WEBVTT 1 00:31:04.439 --> 00:31:07.499 Sorry about that, um, unmuted now. 2 00:31:09.419 --> 00:31:16.858 Okay, well, okay. 3 00:31:16.858 --> 00:31:22.288 So, I has expected value of X greater or equal to a time. So. 4 00:31:22.288 --> 00:31:25.919 Actually, the probability that access. 5 00:31:25.919 --> 00:31:33.239 Very good. Hey, so probably the random credit equal to a is less article to expected value. That's. 6 00:31:33.239 --> 00:31:37.858 And this was called the, um. 7 00:31:41.249 --> 00:31:47.519 Cough and equality. Okay. 8 00:31:51.028 --> 00:31:58.979 Yeah, okay I'll leave the chat window open just in case for the future. Okay the 1st 1. 9 00:31:58.979 --> 00:32:06.929 Now, maybe we know some more information, like, maybe we also know the standard deviation if we know if we have more. 10 00:32:06.929 --> 00:32:10.378 We can compute more so, um. 11 00:32:26.969 --> 00:32:32.788 So, what if we also know Sigma to standard deviation. 12 00:32:33.808 --> 00:32:39.598 And, um, now in this case, Alex can be negative. So. 13 00:32:40.679 --> 00:32:47.398 And be negative so now, this is the champion chef any quality. 14 00:32:56.969 --> 00:33:00.868 So, this Express is the probability that X could be all. 15 00:33:00.868 --> 00:33:04.528 From the main in both ways either way. Um. 16 00:33:09.778 --> 00:33:14.848 So, the probability that it's more than a away from the mean on either side. 17 00:33:18.209 --> 00:33:27.659 Squared is obviously not negative. Positive. Okay. So what this is saying is that if the Sigma is smaller. 18 00:33:27.659 --> 00:33:31.199 Then the probability of being in a way, just entail. 19 00:33:31.199 --> 00:33:38.848 Is, um, larger is smaller. Okay. So how would we prove that? Um. 20 00:33:47.699 --> 00:33:54.568 Okay, well. 21 00:33:58.439 --> 00:34:02.608 Let's try something, um. 22 00:34:04.078 --> 00:34:07.769 D equals X minus mean. 23 00:34:09.268 --> 00:34:14.699 So so the probability. 24 00:34:16.768 --> 00:34:21.028 And X minus mean, value lesson. 25 00:34:21.028 --> 00:34:24.599 Hey. 26 00:34:28.889 --> 00:34:35.878 A, it was the probability that T squared where it okay. Squared. 27 00:34:36.898 --> 00:34:45.329 Okay, um, no, D squared is positive. Um, cause it's a square. 28 00:34:45.329 --> 00:34:55.378 Now, um, so what we can do here is, I can use the previous, any quality that I just worked on. 29 00:34:55.378 --> 00:35:07.528 Um, and let me just make certain that recording stuff and so on. Okay. 30 00:35:10.619 --> 00:35:21.539 Appear to be so okay yeah. For your remote people, I'm going to try and leave the chat window open just in case this future problems. 31 00:35:21.539 --> 00:35:24.719 So. 32 00:35:24.719 --> 00:35:32.248 Okay. 33 00:35:35.128 --> 00:35:46.079 Wow. 34 00:35:46.079 --> 00:35:55.949 Maybe not your question up here? No. Okay. Okay. So what I'm doing now is I'm working my way through the Chevy chef. 35 00:35:55.949 --> 00:36:02.938 Any quality and and just remind you what the Chevy ship any quality. 36 00:36:04.228 --> 00:36:11.248 Um, we know the mean and standard deviation and this will this will give us a bound. 37 00:36:11.248 --> 00:36:15.809 On the probability that we're more than some number away from the mean. 38 00:36:15.809 --> 00:36:25.409 Now, some given number a, and the probability of that in either direction is at worst Sigma squared over a squared. So, how do I prove that. 39 00:36:25.409 --> 00:36:33.119 Is all defined D is X - and here we got the Pro build x minus critical to hey. 40 00:36:33.119 --> 00:36:36.179 So that's probably a B squared a squared. 41 00:36:36.179 --> 00:36:42.208 And, um, so what is that? Well. 42 00:36:44.369 --> 00:36:50.668 D, squared is gone negative and this here by the is going to be less than or equal to the. 43 00:36:50.668 --> 00:36:53.969 Expected value of D squared. 44 00:36:55.349 --> 00:36:59.818 Provided by a squared, so. 45 00:37:05.398 --> 00:37:11.068 By the previous inequality, the, um, our call thing. Okay. So. 46 00:37:11.068 --> 00:37:21.688 So, um, so what we have here is the probability. 47 00:37:21.688 --> 00:37:24.869 Okay, with X, minus greater and a. 48 00:37:26.099 --> 00:37:32.998 Is less than or equal to the spectrum value D squared over a squared. 49 00:37:32.998 --> 00:37:37.648 Now, D Square, um, were again T, equals X . 50 00:37:37.648 --> 00:37:44.668 You okay, so expected value of D squared. 