WEBVTT 1 00:00:26.850 --> 00:00:32.969 What happened here. 2 00:00:34.079 --> 00:00:39.869 Something. 3 00:00:46.945 --> 00:01:10.465 Okay. 4 00:01:19.890 --> 00:01:24.209 Hello. 5 00:01:24.209 --> 00:01:28.260 Let's see, I hope the screen is sharing at least. 6 00:01:28.260 --> 00:01:31.290 And that's just to see. 7 00:01:38.549 --> 00:01:41.819 Okay. 8 00:01:52.290 --> 00:01:55.469 Okay, so. 9 00:01:57.120 --> 00:02:01.829 So this would be. 10 00:02:10.500 --> 00:02:19.710 1 and Thursday. Okay. Um, the 1st reminder that the. 11 00:02:19.710 --> 00:02:23.189 2nd exam is next Thursday. 12 00:02:23.189 --> 00:02:26.280 Material up to chapter 6. 13 00:02:26.280 --> 00:02:35.460 May contain material from before the 1st test so going to concentrate on recent stuff, but the Emmy write this down. 14 00:02:43.319 --> 00:02:55.650 Okay, okay. 15 00:02:55.650 --> 00:02:58.740 Okay. 16 00:02:58.740 --> 00:03:07.919 So stuff I want to do today, I'm continuing on with likelihood stuff and normal variable stuff and so on. And, um. 17 00:03:07.919 --> 00:03:12.449 Just, um, and I'll jump back and forth a little. 18 00:03:12.449 --> 00:03:18.840 Um, just to remind you about some stuff involving normal variables. So the, um. 19 00:03:22.650 --> 00:03:34.530 Okay, so Webex of course this is for me and equals 0 on the standard deviation equals 1. 20 00:03:34.530 --> 00:03:40.919 Um, 1st, yes, um. 21 00:03:45.150 --> 00:03:49.560 Okay, yes. 22 00:03:59.849 --> 00:04:05.789 Okay, sure and this is for. 23 00:04:09.000 --> 00:04:14.969 Okay, and that's your thing that looks like that. And just to remind you. 24 00:04:14.969 --> 00:04:20.310 Why this is so useful this is a review for a minute or 2. um. 25 00:04:22.019 --> 00:04:22.709 So, 26 00:04:26.334 --> 00:04:37.915 okay, 27 00:04:38.425 --> 00:04:39.415 like this. 28 00:04:44.728 --> 00:04:48.028 Or large numbers, and so on. 29 00:04:48.028 --> 00:04:53.639 Okay, and a review I showed you before, um. 30 00:04:54.959 --> 00:05:01.288 Prove that dinner girl Equifax D x equals 1. 31 00:05:02.608 --> 00:05:06.028 I did this before on just review it. So let a call the undergo. 32 00:05:06.028 --> 00:05:19.319 Um, okay, well they also equals I could do Y, instead of X. um. 33 00:05:24.598 --> 00:05:31.228 I just changed my variables. Um, so now, if I'm so not, but I can multiply the to them. I get a squared. 34 00:05:31.228 --> 00:05:35.459 Hey, calls. 35 00:05:42.418 --> 00:05:45.959 Okay, so now I'll modify the 2 of them and now I've got this. 36 00:05:45.959 --> 00:05:53.158 Um, I go too fast. Let me know. 37 00:05:53.158 --> 00:05:57.778 And change, I'm sorry. 38 00:05:57.778 --> 00:06:03.358 Oh, I multiplied the 1st, 1 by the 2nd 1. 39 00:06:04.499 --> 00:06:16.048 No, 1 reply them. Okay. 40 00:06:21.928 --> 00:06:28.738 Hmm okay. 41 00:06:35.189 --> 00:06:40.619 I do a black maybe between okay now we change variables. 42 00:06:47.399 --> 00:06:56.819 I hear the range that all different. Are. 43 00:06:58.319 --> 00:07:02.968 And, okay. 44 00:07:04.559 --> 00:07:07.709 So, now it will turn out you have this, um. 45 00:07:07.709 --> 00:07:16.079 To Colby in whatever you call it sometimes it's called a, and actually, um. 46 00:07:17.879 --> 00:07:21.028 So, all your derivatives. 47 00:07:22.108 --> 00:07:26.519 Um, our, so on. 48 00:07:31.769 --> 00:07:35.579 Okay, cause so this will be say the, um. 49 00:07:39.418 --> 00:07:44.218 Dx Y, divided by data, you know. 50 00:07:44.218 --> 00:07:54.209 And that will be, um. 51 00:08:01.858 --> 00:08:08.098 Hello. 52 00:08:09.713 --> 00:08:10.314 We are. 53 00:08:24.809 --> 00:08:28.199 I can say to turn out to be. 54 00:08:28.199 --> 00:08:31.559 Turned out to be Co said. 55 00:08:31.559 --> 00:08:35.578 Our go. 56 00:08:41.278 --> 00:08:49.528 Julie, um, and that won't be. 57 00:08:53.999 --> 00:08:59.698 Okay, so now, what we would have is D, x D. Y. 58 00:08:59.698 --> 00:09:05.399 We'll be R. D. R. D. 59 00:09:06.568 --> 00:09:11.849 So now a squared, if I scroll back up here, it's this I change my variables. 60 00:09:14.009 --> 00:09:18.719 And it's going by, um, either the minus, um. 61 00:09:56.339 --> 00:10:03.298 To infinity, let's say. 62 00:10:06.298 --> 00:10:12.058 Is there a reason why I'm what? Yes. 63 00:10:12.058 --> 00:10:18.119 To make it a workable cause if I can scroll, but I see I didn't know how to integrate that. Okay. 64 00:10:25.019 --> 00:10:30.448 You know how to integrate it that's the. 65 00:10:31.739 --> 00:10:40.318 1 equals 1. 66 00:10:40.318 --> 00:10:46.469 It says illegal probability. Pdf. Okay. 67 00:10:48.683 --> 00:11:06.234 Okay. 68 00:11:11.009 --> 00:11:21.688 So now I can integrate it. Okay. Okay. 69 00:11:21.688 --> 00:11:25.589 That's the 1 thing now 2 variable. 70 00:11:31.469 --> 00:11:46.349 Okay, the 1 over 2 Pi vanished. Um, because. 71 00:11:46.349 --> 00:11:50.129 Inner grows 0 to 2 pie. 72 00:11:50.129 --> 00:11:55.379 A close to pie. That's what canceled it out. 73 00:12:00.683 --> 00:12:16.163 Okay. 74 00:12:18.538 --> 00:12:28.828 Too variable, um. 75 00:12:35.818 --> 00:12:42.119 The variable and, um. 76 00:12:42.119 --> 00:12:46.109 That will have half of X and Y. 77 00:12:46.109 --> 00:12:50.219 Um. 78 00:12:57.774 --> 00:13:16.703 Okay, 79 00:13:17.004 --> 00:13:18.024 X and Y. 80 00:13:18.749 --> 00:13:23.308 And row correlation coefficient. 81 00:13:28.798 --> 00:13:35.038 And it is clickable 1, talked about it before just to remind you. 82 00:13:35.038 --> 00:13:43.349 If Roe equals 0, there's no relation to next. And why is row approach is 1 they're positively connected to each other. 83 00:13:43.349 --> 00:13:47.219 And a counter chart of this. 84 00:13:48.719 --> 00:13:51.928 A counter plot would look something like this. Um. 85 00:13:54.839 --> 00:13:59.668 And why it's not 1. 86 00:13:59.668 --> 00:14:04.019 Um, this would be real greater than 0. 87 00:14:05.129 --> 00:14:10.048 Um, a Contra plot with row lessons arrow. 88 00:14:10.048 --> 00:14:16.979 I looked at something like that equals 0, it's going to look something like that. 89 00:14:18.958 --> 00:14:23.698 Okay, so on okay now, um. 90 00:14:23.698 --> 00:14:33.389 What I want to do is say, um, Charles is showing you techniques, I'm showing you the specific result, but more generally, I'm showing you techniques of working with. 91 00:14:33.389 --> 00:14:37.889 So, um, right this down. 92 00:14:44.219 --> 00:14:49.948 Specific results in general techniques. 93 00:14:55.438 --> 00:14:58.558 Okay, okay now. 94 00:14:58.558 --> 00:15:02.428 What we ought to have what I'm going to do now. 95 00:15:02.428 --> 00:15:07.349 Is I'm going to prove that if we integrate and Y. 96 00:15:07.349 --> 00:15:12.658 Y, um, we're going to get half of X. 97 00:15:12.658 --> 00:15:20.548 Um, the 2 variable, so the 2 variable, 1, if I do the marginal density function, I'll get the 1 variable calcium. 