WEBVTT 1 00:00:14.548 --> 00:00:21.089 Okay. Um, so good afternoon probability class. 2 00:00:21.089 --> 00:00:25.739 22 for, I think, um. 3 00:00:25.739 --> 00:00:29.850 Broadcast to get on Webex then I think I'm, um. 4 00:00:29.850 --> 00:00:34.530 Recording it though, 1 can never be certain with these things. 5 00:00:35.579 --> 00:00:40.740 As I've said, you're always welcome to record me yourself. So. 6 00:00:59.784 --> 00:01:04.224 Okay, okay. 1st for the exam um. 7 00:01:04.469 --> 00:01:11.609 Is an extra office hour is holding Wednesday 2 to 3 in addition to a normal office hour tomorrow. 8 00:01:11.609 --> 00:01:17.099 And the hunting and Hon will Proctor the exam and people with extra time. 9 00:01:17.099 --> 00:01:26.579 Be taken out over to my lab. Okay, so you asked for some reviews. So, let's just go through the last. Um. 10 00:01:26.579 --> 00:01:32.430 May be a few classes and see what they have to say. 11 00:01:35.400 --> 00:01:41.159 I don't know, um. 12 00:01:44.730 --> 00:01:51.510 Next summer oops. 13 00:01:52.620 --> 00:01:56.189 3, I don't think I've done that. Um. 14 00:01:58.079 --> 00:02:02.489 Okay, so that yeah, basically I did the ta's handle the homework, so. 15 00:02:02.489 --> 00:02:06.180 Yeah, you want to ask them questions about that so. 16 00:02:06.180 --> 00:02:12.689 But, um, but to go see in with 2 variables, um. 17 00:02:12.689 --> 00:02:19.409 You know, I've done several classes, but I could look at maybe some of these, let's see archive and. 18 00:02:19.409 --> 00:02:23.729 Go back to where we want to start from. I don't know some were like. 19 00:02:24.750 --> 00:02:30.870 And go back a month and a half or so, what do we have here? Okay, let's go back. Um. 20 00:02:32.219 --> 00:02:42.150 After the exam, so okay, so, Matt lab, we've just enrichment material. It's a very useful tool. 21 00:02:42.150 --> 00:02:51.689 And I'd like to, um, I may have showed your ran some stuff may run some simple things on that. And on today's, um. 22 00:02:51.689 --> 00:03:00.030 Thing by the way, if you have trouble hearing me, I can attempt to use the microphone. I don't know if it's always in advance, but it. 23 00:03:00.030 --> 00:03:04.469 So you can hear me. Okay, so this is a powerful tool. 24 00:03:04.469 --> 00:03:08.129 I work with numbers and I gave my opinion here. 25 00:03:08.129 --> 00:03:17.189 Um, review density, um, and again I'm going to slow or too fast. Let me know. So. 26 00:03:17.189 --> 00:03:21.300 This is an intuitive tutorial that I made up on. 27 00:03:21.300 --> 00:03:26.550 So you could understand how you transform, because you may transform variables. 28 00:03:26.550 --> 00:03:36.180 A random variable may be in feet. I'm tossing a dart onto the floor coordinates our peak. You may want to convert coordinates to centimeters. 29 00:03:36.180 --> 00:03:42.659 Then what happens to the density function and so on. So. 30 00:03:42.659 --> 00:03:56.520 In this case, well, you'd have like a Chicago, which is what, in this case, it's a 1 by 1 matrix of the various derivatives, and to see scale it and you could check it 2 variables. The same thing. 31 00:03:56.520 --> 00:03:59.580 I gave an example we're converting from. 32 00:03:59.580 --> 00:04:02.789 Say, 82 centimeters, um. 33 00:04:02.789 --> 00:04:07.740 Assuming 30 centimeters a foot approximately and. 34 00:04:07.740 --> 00:04:13.650 They have the determinant and convert the book goes through more detailed examples. 35 00:04:13.650 --> 00:04:25.620 Where it's not just Cartesian, it's Cartesian and say to polar notation. And I use something like that, showing how you'd find integral for the calcium. Let's say. 36 00:04:27.658 --> 00:04:33.059 Some review of it, that's what it looks like and so on, um. 37 00:04:33.059 --> 00:04:38.098 Probably 2, lots of examples. 38 00:04:38.098 --> 00:04:42.809 Um, for gallerson, you cannot integrate it. Exactly. Explicitly. 39 00:04:42.809 --> 00:04:46.588 And the shorthand notation for partial under girls, um. 40 00:04:48.269 --> 00:04:55.228 Big and so on engineers use an expression queue for the right tail of the thing. Um. 41 00:04:58.918 --> 00:05:06.178 2, random variables um, I don't know not 2 examples I could work out again. 42 00:05:06.178 --> 00:05:11.879 Um, conditionals, while we were working out and saying. 43 00:05:11.879 --> 00:05:17.249 Um, there's a big thing, um, with. 44 00:05:18.538 --> 00:05:23.218 For this influencing thing, I'll talk about more if you want me to go slower on something. 45 00:05:23.218 --> 00:05:32.038 I don't know with again, too variable calcium I mean, there's lots of ways you can have a random variable. 46 00:05:32.038 --> 00:05:38.249 Density function with 2 variables, but if each variable separately is a Gaussian. 47 00:05:38.249 --> 00:05:48.329 And then we would define these 2 variable gaps. This is 1 way you could, you could imagine lots of other ways. This is we're going to define this as our 2 variable calcium. 48 00:05:48.329 --> 00:06:01.379 Role is the correlation coefficient of how the 2 variables track each other linearly from -1 to 1. this assumes the individual means our 0 and individual standard deviations are 1. 49 00:06:01.379 --> 00:06:06.838 Um, estimation. 50 00:06:06.838 --> 00:06:16.649 Talk about it more. Um, I showed lots of examples. 51 00:06:18.988 --> 00:06:29.668 Random 2, random girls were independent than the joint density function as a product of the individual density functions. Uh, transformations. I've talked about a couple of minutes ago. 52 00:06:29.668 --> 00:06:39.088 Um, we got into working with vectors pairs and vectors of random variables. Um. 53 00:06:41.668 --> 00:06:45.689 Estimation is a really big topic. In fact. 54 00:06:45.689 --> 00:06:52.168 What I've got to help you understand estimation more. There are some videos from red key. I yeah. Question. 55 00:07:04.319 --> 00:07:08.278 Yes, if you wish to have a fancy calculator, it's allowed then. So. 56 00:07:08.278 --> 00:07:13.139 If you email me, I'll stop it on to the webpage so everyone can see that. So. 57 00:07:14.338 --> 00:07:18.418 Um, other questions. 58 00:07:18.418 --> 00:07:21.778 I hope you don't need it, but you're welcome to bring it up. So. 59 00:07:23.009 --> 00:07:29.728 Um, the estimations a big thing, and that won't noisy channel. 60 00:07:29.728 --> 00:07:33.809 X is transmitted noise is added to it. Why is received. 61 00:07:33.809 --> 00:07:40.468 We want to estimate what the value of X was. If we see a particular Y. 62 00:07:40.468 --> 00:07:46.588 And it's a little more complicated that it looks at times. Um. 63 00:07:46.588 --> 00:07:51.238 1st question is, what do we know about the trans? The. 64 00:07:51.238 --> 00:07:57.778 Source random variable X do we even know what its density function is? We might, or we might not. 65 00:07:57.778 --> 00:08:02.038 And if we, the more we know the more we can do, so. 66 00:08:02.038 --> 00:08:10.528 So, if we know that's the communications, then we call them back. Um, Matt, maximum estimator. 