WEBVTT 1 00:00:13.288 --> 00:00:16.349 Hello. 2 00:00:16.349 --> 00:00:20.280 Hello. 3 00:00:24.719 --> 00:00:29.519 Hello. 4 00:00:43.049 --> 00:00:46.740 Okay. 5 00:00:46.740 --> 00:00:51.509 Hello. 6 00:00:51.509 --> 00:00:54.929 Hello. 7 00:00:58.049 --> 00:01:03.899 Okay. 8 00:01:03.899 --> 00:01:07.650 A lot more. 9 00:01:12.060 --> 00:01:16.469 Okay. 10 00:01:16.469 --> 00:01:23.670 Okay. 11 00:01:23.670 --> 00:01:26.700 Okay. 12 00:01:40.530 --> 00:01:43.829 Decision. 13 00:01:51.299 --> 00:01:54.689 Okay. 14 00:01:55.950 --> 00:02:01.379 Okay. 15 00:02:01.379 --> 00:02:04.530 Okay. 16 00:02:05.579 --> 00:02:09.120 Hello. 17 00:02:20.159 --> 00:02:23.849 Thank you. 18 00:02:23.849 --> 00:02:28.710 Oh, hey, hardware's actually. 19 00:02:28.710 --> 00:02:32.340 Pretending to work. Oh. 20 00:02:32.340 --> 00:02:37.409 This is. 21 00:02:37.409 --> 00:02:40.860 And. 22 00:02:44.639 --> 00:02:52.770 Hello. 23 00:02:52.770 --> 00:03:01.770 Okay, and again, keep if you're in contact with somebody, trying to watch it remotely. And if there's a problem, tell me because. 24 00:03:01.770 --> 00:03:05.069 I have no way to tell what's happening. 25 00:03:05.069 --> 00:03:12.389 Where we are, is, we're in the textbook. 26 00:03:12.389 --> 00:03:15.389 On Garcia. 27 00:03:16.469 --> 00:03:19.919 Text chapter or. 28 00:03:19.919 --> 00:03:23.879 And I'm continuing on working on the stuff from the. 29 00:03:23.879 --> 00:03:32.580 Okay, so do. 30 00:03:32.580 --> 00:03:36.629 I'm just a review of the last thing that. 31 00:03:36.629 --> 00:03:41.849 So, last time actually. 32 00:03:41.849 --> 00:03:48.479 Here where this off a little. 33 00:03:52.500 --> 00:03:55.800 In there. 34 00:04:00.090 --> 00:04:03.449 Okay. 35 00:04:04.560 --> 00:04:09.150 So, if you're wondering why it is complicated mess. 36 00:04:09.150 --> 00:04:14.909 Is that my iPad is best for taking notes. I'm using good notes. 37 00:04:14.909 --> 00:04:18.660 However, if I run a Webex on the iPad. 38 00:04:18.660 --> 00:04:26.098 It's got some issues and I don't see how to share the screen from a non webx type. 39 00:04:26.098 --> 00:04:32.098 Screen on the Webex on the iPad, I can use the whiteboard feature in the. 40 00:04:32.098 --> 00:04:36.209 iPad, but that's not as good as good notes does only 1 page. 41 00:04:36.209 --> 00:04:43.199 Why, I'm not taking notes on that's I think that running Linux. 42 00:04:43.199 --> 00:04:48.749 Is that if I was projecting from that, as I'm projecting from the iPad now. 43 00:04:48.749 --> 00:04:55.379 The projector with force to display on the think pad into a different resolution. 44 00:04:55.379 --> 00:04:58.978 And that confuses the touch screen. It's a touch screen. 45 00:04:58.978 --> 00:05:07.319 Tablet, and the result of that is that when I cut use the pencil or stylus to touch the screen. 46 00:05:07.319 --> 00:05:14.728 When I'm projecting where I touch the screen and where it thinks I'm touching the screen where it makes the mark are different. 47 00:05:14.728 --> 00:05:20.968 And that makes it impossible to. Right. So, I do this hack with a couple of different things. Um. 48 00:05:20.968 --> 00:05:25.048 If I was at home, I have another thing that has a. 49 00:05:25.048 --> 00:05:33.629 The street, the p73, and then with then I can share you as a program called play. 50 00:05:33.629 --> 00:05:36.658 That can share the. 51 00:05:36.658 --> 00:05:43.559 iPad screen into a window on the think pad and then I can tell Webex to. 52 00:05:43.559 --> 00:05:52.048 Display the whole screen on the think pad and that works until something crashes. They halfway through the lecture. 