an error or a 2nd, and there's a lot of bits 1B bits. 672 01:19:49.020 --> 01:19:53.729 An average 1 and a 1B spent will go bad. That's an average of a bit of 2nd. 673 01:19:53.729 --> 01:20:07.350 But, in fact, how many what's the probability that say more than 5 minutes went bad? You just pass on distribution you could use you could use binomial here, but by nobody within equals, tend to the 9. 674 01:20:07.350 --> 01:20:16.979 And not auto, evaluate the nice factorial. Well, you use the product is techniques but the point is. 675 01:20:16.979 --> 01:20:21.899 Once you admit you're going to use techniques 1 could technique is a, is a distribution. 676 01:20:21.899 --> 01:20:27.300 So, and this would be here, so an average of 1 bit. 677 01:20:28.319 --> 01:20:31.680 1 error occurring in a 2nd, is the probability. There are these 5. 678 01:20:33.449 --> 01:20:38.069 So talking about large models, lots of events, but they're rare. 679 01:20:38.069 --> 01:20:44.579 So, the expected number of events is some small natural number falls to vintage or so. 680 01:20:44.579 --> 01:20:51.359 That's what's happening here. Okay. And you can browse through this, yourself in a form. 681 01:20:51.359 --> 01:20:54.989 Not worth talking about, Ah, it's so simple. 682 01:20:54.989 --> 01:20:58.949 And if I'm going to skip, so. 683 01:20:58.949 --> 01:21:03.060 Okay. 684 01:21:03.060 --> 01:21:09.720 And I may have mentioned, do 334 and so on. 685 01:21:09.984 --> 01:21:22.555 On Monday, so basically, what we did today is we extended discrete, random variables that we saw some common ones did some algebra computing. Some means and variances. 686 01:21:22.885 --> 01:21:26.755 We also saw some conditional means conditional variances. 687 01:21:27.000 --> 01:21:36.000 And obvious extension it means and variances. So, the variance is the spread away from the mean. 688 01:21:36.000 --> 01:21:46.439 Of the thing, and so we're continuing on what we'll see more in this chapter or more discreet, random variables. I'm not going to do every. 689 01:21:46.439 --> 01:21:50.399 Probability distribution in the book what I. 690 01:21:50.399 --> 01:21:55.770 Aim to do is give you a sampling of the important ones. And so that. 691 01:21:55.770 --> 01:21:59.399 If you have to learn another 1, you can learn it. 692 01:21:59.399 --> 01:22:04.739 Any case, it's, you know. 693 01:22:04.739 --> 01:22:08.220 Nice weather out there, so if you're allowed to go outside. 694 01:22:08.220 --> 01:22:18.090 Go outside, get a little share exercise and I'll see you Monday and meanwhile, how are my solar panels doing? 695 01:22:18.090 --> 01:22:21.689 What's getting later in the day but. 696 01:22:21.689 --> 01:22:26.430 Come on. 697 01:22:26.430 --> 01:22:34.020 At see, so far today they've generated a 12 kilowatt hours more. 698 01:22:34.020 --> 01:22:41.189 Then I've used going into the evening when I'm using electricity and there's no, but so far this week. 699 01:22:41.189 --> 01:22:51.449 It's generated 10 kilowatt out week, going back, starting midnight, Monday morning. It's generated 10 kilowatt hours more than I've used. This is 1st week in March. 700 01:22:51.449 --> 01:22:58.229 Oh, okay. I'll hang on for a minute. Herself has any questions other than that. 701 01:22:58.229 --> 01:23:03.359 See, you Monday, so we did up. 702 01:23:03.359 --> 01:23:09.060 Get up to page 126. I'll put that on the blog so. 703 01:23:10.109 --> 01:23:14.729 I said, I have a question about that. We'll tell the people what's on the screen right now. 704 01:23:14.729 --> 01:23:22.710 Okay, how does that? How does that summation? Equal 1 here. Okay legal 1 because they're away from my hands. 705 01:23:22.710 --> 01:23:27.510 Um. 706 01:23:28.800 --> 01:23:32.039 Come on scroll. Okay. 707 01:23:34.350 --> 01:23:37.619 Okay, I'll write it down again here. 708 01:23:54.899 --> 01:23:58.350 Well, that's, um. 709 01:24:10.529 --> 01:24:15.390 And so it's advice so you suddenly. 710 01:24:15.390 --> 01:24:18.630 And. 711 01:24:18.630 --> 01:24:23.640 So, you might call that assaulting P and minus 1 K minus 1 or something. 712 01:24:24.899 --> 01:24:28.380 And you sum that you're going to get 1 because okay. 713 01:24:28.380 --> 01:24:36.449 Now, you got to check the edge conditions with K equals 0T, etc. Etc. I was. 714 01:24:36.449 --> 01:24:39.569 Skipping them, but they actually do work out so. 715 01:24:39.569 --> 01:24:45.029 Does that make sense? 716 01:24:45.029 --> 01:24:48.630 What's the, what's the P and minus 1? 7 minus 1? 717 01:24:48.630 --> 01:24:52.439 Oh, this is a this is a. 718 01:24:56.939 --> 01:25:02.100 Silence. 719 01:25:02.100 --> 01:25:05.189 Oh, so the all probability has to be cool. 720 01:25:05.189 --> 01:25:09.090 for and minus one point yeah . 721 01:25:09.090 --> 01:25:12.479 Okay, okay, let me see. 722 01:25:12.479 --> 01:25:16.619 Thank you sure anyone else. 723 01:25:22.350 --> 01:25:27.659 Well, feel free to hang up. I'll stay around then for a couple of minutes, or, as long as there's still. 724 01:25:27.659 --> 01:25:33.119 You know, a number of people still signed in just in case you still think of a question, but feel free hang up and. 725 01:25:33.119 --> 01:25:36.180 Cook soccer. 726 01:25:50.699 --> 01:25:54.449 Silence. 727 01:25:56.939 --> 01:26:02.760 Okay, okay, well, the time to the home bye see you Monday. 728 01:26:12.869 --> 01:26:15.869 Silence. 729 01:26:17.460 --> 01:26:38.579 Silence. 730 01:26:42.149 --> 01:26:46.260 Silence. 731 01:26:50.250 --> 01:26:55.529 Silence. 732 01:27:06.539 --> 01:27:15.029 Silence. 733 01:27:18.390 --> 01:27:26.220 Silence.