67 00:08:10.528 --> 00:08:15.569 If we don't know, then we started with some, the density function. 68 00:08:15.569 --> 00:08:20.699 For access flat, which is not logically possible so it has to integrate to 1, but. 69 00:08:20.699 --> 00:08:27.988 We assume it any way, and it seems to work an example for that would be, um. 70 00:08:27.988 --> 00:08:31.528 You want to play the stock market and. 71 00:08:31.528 --> 00:08:37.859 You want to estimate see well, here are you going forward? You see the price of Google. 72 00:08:37.859 --> 00:08:44.489 Stop today, and you want to estimate it what it will be tomorrow, because you're buying options or something. 73 00:08:44.489 --> 00:08:48.958 Or even have the vector of google's stock every day for the last 10 years. 74 00:08:48.958 --> 00:08:56.999 Well, you don't actually have it just a, you don't have a density function for the Google stock apart from what you observe. 75 00:08:56.999 --> 00:09:01.979 Well, that's another question so we'd go for some maximum likelihood estimator. 76 00:09:01.979 --> 00:09:06.629 So, okay, um. 77 00:09:06.629 --> 00:09:10.048 Yeah, that's my point 7 there. Um. 78 00:09:11.849 --> 00:09:14.999 So well, here's an example I actually did not. 79 00:09:14.999 --> 00:09:18.869 I put it on my blog. I didn't actually do it in class. 80 00:09:18.869 --> 00:09:23.369 Um, so. 81 00:09:23.369 --> 00:09:28.619 Let me spend some time on this. I was going to and I somehow jumped over it. Um. 82 00:09:30.328 --> 00:09:34.948 So, we're testing a fair, so I'll get these likelihood things through all and also. 83 00:09:34.948 --> 00:09:39.389 That I started to mention is I put some links and for some videos. 84 00:09:39.389 --> 00:09:44.158 By, um, professor, rich Rad, keep talking about these estimators. 85 00:09:44.158 --> 00:09:50.428 So, he has a nice when he teaches the course he has a set of videos. He puts them on YouTube. Um. 86 00:09:50.428 --> 00:09:53.489 And you're welcome to look at them also. 87 00:09:53.489 --> 00:09:58.019 They don't follow the textbook. Precisely. He has his own. 88 00:09:58.019 --> 00:10:06.989 Organizing scheme, and I prefer to follow the textbooks, so you can go to the textbook for reference. So but his videos are very good. 89 00:10:06.989 --> 00:10:11.639 And so I'm linking later down on the blog for today I linked to them a lot. 90 00:10:11.639 --> 00:10:15.538 Um, any case. 91 00:10:17.698 --> 00:10:25.558 So, um, I had to work through this a little, maybe to show estimators and. 92 00:10:26.759 --> 00:10:30.568 Oh, that's some of this over here. 93 00:10:30.568 --> 00:10:37.649 This 1, too this was hand out actually. 94 00:10:38.698 --> 00:10:45.178 Um, 1924. okay, go back here. 95 00:10:45.178 --> 00:10:49.259 So things down here, um. 96 00:10:59.519 --> 00:11:03.928 Um, that's a recurring cost. 97 00:11:03.928 --> 00:11:11.788 Okay, and let's make it fair. I mean, I could make all of this stuff more complicated, but, um. 98 00:11:11.788 --> 00:11:20.158 Okay, we got 2 random variables. So you do a random experiment can have several random variables come out of it. Um. 99 00:11:20.158 --> 00:11:31.349 Okay, and so if I just use this as a chance to review some, what some of the terminology is, um. 100 00:11:33.328 --> 00:11:39.328 The random experiment is tough. 101 00:11:39.328 --> 00:11:45.928 That's very coin. Okay. And we will observe, um. 102 00:11:48.989 --> 00:11:57.899 2 random variables okay. From each experiment. Um. 103 00:12:06.089 --> 00:12:11.038 Okay, and just, um. 104 00:12:12.058 --> 00:12:16.288 Sorry, it was this 1 down here. Oh, okay. Um. 105 00:12:21.658 --> 00:12:26.068 And you could have the outcomes actually could be the various, um. 106 00:12:29.908 --> 00:12:33.298 Are the Tom basically. 107 00:12:33.298 --> 00:12:39.778 8 ways, you know, 3 coins, you know, come up. 108 00:12:41.729 --> 00:12:47.548 And so the 2 random variables, um, would be. 109 00:12:48.928 --> 00:12:52.499 Huh. 110 00:12:52.499 --> 00:12:58.798 So, let's say, oh. 111 00:12:58.798 --> 00:13:01.918 Hey. 112 00:13:01.918 --> 00:13:05.249 The. 113 00:13:06.538 --> 00:13:13.739 The 1st, 1, so. 114 00:13:13.739 --> 00:13:17.249 Yes, so hold on. 115 00:13:17.249 --> 00:13:20.879 Oh, good start I think. 116 00:13:20.879 --> 00:13:24.119 Hello. 117 00:13:24.119 --> 00:13:27.298 Okay. 118 00:13:27.298 --> 00:13:31.198 Outcomes here here, um. 119 00:13:33.839 --> 00:13:38.099 Outcome and why. 120 00:13:38.099 --> 00:13:42.119 Um, could be. 121 00:13:44.399 --> 00:13:47.399 Tell tale tale. Okay. 122 00:13:47.399 --> 00:13:50.609 Hello head. 123 00:13:50.609 --> 00:13:54.568 No, okay. 124 00:13:54.568 --> 00:13:57.599 Hello hey. 125 00:13:57.599 --> 00:14:04.318 And, um, add tail tale, head tale head. 126 00:14:04.318 --> 00:14:07.708 Hello? Hello? 127 00:14:07.708 --> 00:14:10.918 Yeah. 128 00:14:10.918 --> 00:14:14.578 Go head and head. 129 00:14:14.578 --> 00:14:19.499 Okay, okay. Okay. So the number of. 130 00:14:19.499 --> 00:14:23.099 Here it was. 131 00:14:23.099 --> 00:14:26.308 0, and. 132 00:14:26.308 --> 00:14:29.489 The position of the 1st head was. 133 00:14:29.489 --> 00:14:33.568 Say, Voicera. Oh, okay. This won't. 134 00:14:33.568 --> 00:14:37.019 On 3 a. 135 00:14:37.019 --> 00:14:40.499 Well. 136 00:14:40.499 --> 00:14:46.288 2 to. 137 00:14:46.288 --> 00:14:49.589 On and had. 138 00:14:49.589 --> 00:14:52.649 Okay. 139 00:14:52.649 --> 00:14:58.139 Oh, wow. 140 00:14:58.139 --> 00:15:02.278 Sorry. 141 00:15:02.278 --> 00:15:08.129 2, I'm counting from 1. um, 1 had. 142 00:15:08.129 --> 00:15:11.999 Starting and position on. 143 00:15:11.999 --> 00:15:17.639 To head, it's in position 1, 2 5. 144 00:15:17.639 --> 00:15:20.879 Mass function, uh, um. 145 00:15:22.019 --> 00:15:25.948 X could be 0, 1, 2 and 3. 146 00:15:25.948 --> 00:15:29.068 X being as they're all. 147 00:15:29.068 --> 00:15:32.308 Um, again, I. 148 00:15:32.308 --> 00:15:35.698 I can't split the screen. I can go back and forth. 149 00:15:35.698 --> 00:15:39.869 So, next being a 0 happens 1. 150 00:15:39.869 --> 00:15:43.288 Time thanks. 151 00:15:43.288 --> 00:15:46.769 You know, 1, 1. 152 00:15:46.769 --> 00:15:50.339 2, 3 times. 153 00:15:50.339 --> 00:15:53.788 Explain it. 154 00:15:53.788 --> 00:15:57.599 2 has happened 3 times. 155 00:15:57.599 --> 00:16:01.259 Text me. 156 00:16:01.259 --> 00:16:06.808 Sure. 157 00:16:06.808 --> 00:16:09.839 3, 3 and 1. 