98 00:15:20.548 --> 00:15:24.568 I just want wanted a side note here. Um. 99 00:15:26.999 --> 00:15:33.629 So, you 1 can imagine, um. 100 00:15:36.839 --> 00:15:44.548 You know, many ways we'll have a 2 variable. Um. 101 00:15:46.619 --> 00:15:51.688 You know, PDF, you know, with some okay. 102 00:15:55.019 --> 00:15:58.739 You know, here is a specific. 103 00:16:02.099 --> 00:16:05.578 So 1 okay, so. 104 00:16:05.578 --> 00:16:13.708 Um, okay, now, let me just see if I got the, um. 105 00:16:15.418 --> 00:16:21.538 I got it so vaguely correct here and. 106 00:16:23.004 --> 00:16:47.303 Okay. 107 00:17:01.048 --> 00:17:05.068 Sure, it is in the book, but I think I have it um. 108 00:17:05.068 --> 00:17:11.638 Okay, so let me show you this thing. 109 00:17:11.638 --> 00:17:14.939 Here, how how about how this works? Um. 110 00:17:16.679 --> 00:17:20.788 Okay, I'm going to show that, um, again. 111 00:17:27.598 --> 00:17:32.249 Okay. 112 00:17:32.249 --> 00:17:42.328 Sure, I got it. Right. Okay. Show that. 113 00:17:43.709 --> 00:17:49.229 Okay, so. 114 00:17:49.229 --> 00:17:58.888 Yes, 2 pie -1-rose Square. 115 00:17:58.888 --> 00:18:07.588 Square root there. I believe. 116 00:18:07.588 --> 00:18:13.469 I'm trying to find what page it is in the textbook here. 117 00:18:21.568 --> 00:18:25.949 I'll I'll look for the page number, so I can quote it to you then. Um. 118 00:18:28.558 --> 00:18:32.278 Yeah. 119 00:18:32.278 --> 00:18:36.659 Yeah, let me I'm just looking for the trying to get the page number here. 120 00:18:55.439 --> 00:19:00.538 Yeah, okay. This would be equation 518 on page 253. 121 00:19:06.088 --> 00:19:11.699 Okay, yeah, your question you had a question. 122 00:19:11.699 --> 00:19:15.419 Okay, okay. Um. 123 00:19:17.219 --> 00:19:24.659 So so I'm integrating this, um, I can split it up to. 124 00:19:33.898 --> 00:19:36.989 C to the, um. 125 00:19:36.989 --> 00:19:50.219 Minus, um, okay split. I put that up into that. Now, integrate this by. 126 00:19:50.219 --> 00:19:56.219 Why okay, um, so what I'm going to do is. 127 00:19:57.989 --> 00:20:03.239 Keep the square on, um. 128 00:20:07.288 --> 00:20:11.638 And what I can do there is, I can. 129 00:20:12.689 --> 00:20:21.568 At added and subtracted. 130 00:20:21.568 --> 00:20:26.009 And this thing here is a row X . 131 00:20:26.009 --> 00:20:29.909 Y, squared minor throw square deck Square. 132 00:20:30.989 --> 00:20:34.108 Um, so this thing here. 133 00:20:37.709 --> 00:20:40.709 That's saying is this is going to be. 134 00:20:40.709 --> 00:20:49.169 And I'm sorry, a good note doesn't display 2 pages at once. But, um. 135 00:20:49.169 --> 00:20:56.189 This will be oh, and we're going to minus in front. Don't forget that. And so this thing here that I boxed. 136 00:20:56.189 --> 00:21:00.419 Oh, let me call it be is going to be, um. 137 00:21:04.679 --> 00:21:08.788 Um. 138 00:21:17.219 --> 00:21:20.788 Minus X squared plus, um. 139 00:21:25.858 --> 00:21:29.848 Plus here, because I got a minus a . 140 00:21:31.199 --> 00:21:36.689 And or girl. 141 00:21:38.159 --> 00:21:45.838 To the minus sorry. 142 00:21:55.259 --> 00:22:00.479 Okay, if I have that correctly. 143 00:22:00.479 --> 00:22:05.189 So. 144 00:22:07.888 --> 00:22:12.388 Better to do a change of variables. Um. 145 00:22:12.388 --> 00:22:16.138 So let's say W, equals Y, over. 146 00:22:18.628 --> 00:22:26.939 Go to 1-row squared. So now this thing here. 147 00:22:26.939 --> 00:22:36.388 It's going to be either or minus W W squared over 2 and now. 148 00:22:37.469 --> 00:22:45.118 Will be 3rd 1-row squared. Let me write that more clearly. 149 00:22:49.318 --> 00:22:53.038 So that, um, so this thing here. 150 00:22:53.038 --> 00:22:56.939 Becomes that and get another color. 151 00:22:56.939 --> 00:23:00.358 This thing here. 152 00:23:00.358 --> 00:23:04.439 Becomes, um, and to grow. 153 00:23:04.439 --> 00:23:09.868 Good 1-row squared is a . 154 00:23:09.868 --> 00:23:13.078 W squared W. 155 00:23:13.078 --> 00:23:18.028 Over 2 and that here and the Sander girl. 156 00:23:18.028 --> 00:23:23.729 That'll be square and go to 1-row squared because this thing here. 157 00:23:23.729 --> 00:23:28.439 21, and then so be will be. 158 00:23:29.578 --> 00:23:40.288 Um, well, that and that cancel out and B will be on top is the next square to actually. 159 00:23:40.288 --> 00:23:43.679 1-row squared this will net out to be. 160 00:23:43.679 --> 00:23:47.519 Minus sorry. 161 00:23:49.769 --> 00:23:57.449 Of course over. 162 00:24:05.459 --> 00:24:08.939 I don't want to get a square root a 2 pie here. Ummm. 163 00:24:10.528 --> 00:24:14.548 There's somewhere else we lost, uh, to tie, but basically, um. 164 00:24:21.298 --> 00:24:28.048 Okay, I see. 165 00:24:29.398 --> 00:24:34.288 But, basically what I showed. 166 00:24:34.288 --> 00:24:39.088 That's an interesting 1 here happened here. 167 00:24:42.419 --> 00:24:47.848 So, what I showed. 168 00:24:47.848 --> 00:24:52.769 Is that the under girl. 169 00:24:58.709 --> 00:25:02.669 X, which is good. Okay. Um. 170 00:25:02.669 --> 00:25:11.729 Yeah, 2 square to 25 fill out somewhere. That's the idea. Okay now um. 171 00:25:11.729 --> 00:25:16.318 So, if we've got that, then we can find, um. 172 00:25:16.318 --> 00:25:21.898 conditionals, so, because f, uh, say. 173 00:25:22.979 --> 00:25:28.078 I give an X is why. 174 00:25:28.078 --> 00:25:32.189 Bye bye. Okay. 175 00:25:33.959 --> 00:25:38.009 So, we can find that and. 176 00:25:39.568 --> 00:25:42.929 So f, effects of why was got it right? 177 00:25:46.433 --> 00:25:59.513 Okay, bye. 178 00:26:04.618 --> 00:26:09.028 And that will be, um. 179 00:26:15.538 --> 00:26:21.719 And, um, is this something or other up here? 180 00:26:21.719 --> 00:26:34.648 Okay. 181 00:26:34.648 --> 00:26:41.159 Let me call it see here some big C. we're sequels. 182 00:27:16.169 --> 00:27:19.858 Um, and then this. 183 00:27:19.858 --> 00:27:25.288 Close out. Okay. 184 00:27:27.443 --> 00:27:28.134 Minus 185 00:27:29.243 --> 00:27:42.534 okay. 186 00:27:42.868 --> 00:27:46.588 So, the whole thing, the condition for that plugs in. 187 00:27:46.588 --> 00:27:51.028 And the whole, so the whole thing for the conditional is, um. 188 00:27:56.338 --> 00:28:00.898 This was I was doing an f. Y. you have an accent Chile. 189 00:28:07.648 --> 00:28:12.148 Yeah, why you have an X what's going on here? 190 00:28:24.719 --> 00:28:30.118 Why give an X is. 191 00:28:36.388 --> 00:28:39.509 Minus, um. 192 00:28:52.858 --> 00:28:56.338 Okay, now, um. 193 00:28:57.989 --> 00:29:05.848 Okay, I think sorry, X squared row squared. 