53 00:05:52.048 --> 00:05:55.288 But the screen sharing thing does not work. 54 00:05:55.288 --> 00:06:01.079 From the iPad to this thing pad because this is integrated graphics not a discreet. So. 55 00:06:01.079 --> 00:06:04.288 Okay, fine. 56 00:06:04.288 --> 00:06:11.488 The new thing I did Monday was something because this is just a review was the cumulative. 57 00:06:13.108 --> 00:06:17.699 Distribution function. 58 00:06:19.559 --> 00:06:23.759 It's called a. 59 00:06:24.959 --> 00:06:29.278 And it's typically. 60 00:06:29.278 --> 00:06:35.189 Some sort of capital. 61 00:06:35.189 --> 00:06:42.329 And we might right here, this is the RV, the random variable if we need to. 62 00:06:42.329 --> 00:06:45.509 And this is the value. 63 00:06:45.509 --> 00:06:48.689 And that's defined as the probability. 64 00:06:48.689 --> 00:06:54.059 That that, that, that random variables less or equal to the given value. 65 00:06:54.059 --> 00:07:01.048 Um, and just as an example, um. 66 00:07:04.468 --> 00:07:10.619 If we have the to the coin toss thing, just to remind you say, calling cost. 67 00:07:10.619 --> 00:07:16.528 Let's do, and it goes to, let's say P while someone half. 68 00:07:17.759 --> 00:07:22.949 So, the probability of a 0, a new will do a number of heads, probably a 0 head. 69 00:07:22.949 --> 00:07:26.129 Quarter 1 head is 1. yeah. 70 00:07:26.129 --> 00:07:31.528 Quarter, but the CDF is. 71 00:07:31.528 --> 00:07:35.098 Hello. 72 00:07:35.098 --> 00:07:38.999 X. 73 00:07:38.999 --> 00:07:43.619 Good here. 74 00:07:43.619 --> 00:07:50.999 Um, it goes up here to 1 quarter. 75 00:07:52.619 --> 00:07:56.548 It goes over. 76 00:07:56.548 --> 00:08:00.209 3 quarters. 77 00:08:00.209 --> 00:08:07.499 Over here. 78 00:08:07.499 --> 00:08:10.619 1, okay. 79 00:08:10.619 --> 00:08:14.668 Now, I did know, I'm showing you some new stuff here. 80 00:08:14.668 --> 00:08:25.199 This notation with a solid circle, and the empty circle means that if you write on the line there, it's the solid circle value. 81 00:08:26.249 --> 00:08:33.359 So, the CDF and so some properties are that the very small values Equifax. 82 00:08:33.359 --> 00:08:37.499 Is 0, very large values it's 1. 83 00:08:37.499 --> 00:08:40.798 And in between with a discreet. 84 00:08:40.798 --> 00:08:51.239 Probability random variable here it takes jobs up and at the here at the where it takes the job, its value is the higher thing. That's what the solid. 85 00:08:51.239 --> 00:08:54.389 So. 86 00:08:54.389 --> 00:08:58.109 Write down some properties here. 87 00:08:59.399 --> 00:09:02.879 Hello. 88 00:09:02.879 --> 00:09:09.688 So, the. 89 00:09:09.688 --> 00:09:13.499 As it goes to minus infinity. 90 00:09:13.499 --> 00:09:19.589 0, the amount of experts to plus infinity. 91 00:09:19.589 --> 00:09:22.589 Yeah, for the next 1. 92 00:09:22.589 --> 00:09:29.158 And it's text less than why. 93 00:09:29.158 --> 00:09:32.639 That after that. 94 00:09:32.639 --> 00:09:36.839 On the calendar. 95 00:09:38.639 --> 00:09:42.839 Okay, but that's just the real this stuff was new here. That's the review. 96 00:09:42.839 --> 00:09:50.729 Okay, now I want to bring in a new idea, which are continuous random variables. 97 00:09:50.729 --> 00:09:55.109 Okay, so next topic and this is. 98 00:09:56.129 --> 00:10:02.969 Oh, the motivation for the CDF. Let me write that down. It's always good to see why we're doing things. 