158 00:16:09.839 --> 00:16:13.078 Okay, we can look at the Y. 159 00:16:16.979 --> 00:16:20.399 I scroll back here. Why is it? 160 00:16:20.399 --> 00:16:24.328 The 0, 1. 161 00:16:24.328 --> 00:16:27.509 So, on. 162 00:16:27.509 --> 00:16:30.808 Or. 163 00:16:30.808 --> 00:16:35.668 Bye bye. 164 00:16:42.389 --> 00:16:46.139 This is 2. 165 00:16:47.428 --> 00:16:51.089 Ice, and it's the 3. 166 00:16:51.089 --> 00:16:54.568 Once so there are a, once. 167 00:16:54.568 --> 00:16:59.399 So, for once. 168 00:16:59.399 --> 00:17:05.608 For twice, and once these are probably mass functions for. 169 00:17:05.608 --> 00:17:10.709 Sample, um, okay. 170 00:17:10.709 --> 00:17:14.999 Hey. 171 00:17:14.999 --> 00:17:18.449 And so we can get. 172 00:17:18.449 --> 00:17:23.249 Probability. 173 00:17:23.249 --> 00:17:26.909 Just divide by 8 and so on. 174 00:17:27.743 --> 00:17:28.223 So, 175 00:17:35.364 --> 00:19:18.473 okay. 176 00:19:22.378 --> 00:19:28.288 Care. 177 00:19:32.519 --> 00:19:38.159 Okay, um. 178 00:19:47.159 --> 00:19:59.489 I was wondering why X big number of heads live position in the 1st, set of tossing the 3 coins and counting. 179 00:19:59.489 --> 00:20:03.659 Okay, so if there were 2 heads. 180 00:20:03.659 --> 00:20:11.398 For 1st. 181 00:20:13.919 --> 00:20:19.439 2 or 1, it could be 2212 and 2 1 twice. 182 00:20:19.439 --> 00:20:25.919 Oh, up here 0, 2, 1 0. 183 00:20:31.618 --> 00:20:35.128 The 1st, 1 in position 1. 184 00:20:39.209 --> 00:20:49.949 If I add up all of the okay. Okay. So now, um. 185 00:20:52.588 --> 00:20:55.739 And then the probability, so just be divided by. 186 00:20:58.558 --> 00:21:02.848 So, 1, 8, so. 187 00:21:02.848 --> 00:21:06.058 Whatever just divide those things by yeah. 188 00:21:06.058 --> 00:21:10.019 Okay, so now, um, now we could. 189 00:21:10.019 --> 00:21:14.189 2 things like the. 190 00:21:14.189 --> 00:21:19.798 Conditional probability to say. 191 00:21:20.969 --> 00:21:29.638 Additional, so probably given, why isn't probably under why we got to join probability. Okay. 192 00:21:29.638 --> 00:21:38.489 So, probably that's given why okay will be the probability X and Y, divide it by probability. And why. 193 00:21:39.929 --> 00:21:47.848 Um, well, we can look at this thing here and get the probabilities of different access. 194 00:21:47.848 --> 00:21:50.878 So, did I do it up here? 195 00:21:52.588 --> 00:21:56.939 Yes, I now we could do something like, I don't know. 196 00:22:00.239 --> 00:22:05.189 Just do an example X equals 1. 197 00:22:05.189 --> 00:22:16.709 Well, probably X equals 1 and why was 1. 198 00:22:16.709 --> 00:22:20.398 It's the right here, it will be 1, 8. 199 00:22:20.398 --> 00:22:25.558 And the probability that why was 1 is, um. 200 00:22:25.558 --> 00:22:32.909 1, half. 201 00:22:35.038 --> 00:22:39.388 It's divided by 1 half. 202 00:22:39.388 --> 00:22:45.269 Order okay. Um. 203 00:22:45.269 --> 00:22:49.138 And. 204 00:22:49.138 --> 00:22:52.979 I look up and we could. 205 00:22:52.979 --> 00:22:56.699 Go to other samples here. 206 00:22:56.699 --> 00:22:59.909 Here and the. 207 00:22:59.909 --> 00:23:08.219 Weekly then, um, here I Thanksgiving, why for all the cases here? So actually close 1 Y, equals 1. 208 00:23:08.219 --> 00:23:14.098 Um, right there that's the 1 quarter I did for example. Um. 209 00:23:15.419 --> 00:23:25.618 Okay, so so I was just working out some things on the table to the right there. Now. Um. 210 00:23:28.679 --> 00:23:32.009 So we observe maybe why was 1. 211 00:23:32.009 --> 00:23:36.568 And then what should we guess what, what should we guess for X. 212 00:23:36.568 --> 00:23:37.973 And this example, 213 00:23:38.243 --> 00:23:38.723 well, 214 00:23:38.753 --> 00:23:39.203 um, 215 00:23:50.304 --> 00:23:50.844 well, 216 00:23:50.844 --> 00:23:51.503 for example, 217 00:23:51.503 --> 00:23:52.583 we find the problem here. 218 00:23:52.583 --> 00:24:00.503 X equals 1 given Y, equals 1 is 1 quarter we could do other things the probability given wink with 1. 219 00:24:01.769 --> 00:24:06.838 Just work out exactly the same. And then we would get stuff like. 220 00:24:08.729 --> 00:24:19.888 I would show on the right here, um, for finding the map and the maximum likelihood estimator. So. 221 00:24:22.169 --> 00:24:33.088 Okay, so that might help you understand these estimators cause he estimators are going to come up again and again so central I mentioned, um. 222 00:24:33.088 --> 00:24:36.598 I've mentioned it several times. I'll repeat that. 223 00:24:36.598 --> 00:24:46.378 It's that with most other distributions, as then gets larger, they start looking like a central and a Gaussian distribution really quickly. 224 00:24:47.519 --> 00:24:54.358 Okay, um, next class, 22 vectors and, um. 225 00:24:56.909 --> 00:25:00.778 Get into vectors and so on. Okay. Um. 226 00:25:02.608 --> 00:25:06.298 So, that was a quick review of some of the things. Um. 227 00:25:10.439 --> 00:25:13.499 Let me mention another review so, um. 228 00:25:13.499 --> 00:25:19.919 You got your variances and so on, you expected value of X. 229 00:25:19.919 --> 00:25:25.828 And then you've got the expected value of X Square, and you get things like the variance of X. 230 00:25:26.848 --> 00:25:31.499 Effective value of X, minus X value of X. 231 00:25:32.878 --> 00:25:45.719 Squared and so on, and that turns out to be things like that. Um, with 2, random variables, you could get a CO variance and. 232 00:25:48.659 --> 00:25:51.023 Who random Barry was gonna code variants, 233 00:26:02.513 --> 00:26:07.134 and it will track how the mobility relate to each other and so. 234 00:26:07.409 --> 00:26:16.108 And if we expand that out to expect the value of X times, Y, minus the expected value. 235 00:26:17.189 --> 00:26:23.128 I think something like that. 236 00:26:24.568 --> 00:26:29.939 It's time to why so. 237 00:26:34.858 --> 00:26:40.348 And, um, any divide through by the standard deviation to get a, um. 238 00:26:40.348 --> 00:26:55.108 Correlation coefficient so well, you'd list you'd, um, if it's a finite thing, you'd list all of the, um. 239 00:26:57.419 --> 00:27:02.308 Well, let me give an example, say, let's give an example, let's say. 240 00:27:04.469 --> 00:27:09.088 Well, well, let me, do you say the specific say the finite thing. 241 00:27:09.088 --> 00:27:12.749 Let me say, we've got say 2, um. 242 00:27:14.878 --> 00:27:19.378 Like, coins or something, um, and what's going to happen. 243 00:27:19.378 --> 00:27:24.598 Is that edge tails heads or tails. 244 00:27:24.598 --> 00:27:31.648 Um, they're going to come heads, let's say. 245 00:27:33.358 --> 00:27:38.219 Point 3 or tales point 3 and, um. 246 00:27:38.219 --> 00:27:41.788 This point 6 and, um. 247 00:27:47.128 --> 00:27:53.459 Yeah, okay so we're tossing 2 coins and they tend to track each other a little. Okay. 248 00:27:53.459 --> 00:27:59.578 And then we're going to let say tales is a 0 and heads as a 1. 