51 00:37:46.739 --> 00:37:52.139 Um, that's the that's the variants actually so. 52 00:37:56.248 --> 00:38:01.259 And why don't we Sigma squared. 53 00:38:01.259 --> 00:38:05.039 Um, because. 54 00:38:06.148 --> 00:38:10.708 It's, it's the variance I can work it through in more detail. If you want um. 55 00:38:12.599 --> 00:38:19.139 So, and so this 1 that out to, um. 56 00:38:19.139 --> 00:38:25.920 The probability equal to. 57 00:38:28.050 --> 00:38:32.849 So, um, so let's go back to my RPI student example. 58 00:38:35.639 --> 00:38:41.909 They are students saying, let's assume the average height is 5. 59 00:38:41.909 --> 00:38:46.110 And now we're going to add, let's say the standard deviation is 1 foot. 60 00:38:47.309 --> 00:38:51.030 So this means 2 thirds of students. 61 00:38:53.519 --> 00:38:58.289 Right access. 62 00:38:58.289 --> 00:39:01.349 4 x less equal 6. okay. 63 00:39:03.719 --> 00:39:07.110 Um, if they're normally distributed. 64 00:39:09.960 --> 00:39:19.500 In any case, so the probability that the students are more than a foot away from the mean. 65 00:39:19.500 --> 00:39:24.960 Minus say, 5, let's say grid equal to 1. 66 00:39:26.309 --> 00:39:31.380 It's going to be less than or equal to the standard deviation which is 1 divided by. 67 00:39:31.380 --> 00:39:36.449 In this case, 1, that doesn't say anything. Um, okay. That's certainly. 68 00:39:36.449 --> 00:39:40.380 It's a very quick thing. Well, let's go up to 2. let's say. 69 00:39:42.599 --> 00:39:46.110 Let's say the probability X. -5. 70 00:39:46.110 --> 00:39:49.590 Is greater than 2 feet. 71 00:39:52.469 --> 00:39:56.519 Center equal to Sigma squared over 4 equals 1 quarter. 72 00:39:58.110 --> 00:40:06.420 So, what this means is that if the if I got a group of students and your average height is 5 feet, and the standard deviation is 1. 73 00:40:06.420 --> 00:40:17.039 Without knowing anything more than the probability that you're more than 2 feet off of the mean outside 3 to 7 feet is at most a quarter. 74 00:40:17.039 --> 00:40:22.920 Which is a very loose found, but this applies to any possible distribution. 75 00:40:22.920 --> 00:40:30.869 So very loose bound. 76 00:40:34.440 --> 00:40:44.309 But it assumes nothing about the distribution. 77 00:40:47.820 --> 00:40:53.099 Except the mean is 5. 78 00:40:53.099 --> 00:40:56.880 And this thing, this 1 in this case, so. 79 00:40:58.829 --> 00:41:04.469 And the distribution, um, here would be a possibility. 80 00:41:07.619 --> 00:41:14.309 Is that, um, excellent probability we might have, um. 81 00:41:15.329 --> 00:41:28.650 For example, you know, half the students might be high 3 and half might be high 7 or something. 82 00:41:28.650 --> 00:41:35.130 Um, or something weird well, then the signal doesn't work out, but you got the idea. So. 83 00:41:36.235 --> 00:41:51.085 Okay, now, if we know more about the distribution, the bounce can be tighter or something. Well. 84 00:41:52.980 --> 00:41:56.219 That's actually not a particularly good example, but, um. 85 00:41:56.219 --> 00:42:04.110 Not a good example, but Chelsea idea. 86 00:42:12.719 --> 00:42:16.500 And if we know the distribution itself, we can do better. 87 00:42:23.909 --> 00:42:28.380 We can do better. 88 00:42:34.199 --> 00:42:40.019 Okay, so and there's other bounds and so on. So. 89 00:42:41.880 --> 00:42:45.570 Okay. 90 00:42:45.570 --> 00:42:58.469 Okay, um. 91 00:43:01.199 --> 00:43:06.119 Topic, um, transform methods. 92 00:43:14.789 --> 00:43:18.480 And this is like, I'll give you a page, um. 93 00:43:23.130 --> 00:43:29.699 1, 2, 3, 4. 94 00:43:32.190 --> 00:43:37.949 So, basically you've seen transforms the class transforms and stuff like that. 95 00:43:37.949 --> 00:43:43.289 Okay um, so this is similar. 96 00:43:46.619 --> 00:43:50.880 Uh. 97 00:43:52.349 --> 00:44:02.280 Transforms and the motivation is that they sometimes make it easier to compute some stuff. So. 98 00:44:02.280 --> 00:44:08.820 What I'm going to do is I'm going to give you a sample though. No, I'm not going to give you all of them. Um. 99 00:44:10.590 --> 00:44:16.590 So, um. 100 00:44:20.789 --> 00:44:22.284 And it's gonna hit 1 or 2. 101 00:44:22.284 --> 00:44:22.675 um, 102 00:44:22.704 --> 00:44:23.184 here, 103 00:44:37.974 --> 00:44:38.905 let me give you 1 called, 104 00:44:38.905 --> 00:44:38.934 uh, 105 00:44:38.934 --> 00:44:41.034 probability generating function. 