194 00:29:05.848 --> 00:29:09.509 Okay, and there should be a row in here. 195 00:29:12.209 --> 00:29:23.788 Okay, so so what I have is the conditional probability density function of X given why. 196 00:29:23.788 --> 00:29:28.979 Works and Y, are 2 cosines that are correlated with the correlation coefficient of W. 197 00:29:28.979 --> 00:29:33.749 Oh, I can simplify to touch more the thing up here. 198 00:29:38.519 --> 00:29:43.199 That equals, um, row X minus Y squared. 199 00:29:43.199 --> 00:29:47.398 Okay, so the more simplified thing would be. 200 00:29:55.318 --> 00:30:03.659 You hit the minus. Okay. 201 00:30:03.659 --> 00:30:09.568 So, how is that useful? Is that. 202 00:30:09.568 --> 00:30:17.489 Um, we want to, um. 203 00:30:18.838 --> 00:30:23.939 So, we've got a noisy channel and so on. So, how is this useful? Um. 204 00:30:36.419 --> 00:30:44.818 No easy com, channel X and. 205 00:30:44.818 --> 00:30:49.078 Out, um. 206 00:30:53.068 --> 00:31:00.689 We say why it's X. 207 00:31:00.689 --> 00:31:05.009 Okay, now, um. 208 00:31:05.009 --> 00:31:08.308 I scroll back a little here. 209 00:31:08.308 --> 00:31:12.269 Oh, okay. Um. 210 00:31:14.038 --> 00:31:18.598 Now, there's different ways we could do this. I introduced them on Monday. 211 00:31:18.598 --> 00:31:25.558 Um, okay or different. 212 00:31:28.378 --> 00:31:37.019 There's different ways, depending on say what we know perhaps. 213 00:31:41.638 --> 00:31:48.479 So this 1 here, um, f, uh. 214 00:31:48.479 --> 00:31:53.729 I give an ax use ax. 215 00:31:53.729 --> 00:32:01.858 Prior okay, so, um, um. 216 00:32:01.858 --> 00:32:06.388 So, what the, what we would have here is something called. 217 00:32:07.528 --> 00:32:11.939 That was the maximum a method. 218 00:32:18.568 --> 00:32:22.798 So, we're given an f of Y. 219 00:32:23.939 --> 00:32:29.098 Try next to maximize that. Okay. 220 00:32:30.179 --> 00:32:37.828 Um, we want to find the X that has the greatest probability of generating that Y, that we saw. 221 00:32:37.828 --> 00:32:47.068 Okay, so I, I gave you the definition last time I'll work with the thing so if I can scroll back without making you 2, seasick. 222 00:32:47.068 --> 00:32:53.459 So we want to find so why is known um. 223 00:32:53.459 --> 00:33:02.429 So, I received, and they're both continuous. I'm doing this with continuous calcium distribution case. Um. 224 00:33:03.538 --> 00:33:11.759 So, we see a Y, that's, um, 3, what X what's the most probable X? What was the is most likely to give that why. 225 00:33:11.759 --> 00:33:21.868 Um, and looking at that formula up there, so we want to maximize that form. The f. Y, give an X. yes. 226 00:33:27.328 --> 00:33:33.298 Why is it. 227 00:33:37.409 --> 00:33:42.989 Well, we know X, so let's suppose. 228 00:33:42.989 --> 00:33:46.318 You know, we're adding noise, um. 229 00:33:51.388 --> 00:33:58.499 Um, well, half of why given x X is in the equation, I just have half of why. 230 00:33:58.499 --> 00:34:01.528 By itself, X will not be in that equation. 231 00:34:02.939 --> 00:34:13.108 There's a difference right there. And the meeting is that, I mean, that's the po's X was 5. 232 00:34:13.108 --> 00:34:17.909 Oh, okay so these are all these are both normal. Okay. Sigma 1 means 0. 233 00:34:17.909 --> 00:34:27.568 If I don't know, X, the most likely value of why it's going to be 0 also. Okay. But suppose I know that X was 3. 234 00:34:27.568 --> 00:34:34.018 Then, why, and why is X plus some noise? Let's say the most likely value for why. 235 00:34:34.018 --> 00:34:37.798 Is also 3, so if we know X. 236 00:34:37.798 --> 00:34:40.858 Know what else? Or? Maybe not 3, but something. 237 00:34:40.858 --> 00:34:45.179 So, if we know X is very big or very small. 238 00:34:45.179 --> 00:34:49.648 Then, why is probably going to be correspondingly big or small. 239 00:34:49.648 --> 00:34:56.699 Okay, so if I take that reasoning and turn it backwards. 240 00:34:58.079 --> 00:35:01.648 If I see a, why that's very big. 241 00:35:01.648 --> 00:35:06.688 Then it's, you know, I'm guessing common sense. The X was pretty big also. 242 00:35:06.688 --> 00:35:10.289 If I see a, why that's very small. 243 00:35:10.289 --> 00:35:16.228 Than common sense, X was probably very small also because X and Y are correlated with each other. 244 00:35:16.228 --> 00:35:23.938 Yeah, that this makes sense. So what I'm doing here is taking that common sense thing and quantifying it. 245 00:35:23.938 --> 00:35:29.278 Giving you the formula, so if that makes sense, then. 246 00:35:29.278 --> 00:35:33.358 So, if you can see what I'm trying to do. 247 00:35:33.358 --> 00:35:36.898 Before I do the actual math that. 248 00:35:36.898 --> 00:35:40.228 I'm transmitting an X I'm receiving a why. 249 00:35:40.228 --> 00:35:47.398 And what I receive is correlated with what was sent, but it's not exactly what was said. 250 00:35:47.398 --> 00:35:56.639 You know, there's some noise in there. So what I want to do is I want to make the best guess, as to what was sent. 251 00:35:56.639 --> 00:36:01.349 Based on what I received, so. 252 00:36:03.028 --> 00:36:06.929 So that's an English what I'm trying to do and then the next thing is the math. 253 00:36:08.608 --> 00:36:12.179 So does that make sense? 254 00:36:12.179 --> 00:36:20.818 Okay, so here's the equation for the conditional probability density of why give an X. 255 00:36:22.289 --> 00:36:27.449 So, if I want to, so what I want is. 256 00:36:27.449 --> 00:36:32.728 So, I saw a particular why I want to find what is the X. 257 00:36:35.009 --> 00:36:40.588 Um, which maximizes the probability of seeing that why. 258 00:36:40.588 --> 00:36:45.418 And this thing by the way, used to get that formula. 259 00:36:45.418 --> 00:36:51.688 I want to maximize it. I want to find the value of X that makes this thing as big as possible. 260 00:36:51.688 --> 00:36:55.949 Well, that's the thing that makes the, um. 261 00:36:55.949 --> 00:37:00.329 The exponent after the minus sign as small as possible. 262 00:37:00.329 --> 00:37:05.759 And so, so if I can make this expanded to 0, that's the best I can do. 263 00:37:05.759 --> 00:37:08.969 Cause there's a square up there, um. 264 00:37:08.969 --> 00:37:14.998 And I could make that exponent 0, if row X minus Y equals. 0. so. 265 00:37:33.659 --> 00:37:39.628 Okay um, so it turns out it actually. 