99 00:10:03.989 --> 00:10:07.828 Um. 100 00:10:08.908 --> 00:10:12.149 The motivation for the CDF. 101 00:10:12.149 --> 00:10:17.068 It handles discreet. 102 00:10:19.229 --> 00:10:24.298 Continuous and mix. 103 00:10:25.708 --> 00:10:31.408 Variables it handles them all, but it's a unifying idea. 104 00:10:31.408 --> 00:10:38.788 Okay, so now I want to talk about continuous, let's say. 105 00:10:40.288 --> 00:10:46.678 Um, random variable, so. 106 00:10:48.089 --> 00:10:53.308 The next page is always if I scroll too quickly and tell me. 107 00:10:53.308 --> 00:10:59.969 Okay, the random variables need a random experiment. 108 00:10:59.969 --> 00:11:07.078 Um, there are a random experiment is pick a point. 109 00:11:07.078 --> 00:11:13.229 uniformally on the line. 110 00:11:13.229 --> 00:11:21.269 Hello hey. 111 00:11:21.269 --> 00:11:25.979 Okay, so we've got something here. 112 00:11:25.979 --> 00:11:30.298 And we pick points, um. 113 00:11:30.298 --> 00:11:34.708 Point 40, 40. okay. Now. 114 00:11:34.708 --> 00:11:37.979 Because it yes. 115 00:11:37.979 --> 00:11:41.519 Okay. 116 00:11:41.519 --> 00:11:44.639 Take. 117 00:11:44.639 --> 00:11:48.719 You casting this versions on my hand writing. 118 00:11:48.719 --> 00:11:55.979 Yeah okay. That's good. 119 00:11:58.288 --> 00:12:12.208 There's a reason I type notes into the wiki, because you can read my hand. I've discovered that these hand writing recognition things can actually. 120 00:12:12.208 --> 00:12:15.749 Recognize the handwriting about as well as I can. 121 00:12:15.749 --> 00:12:19.798 Okay, so. 122 00:12:19.798 --> 00:12:22.918 So, what's the probability? Um. 123 00:12:22.918 --> 00:12:30.509 So, here, the probability that the random variable acts as last equal, some value. 124 00:12:30.509 --> 00:12:36.119 Well, it's, it's value that from 0 to 1. okay so it's 0. 125 00:12:36.119 --> 00:12:40.379 Next lesson 0, X if. 126 00:12:40.379 --> 00:12:44.489 There was the 1 and 1. 127 00:12:44.489 --> 00:12:48.719 1, okay, so the. 128 00:12:48.719 --> 00:12:52.288 And that's equal to, um. 129 00:12:54.778 --> 00:12:58.438 That's equal to the yeah. 130 00:12:59.519 --> 00:13:04.109 Okay, and if we plot it, it would look like this um. 131 00:13:10.198 --> 00:13:13.408 That's 1. that's 1. okay. 132 00:13:13.408 --> 00:13:22.168 That's the okay, okay. For the uniform random variables of 0 to 1. 133 00:13:22.168 --> 00:13:26.068 Um. 134 00:13:26.484 --> 00:13:27.293 So 135 00:13:36.024 --> 00:13:54.714 here's 136 00:13:54.714 --> 00:13:54.864 a. 137 00:13:56.249 --> 00:14:00.178 And the is the probability mass function. 138 00:14:00.178 --> 00:14:05.578 Okay. 139 00:14:08.428 --> 00:14:14.188 And that is going to be. 140 00:14:16.469 --> 00:14:20.009 Hello. 141 00:14:20.009 --> 00:14:24.749 That's the probability. 142 00:14:26.188 --> 00:14:30.808 Density function. 143 00:14:34.198 --> 00:14:38.639 And that uses little app. So That'll be like a little there. 144 00:14:40.469 --> 00:14:44.068 Can you go back here? 145 00:14:44.068 --> 00:14:50.428 Okay, tell me when. 146 00:14:53.698 --> 00:14:59.639 Hello. 147 00:15:03.298 --> 00:15:09.298 Okay, so the density function is a little left. 148 00:15:09.298 --> 00:15:14.969 Laugh or whatever that's the name of the random variable. 149 00:15:16.619 --> 00:15:21.749 And that's a specific value. 150 00:15:22.769 --> 00:15:27.479 So, okay. 151 00:15:27.479 --> 00:15:34.379 So, for the continuous, random variable uniform. 