106 00:44:48.119 --> 00:44:51.690 Generating function, um. 107 00:44:51.690 --> 00:44:55.050 So this is, um. 108 00:45:00.599 --> 00:45:04.650 Um, just basically assumes that the random variable is not is. 109 00:45:04.650 --> 00:45:10.469 Negative so. 110 00:45:11.820 --> 00:45:18.360 Whatever call it K and it's where we typically do it for discreet. 111 00:45:19.530 --> 00:45:31.260 Okay, so, um, g, for generating function and it's defined as the expected value. 112 00:45:31.260 --> 00:45:36.690 Of the to the end. Okay. Um. 113 00:45:36.690 --> 00:45:39.929 It's a random variable it's called. 114 00:45:39.929 --> 00:45:44.730 And, um. 115 00:45:44.730 --> 00:45:52.409 That's a sum of all the, hey 0 to infinity the probability of getting a K times. 116 00:45:52.409 --> 00:45:56.579 Okay, um. 117 00:45:56.579 --> 00:46:02.099 Do an example say, um, geometric and variable. 118 00:46:03.210 --> 00:46:11.099 So they do some geometric random variable. 119 00:46:11.099 --> 00:46:16.139 There's a probability okay. Equals. Um. 120 00:46:17.309 --> 00:46:20.849 I'm going to do a specific example. Um. 121 00:46:20.849 --> 00:46:29.639 I think this makes it easier to understand. So I'm going to say, for example, that in the book so it'll be a geometric random variable with. 122 00:46:29.639 --> 00:46:34.260 Equals 1 half so. 123 00:46:34.315 --> 00:46:48.864 It's like, I'm tossing a coin and fair coin tosses, not correlated. And random variable is I cost until I get ahead. And the random variable is which costs? 124 00:46:48.894 --> 00:46:50.485 Do I see the 1st ad? 125 00:46:50.730 --> 00:46:56.639 50 50 I see it on the 1st pass. 25%. I see it on the 2nd task. 126 00:46:56.639 --> 00:47:00.000 And so on down the line okay so. 127 00:47:02.099 --> 00:47:09.389 So the probability, okay equals 1 over to the K call. So. 128 00:47:11.760 --> 00:47:19.079 Okay 1. okay. So the generating function. 129 00:47:20.309 --> 00:47:24.239 He is going to be the sum of all the to the K. 130 00:47:24.239 --> 00:47:30.059 See to the K so it goes the sum of all of the. 131 00:47:34.530 --> 00:47:41.070 Let me write things. Let me write this. I got it right very carefully. So you can read it. Um. 132 00:47:43.949 --> 00:47:47.969 That here is a tube this here is a Z. 133 00:47:58.739 --> 00:48:01.829 Okay, um. 134 00:48:06.420 --> 00:48:17.760 And, um, equals okay. 135 00:48:17.760 --> 00:48:23.670 Now, what can I do with it? Um. 136 00:48:26.880 --> 00:48:37.829 Well, let us see here um. 137 00:48:46.889 --> 00:48:50.400 Well, let me go back to my definition where, Gee. 138 00:48:50.400 --> 00:49:02.190 For January do derivative so, um. 139 00:49:04.260 --> 00:49:07.440 Well, the 1st thing is G0. 140 00:49:07.440 --> 00:49:10.469 Equals p0 so. 141 00:49:10.469 --> 00:49:14.099 Zeroes probability is just. 142 00:49:14.099 --> 00:49:20.880 Value to you at 0, because it's like anything else in, you know, C2 the K for cake was 1 off. So. 143 00:49:20.880 --> 00:49:24.840 0, so we can find. 144 00:49:24.840 --> 00:49:33.570 If I do a derivative, say D over D. G. G. 145 00:49:39.385 --> 00:49:50.755 Okay, um. 146 00:49:53.849 --> 00:49:58.079 Okay. 147 00:49:58.079 --> 00:50:01.409 K times either the K. -1. 148 00:50:01.409 --> 00:50:07.829 And if K is 0, it's 0 So I might as well start to secure equals 1 to infinity. 149 00:50:09.989 --> 00:50:17.369 So, um, so, D, over GFC at z0. 150 00:50:17.369 --> 00:50:23.429 Is going to be 1 think about it. 151 00:50:23.429 --> 00:50:31.349 Because if K equals 10 K -1, that's 1 case greater than 1. 152 00:50:31.349 --> 00:50:37.440 See, I'm doing a sequel 0, so anything greater than 1 zeros out. So. 153 00:50:38.610 --> 00:50:44.070 So, basically P1 is the derivative evaluated in 0 and so on. 154 00:50:48.690 --> 00:50:53.190 So that is, um. 155 00:50:53.190 --> 00:50:56.219 And in general, we can get. 156 00:50:59.460 --> 00:51:03.719 K is going to be the case derivative. 157 00:51:05.130 --> 00:51:17.940 I just want neatly actually divided by, um. 158 00:51:17.940 --> 00:51:21.780 And evaluated sequence. 159 00:51:21.780 --> 00:51:24.780 So the 1st thing we can do is generating function. 160 00:51:24.780 --> 00:51:28.829 Is we can we can pull out all of the original probabilities. 