266 00:37:39.628 --> 00:37:43.199 We're going to guess the next it's so much bigger than why um. 267 00:37:48.869 --> 00:37:52.079 Is bigger than it's greater than why here? Um. 268 00:37:53.458 --> 00:37:56.909 The reason it takes some thinking, um. 269 00:37:58.438 --> 00:38:01.829 But the noise will tend to make, um. 270 00:38:03.478 --> 00:38:06.778 Things sort of regret to the means so. 271 00:38:06.778 --> 00:38:11.760 Experts mangled with some noise and then to produce why. 272 00:38:11.760 --> 00:38:17.880 So, it's just that why will tend to be closer to 0 um. 273 00:38:17.880 --> 00:38:22.710 So, we're inferring, we're sort of back inference saying. 274 00:38:22.710 --> 00:38:27.780 The X that was most likely to cause this is I'm bigger than why. 275 00:38:30.659 --> 00:38:36.179 And, um, if I can. 276 00:38:37.289 --> 00:38:41.010 It's a page number for you, let's say. 277 00:38:44.429 --> 00:38:48.570 Okay, um. 278 00:39:04.105 --> 00:39:05.394 Okay, 279 00:39:22.735 --> 00:39:23.094 um, 280 00:39:23.094 --> 00:39:24.385 page 332. 281 00:39:25.889 --> 00:39:29.190 Hello sorry um. 282 00:39:29.190 --> 00:39:36.329 A, 2nd, here. 283 00:39:40.170 --> 00:39:47.099 Page. 284 00:39:59.460 --> 00:40:03.360 Uh, okay, I didn't here. Um. 285 00:40:09.989 --> 00:40:16.530 So this is actually what I did was the okay. 286 00:40:21.599 --> 00:40:26.309 This was the maximum likelihood Estimator here shortly. Um. 287 00:40:29.099 --> 00:40:33.780 So. 288 00:40:44.010 --> 00:40:47.760 So, this 1 was the maximum likelihood estimator. Um. 289 00:40:59.070 --> 00:41:03.659 And it's 2 different ones, depending on what we know. Okay. 290 00:41:03.659 --> 00:41:09.090 This would be page, um, 333 so that's. 291 00:41:09.090 --> 00:41:20.579 Example, okay um, the maximum method, um. 292 00:41:22.860 --> 00:41:30.210 Is yeah, I just a sec. 293 00:41:39.510 --> 00:41:45.389 Okay, just a 2nd, is if we want to maximize, um. 294 00:41:46.800 --> 00:41:51.300 Given why you don't Webex isn't working now. 295 00:41:53.280 --> 00:42:00.300 Okay, thank you. 296 00:42:05.340 --> 00:42:11.429 Here again. 297 00:42:14.190 --> 00:42:18.599 For some reason I start sharing and then it stops after. Well, how about now? 298 00:42:22.650 --> 00:42:27.690 I cannot hear okay, thank you. 299 00:42:27.690 --> 00:42:40.889 I think there's a problem with how webx is implemented on iPads or something. Like I said, it shares and then it stops. So keep telling me when it stops. Oh. 300 00:42:43.650 --> 00:42:49.559 Okay, any case. 301 00:42:55.644 --> 00:42:56.394 Okay, 302 00:42:56.394 --> 00:42:56.934 so 303 00:42:58.195 --> 00:43:12.684 okay, 304 00:43:13.224 --> 00:43:13.764 so. 305 00:43:14.250 --> 00:43:20.909 Now, to do some more with this, I want to go back and review something that I didn't cover. 306 00:43:20.909 --> 00:43:24.030 Um. 307 00:43:24.030 --> 00:43:27.780 A, while back, which was. 308 00:43:27.780 --> 00:43:32.369 Let me get you the page number and then I'll show you. 309 00:43:42.389 --> 00:43:50.880 Hmm. 310 00:44:04.739 --> 00:44:14.070 Okay, um, oh. 311 00:44:21.150 --> 00:44:26.760 Okay, um, something called conditional expectation around page 260. 312 00:44:27.085 --> 00:44:40.914 Hey, so we're gonna review page 268 um. 313 00:44:45.090 --> 00:44:53.730 And so what we do is what we've got. 314 00:44:53.730 --> 00:45:02.369 X and we've got Y2 random variables and they're related and we have the expected value of why given X. 315 00:45:03.659 --> 00:45:10.349 So, for example, it's like again, access the input wise, the output, and there's some noise involved. 316 00:45:10.349 --> 00:45:16.409 And what we want to do is we want to show. 317 00:45:18.900 --> 00:45:24.449 Expected value of why is the expected value of. 318 00:45:28.739 --> 00:45:32.610 Um, okay, so. 319 00:45:35.340 --> 00:45:39.510 So the way you read, this is expected value of why give an X. 320 00:45:39.510 --> 00:45:42.840 For any value of X, there's an expected value of why. 321 00:45:42.840 --> 00:45:47.280 And then, so that expression and why. 322 00:45:47.280 --> 00:45:52.860 So, now I can find its expected value so I want to show this sort of thing here. 323 00:45:54.119 --> 00:45:59.309 That I might do this at later. Okay. Um. 324 00:46:02.369 --> 00:46:06.659 So, how do we do this? And, um. 325 00:46:11.369 --> 00:46:22.019 I'll just rewrite it here. Okay. 326 00:46:22.019 --> 00:46:29.940 So, um, and we're given conditionals and so on, um. 327 00:46:36.269 --> 00:46:46.860 Fair enough of acts we're given things like that. Okay. Now, how can we do this? Um. 328 00:47:06.599 --> 00:47:17.579 Well, in value why give an X is, um. 329 00:47:19.139 --> 00:47:24.989 How do we do this? Um, let's see. 330 00:47:24.989 --> 00:47:30.900 Oh, to do this, let me just work my way up to it. Um. 331 00:47:33.539 --> 00:47:37.619 Expected value say g of X is the enter girl. 332 00:47:37.619 --> 00:47:49.289 Um, okay, just for any how we get expected value of the function. 333 00:47:49.289 --> 00:47:52.800 Backs okay. 334 00:47:52.800 --> 00:48:00.329 Expected value of the value of. 335 00:48:02.760 --> 00:48:12.599 I gave an X well, B. 336 00:48:14.460 --> 00:48:19.170 Okay. 337 00:48:19.170 --> 00:48:28.829 All kinds of ex? Um, just a sec. I'm confusing myself actually. Um, sorry. 338 00:48:51.864 --> 00:48:53.155 Are you okay? 339 00:48:57.985 --> 00:49:23.514 Hi. 340 00:49:24.840 --> 00:49:28.019 Hello. 341 00:49:28.019 --> 00:49:34.050 Okay, now, um. 342 00:49:41.820 --> 00:49:51.659 That equals, um. 343 00:49:55.289 --> 00:50:00.329 Yeah. 344 00:50:03.360 --> 00:50:11.280 That equals the integral of. 345 00:50:12.329 --> 00:50:22.710 Y, okay, at this point, I mean, in this expression, X is fixed where except value why we're integrating over, why. 346 00:50:22.710 --> 00:50:27.210 The whole thing, um, darker. 347 00:50:32.309 --> 00:50:44.190 Um, okay, um. 348 00:50:45.360 --> 00:50:49.170 This thing here that why. 349 00:50:50.250 --> 00:50:54.780 Um, so. 350 00:50:56.429 --> 00:50:57.864 And what do we have? 351 00:50:57.894 --> 00:50:58.585 Um, 352 00:51:21.474 --> 00:51:21.894 sorry. 353 00:51:24.960 --> 00:51:28.530 Why gave an extra app hub. 354 00:51:28.530 --> 00:51:33.329 Ex com, why? So? So we have, um. 355 00:51:43.860 --> 00:51:47.070 Just moving stuff around a little and this thing. 356 00:51:47.070 --> 00:51:52.260 Goes half of why. 357 00:51:52.260 --> 00:51:55.380 And it nets out to, um. 358 00:51:55.380 --> 00:52:08.639 So, I showed was expected value of the expected value. 359 00:52:09.780 --> 00:52:18.030 While I give an ex expected a, you know, why it's called a conditional probability thing. Um. 360 00:52:19.559 --> 00:52:22.619 It's called conditional expectation. 361 00:52:30.690 --> 00:52:34.079 Okay, so, um. 362 00:52:34.079 --> 00:52:37.559 So then what I want to do, yes. 363 00:52:41.039 --> 00:52:47.309 Okay. 364 00:52:49.050 --> 00:53:00.659 Why are you. 365 00:53:02.639 --> 00:53:14.940 Okay. 366 00:53:16.230 --> 00:53:23.190 Well, if you're asking for motivation, I'm doing it cause it gets me to the answer. If you're asking is it legal then? 367 00:53:23.190 --> 00:53:32.550 Um, well, the bigger motivation is, it's something I use later in chapter 6. 