152 00:15:34.379 --> 00:15:40.438 For the uniform. 153 00:15:42.599 --> 00:15:48.448 So, again, this is the 0 X and 1. 154 00:15:49.499 --> 00:15:53.249 Clicks that's a good 1. 155 00:15:54.328 --> 00:15:59.129 So, here the PDF, the density function. 156 00:15:59.129 --> 00:16:05.818 Okay, that's defined as a derivative of app effects. 157 00:16:06.869 --> 00:16:12.149 If it exists. 158 00:16:12.149 --> 00:16:17.639 Okay, well, it's not just for the uniform with for all of them here, but that aside. 159 00:16:17.639 --> 00:16:23.399 Okay, so we have the function. The profiles were up to a certain point. 160 00:16:23.399 --> 00:16:27.658 And the slope of that, the density function, if you can do a derivative. 161 00:16:27.658 --> 00:16:34.048 If it takes a jump, then it doesn't make so. 162 00:16:35.519 --> 00:16:41.578 If access jumps, then it affects doesn't exist there. 163 00:16:45.089 --> 00:16:49.048 There okay. 164 00:16:49.048 --> 00:16:54.328 So, and this is for continuous, random variables. So. 165 00:16:55.558 --> 00:16:58.739 Okay. 166 00:16:58.739 --> 00:17:08.159 Okay. 167 00:17:11.759 --> 00:17:14.939 So the uniform. 168 00:17:14.939 --> 00:17:20.578 Again, this is, this is the aquafax here. 169 00:17:20.578 --> 00:17:24.118 So, what, um, the. 170 00:17:24.118 --> 00:17:29.969 The PDF the density function that's that is going to be this. 171 00:17:29.969 --> 00:17:33.659 Because this here is. 172 00:17:33.659 --> 00:17:38.848 And so, by DX equals 1, and this will be 1 up here. 173 00:17:38.848 --> 00:17:45.538 So very simple. 174 00:17:45.538 --> 00:17:51.659 That's 1 this 1 so the continuous, um. 175 00:17:51.659 --> 00:17:56.459 That uniform distribution on intervals 0 to 1. this is the density function. 176 00:17:56.459 --> 00:18:00.209 But what it means is that, um. 177 00:18:04.169 --> 00:18:10.739 The probability that the random variable is from some value, say, et cetera. 178 00:18:14.459 --> 00:18:17.729 T, X is f. 179 00:18:18.778 --> 00:18:23.909 1st small so. 180 00:18:23.909 --> 00:18:29.818 So, the density function is how that. 181 00:18:29.818 --> 00:18:35.368 The random variable is there so on so, for the. 182 00:18:36.868 --> 00:18:42.808 And let me do lots of examples. So you can see this, um. 183 00:18:42.808 --> 00:18:49.739 So let's say the experiment is. 184 00:18:49.739 --> 00:18:53.489 Rotate a pointer. 185 00:18:54.868 --> 00:18:59.788 And and measure the angle. 186 00:18:59.788 --> 00:19:08.128 Data and maybe so we might have 0 ethical the data. 187 00:19:08.128 --> 00:19:14.278 Um. 188 00:19:15.898 --> 00:19:19.078 Should I use degrees or radius, which to people prefer. 189 00:19:21.298 --> 00:19:25.169 Sorry, it's okay. 190 00:19:26.818 --> 00:19:31.888 Okay, so that's and let's say this is uniform. We're spinning it. 191 00:19:31.888 --> 00:19:37.919 So, the, um, the cumulative thing. 192 00:19:37.919 --> 00:19:41.519 It's going to look like this, um. 193 00:19:42.838 --> 00:19:48.929 Okay, so. 194 00:19:52.409 --> 00:19:56.009 Right. 195 00:19:57.598 --> 00:20:01.409 Hmm. 196 00:20:01.409 --> 00:20:05.699 So the probability. 197 00:20:05.699 --> 00:20:09.388 That. 198 00:20:09.388 --> 00:20:14.219 They ran a variable date is last equal to something. That's this column. 199 00:20:14.219 --> 00:20:19.378 A, let's say equal to 0. 200 00:20:19.378 --> 00:20:24.179 If they'd have less, it goes 0, it's a over 2 Pi. 201 00:20:24.179 --> 00:20:27.749 0, ethical fatalistic number 1. 202 00:20:27.749 --> 00:20:33.058 A, Julie. 