161 00:51:28.829 --> 00:51:33.059 So the case from probability, a K for K. 162 00:51:33.059 --> 00:51:38.489 And we have to do the case script to evaluate it to the 0. that's the 1st thing that we can do. 163 00:51:38.489 --> 00:51:43.619 But we can do more, um, we can also find the expected value. 164 00:51:43.619 --> 00:51:54.539 So so if I do the 1st derivative up here. 165 00:51:55.860 --> 00:52:01.170 Okay, let me read it again to be fun. 166 00:52:01.170 --> 00:52:07.710 So, the generating function, that's the expected value of. 167 00:52:11.280 --> 00:52:14.340 And then the 1st, Aruba to. 168 00:52:16.050 --> 00:52:19.110 It goes to some of all the P. A. K. 169 00:52:19.110 --> 00:52:22.559 K. times see you the K -1. 170 00:52:22.559 --> 00:52:27.570 Well, let's evaluate it at the equals 1 so. 171 00:52:30.059 --> 00:52:35.550 Evaluated equals 1 that's expected value of, um. 172 00:52:35.550 --> 00:52:41.730 That's the sum of all the, um, K times Peter. Okay. 173 00:52:41.730 --> 00:52:44.760 That's the expected value of. 174 00:52:46.079 --> 00:52:52.800 Okay, so we take the 1st, derivative of the generating function and evaluate it at 1. 175 00:52:52.800 --> 00:52:56.730 We have the expected value, so. 176 00:52:56.730 --> 00:53:02.250 And we can also. 177 00:53:02.250 --> 00:53:05.940 Buying higher order moments similarly so. 178 00:53:14.519 --> 00:53:21.989 Are similar expected value case squared so. 179 00:53:21.989 --> 00:53:28.710 So, if we compute the generate this generating function, we can. 180 00:53:28.710 --> 00:53:36.449 We can from it derive the original probabilities again and we can drive moments like the mean and so on. So. 181 00:53:40.739 --> 00:53:44.429 Um, and if I try it say. 182 00:53:44.429 --> 00:53:48.119 Always screw it up, but if I can go back here. 183 00:53:48.119 --> 00:53:53.429 I got 1 over -2. 184 00:53:55.380 --> 00:54:03.360 So, um, go to another page, I think to be safe. 185 00:54:05.130 --> 00:54:13.619 So, we're going to try geometric, um, equals 1 half. 186 00:54:13.619 --> 00:54:19.949 So, okay, it goes to the minus K and the generating function. 187 00:54:19.949 --> 00:54:25.380 I got it right was 1 over 1-2. 188 00:54:25.380 --> 00:54:35.670 That's a 21, huh okay. Um. 189 00:54:37.860 --> 00:54:41.670 So, over Z. 190 00:54:41.670 --> 00:54:49.019 Let's see, now it's going to be -1 half. 191 00:54:49.019 --> 00:54:55.619 I was -1 times 1-C over 2 to the -2. 192 00:54:55.619 --> 00:55:00.840 Think if I thought it right? So it's 1 half. 193 00:55:05.099 --> 00:55:10.980 I don't know, let's give it a try. Okay, so. 194 00:55:10.980 --> 00:55:14.610 I see equals 0. 195 00:55:16.920 --> 00:55:23.369 Then we get 1 half and that's the probability of 1. okay. 196 00:55:24.690 --> 00:55:28.409 It's equals 1, we are going to get. 197 00:55:28.409 --> 00:55:40.679 1, half, we're gonna get infinity or something. 198 00:55:41.699 --> 00:55:45.809 Equals 1, I don't know I'll continue next week. 199 00:55:50.579 --> 00:55:55.650 And it's something funny. So, but we should get the expected value on it. That's all. 200 00:55:56.065 --> 00:56:03.925 Okay. So okay. 201 00:56:03.925 --> 00:56:12.355 Um, yeah, there's several other transforms, I may pick them in 1 more time so I've been showing you so far, just review part which of the lecture. 202 00:56:12.684 --> 00:56:20.905 I showed you some, any qualities that will give you some limits on the probability that your random variable could be way far off. The me. 203 00:56:21.210 --> 00:56:26.460 If we know only the mean and nothing else at all, we can use some mark off any quality. 204 00:56:26.460 --> 00:56:30.420 If we also know the standard deviation in addition to the me. 205 00:56:30.420 --> 00:56:34.829 Then we can have the 70, 70 quality, which is tighter. 206 00:56:34.829 --> 00:56:39.090 If we have more information, we can get even tighter. Like, we know the distribution. 207 00:56:39.090 --> 00:56:43.409 Okay, and then I gave a sample of there just as. 208 00:56:43.409 --> 00:56:49.289 And calculus there are for you transform, slip plus transforms and so on. 209 00:56:49.289 --> 00:56:53.849 In probability you can do transforms of your probabilities. 210 00:56:53.849 --> 00:56:59.699 I gave you 1 example of this geometric a probability generating function. It's called. It's for discreet. 211 00:56:59.699 --> 00:57:14.610 Random variables that are non negative and I gave an example of that and given that you can re, derive your probabilities that went into it. And you can also compute mass properties moments like expected value and so on. 212 00:57:14.610 --> 00:57:23.039 Okay, okay another topic is reliability. 