368 00:53:33.900 --> 00:53:41.159 So, um. 369 00:54:05.034 --> 00:54:05.755 I beg your pardon? 370 00:54:08.130 --> 00:54:12.960 This was in the textbook on page 268 I believe. 371 00:54:14.219 --> 00:54:21.420 So, that's why they 1st presented and we're gonna use it later on. 372 00:54:21.420 --> 00:54:25.800 Um, um. 373 00:54:31.440 --> 00:54:37.590 Oh, okay. Um. 374 00:54:47.610 --> 00:54:52.320 So. 375 00:54:52.320 --> 00:55:06.059 Okay, so if I remind you where our place in the universe here is the universe of this course is we're back to, like, page 3. 376 00:55:06.059 --> 00:55:14.670 34, I think, and we're talking about estimators. 377 00:55:17.730 --> 00:55:21.449 So, basically in goes X. 378 00:55:21.449 --> 00:55:25.769 Outcomes why, um. 379 00:55:25.769 --> 00:55:29.820 We see why. 380 00:55:31.320 --> 00:55:34.440 What was. 381 00:55:34.440 --> 00:55:37.739 Okay, that's our broader question. 382 00:55:37.739 --> 00:55:42.119 Um, and. 383 00:55:43.710 --> 00:55:48.929 Noisy communication channels, for example, so we saw some, um. 384 00:55:48.929 --> 00:55:52.650 We saw some linear estimators, um. 385 00:56:03.449 --> 00:56:07.860 Um, M. A. P. M. L. where. 386 00:56:07.860 --> 00:56:14.460 Or he would guess X was some constant times Y, or whatever for some. C. okay. 387 00:56:16.500 --> 00:56:24.090 And these things depend on what we know, like, if we may not know the density function of X. 388 00:56:24.090 --> 00:56:29.969 If we do, if we know more, we can do more. Okay so now we're going to see. 389 00:56:29.969 --> 00:56:35.460 Uh, now we're going to see a more a more general answer. 390 00:56:36.750 --> 00:56:42.389 Okay, um, we're going to see something is. 391 00:56:44.789 --> 00:56:48.239 A more powerful estimator. 392 00:56:50.130 --> 00:56:54.480 It's going to be nonlinear perhaps. Um, so. 393 00:57:02.550 --> 00:57:07.170 Will not just be, uh, some constant time so. 394 00:57:07.170 --> 00:57:10.590 And and we're going to see something called, um. 395 00:57:11.730 --> 00:57:20.909 A minimum mean square error minimum. 396 00:57:20.909 --> 00:57:24.090 Mean square error. 397 00:57:27.599 --> 00:57:32.909 Estimator okay um. 398 00:57:32.909 --> 00:57:38.730 And it's more complicated, but it will give us a better estimate. Perhaps. Okay. 399 00:57:38.730 --> 00:57:45.780 So, um, so basically. 400 00:57:47.489 --> 00:57:53.489 Um, I guess for X. 401 00:57:56.010 --> 00:58:02.039 Will be a function of why. 402 00:58:02.039 --> 00:58:06.900 And it's going to be called g. Y. okay. 403 00:58:10.800 --> 00:58:15.840 Be more general question. Okay. Um. 404 00:58:15.840 --> 00:58:20.880 And what we're going to do is we've got a way to evaluate it. 405 00:58:20.880 --> 00:58:25.949 Um, so basically. 406 00:58:33.690 --> 00:58:38.699 A computed x x X with a hat on it. Okay. 407 00:58:38.699 --> 00:58:42.239 Um, okay, so. 408 00:58:43.829 --> 00:58:49.199 So, we see why compute X had. 409 00:58:49.199 --> 00:58:52.920 But now there's an error. Okay. 410 00:58:57.420 --> 00:59:02.340 Okay, so what we compute X hat is not probably not. 411 00:59:02.340 --> 00:59:06.989 What X actually was, it was an error X minus X. 412 00:59:06.989 --> 00:59:14.309 And, um, and the error is squared. 413 00:59:17.730 --> 00:59:24.659 Is X minus X hat squared and it has an expected value. 414 00:59:27.329 --> 00:59:35.730 Um, and X x x equals X. 415 00:59:38.309 --> 00:59:42.570 Bye or other, um. 416 00:59:42.570 --> 00:59:51.269 Okay, and we want to minimum, so that's the expected value of our squared error. 417 00:59:51.269 --> 00:59:56.219 And, um, we want to. 418 00:59:58.199 --> 01:00:01.590 Want to pick g. 419 01:00:01.590 --> 01:00:06.840 To minimize that. Okay. I set expectation. 420 01:00:06.840 --> 01:00:12.329 So, it's a general question, I'm saying, I'm not saying that, um. 421 01:00:12.329 --> 01:00:16.349 G is linear or something. It could be anything, but. 422 01:00:16.349 --> 01:00:29.340 The thing is, so I know, um, X, I know correlations. I know whatever I know. And the more I put into it, the better the guests for g and. 423 01:00:29.340 --> 01:00:34.260 So, this is called a minimizing the mean squared error. 424 01:00:34.260 --> 01:00:40.079 Now, why do we pick the square of the error? 425 01:00:40.079 --> 01:00:44.460 Instead of just minimizing, say, the absolute value of the error. Um. 426 01:00:46.440 --> 01:00:52.260 The real reason, and they don't actually write this down in books. So much is it makes the mathematics easier. 427 01:00:52.260 --> 01:01:00.030 So, um, why is the square. 428 01:01:09.210 --> 01:01:15.900 Make some math easier. Okay. Um. 429 01:01:15.900 --> 01:01:20.429 I mean, that's the number of things if you look into statistics and so on or. 430 01:01:20.429 --> 01:01:28.829 But you'll see some formula and, you know, you may puzzle yourself a little as to why does the formula. 431 01:01:28.829 --> 01:01:35.489 Have this form, and the answer might be simply that. 432 01:01:35.489 --> 01:01:40.619 It's easier to work with it doesn't always have a deep. 433 01:01:40.619 --> 01:01:45.960 Rational explanation just the fact that it's. 434 01:01:45.960 --> 01:01:53.099 We can work with that, and it, um, gives us results that we think look okay. 435 01:01:54.179 --> 01:01:59.579 Or I'm being acidic sometimes I said these methods give results that you capture wrong. 436 01:01:59.579 --> 01:02:05.639 So, um, you know, there's a joke. 437 01:02:07.139 --> 01:02:11.369 That with some, you know, systems or formulas or programs. 438 01:02:11.369 --> 01:02:14.789 You know, they can be so simple that they're obviously. 439 01:02:14.789 --> 01:02:20.400 Have no errors are they going to be so complicated that there are no obvious errors. 440 01:02:20.400 --> 01:02:25.139 So, okay and any K. 441 01:02:25.139 --> 01:02:35.639 So, I'm showing you now a more general way to find an estimator. So, what an Estimator is, is we see why we want to estimate what X was. 442 01:02:35.639 --> 01:02:38.699 And we need some way. 443 01:02:38.699 --> 01:02:45.449 Who, um, compute how good our Estimator is and what I'm doing here. 444 01:02:45.449 --> 01:02:52.320 Is that I have my form sorry some Estimator function at G. 445 01:02:52.320 --> 01:02:58.110 And I'll look at the squared error between g of why, and what X really was. 446 01:02:58.110 --> 01:03:02.099 And minimize the expected value. So this requires knowing stuff. 