249 00:27:59.578 --> 00:28:05.729 So, um, X is the 1st coin and why is the 2nd coin? 250 00:28:05.729 --> 00:28:12.959 Okay, so expected value of X equals. Okay. Well, 1st, we do the marginals. Let's say. 251 00:28:12.959 --> 00:28:21.388 Um, yes, um. 252 00:28:21.388 --> 00:28:29.338 Some this way, we're getting 25.5 we signed this way, we get point 5.5. okay. 253 00:28:29.338 --> 00:28:32.459 Um, probabilities here, so. 254 00:28:34.318 --> 00:28:38.848 Okay, so the so the expected value of X. 255 00:28:38.848 --> 00:28:46.439 Is going to be at point 5 times 0.5 times 1 equals point 5. so this 1 half. Okay. 256 00:28:46.439 --> 00:28:51.148 Executive value of Y, equals 1 half um. 257 00:28:53.489 --> 00:29:04.078 Expected value of X. Y, well, um, the only 1 that's done zeros heads heads and that's going to be point 3. 258 00:29:04.078 --> 00:29:14.788 Cause it's, it's point 3 times 1+point 2 time 0+point, 2 time 0+point 3 times. 0.3. okay. Um. 259 00:29:14.788 --> 00:29:19.919 Expected value of X and half so the covariant. 260 00:29:22.558 --> 00:29:26.669 Of the expected value of X Y, minus expected value of X. 261 00:29:26.669 --> 00:29:33.778 Value of Y equals, um, point X Y, Z point 3. 262 00:29:33.778 --> 00:29:37.199 Um, point 2, 5. 263 00:29:39.749 --> 00:29:44.159 505 and so it's positive. 264 00:29:44.159 --> 00:29:51.479 So so, X and Y are correlated. 265 00:29:54.298 --> 00:30:04.259 Oh, like we look at this table here has head should occur a quarter of the time, but in fact, it's occurring point 3 of the time. It's a little more than an odd to. 266 00:30:04.259 --> 00:30:14.489 Okay, so there's a policy entails tales should occur a quarter of the time. In fact, in my example, that's occurring point 3 at the time, which is too much. 267 00:30:14.489 --> 00:30:19.558 Heads tails that occur quarter at the time, but it's only occurring point too. It's too little. 268 00:30:19.558 --> 00:30:34.403 Yes, well, I'm taking all the values possible values X Y, heads the 1 tail, all the values of X. Y, and they're probably Bill and multiplying them by their probabilities and something. 269 00:30:44.848 --> 00:30:48.659 Well, okay, let me write this out in more detail for it for anything. 270 00:30:48.659 --> 00:30:58.888 Um, I expected value of, um, queue let's say. 271 00:30:58.888 --> 00:31:03.568 Okay, well some of all the. 272 00:31:04.888 --> 00:31:13.528 Possible values for Q. 273 00:31:14.759 --> 00:31:19.979 The probability. Okay. 274 00:31:21.388 --> 00:31:27.689 So, um, I mean, that's nail it down by looking or integral if it's continuous. So. 275 00:31:27.689 --> 00:31:31.558 You know, so toss the coin. 276 00:31:33.179 --> 00:31:36.659 So X, and the probability X is 0, probably 1 half. 277 00:31:36.659 --> 00:31:45.838 Access 1 is 1 half factor value of X equals 0 times 1 half +1 times 1 half equals 1 half. Okay. 278 00:31:47.189 --> 00:31:53.909 For X, Y, we've got X Y, and the probability. 279 00:31:53.909 --> 00:32:00.239 So, um, oh, let's do it list and list. So X Y, X. Y so X is 0. 280 00:32:00.239 --> 00:32:05.068 And there all the probabilities points. Um, 0. 281 00:32:06.509 --> 00:32:09.598 Okay, let let me take this more detail. So. 282 00:32:11.338 --> 00:32:17.548 Come on what's going on here. Okay. Now. 283 00:32:19.919 --> 00:32:23.969 For expected value of X and Y, this is X times. Y, okay. 284 00:32:23.969 --> 00:32:33.929 Is X times Y, so, let's just listed out X. Y, X. Y, and it'll probably bill. It's like. 285 00:32:33.929 --> 00:32:41.818 If X is 0 and 0 x0 0 and 1010001 and 1 is 1. okay. 286 00:32:41.818 --> 00:32:50.098 Uh, if I go back to the table back here, um, heads and heads is point 3 and tails at point 3 and then, you know, this. So. 287 00:32:50.098 --> 00:32:57.419 I go back here, you're on 0, the probabilities point. 3.2.2.3. 288 00:32:57.419 --> 00:33:01.648 Okay, so the expected value of X Y. 289 00:33:01.648 --> 00:33:04.858 I take this call, I take, um. 290 00:33:07.318 --> 00:33:12.929 Uh, this call, um, times this column okay. Is, um. 291 00:33:12.929 --> 00:33:19.528 0+0+0+point 3.3 so the expected value of X times wise point 3. 292 00:33:28.378 --> 00:33:32.398 Make sense. Okay here. 293 00:33:34.229 --> 00:33:41.249 X Y, will be done so valuable back. 294 00:33:41.249 --> 00:33:46.769 Why say that again please. 295 00:33:50.788 --> 00:34:04.048 X and Y, it's just multiplying the values into I mean, I could have functions of random variables. If I buy 1 brand new value has got some number 2nd, random variable. I can define the 3rd, random variable. It's a product of the 1st 2. that's what I just did. 296 00:34:04.048 --> 00:34:10.199 I can do that, so, and I got a function, so I could get a random variables X squared. 297 00:34:10.199 --> 00:34:16.018 The 2nd time called sign of why any mathematical function I want to. 298 00:34:16.018 --> 00:34:22.588 Oh, of random variables and then I could find expected value of this function of the random variable effects. The book. 299 00:34:22.588 --> 00:34:25.949 Goes into that maybe more than I did. So, does that make sense? 300 00:34:25.949 --> 00:34:37.708 Okay, so. 301 00:34:39.509 --> 00:34:51.539 Okay, so I was talking about, um. 302 00:34:55.798 --> 00:35:01.108 I was talking well, for 2, random variables we've got correlations and so on. Okay. 303 00:35:01.108 --> 00:35:10.409 Um, now. 304 00:35:10.409 --> 00:35:16.289 Take this 1 more step, they got the 2 coins that are somehow linked together. 305 00:35:17.338 --> 00:35:32.159 Um, the variance excited value was a half. Okay the expected value of X squared. 306 00:35:32.159 --> 00:35:37.259 Turns out to be 1 half also. Um, I can. 307 00:35:37.259 --> 00:35:44.938 Somebody asked, I'll go through it otherwise. So it's a half -1. half times 1 half equals a half 1 quarter. 308 00:35:44.938 --> 00:35:51.509 Okay um, so. 309 00:35:52.528 --> 00:35:56.878 And standard deviation square root of variance. 310 00:35:58.829 --> 00:36:02.099 Um. 311 00:36:02.099 --> 00:36:11.219 And let's see now, and the correlation coefficient so we get the CO variance of the 2 of them and the code variants. I worked out. 312 00:36:11.219 --> 00:36:16.139 As, um, that's what I said in the middle of it, I guess. 313 00:36:16.139 --> 00:36:19.168 So, the covariant, um. 314 00:36:22.199 --> 00:36:31.798 Wait, a 2nd, uh, secondary was point 3. 315 00:36:31.798 --> 00:36:35.518 And this was, um, point 5.5. 