213 00:57:29.489 --> 00:57:32.849 This will be actually section 4.8. 214 00:57:32.849 --> 00:57:37.829 And will be page 189 or something. 215 00:57:39.840 --> 00:57:47.760 Because a lot of the time we want to work with, um, parts that fail, um. 216 00:57:47.760 --> 00:57:55.409 Like, if I look up at the ceiling, these I think are not, I think they're old fashioned incandescent lights. 217 00:57:55.409 --> 00:58:00.300 I can tell from the callers, so they're old fashioned incandescent lights. I think. 218 00:58:00.300 --> 00:58:09.659 They burn out. Okay. Um, back when electric lights or 1st, pop to the rights in the 930. so it was an industry group. 219 00:58:09.659 --> 00:58:14.130 That settled on a standard of 1000 hours for a light bulb. 220 00:58:15.179 --> 00:58:20.219 And the pressured manufacturer is not to make light bulbs to lasted more than a 1000 hours. 221 00:58:20.219 --> 00:58:24.480 Um, so in any case they have, um. 222 00:58:24.480 --> 00:58:28.170 You know, we're interested in how long is it going to last? Because. 223 00:58:28.170 --> 00:58:35.280 Well, not just the cost of the lightbulb if it burns out on the ceiling up here, it's going to cost a bit of money to replace it. 224 00:58:35.280 --> 00:58:39.449 Um, I don't know how they do. I guess there's a. 225 00:58:39.449 --> 00:58:54.420 There's a cat walk up there you can reach out to her and get the light bulb perhaps or else you put a ladder up from the floor. I don't know what to do, but it's 1 of the 12 of the milk right now. Well, those are probably just turned off with those 2. maybe burnt out. Okay. 226 00:58:54.420 --> 00:58:57.570 So, I want to work with probabilities um. 227 00:58:57.570 --> 00:59:05.159 And so we've got some, so we've got something so we might define, um. 228 00:59:08.760 --> 00:59:12.630 Liability, and that's going to be the probability. 229 00:59:16.500 --> 00:59:21.539 That whatever, um, our object or whatever. 230 00:59:25.320 --> 00:59:29.670 Is still alive at time. 231 00:59:29.670 --> 00:59:35.489 So, the random variable is when a die, so the random variable key. 232 00:59:35.489 --> 00:59:39.599 Equals time. Okay. 233 00:59:39.599 --> 00:59:47.940 And, of course, we have, we assume we have the probability distribution. We have the density function of the kill, the distribution function. 234 00:59:47.940 --> 00:59:54.809 Reliability, that's a probability that it's time it dies. 235 00:59:54.809 --> 01:00:05.699 More than anything is continuous greater than greater equal in this case doesn't actually matter. And of course, this is 1-for distribution function. So. 236 01:00:09.449 --> 01:00:14.909 So, it's it's a compliment of the CDF, but just more convenient to look at the liability. Perhaps. 237 01:00:14.909 --> 01:00:20.159 Okay, now what this means is that the derivative. 238 01:00:22.469 --> 01:00:28.889 Is this minus the density function? So, and we'll see. 239 01:00:28.889 --> 01:00:37.800 Um, okay. Um, so if it's, um. 240 01:00:37.800 --> 01:00:41.039 Exponential, let's say. 241 01:00:46.289 --> 01:00:50.130 Say, it's exponential, so whatever. 242 01:00:56.969 --> 01:01:01.409 Let's just throw 1 overlap and or whatever. I forget. Um. 243 01:01:01.409 --> 01:01:11.280 Okay, um, something like that. 244 01:01:11.280 --> 01:01:15.239 Okay, so that then. 245 01:01:15.239 --> 01:01:21.030 So this is the, um. 246 01:01:22.590 --> 01:01:28.769 Probably failing at any given. So, is this okay going down like that? 247 01:01:30.690 --> 01:01:33.900 And then the cumulative thing. 248 01:01:36.150 --> 01:01:45.570 The T, or some, uh, keys error, let's say, um. 249 01:01:49.110 --> 01:01:54.150 Yeah, so let's do a quick check on this so that. 250 01:01:58.230 --> 01:02:02.730 To be. 251 01:02:02.730 --> 01:02:06.030 Okay. 252 01:02:08.940 --> 01:02:23.610 Okay um, so up here for the Lambda up on top and the bottom yeah. Okay. And then this looks like, um. 253 01:02:25.110 --> 01:02:29.130 Arrows 15 0, then it's, um. 254 01:02:29.130 --> 01:02:33.179 So, that's going off like that. Okay. Um. 255 01:02:33.179 --> 01:02:37.349 And then the liability is the 1-that. 256 01:02:38.820 --> 01:02:44.670 It was something going down something like that. Oh, okay. Probably it's still a lot of good time too. 257 01:02:44.670 --> 01:02:49.380 So that's, um, okay and that's going to be the . 