447 01:03:02.099 --> 01:03:10.380 It requires obviously knowing what the density functions for X and Y are and the joint density functions, et cetera. Okay. 448 01:03:10.380 --> 01:03:20.340 So, um, um. 449 01:03:20.340 --> 01:03:24.719 So, I'm going to work my way up to this to help you understand this broad idea. 450 01:03:24.719 --> 01:03:28.590 Um, so. 451 01:03:31.380 --> 01:03:35.039 Okay, so again we've got X. we've got Y. 452 01:03:35.039 --> 01:03:39.539 We've got 3 of why, and we want to minimize. 453 01:03:39.539 --> 01:03:43.769 Expected value by varying g, expected value. 454 01:03:43.769 --> 01:03:47.099 X minus g of why squared. 455 01:03:47.099 --> 01:03:50.579 Okay, so here it'd be a really simple. 456 01:03:56.400 --> 01:04:07.590 Let me make it a constant. Okay. Hey, Chris. I may. Okay. 457 01:04:10.619 --> 01:04:17.010 So, I'm going to ignore what why is I'm just going to always make the same gas for a, for some a. 458 01:04:17.010 --> 01:04:26.610 Okay um, and then the question is, um. 459 01:04:32.010 --> 01:04:35.730 What's the best day? Um, well. 460 01:04:38.849 --> 01:04:45.090 So the best day minimize as expected value. 461 01:04:46.139 --> 01:04:51.090 X minus a weird. I don't let me write it more clearly. 462 01:04:59.070 --> 01:05:05.849 Okay. Um, what do you think is going to be the best value of any idea? 463 01:05:09.659 --> 01:05:12.929 That's working out and see, so. 464 01:05:20.639 --> 01:05:25.469 Okay um, so Andy. 465 01:05:26.849 --> 01:05:33.750 Um, sorry. 466 01:05:57.929 --> 01:06:02.880 Okay, um. 467 01:06:06.420 --> 01:06:10.050 Okay, I want to minimize that. Um. 468 01:06:17.070 --> 01:06:21.480 I'll add and subtract to square the expected value. Um. 469 01:06:36.179 --> 01:06:41.880 Okay, um, and this thing here. 470 01:06:44.909 --> 01:06:49.199 Equals expected value of X minus, say it's squared. 471 01:06:50.400 --> 01:06:55.679 Okay, so I want to minimize this whole expression. The stuff at the front doesn't involve a. 472 01:06:55.679 --> 01:07:01.079 And that's minimize. 473 01:07:01.079 --> 01:07:04.409 Hey, what's expected value the facts. 474 01:07:05.550 --> 01:07:10.139 So, this, um, minimum mean square error estimator. 475 01:07:10.139 --> 01:07:16.920 If the estimators, it's gonna be a constant then we'll make that constant the mean value of X. and that's the best thing. 476 01:07:16.920 --> 01:07:23.219 Oh, okay. 477 01:07:29.190 --> 01:07:33.329 Um. 478 01:07:33.329 --> 01:07:37.619 Now, I can go a little more complicated instead of being. 479 01:07:37.619 --> 01:07:42.269 The Estimator being a constant I can make it a linear function. A Y. 480 01:07:42.269 --> 01:07:49.679 And, um, okay now a little more complicated. 481 01:08:00.119 --> 01:08:05.489 We're going to allow g. Y. 482 01:08:05.489 --> 01:08:09.659 To me, a Y, plus B. so now it's linear. 483 01:08:11.489 --> 01:08:17.279 Okay, um, and what are the best. 484 01:08:26.520 --> 01:08:31.529 What is the best day in? B? Well, um. 485 01:08:59.760 --> 01:09:06.449 So our error, um, so plus B. 486 01:09:06.449 --> 01:09:12.510 So we want to minimize X minus, um. 487 01:09:18.420 --> 01:09:24.720 Square and minimize by varying. N. P. okay. So, um. 488 01:09:31.079 --> 01:09:38.100 Um, so well, the 1st thing is. 489 01:09:42.180 --> 01:09:45.420 What's the best P. 490 01:09:48.689 --> 01:09:52.109 Given everything else. 491 01:09:58.079 --> 01:10:01.350 And that's going to be actually the. 492 01:10:16.859 --> 01:10:26.399 It's just going to be, um, X minus for this question we're going be in the best is gonna be X minus expected value of X, minus. Say why actually. 493 01:10:26.399 --> 01:10:41.189 Okay um, okay, so now we got the best be. 494 01:10:42.779 --> 01:10:47.970 Um, so now our error. 495 01:10:49.050 --> 01:10:56.789 Is, um, X minus a Y. 496 01:10:58.289 --> 01:11:08.579 Minus, um, squared. 497 01:11:10.170 --> 01:11:13.770 And we want to pick the best a so, um. 498 01:11:14.850 --> 01:11:17.909 Huh well. 499 01:11:19.920 --> 01:11:29.100 Rewrite it, um, and then . 500 01:11:30.930 --> 01:11:39.750 Okay, okay. Um. 501 01:11:40.920 --> 01:11:50.699 So so I want to vary 8, uh, minimize that, eh. 502 01:11:52.890 --> 01:11:56.310 That okay, um. 503 01:11:57.869 --> 01:12:05.310 It's quadratic in a so, the obvious thing is to a derivative. Um, okay. 504 01:12:15.180 --> 01:12:18.210 You know, so don't worry about it here. 505 01:12:20.010 --> 01:12:24.000 0, okay, so, um. 506 01:12:28.319 --> 01:12:34.020 Sure, so di, da. 507 01:12:34.020 --> 01:12:38.430 Um, and since, um. 508 01:12:42.329 --> 01:12:47.399 Since it's getting a little late I'll write, I'll write what the answer is. I'm the best day. 509 01:12:47.399 --> 01:12:51.840 The star for the best day will turn out to be, um. 510 01:12:53.550 --> 01:12:59.010 X Y, we're, we're using. 511 01:13:00.029 --> 01:13:05.340 Definitions of Sigma axis, you know, effective value X squared squared. 512 01:13:05.340 --> 01:13:09.060 And so on, okay, um. 513 01:13:14.220 --> 01:13:18.930 You know, where, where that uses. 514 01:13:22.199 --> 01:13:29.489 Take my axe sequel where. 515 01:13:29.489 --> 01:13:35.250 Et cetera. Okay. Reasonable point to stop here. 516 01:13:35.250 --> 01:13:39.569 Um, so again, we're in a, this is a top. 517 01:13:39.569 --> 01:13:43.649 Page 335. 518 01:13:47.310 --> 01:13:50.460 Oh, probably got another day on, um. 519 01:13:52.050 --> 01:13:55.380 Chapter 6 or so. 520 01:13:55.380 --> 01:14:00.000 And There'll be nothing on the last class on the exam and so on. 521 01:14:00.000 --> 01:14:09.329 Okay, so, where this fits in the grand scheme of things is we're working with vectors, um, vectors of. 522 01:14:09.329 --> 01:14:13.319 Variables were doing things through them. This particular thing. 523 01:14:13.319 --> 01:14:20.819 What I've been doing, um, today is I started by working with, um, reviewing some calcium. 524 01:14:20.819 --> 01:14:26.670 I showed that the density function for the 1 variable calcium integration. The 1. 525 01:14:26.670 --> 01:14:34.109 Which is good. It should. Then I talked to 2 has the parameter the correlation coefficient. 526 01:14:34.109 --> 01:14:40.619 And I showed if we took that and integrated out, why to get the marginal. 527 01:14:40.619 --> 01:14:43.619 Density function and axe that is, in fact, was. 528 01:14:43.619 --> 01:14:49.739 So, density function for X, where I assumed that the means were there on the segments for 1 but I had that. 529 01:14:49.739 --> 01:15:01.170 So the parameter, the correlation coefficient. So I showed that and then given that, I could find the conditional density function of why give an extra X given widely analogous. 530 01:15:01.170 --> 01:15:08.279 For the 2 variable calcium and then we're, I'm using some of this as I'm now showing more estimators. 