316 00:36:35.518 --> 00:36:43.498 Point 25.05 I did and covanents could be plus or - and so on. Okay. Then, um. 317 00:36:44.608 --> 00:36:55.528 Correlation coefficient um. 318 00:36:55.528 --> 00:36:58.978 Row or something so that would be, um. 319 00:36:58.978 --> 00:37:11.309 Co variants divided by the 2 standard deviations I think, although my mind is studying the dark blue. At this point. I need infectious. 320 00:37:11.309 --> 00:37:14.759 So point 05, divided by, um. 321 00:37:17.398 --> 00:37:20.668 25, I think, um. 322 00:37:21.748 --> 00:37:25.378 Equals point Tom to. 323 00:37:25.378 --> 00:37:29.489 And point 2 and again, this is. 324 00:37:29.489 --> 00:37:33.688 And point 2 is rather a small point, too is small. 325 00:37:35.789 --> 00:37:41.699 So, um, okay, uh, so if we. 326 00:37:41.699 --> 00:37:45.688 Um, had the 2 coins being more tightly linked. 327 00:37:45.688 --> 00:37:50.128 Then this would be larger so, um. 328 00:37:59.909 --> 00:38:12.989 Okay. 329 00:38:12.989 --> 00:38:18.329 So, I could do something like this edge tails heads or tails. 330 00:38:18.329 --> 00:38:26.670 Um, I suppose it was point 4.1. okay, so. 331 00:38:28.349 --> 00:38:32.610 So the marginal point, 5.5.5.5. 332 00:38:32.610 --> 00:38:35.940 Expected value called this X and call this. Why. 333 00:38:35.940 --> 00:38:41.340 Is excess 1 half and so on. So here is the covariant um. 334 00:38:43.500 --> 00:38:50.219 And the variants effects is 1 quarter. Um. 335 00:38:53.340 --> 00:38:56.730 I think correct me if I'm like me. So here's the CO variance. 336 00:38:56.730 --> 00:39:07.590 Is, um, and, um, point 8. 337 00:39:09.989 --> 00:39:15.630 Okay, so the correlation coefficient is point 8 divided by. 338 00:39:15.630 --> 00:39:27.030 Um, sorry. 339 00:39:29.820 --> 00:39:38.760 Try this again. 340 00:39:38.760 --> 00:39:42.780 Except the value of X Y, was point 8. 341 00:39:43.949 --> 00:39:50.429 Co variant equals point 8-um. 342 00:39:54.090 --> 00:40:01.170 Point 25 equals point 55 the correlation coefficient. 343 00:40:03.269 --> 00:40:07.320 Point 55 divided by, um. 344 00:40:11.309 --> 00:40:15.420 A nasty feeling I'm doing something wrong here, but, um. 345 00:40:20.550 --> 00:40:25.920 Yeah, of course wait for. 346 00:40:27.510 --> 00:40:35.940 This is for. 347 00:40:38.730 --> 00:40:48.599 Point 15 divided by point. 5 equals point 3 so it's larger and so on. So so here, they're more strongly. So. 348 00:40:48.599 --> 00:40:55.679 Correlation quite usually fairly small to get anywhere up near 1. you've got an incredibly strong correlation. So. 349 00:40:55.679 --> 00:40:59.940 And I might have made a mistake here. You're welcome to call me on. 350 00:40:59.940 --> 00:41:09.179 Okay, so does this make sense somewhat. 351 00:41:25.650 --> 00:41:30.269 Just a 2nd, 15 divided by 2. 5. 352 00:41:30.269 --> 00:41:34.289 Calls, um, point 6. 353 00:41:36.449 --> 00:41:41.460 I think probably. 354 00:41:44.039 --> 00:41:50.519 Okay, um, so that's basically review of some of the, um. 355 00:41:51.570 --> 00:41:59.849 Stuff so okay, let me hit, uh, some new stuff in chapter 7 a little, um. 356 00:42:05.460 --> 00:42:08.909 Which would be, um, age, whatever. 357 00:42:08.909 --> 00:42:19.170 359 roughly. Okay. I'm going to hit some highlights of that. Um. 358 00:42:19.170 --> 00:42:23.429 We're getting into estimation again, which is a big thing. Um. 359 00:42:26.190 --> 00:42:35.340 And so I'll be having a lot more and getting into statistics, which I'll talk about to find that later. Um, so just, um. 360 00:42:35.340 --> 00:42:41.190 So, you've got an random variables X1 up to X and. 361 00:42:41.190 --> 00:42:45.179 Some, some of all the. 362 00:42:46.619 --> 00:42:50.190 Okay, the expected value of. 363 00:42:50.190 --> 00:43:00.570 There's a sum of all the expected values of the, because you can some expected values. You can always some expected values doesn't matter if the variables are correlated or not. 364 00:43:00.570 --> 00:43:06.809 Okay um, and if we want. 365 00:43:07.829 --> 00:43:15.420 The variants, um, so. 366 00:43:18.420 --> 00:43:21.929 The variants of, um. 367 00:43:27.480 --> 00:43:30.570 Well, we can use the formula, um. 368 00:43:32.969 --> 00:43:38.190 Basically is expected value of the van minus sector value of this event. 369 00:43:40.769 --> 00:43:45.630 Um, which nets out to, um. 370 00:43:58.860 --> 00:44:07.710 Okay, and, um, um. 371 00:44:09.869 --> 00:44:11.155 That good value 372 00:44:27.235 --> 00:44:28.405 and well, 373 00:44:28.434 --> 00:44:30.594 I'm gonna skip a step here and, 374 00:44:30.594 --> 00:44:31.045 um. 375 00:44:31.320 --> 00:44:38.610 It falls fairly easy. Some of all the separate variances. 376 00:44:38.610 --> 00:44:42.449 Plus, the, some of all the covariances so. 377 00:45:06.510 --> 00:45:10.079 Oh, oh, oh, oh, um. 378 00:45:11.905 --> 00:45:26.244 Hello. 379 00:45:27.510 --> 00:45:35.280 Hello? Hello? Hello? 380 00:45:40.320 --> 00:45:46.260 Hello? Hello? 381 00:45:46.260 --> 00:45:58.469 Hello? Hello? Hello? Hello? Hello? Hello? 382 00:45:58.469 --> 00:46:02.250 Hello. 383 00:46:04.139 --> 00:46:10.019 Hello? Hello? Hello? 384 00:46:10.019 --> 00:46:14.400 Hello? Hello? Hello? Hello? 385 00:46:18.989 --> 00:46:22.739 I'm. 386 00:46:24.360 --> 00:46:29.250 Great. 387 00:46:30.780 --> 00:46:35.130 Hello? Hello? Hello? 388 00:46:37.619 --> 00:46:41.940 Okay. 389 00:46:41.940 --> 00:46:45.599 Alright. 390 00:46:45.599 --> 00:46:48.599 Yeah. 391 00:46:58.260 --> 00:47:01.829 Hello. 392 00:47:03.329 --> 00:47:08.219 Oh, oh. 393 00:47:08.219 --> 00:47:11.760 Hello. 394 00:47:15.659 --> 00:47:18.719 Oh, oh, oh. 395 00:47:27.355 --> 00:47:30.655 Okay. 396 00:47:43.500 --> 00:47:50.159 Hello? Hello? 397 00:53:20.579 --> 00:53:23.730 Connecting connecting. 398 00:53:25.260 --> 00:53:35.250 Their screen and it's recording and just for fun. 399 00:53:36.690 --> 00:53:39.750 Well, 2 of you guys hi. Um. 400 00:53:40.860 --> 00:53:49.170 Oh, okay so the sample mean has a mean. Okay. 401 00:53:49.170 --> 00:53:53.909 Now, this starts making your brain go, but, um. 402 00:54:10.559 --> 00:54:15.780 Every time we take a different sample, we have a different sample mean, all these different sample means. 403 00:54:15.780 --> 00:54:18.929 Find their mean, okay sample. 404 00:54:18.929 --> 00:54:22.320 Hello. 405 00:54:22.320 --> 00:54:25.409 Does it mean in. 406 00:54:27.414 --> 00:54:40.405 Okay. 407 00:54:40.829 --> 00:54:44.639 Valley. 408 00:54:45.900 --> 00:54:56.849 Okay, okay. 409 00:54:58.050 --> 00:55:01.440 So so the, uh. 410 00:55:01.440 --> 00:55:05.639 Um, so if I. 411 00:55:05.639 --> 00:55:09.119 I need to get get the meat at all. 412 00:55:26.519 --> 00:55:30.059 On the 30. 413 00:55:30.059 --> 00:55:36.690 Yes. 414 00:55:36.690 --> 00:55:40.829 If you're, I mean. 415 00:55:44.155 --> 00:56:13.373 Okay. 416 00:56:15.235 --> 00:56:30.085 He could say or something. 