258 01:02:49.380 --> 01:02:55.800 Um, the T. F. T. is 0. that's either a -0. that's a 1. 259 01:02:55.800 --> 01:02:59.789 And if tea's infinity, it goes down to 0. 260 01:02:59.789 --> 01:03:08.309 Okay um, and the derivative of the liability is decreasing drip. Reliability is going to be negative. 261 01:03:08.309 --> 01:03:11.849 So minus f. T that makes sense. 262 01:03:11.849 --> 01:03:14.940 Okay, um. 263 01:03:18.389 --> 01:03:24.000 Now, a big thing that we may want is the meantime to failure. 264 01:03:24.925 --> 01:03:25.525 Okay, 265 01:03:25.554 --> 01:03:26.034 um, 266 01:03:39.835 --> 01:03:42.655 so the slide 12 manufacturers consortium, 267 01:03:42.655 --> 01:03:42.925 the 930. 268 01:03:44.190 --> 01:03:48.329 1, to specify the meantime to fail, there should be a 1000 hours. 269 01:03:49.380 --> 01:03:53.940 Okay, so what this is just expected value of the key. 270 01:03:55.050 --> 01:04:02.250 Okay, now, um, how do we find that? Um. 271 01:04:09.389 --> 01:04:17.489 Is actually, um, taking a short time and so this, this will not out to something like the integral D. 272 01:04:17.489 --> 01:04:23.159 Liability actually arrived on Tuesday or something. 273 01:04:23.159 --> 01:04:30.690 In any case, um. 274 01:04:30.690 --> 01:04:38.639 Contingent, but this is the way you get expected value because we like to talk about reliability say, at a certain time, um. 275 01:04:38.639 --> 01:04:42.840 Now, you may want to have a conditional. 276 01:04:42.840 --> 01:04:46.559 Lifetime as a bulk, so. 277 01:04:54.809 --> 01:05:02.159 A conditional, um, security distribution function. I E. 278 01:05:05.369 --> 01:05:10.469 What's the bulbs, you know, future? Um. 279 01:05:10.469 --> 01:05:16.320 Probably in the future given. 280 01:05:16.320 --> 01:05:23.010 It's still alive at time key. Okay. 281 01:05:23.010 --> 01:05:28.230 Hey. 282 01:05:28.230 --> 01:05:32.340 Or, okay, so. 283 01:05:35.280 --> 01:05:39.869 So, we may want to say to continue to consumer distribution fonts that. 284 01:05:39.869 --> 01:05:46.650 Big, um, on some value, um. 285 01:05:48.869 --> 01:05:52.980 Doesn't matter what we want to call it, call it X or something. 286 01:05:54.539 --> 01:06:00.119 Given, um, given that it's. 287 01:06:02.670 --> 01:06:05.880 So, what this would be would be the clickable the function. 288 01:06:05.880 --> 01:06:10.710 For the lifetime given, it's still alive at time key. 289 01:06:10.710 --> 01:06:16.260 So, what's it came to think and the way we can find that is that, um. 290 01:06:16.260 --> 01:06:19.530 Well, the definition here, that's the, um. 291 01:06:24.119 --> 01:06:28.530 So, Jeff is skilled is the probability that, um. 292 01:06:32.730 --> 01:06:40.079 I'll build the random variables X. 293 01:06:41.670 --> 01:06:46.889 Um, I don't know big key is less than equal to X. 294 01:06:46.889 --> 01:06:52.590 And, um, and then given that, it's at least equal to T. 295 01:06:54.539 --> 01:07:01.320 Okay um, so it's a conditional CDF that I think. 296 01:07:01.320 --> 01:07:13.230 I showed you a little of it before. So how would I find something like that? Um. 297 01:07:15.750 --> 01:07:20.550 Let me just rehab so what I'm doing is. 298 01:07:20.550 --> 01:07:26.280 Is what I want is the Kim, the distribution function. 299 01:07:26.280 --> 01:07:31.650 On, um, given that, um. 300 01:07:33.329 --> 01:07:43.079 Brand about so, given that it's, um, that it's still alive at time key. And again, as I said that I just worked out that that is, um. 301 01:07:44.280 --> 01:07:53.940 The probability that it's, um, it's an X given that it's, um. 302 01:07:54.960 --> 01:08:05.550 Later than key, and, um, we can use the conditional things um, just as an aside here. 303 01:08:06.960 --> 01:08:16.229 Given me to say, probably say and be my problem. Okay. 304 01:08:16.229 --> 01:08:22.109 An, aside here, the last thing up there is, um. 305 01:08:24.659 --> 01:08:32.640 The probability that, um, variables some key to. 306 01:08:32.640 --> 01:08:38.850 X divided by the probability that is greater than, um, T. 307 01:08:38.850 --> 01:08:42.029 1-f T. 308 01:08:43.649 --> 01:08:48.779 And the thing here, this is assuming that the random variables cheese very, very cool. 309 01:08:48.779 --> 01:08:52.289 And the thing at the top is, um. 310 