531 01:15:08.279 --> 01:15:14.699 And again, the estimators are in goes acts outcomes. Why say? From a noisy channel or something? 532 01:15:14.699 --> 01:15:18.600 And we want to see different techniques for. 533 01:15:18.600 --> 01:15:26.609 Estimating what's the best value of X given that we saw some particular? Y, and we've seen, like, 3 different estimators. 534 01:15:26.609 --> 01:15:31.739 What's the difference between them is some of them require more knowledge and others. 535 01:15:31.739 --> 01:15:37.859 And the maximum story requires knowing the density index maximum likelihood does not and. 536 01:15:37.859 --> 01:15:43.109 Half basically, but it works usually and now we're saying. 537 01:15:43.109 --> 01:15:48.689 This minimum mean square error estimator, which is more complicated, but will be better. 538 01:15:48.689 --> 01:15:52.979 Is it enough better to be worth the hassle? Who knows? 539 01:15:52.979 --> 01:15:58.170 It depends on your functions. Okay have a good weekend. See you Monday. 540 01:15:58.170 --> 01:16:03.090 Oh, is on next week? Thursday? Yes. 541 01:16:04.590 --> 01:16:14.340 Until what is it it's just gonna be everything off of Monday. That's fine. No, I won't have Monday on it. So, and I may not have the day on. 542 01:16:14.340 --> 01:16:19.890 So, I'm talking over with the later today, so create the exam. 543 01:16:19.890 --> 01:16:26.550 So, up to my knowledge, they also create the homework and take that. Yes, they do. 544 01:16:26.550 --> 01:16:30.930 So, I'm guessing the closest thing we're gonna have to. 545 01:16:30.930 --> 01:16:35.010 Tactics exam is the actual homeworks that we've been doing. Probably yes. 546 01:16:36.390 --> 01:16:44.970 That's the textbook seems to be a lot more complicated. Yeah, that's 1 of my value added to simplify the textbook for you. 547 01:16:44.970 --> 01:16:49.380 So, um. 548 01:16:49.380 --> 01:16:55.560 It's the best book out there, it's, it has a lot of examples, which is why I can. 549 01:16:55.560 --> 01:17:00.750 But it will be will be up to the Monday. 550 01:17:00.750 --> 01:17:07.800 Yeah, I think so. Yes. So today is going to be for like, maybe finals if you have. 551 01:17:08.819 --> 01:17:15.689 Well, you don't have to take the client, we feel like your grade from exam 1 and 2. 552 01:17:15.689 --> 01:17:21.869 Well, it's also possible you'd like the topics so much they may want to write the final just to see what's in it. I don't think I want to. 553 01:17:21.869 --> 01:17:32.609 Also, like the topic come on the topic it's just that if I do well, in all the partners, I have possibility of not taking by not and just go. 554 01:17:32.609 --> 01:17:37.439 Okay, well, that's good. My experience is very few students will write the final. 555 01:17:38.609 --> 01:17:42.029 So, that's what happened last year, but, you know, it's out there for you. 556 01:17:43.560 --> 01:17:48.479 But, um, up to 6. 557 01:17:49.500 --> 01:17:54.060 I, I, I looked at the fall. 558 01:17:54.060 --> 01:17:59.310 And just kind of look at the, uh. 559 01:17:59.310 --> 01:18:04.380 So, if that's where is it, I feel. 560 01:18:04.380 --> 01:18:07.470 Good. 561 01:18:07.470 --> 01:18:11.939 Well, you know, you're spending a lot of effort on it, so I'm guessing you'll do. Well. 562 01:18:11.939 --> 01:18:16.560 That's not a promise, but it's a guess probabilistically. Okay. 563 01:18:16.560 --> 01:18:20.430 Sometimes you spend a lot more effort. 564 01:18:21.510 --> 01:18:28.470 Well, no, that's actually I'm being serious now. I'm like, sometimes it's important to spend time on the things that are difficult for you and this. 565 01:18:28.470 --> 01:18:33.479 This is a serious point. I'm not sure. Okay. Yes. Oh. 566 01:18:33.479 --> 01:18:37.050 You know, to, because it's useful, you know. 567 01:18:37.050 --> 01:18:48.895 You know, the course and useful if you want to try to crack the lottery or the stock market, the dream well, understanding the techniques that I'm telling you, they were years ago. So they don't work anymore. 568 01:18:48.895 --> 01:18:52.255 That's why people talk about them, but maybe there's techniques out there. 569 01:18:53.670 --> 01:18:57.479 Maybe well, you've got to look for him come on, you know, it's not easy. 570 01:18:57.479 --> 01:19:01.380 Hello. Hi so the exam to is. 571 01:19:01.380 --> 01:19:08.399 Mainly just on chapters 5 and 6? Yeah. Okay. But not, uh, I'll, I'll make it clear nothing on. 572 01:19:08.399 --> 01:19:17.340 From today nothing on to that. No normal. No worries. No worries next Monday, though. Okay, so, but up through last month yeah so just. 573 01:19:17.340 --> 01:19:29.069 From after spring break totally. Well, there could be some stuff from before it's too early to forget it now. You know, you wait till may before you forget that's not good. Cause everything's just gone from the last exams. 574 01:19:29.069 --> 01:19:33.960 Yeah, I wonder if we will concentrate on stuff since the 1st exam. 575 01:19:33.960 --> 01:19:38.399 But that stuff may use some early stocks that. 576 01:19:38.399 --> 01:19:44.279 And again, the experience is and I think, what is that. 577 01:19:44.279 --> 01:19:50.609 We try to set it, so you are not context. Okay. 578 01:19:50.609 --> 01:19:55.590 Now, we're not always successful, but that's the goal. We're not trying to make it. 579 01:19:55.590 --> 01:20:04.710 A spread to finish and there was that fun scatterplot that I did show. There was no relation between speed and grade for the grades for the 1st exam. 580 01:20:04.710 --> 01:20:09.510 Is there a guy who got like, a 100 from like, 20 minutes? Probably yes. 581 01:20:09.510 --> 01:20:17.489 Um, just and well, then, sometimes, but such a person then spend the rest of the time double checking everything. 582 01:20:17.489 --> 01:20:21.239 But that can be bad and open when I was a student. 583 01:20:21.239 --> 01:20:25.109 I have some certain the answer is, I learned goal was my 1st guess. 584 01:20:25.109 --> 01:20:29.880 It's more likely to be added so, I mean, I fixed 1 lands that before. 585 01:20:29.880 --> 01:20:34.020 Yeah, you want to look for stupid mistakes. 586 01:20:34.020 --> 01:20:38.640 Okay, but that's the main reason for double checking it. 587 01:20:38.640 --> 01:20:47.010 And other than that, I can't remember what the average period was for the 1st exam, but it was reasonably good. I thought and children. So. 588 01:20:47.010 --> 01:20:53.159 Okay, so if you're trying to try to get a good letter great step 1 is hand in all the homeworks. 589 01:20:53.159 --> 01:21:01.470 And that's the biggest, um, explainer of differences. If you get a 0, because you didn't hand it in. That's the biggest effect. 