417 00:56:30.539 --> 00:56:33.539 And and. 418 00:56:33.539 --> 00:56:37.590 So, yeah. 419 00:56:37.590 --> 00:56:40.619 I mean, it should, should. 420 00:56:40.619 --> 00:56:44.099 Hello hi if you. 421 00:56:44.099 --> 00:56:48.360 You think the 3rd thing would make sense. 422 00:56:48.360 --> 00:56:51.840 Okay um, so now back to this, um. 423 00:56:51.840 --> 00:56:59.820 So, we then we add random variables that are mean, is the sample main sample mean. 424 00:56:59.820 --> 00:57:06.510 Its mean is the population mean, but it does bounce around somewhat. Okay. Um. 425 00:57:07.650 --> 00:57:16.409 Is a precise term 2nd value. Does that mean bounce around? 426 00:57:21.900 --> 00:57:27.210 Um, because the thing is that every different sample will have a different sample mean. 427 00:57:27.210 --> 00:57:32.730 So, they're all gonna be different. So, um, so what I want. 428 00:57:34.019 --> 00:57:40.440 Is the variance the variance of the population mean. 429 00:57:41.579 --> 00:57:44.820 Okay, and that if the. 430 00:57:46.019 --> 00:57:55.440 It's the or not correlated then the variance. 431 00:57:55.440 --> 00:58:00.929 It goes a sum equals end times the variance of 1 of the exercise. 432 00:58:02.130 --> 00:58:07.019 Hello, because though. 433 00:58:07.019 --> 00:58:10.230 Bill that okay. Um. 434 00:58:12.179 --> 00:58:18.300 Sorry, I'm wrong. Um, I'm going completely now. Um. 435 00:58:23.820 --> 00:58:31.469 Variants of the sample mean, because the variance, and it gives the definition 1 over and. 436 00:58:31.469 --> 00:58:35.610 Times the top of all the exercise. 437 00:58:37.110 --> 00:58:42.780 Um, now, burian says units are squares that comes out as 1 over and squared. 438 00:58:42.780 --> 00:58:45.929 Variance of the sum of all the exercise. 439 00:58:47.250 --> 00:58:58.559 In red, um, so is this this is worth noticing here okay. 440 00:58:58.559 --> 00:59:04.050 I pulled a constant factor outside the variant, um, it squares. 441 00:59:04.050 --> 00:59:13.230 Um, because the units of variance of squares sort of squared lines. Okay. So, um. 442 00:59:14.880 --> 00:59:22.019 So, the event is this and and the variance of the song. 443 00:59:22.019 --> 00:59:32.219 Is it some of the variances it goes 1 over and square times and times Sigma squared. 444 00:59:32.219 --> 00:59:38.280 Equals, um, squared over and. 445 00:59:40.320 --> 00:59:46.380 So this says that a larger sample means the sample mean is a smaller variance. 446 00:59:48.360 --> 00:59:52.920 Larger sample, which is a larger. 447 00:59:54.000 --> 01:00:00.090 And smaller variance. 448 01:00:02.699 --> 01:00:06.119 Of the, um, sample mean. 449 01:00:10.380 --> 01:00:14.730 So, I, I think you'd probably mostly seen that before. So. 450 01:00:14.730 --> 01:00:21.989 Um, so the sample mean does not jump around as much. 451 01:00:21.989 --> 01:00:25.289 When the sample is larger, um. 452 01:00:26.699 --> 01:00:31.199 If I take samples of 4 students and. 453 01:00:31.199 --> 01:00:38.010 1234 take your average high 1234, your average, your average, your average. 454 01:00:38.010 --> 01:00:41.219 Then we got maybe 5 samples here. 455 01:00:41.219 --> 01:00:45.719 And the 5 sample means we'll have a reasonable spread. 456 01:00:45.719 --> 01:00:49.590 But that's where each sample of the 4 students, if I take. 457 01:00:49.590 --> 01:00:53.099 Samples of a 100 students go outside. 458 01:00:53.099 --> 01:00:56.489 But some sort of laser scanner outside and. 459 01:00:56.489 --> 01:01:00.960 Measure blocks of 100 students and get the hike. 460 01:01:00.960 --> 01:01:04.710 The main height of each sample of a 100 students. 461 01:01:04.710 --> 01:01:09.989 Those the means of those large samples will jump around a lot less. 462 01:01:09.989 --> 01:01:14.849 So more formally the variance of the sample mean. 463 01:01:14.849 --> 01:01:21.840 Um, is the population varies divided by N. 464 01:01:21.840 --> 01:01:27.389 Um. 465 01:01:28.800 --> 01:01:36.900 And that's worth, um, so, let me, uh, I want to hit this in more detail. 466 01:01:38.820 --> 01:01:46.289 Because, um. 467 01:01:46.289 --> 01:01:54.179 It's a little confusing perhaps. Um, let me put it in words. 468 01:01:56.820 --> 01:01:59.909 Pick a different example. 469 01:02:04.260 --> 01:02:11.639 Well, it'd be a nice example or something. Um. 470 01:02:32.699 --> 01:02:37.110 Spring Robins have come back, so that's observe robin's. 471 01:02:38.849 --> 01:02:42.000 These are people these are birds for 2 other countries. 472 01:02:42.000 --> 01:02:49.889 Okay, um, is the number of minutes. 473 01:02:53.639 --> 01:02:56.849 For number I thing. 474 01:02:56.875 --> 01:03:11.034 Things today. Okay. And maybe I'm facing an average of, I don't know, 30 minutes or something, so you might have being, you know, 30,283,520, 40. 475 01:03:15.389 --> 01:03:19.500 32 whatever. Okay. Um. 476 01:03:19.500 --> 01:03:23.400 And so the whole population. 477 01:03:28.679 --> 01:03:36.900 Say is, I don't know how many Robins there are in the Albany area. Maybe say, 10,000. 478 01:03:36.900 --> 01:03:43.800 Problems okay and maybe the, um. 479 01:03:45.119 --> 01:03:48.929 So you've got from I, from 1 to 10,000. 480 01:03:48.929 --> 01:03:52.619 And we might have the population mean. 481 01:03:54.389 --> 01:04:00.119 It was 30 minutes so the, the average Robin was singing 30 minutes let's say. 482 01:04:00.119 --> 01:04:04.980 A day and we compute the standard deviation. 483 01:04:04.980 --> 01:04:08.460 Was maybe 10 minutes. Okay. 484 01:04:10.800 --> 01:04:15.929 So, um, it's, uh, it's a, and let's say. 485 01:04:18.449 --> 01:04:21.750 The state okay. Um. 486 01:04:24.239 --> 01:04:30.869 I'm saying that. Okay. Okay. Um. 487 01:04:30.869 --> 01:04:38.670 So, we look at so random variable is we pick a random Robin and how many. 488 01:04:39.719 --> 01:04:46.260 And we pick how many minutes? It's things. Okay. 489 01:04:46.260 --> 01:04:50.849 And so it's things with an average of 30 minutes a day standard deviation at 10. 490 01:04:50.849 --> 01:04:55.050 Which means you can do a Gaussian here. Um. 491 01:04:57.719 --> 01:05:02.519 30 and 20 and 40. okay. 492 01:05:02.519 --> 01:05:06.719 So this is and this is the, um. 493 01:05:06.719 --> 01:05:14.610 Perfect, that's that. And then so this is a chance to review random a Gaussian random variables. 494 01:05:14.610 --> 01:05:22.530 And so let's give or take a 3rd here and a 3rd here, and I can't remember what the next ones are. Um. 495 01:05:24.119 --> 01:05:28.139 But in any case, so the probability. 496 01:05:28.139 --> 01:05:34.500 That it's saying, say, um, more than, um, 40 minutes. 