01:08:59.430 --> 01:09:07.890 Okay, no access here. It doesn't matter. So, this is the cable distribution function on the, um, thing that. 311 01:09:07.890 --> 01:09:12.750 Um. 312 01:09:14.220 --> 01:09:18.239 X minus, um, that's f. F. T. in here. 313 01:09:18.239 --> 01:09:23.819 Okay, so, let me go work through an example. So this is a. 314 01:09:23.819 --> 01:09:32.579 For the light bulb, let's say on, um, this is the probability that it's still alive at time X. 315 01:09:32.579 --> 01:09:36.930 Given it was still alive at time key. 316 01:09:36.930 --> 01:09:40.260 We write that down in words, let's say. 317 01:09:40.260 --> 01:09:45.899 Okay. 318 01:09:46.949 --> 01:09:53.760 At time equals ex, given. 319 01:09:56.880 --> 01:10:02.250 It was alive team. 320 01:10:02.250 --> 01:10:14.399 Okay um, work on an example, say. 321 01:10:19.020 --> 01:10:28.439 Okay, so simple example here, I say, exponential. 322 01:10:28.439 --> 01:10:32.699 Let's say a map of. 323 01:10:33.840 --> 01:10:42.569 X equals, um, okay. 324 01:10:52.020 --> 01:10:58.500 Sorry, um, I'm just going to. 325 01:10:58.500 --> 01:11:09.869 Right. 326 01:11:14.069 --> 01:11:20.069 Yeah. Okay. So much like drink today. Okay. Um. 327 01:11:23.939 --> 01:11:28.529 Okay, so. 328 01:11:28.529 --> 01:11:37.619 What's the, um, yeah, I'm just gonna make 1 equals 1 make it easy. 329 01:11:39.060 --> 01:11:43.199 So, half of act, 1-minus X. 330 01:11:43.645 --> 01:11:58.435 Okay, so, let's say was alive at time, say 2 or something. Okay. Um, so if I go back to the previous thing. 331 01:11:58.649 --> 01:12:03.270 Then then our distribution thing. 332 01:12:06.539 --> 01:12:14.550 Um, okay, given that, it's still live it to this would be, um. 333 01:12:17.699 --> 01:12:22.710 Let's see 1-the minus X minus. Um. 334 01:12:28.260 --> 01:12:42.989 Okay, but I said it was 2 so this would be. 335 01:12:47.970 --> 01:12:53.520 Okay, um, in the -2 is. 336 01:12:55.260 --> 01:12:58.590 Give or take point 1 something. Um. 337 01:13:17.609 --> 01:13:23.699 Or something like that, so still decaying exponentially. So. 338 01:13:28.079 --> 01:13:31.680 I mangled something here, but that's the basic idea. So. 339 01:13:44.250 --> 01:13:49.770 So, what we're learning here is reliability so take the light bulbs. 340 01:13:49.770 --> 01:13:56.729 We have the density function, the CDF, but it's more interesting. 341 01:13:56.729 --> 01:13:59.939 With reliability to think about things like. 342 01:13:59.939 --> 01:14:03.449 Mean time to fail your, the liability. 343 01:14:03.449 --> 01:14:07.890 And stuff like that, um. 344 01:14:10.170 --> 01:14:14.729 Probability, um, failure, rate functions and so on. 345 01:14:14.729 --> 01:14:20.369 Let me tie this force to, um, I like to tie my courses to the, um. 346 01:14:21.930 --> 01:14:26.789 You know, to what's happening in the world without expressing any political opinions. 347 01:14:26.789 --> 01:14:32.460 I hope you never figure out what my politics actually are, because it's not appropriate to bring them into the classroom. 348 01:14:32.460 --> 01:14:38.279 But people are talking about life, expectancies and death rates and so on. 349 01:14:38.279 --> 01:14:46.829 So, you know, what's your life expectancy at birth? If a baby is born today? How many years should the kid expect to the live? 350 01:14:46.829 --> 01:14:50.399 Um, 80, 83 years give or takes. 351 01:14:50.399 --> 01:14:53.939 Um, the failure rate function are things like. 352 01:14:53.939 --> 01:14:58.140 Death rates and so on. So, what's your life expectancy given? 353 01:14:58.140 --> 01:15:04.260 That you're still alive at 20 because the way things happen is some baby. 354 01:15:04.260 --> 01:15:07.590 Was a very young just at birth or just after birth. 355 01:15:07.590 --> 01:15:13.079 But given that given, let's say that the kid is still alive at 1 year. 356 01:15:13.079 --> 01:15:16.439 Then the failure rate is quite small. 357 01:15:16.439 --> 01:15:22.890 Um, death rate and so on and then up until 20 give or take. 358 01:15:22.890 --> 01:15:27.000 And then what you might call the failure rate for people. 