590 01:21:01.470 --> 01:21:07.710 So, and I told them, you know, the ta's, they're not to try and trick you. 591 01:21:07.710 --> 01:21:10.859 Of course, you know, we may have different opinions about this, but. 592 01:21:10.859 --> 01:21:14.579 Is it not serious things we can test you on without trying to play games? 593 01:21:14.579 --> 01:21:20.520 Oh, yeah, get like 30% and. 594 01:21:21.899 --> 01:21:29.550 I can't remember what I said it said, and it wasn't here. It didn't add up to 100 if I recall. So I'm gonna have to scale things slightly, but. 595 01:21:30.689 --> 01:21:37.770 Yeah, and if here is also tell me almost no, 1, um, comes to their office hours office always posted on here. 596 01:21:37.770 --> 01:21:41.729 So, it's a standard ones I'll tell him next week, we'll add another. 597 01:21:41.729 --> 01:21:50.430 So, yeah, so I'll, I'll tell him maybe put 1 on the weekend. Uh. 598 01:21:50.430 --> 01:21:56.010 You prefer Saturday or Sunday would it be the same as last name? Or would it be. 599 01:21:56.010 --> 01:22:05.909 Well, I know what I do is, I'll tell the, um, that I'd like you to hold the next 2 hours next week say, and I'll say you pick all at the. 600 01:22:05.909 --> 01:22:16.619 Well, what's good for them so, and but I've also told them that if you write them and say you'd like to talk to them some other time, then. 601 01:22:16.619 --> 01:22:23.189 It's supposed to try to accommodate you, but the big thing, like you said, they televise no 1 ever wants to talk to them. So. 602 01:22:23.189 --> 01:22:28.020 Dying for loneliness, so, or they got more time to create homework. 603 01:22:28.020 --> 01:22:32.609 So, it takes time to create those multiple choice. homeworks. 604 01:22:32.609 --> 01:22:37.170 That's why I letting them do it, but it makes it easier to grade it. So. 605 01:22:37.170 --> 01:22:40.680 So, let's I should go to their office hours. They don't have more time to create. 606 01:22:40.680 --> 01:22:52.439 Yeah, whatever we're also trying to motivate. I think this is the front topic, so all right. Thank you. You're welcome. Hi. Yeah, you've been waiting a long time. So I just have a question. 607 01:22:52.439 --> 01:22:57.899 Okay, these are the better people to ask about that since they created it, but show it to me. 608 01:22:57.899 --> 01:23:01.020 So, for this Christian, I'm not really sure. 609 01:23:01.020 --> 01:23:04.260 What is what it specifically this beautiful. 610 01:23:04.260 --> 01:23:07.949 Can I make it Friday? Yeah. Okay. 611 01:23:07.949 --> 01:23:12.210 Make the wrong button, I assume the volume button. I'm sorry it's about. 612 01:23:12.210 --> 01:23:16.260 So, we okay, I guess we destinations. 613 01:23:17.640 --> 01:23:21.659 Oh, this is like the switch box. I will go back. 614 01:23:22.284 --> 01:23:23.244 Okay, 615 01:23:36.984 --> 01:23:37.465 okay. 616 01:23:37.465 --> 01:23:41.125 By after, I assume, no 1 arrived for 30 minutes. Okay. 617 01:23:45.779 --> 01:23:48.930 So, it's a probably no 1 to rise for 30 minutes. 618 01:23:50.010 --> 01:23:54.569 Well, that's like 1 of the examples I did on Monday actually, so. 619 01:23:54.569 --> 01:24:02.970 That requires if I say, doesn't arrive in the next 30 minutes and I did, this must be neither does bus C T multiply the 3 probabilities. 620 01:24:02.970 --> 01:24:09.960 Um, and then enter arrival time so it's like. 621 01:24:09.960 --> 01:24:15.479 You know, the of the function is like, 1-either the minus land T. 622 01:24:15.479 --> 01:24:21.630 Um, there's a compliment of maybe more interesting. You didn't mind assigned to tea, so. 623 01:24:24.300 --> 01:24:34.109 So, for the 1 for the 1st, 1, and then the probability that it didn't a, didn't arrive in the next half hours, it is . 624 01:24:34.109 --> 01:24:38.489 Do you want to write this given an hour and what is it? 625 01:24:38.489 --> 01:24:50.670 30 minutes off after 30 minutes. Talking about I use point 5. it's half an hour. Okay you don't need a calculator for that 1. um, you noticed some simple math I can beat you on the calculator. 626 01:24:50.670 --> 01:24:55.649 If it's simple or not, so okay. Um, yeah. 627 01:24:55.649 --> 01:24:59.100 So then B, the problem with the B. 628 01:24:59.100 --> 01:25:02.789 Didn't arrive the next half hour is either the . 629 01:25:03.414 --> 01:25:14.784 Hey, boss, it's either a minus a half the bus. It's either -1 C bosses. You the -3 has that probably none of them arrived. It's a product of those 2. so now you got to go to a calculator. 630 01:25:14.784 --> 01:25:20.274 Okay, so it's asking for the probability that none of them arrived before 30 minutes. 631 01:25:21.180 --> 01:25:24.210 Well, that's my wording of the last question there so so. 632 01:25:24.210 --> 01:25:34.229 You know, it makes sense. Yeah. Okay. I, I guess I was just confused about the word and what it was specifically. Yeah. You're also allow to ask them if the wording. 633 01:25:34.229 --> 01:25:47.789 Is confusing again, we're not trying to make this an English comprehensive course. Um, we're not trying to trick you on that. I know some exams I can get you on real probability stuff. I don't need to play word games. 634 01:25:47.789 --> 01:25:54.689 So that's my guess and so it's either the minus say. 635 01:25:54.689 --> 01:26:02.699 After I get to the -1 because that that's out to either the -2 and you did a -2. that's the, um. 636 01:26:02.699 --> 01:26:13.319 Okay, that's the correct answer. Point 09, it's, you know, uh, I'm assuming that's not so, whatever you do, the -2 is assuming I didn't make an error doing this mentally and, um. 637 01:26:13.319 --> 01:26:16.920 You know, he is. 638 01:26:16.920 --> 01:26:22.439 2.7, so he squared, but 8 or something and 1 over. 639 01:26:22.439 --> 01:26:29.760 I don't know what it is. I'm guessing it's not enormously off of that. So. 640 01:26:29.760 --> 01:26:38.670 Thank you, you know, actually, you know, with the spoken in the Google, I'm sono actually almost does several bath very simple math. If you speak the question. 641 01:26:38.670 --> 01:26:43.439 Maybe not that fancy, but mm. Hmm. 642 01:26:43.439 --> 01:26:46.770 Gotten better. 643 01:26:48.060 --> 01:26:51.569 You know, the spoken job, somebody years ago here in our room. 644 01:26:51.569 --> 01:26:56.159 The voices, but the computer, the back of the room childhood. 645 01:26:56.159 --> 01:27:03.359 Format to try to make this bigger computer reformat itself. 646 01:27:03.359 --> 01:27:08.369 So. 647 01:27:20.635 --> 01:27:55.104 Okay. 648 01:28:12.750 --> 01:28:25.920 Okay. 649 01:28:27.779 --> 01:28:36.359 Hello. 650 01:28:36.359 --> 01:28:40.560 Oh. 651 01:28:43.380 --> 01:28:49.649 Hello.