497 01:05:34.500 --> 01:05:41.130 This is, um, this will plus Sigma will be about 106 so good. 498 01:05:41.130 --> 01:05:46.289 And so on, okay, now that's for 1 bird now. Um. 499 01:05:48.480 --> 01:05:58.530 Let's take a sample of, um, of 4 birds. Let's say. 500 01:06:01.349 --> 01:06:07.139 Okay um, so we've got say, you know. 501 01:06:07.139 --> 01:06:11.699 Okay, and so, you know, 1st sample might be. 502 01:06:11.699 --> 01:06:22.590 32,253,329 sample 2 might be 40 302,531 sample 3. 503 01:06:22.590 --> 01:06:27.510 29,293,230 and so on okay. 504 01:06:27.510 --> 01:06:30.780 And sample of 4 birds. 505 01:06:30.780 --> 01:06:35.940 We have a population mean, so we have a sample mean. 506 01:06:41.159 --> 01:06:44.969 Okay. 507 01:06:44.969 --> 01:06:51.389 Am I am okay now all the sample means. 508 01:06:54.150 --> 01:07:05.610 I mean, and so the expected value of all the sample means that's the population mean, which is 30 minutes a day. 509 01:07:05.610 --> 01:07:15.840 But the thing is, each sample of 4 birds has a different, um, you know, how are these sample means spread around well, I can do the variance. 510 01:07:15.840 --> 01:07:20.880 Of all the sample means and that's going to be. 511 01:07:22.170 --> 01:07:27.690 Sigma squared over and and the standard deviation all the sample means will be. 512 01:07:27.690 --> 01:07:31.619 Squared Sigma divided by a square root. 513 01:07:32.940 --> 01:07:38.190 So, if an equals 4, then for the sample mean. 514 01:07:38.190 --> 01:07:44.219 The sample Sigma is going to be 15, um, is going to be 5. 515 01:07:45.929 --> 01:07:49.320 Okay, um. 516 01:07:49.320 --> 01:07:54.840 So, if I go back here, so this was the distribution for. 517 01:07:54.840 --> 01:07:59.340 Individually for the whole population for individual birds now. 518 01:07:59.340 --> 01:08:04.019 Samples of 4 birds are going to be something like, um. 519 01:08:08.130 --> 01:08:14.369 It'd be a lot more tightly packed in. Um, so. 520 01:08:16.710 --> 01:08:19.829 Distribution of the sample mean. 521 01:08:21.539 --> 01:08:26.850 It's tighter, it's just and. 522 01:08:26.850 --> 01:08:37.890 If I had a larger sample mean, I'll tell you 100 birds. That would be really like, even tighter. Okay. Does this make intuitive sense? This is. 523 01:08:37.890 --> 01:08:44.399 Um, you have to really intuitively understand these ideas here about. 524 01:08:44.399 --> 01:08:51.899 Sample means so, individual birds, the, um, the random variables. How many minutes they sing today. 525 01:08:51.899 --> 01:09:00.600 Mean, 30 s standard deviation 10. if I take 4 birds 4 by 4 by 4 and take the average. 526 01:09:00.600 --> 01:09:05.340 Of those 4 birds singing time and that will have a standard deviation of. 527 01:09:05.340 --> 01:09:10.979 If I take birds 100 by 100 by 100. here's 100. here's 100. here's 100. 528 01:09:10.979 --> 01:09:16.260 And take the mean of each group call it a sample of 100 birds. 529 01:09:16.260 --> 01:09:20.340 Then that standard deviation is only 1 so. 530 01:09:20.340 --> 01:09:24.479 It gets a lot tighter and the exact formula. 531 01:09:26.100 --> 01:09:29.640 Is that is right here. 532 01:09:29.640 --> 01:09:36.960 The standard deviation for the meaning of the sample is a standard deviation. The whole population divided by square root event. 533 01:09:38.069 --> 01:09:41.760 Okay, this makes sense. 534 01:09:43.770 --> 01:09:48.270 Okay, so now, um. 535 01:09:51.779 --> 01:09:56.609 And this is this is 1 point part of the law of large numbers actually. So. 536 01:10:00.119 --> 01:10:08.729 And you can use Harris things like Chevy, sharp and so on that, I showed you last time. Um. 537 01:10:10.229 --> 01:10:20.100 Okay, so. 538 01:10:21.989 --> 01:10:34.710 So, a question might be unknown. True population mean. 539 01:10:37.050 --> 01:10:48.090 This is new to the sample mean and I, I'll use chevy's have, um, not won't work a 2 line by line. 540 01:10:48.090 --> 01:10:55.409 Yeah, we should have any quality and what we're going to get is that, um. 541 01:10:59.069 --> 01:11:06.989 Um, we'll go ahead and get the produce this. There'll be, um, actually, 7 point equation 7.19 on. 542 01:11:06.989 --> 01:11:14.579 366, so it'd be the probability that my sample mean. 543 01:11:14.579 --> 01:11:23.880 Is, um, is greater than some Epsilon. 544 01:11:23.880 --> 01:11:37.199 Is that cool? Well, this is saying is, the sample mean is bouncing around the true unknown population mean. 545 01:11:37.199 --> 01:11:41.130 But the probability that it's more than Epsilon is. 546 01:11:41.130 --> 01:11:45.689 So, on squared, so. 547 01:11:50.369 --> 01:11:56.970 Squared here, so, in other words, I have the sample mean then the true population mean. 548 01:11:56.970 --> 01:12:02.430 It's not going to be that far away in this bounds. How often the true unknown mean. 549 01:12:02.430 --> 01:12:11.130 Is from the measured sample, meaning as then gets bigger this probability gets smaller. And this is also a very loose bound. Um. 550 01:12:11.130 --> 01:12:21.060 So, there's a real number is much smaller. Okay. 551 01:12:22.529 --> 01:12:27.420 Any case so okay. Um. 552 01:12:30.119 --> 01:12:34.229 So, I have a lot of notes I typed up on the blog. 553 01:12:34.229 --> 01:12:45.869 Using this for sampling and so on, you know, elections get in the news, you want to find out, you ask people, what's it gonna vote? Or are they gonna buy some brand to soap or something? 554 01:12:45.869 --> 01:12:48.960 And so. 555 01:12:50.640 --> 01:12:55.439 You measure a sample and then you'll see things are reported in the papers that. 556 01:12:55.439 --> 01:12:58.470 You know, this is within 5 percentage points. 557 01:12:58.470 --> 01:13:04.260 There are 2 points of the 2 result, and, you know, 95 times out of a 100 or something. 558 01:13:04.260 --> 01:13:09.510 Wow, that statement is based on math like this. Okay. So. 559 01:13:10.590 --> 01:13:20.550 Okay, um, and we're getting things like week, law, large numbers. Um. 560 01:13:31.199 --> 01:13:34.260 We call it large numbers, which is, um. 561 01:13:41.970 --> 01:13:47.039 It then goes to infinity, then the probability that our. 562 01:13:47.039 --> 01:13:52.350 Sample Maine is bigger than is. 563 01:13:55.590 --> 01:14:02.670 So, what this says is the probability of the sample mean, being with an Epsilon of the true mean, goes to 1, it then gets bigger and bigger. 564 01:14:02.670 --> 01:14:15.210 Oh, okay. Um, and a stronger law of large numbers says, um. 565 01:14:19.260 --> 01:14:24.390 The probability minor changes um, this assumes. 566 01:14:28.079 --> 01:14:32.699 Population variance is finite. 