359 01:15:27.000 --> 01:15:35.069 Starts rising, so if you're alive at age 30 that the probability of dying, say in the next year. 360 01:15:35.069 --> 01:15:44.640 If you're got to 40, you probably be dying the next year. It's going to be perhaps double in every 810 years, or whatever your probability of dying in the next year. W. 361 01:15:44.640 --> 01:15:49.590 Well, until you get up to 100 or something, and you have to talk about smaller periods than a year. 362 01:15:49.590 --> 01:15:52.800 So, we can take what I'm talking about now. 363 01:15:52.800 --> 01:15:59.069 And apply it to things like death rates and, like, expectancies and so on that are in the news today. 364 01:15:59.069 --> 01:16:03.630 So, the mathematics would be similar, but. 365 01:16:03.630 --> 01:16:06.899 You know, just the terminology would be different so. 366 01:16:08.100 --> 01:16:13.649 And, you know, talk about life expectancy deaths and stuff like that. 367 01:16:13.649 --> 01:16:20.729 Okay, so what I just to remind you again. So, what we're looking at today is more topics from, um. 368 01:16:20.729 --> 01:16:27.390 From chapter 4, so some new stuff that I showed you, I showed you 2 ways. 369 01:16:27.390 --> 01:16:35.760 To get a estimate of the probability of an observation, being way far off to me, if, you know, only the mean or, you know, all you mean in Sigma. 370 01:16:35.760 --> 01:16:39.329 Again, the more, you know, the tighter your estimate can be. 371 01:16:39.329 --> 01:16:45.659 Gave you a sample of generating functions or several different generating functions or analog. 372 01:16:45.659 --> 01:16:49.949 Is 1 is the but you just put a minus sign in just to confuse people. 373 01:16:49.949 --> 01:17:01.979 Blah, blah, I gave you 1 that's called generating function and then, and I may hit you some more later and then the next 3 new topics today, basically, and the 3rd new topic. 374 01:17:01.979 --> 01:17:05.130 Was this reliability analysis. 375 01:17:05.130 --> 01:17:17.039 So the math is not much different. Just the names are different. So we talk about light bulbs. What's expected time to fail? You're the light bulb that's expected value. What's about probabilities? Light bulbs still alive at time key. 376 01:17:17.039 --> 01:17:21.600 Probably, it's still alive given that it was alive at a certain time. So. 377 01:17:21.600 --> 01:17:27.869 So, what we'll do to, um, continue on Tuesday, coming up in a week, and a half. 378 01:17:27.869 --> 01:17:34.079 Is the 1st test I got a blurb about it on the webpage, so. 379 01:17:34.079 --> 01:17:39.270 That will be in class unless you have a good excuse, but will run it on great scope. 380 01:17:39.270 --> 01:17:43.079 And, um, bring your computers. 381 01:17:43.079 --> 01:17:51.449 And bring, you can bring a 2 sided 8 and a half by 11 inch sheet, which you can produce any way. You like it can be. 382 01:17:51.449 --> 01:17:57.449 It's got to be it cannot do communication. It's got to be a passive sheet. 383 01:17:57.449 --> 01:18:01.470 And again, if some entrepreneurs want to. 384 01:18:01.470 --> 01:18:06.359 Produce the ultimate cheat sheet for the exam. 385 01:18:06.359 --> 01:18:10.829 And solid. Okay, give me a coffee. I'd be curious, but. 386 01:18:10.829 --> 01:18:16.170 Some universities that happens. I spent a year at Berkeley, many, many years ago. 387 01:18:16.170 --> 01:18:24.720 And at Berkeley, for the very large classes, there was an organized group that did notes on the class. This was before we had Webex and so on. So. 388 01:18:24.720 --> 01:18:30.869 So, for the very large class, this group would basically have a stenographer record what the cross said, and they would sell it. 389 01:18:30.869 --> 01:18:35.819 So, right, so now you're welcome to do that. Other than that. 390 01:18:35.819 --> 01:18:43.439 If you have questions, come down and I'll answer them or I'll confuse you or whatever. So I have a good weekend. See, you Tuesday. 391 01:18:43.439 --> 01:18:47.760 Yes. 392 01:18:47.760 --> 01:18:54.119 It's I think it's the 28 it's on the West. We talked about it in class 2 weeks ago. 393 01:18:54.119 --> 01:18:59.460 And I put it off on the, um, on the blurb. So it's, um. 394 01:19:03.300 --> 01:19:08.460 48 Monday. 395 01:19:08.460 --> 01:19:11.760 Hello hello so I'm just trying to. 396 01:19:11.760 --> 01:19:15.539 Make sure I'm understanding quickly. Not their phone. 397 01:19:15.539 --> 01:19:18.600 1, correct. 398 01:19:18.600 --> 01:19:22.140 Because there are some specific I might have lost the. 399 01:19:22.140 --> 01:19:28.890 Let me, excuse me no point continuing this um. 400 01:19:28.890 --> 01:19:33.024 And so it's really quick.