567 01:14:35.130 --> 01:14:41.850 Or you go to Las Vegas, you're looking at cards or something you're trying to observe stuff like that. Okay. 568 01:14:41.850 --> 01:14:48.180 Oh, um. 569 01:14:51.960 --> 01:15:02.609 Yeah, the, um, well, the strong 1 is similar and the central limit theorem is what says as then distribution start looking like, if, um. 570 01:15:09.420 --> 01:15:14.460 Is that is that a population mean? 571 01:15:18.779 --> 01:15:23.729 It comes and. 572 01:15:23.729 --> 01:15:31.050 And goes to infinity, regardless of the underlying probability distribution. So. 573 01:15:34.529 --> 01:15:37.649 How's the. 574 01:15:37.649 --> 01:15:45.989 Okay. 575 01:15:45.989 --> 01:15:55.590 If, uh. 576 01:15:55.590 --> 01:16:00.869 Variances are fine item. 577 01:16:02.159 --> 01:16:05.880 All that all this stuff. 578 01:16:05.880 --> 01:16:09.329 So, okay. 579 01:16:21.960 --> 01:16:25.979 Okay, uh, so that's, um, so. 580 01:16:25.979 --> 01:16:29.069 Limit theorem is. 581 01:16:29.069 --> 01:16:34.020 Page 306 to 9, so. 582 01:16:35.039 --> 01:16:39.810 Okay, so today it's growing, you're starting you off on chapter 7. 583 01:16:39.810 --> 01:16:43.529 Um, I'd like to show, you. 584 01:16:43.529 --> 01:16:47.880 Some stuff from my thing here. 585 01:16:47.880 --> 01:16:51.720 And it's enrichment um, I'll. 586 01:16:56.609 --> 01:17:01.590 Yeah, well. 587 01:17:02.095 --> 01:17:08.274 Maybe, um, next week, if you've got good tools, you can do more work. 588 01:17:08.274 --> 01:17:19.345 I showed mathematic Matlab, which is nice for working with matrices and so on work with stuff like that I have estimation here. He's got some of them here. 589 01:17:19.404 --> 01:17:25.734 We want to look at that point, we're more focused on profitability. 590 01:17:26.460 --> 01:17:32.039 And mobility, we know what the density function as the distributions are. 591 01:17:32.039 --> 01:17:36.659 Like, we know it's a fair coin or a fair dye, and we can keep probabilities. 592 01:17:36.659 --> 01:17:44.939 Statistics goes the other direction we don't know. And we know that we, you know, with the coins that you don't mean to say, it wasn't statistical for the pressure. 593 01:17:44.939 --> 01:17:53.100 We're making observations, we do not know the means and segments and we determine or we may not me. So. 594 01:17:53.100 --> 01:17:57.270 Like, 1 of the historical examples keep mentioning was. 595 01:17:57.270 --> 01:18:01.710 Get us brewery and Ireland around 900 under. 596 01:18:01.710 --> 01:18:07.800 They wanted to get an idea of what the alcohol content of a batch appear was. 597 01:18:07.800 --> 01:18:19.470 So, they measure some, take some samples and measure alcohol contents. So, the mean would be the alcohol content of the whole batch of beer and they're taking some observations and trying to. 598 01:18:19.470 --> 01:18:24.960 Get a guess bound to get the alcohol content of the whole, the whole run. 599 01:18:24.960 --> 01:18:30.300 And so the mathematician name was Carson was deriving some math. 600 01:18:30.300 --> 01:18:37.140 Um, so you see, the statistics is, we do not know the underlying means it's all we want to determine that. 601 01:18:37.140 --> 01:18:40.800 A big example that's in the news is. 602 01:18:40.800 --> 01:18:47.100 Yeah, with drug and the drug company says perhaps the drug curve. 603 01:18:47.100 --> 01:18:54.060 Cure some disease and, you know, pick some, they pick your favorite favorite disease. 604 01:18:54.060 --> 01:18:58.859 And maybe the drug works, but the thing is, people are all different. 605 01:18:58.859 --> 01:19:04.590 So, the drug helps, some people does not help some people and kills every 1000 person. Maybe. 606 01:19:04.590 --> 01:19:11.010 So, is it on the whole good or bad? So the drug companies are doing experiments called clinical trials. 607 01:19:11.010 --> 01:19:15.960 And they're trying to infer on the average is the drug good or bad. 608 01:19:15.960 --> 01:19:22.170 And there's a lot of money involved in that in that. 609 01:19:22.170 --> 01:19:31.979 Getting a new drug through from idea to actually on the market might cost a couple of 1Billion dollars that's building with the beat couple of 10 of the 9 dollars. 610 01:19:31.979 --> 01:19:35.039 So this is statistics, um. 611 01:19:36.720 --> 01:19:43.859 Well, just to throw the macro numbers at the pharmaceutical industry, the total gross worldwide. 612 01:19:43.859 --> 01:19:48.029 Is, um, a 1Trillion dollars a year? 613 01:19:48.029 --> 01:19:51.840 10 of the 9 dollars 10 to 12 dollars a year. 614 01:19:51.840 --> 01:19:58.109 That's a growth worldwide income assets in the United States the 5 times 10 to the 11. 615 01:19:58.109 --> 01:20:06.989 So, a new drug is maybe 10 to the 9 dollars to develop or something. These are serious numbers. We're talking about. 616 01:20:06.989 --> 01:20:17.729 So you're gonna use some statistics now obviously you can do it wrong. You can be biased so there's, there's rules. So these are examples for what's going on. 617 01:20:17.729 --> 01:20:23.250 The statistics here, um, that's what statistics is now. 618 01:20:25.380 --> 01:20:31.319 Interesting property of statistics is some counterintuitive things can happen. 619 01:20:31.319 --> 01:20:36.390 I'll work through these things in more detailed next week, but I mentioned some things happen in statistics and. 620 01:20:36.390 --> 01:20:41.819 You can check them, they do not appear possible, but in fact, they happen. 621 01:20:41.819 --> 01:20:46.319 Some paradox, so so be talking about that and. 622 01:20:46.319 --> 01:20:53.100 Get into the statistics and so on. So you've got estimators and stuff like that testing hypothesis. 623 01:20:56.399 --> 01:21:01.199 Is this food drug good or bad then? Um. 624 01:21:01.199 --> 01:21:07.229 And again, things are going to be variable. So if you find the drugs, if 100 people take the drug. 625 01:21:07.229 --> 01:21:10.529 Then maybe, you know what, you know. 626 01:21:10.529 --> 01:21:16.229 55 get better and 45 show no difference or 40. I want to look at the society get worse. 627 01:21:18.029 --> 01:21:23.430 But if the drug did nothing, if it's a placebo, this thing's gonna be happening. So you're doing. 628 01:21:23.430 --> 01:21:33.090 What's called hypothesis testing and the way they worded is, is this drug is a placebo. How likely are we to see these facts that in fact we saw. 629 01:21:33.090 --> 01:21:36.119 So, talk about stuff like that. 630 01:21:37.409 --> 01:21:45.119 And I've got some other videos on that. Okay, so that's, um, enough stuff for today. Um. 631 01:21:45.119 --> 01:21:49.260 Good luck on the exam and I'll see you in a week. 632 01:21:54.359 --> 01:21:57.420 Mentioned it on last class. 633 01:21:57.420 --> 01:22:02.430 So. 634 01:22:02.430 --> 01:22:11.729 Hello. 635 01:22:13.109 --> 01:22:20.489 Hey, Hello? Hello? Yeah. Oh, let me just send to you on here. 636 01:22:20.489 --> 01:22:21.840 Session in there.