203 00:20:35.788 --> 00:20:41.729 Okay. 204 00:20:41.729 --> 00:20:45.209 And it's 1, it's a very cool. 205 00:20:46.229 --> 00:20:51.509 And then we'll start this 1 again. 206 00:20:51.509 --> 00:20:55.199 The CDM. 207 00:20:57.509 --> 00:21:06.749 Half of the angle at the data. 208 00:21:06.749 --> 00:21:11.729 It was 0, if they are less than or equal to 0. 209 00:21:14.368 --> 00:21:20.669 They are over over to high. 210 00:21:20.669 --> 00:21:26.219 If they're a to listen to by 1. okay. 211 00:21:26.219 --> 00:21:31.919 They integrated into the pie and apply that. It looks like that. 212 00:21:32.969 --> 00:21:36.269 Okay. 213 00:21:39.179 --> 00:21:42.898 The density function. 214 00:21:42.898 --> 00:21:51.209 Half of data equals PII by, do you say. 215 00:21:51.209 --> 00:21:54.358 And that is 0. 216 00:21:55.469 --> 00:21:58.558 1, over 2, 5 0. 217 00:22:00.598 --> 00:22:05.818 There are 2, 5, 1. 218 00:22:07.469 --> 00:22:11.368 Okay, and if we draw it. 219 00:22:11.368 --> 00:22:14.459 Affect something like this. 220 00:22:15.959 --> 00:22:20.818 Sarah pie is 1 there. 221 00:22:22.048 --> 00:22:29.669 So, the here with the random variable goes from 0 to 2 Pi and density function is at 1 is 1 over 2. Pi. 222 00:22:29.669 --> 00:22:33.298 Because the values are spread out more. 223 00:22:33.298 --> 00:22:36.778 So that's the functions going to be lower. 224 00:22:37.828 --> 00:22:42.808 Hello. 225 00:22:49.409 --> 00:22:54.209 And the total area under the density function will be 1. 226 00:22:54.209 --> 00:23:00.179 So. 227 00:23:03.449 --> 00:23:06.749 Right. 228 00:23:22.794 --> 00:23:27.084 And we can go backwards and Britney in particular value. 229 00:23:28.588 --> 00:23:33.419 That's the inner goal. The little lab. 230 00:23:33.419 --> 00:23:39.179 Next 1, infinity of 2 X. 231 00:23:39.179 --> 00:23:43.888 So, and the integral, the density function. 232 00:23:43.888 --> 00:23:49.199 Minus symphony do infinity 1 so. 233 00:23:49.199 --> 00:23:55.709 And so this is a continuous analogue to the mass function. 234 00:23:55.709 --> 00:24:03.298 So, it gives the problem the density is the probability that the random variables is going to be in. 235 00:24:04.709 --> 00:24:11.009 Yeah. 236 00:24:13.769 --> 00:24:20.368 Now, we could also get some mixed things. Um. 237 00:24:23.489 --> 00:24:27.959 There's also. 238 00:24:27.959 --> 00:24:31.709 Mixed brand and variable. 239 00:24:31.709 --> 00:24:36.659 Hello. 240 00:24:36.659 --> 00:24:40.078 And let's say, um. 241 00:24:42.989 --> 00:24:47.578 You know. 242 00:24:47.578 --> 00:24:50.999 Do you want a mover. 243 00:24:52.528 --> 00:24:57.989 50% chance that. 244 00:24:59.098 --> 00:25:05.848 It's here now, 50% I wait. 245 00:25:08.009 --> 00:25:13.169 From 0, to 10 minutes or something. 246 00:25:13.169 --> 00:25:17.278 And that's the uniform way. 247 00:25:19.679 --> 00:25:23.939 So, the CDM. 248 00:25:23.939 --> 00:25:28.828 For the waiting time. Oh, oh, and let me be specific. 249 00:25:28.828 --> 00:25:33.148 That's your experiment the random variable. 250 00:25:35.909 --> 00:25:44.818 Is the waiting time so the CDF looks like this. Um. 251 00:25:47.459 --> 00:25:50.969 Well, let let me draw out on the next page. 252 00:25:52.259 --> 00:26:01.588 So this is mixed. This 50% the waiting time is exactly. 0 and 50%. It's somewhere between 0 and 10 minutes uniformly. 253 00:26:03.538 --> 00:26:12.749 So. 254 00:26:17.338 --> 00:26:23.219 Okay, so what the. 255 00:26:24.568 --> 00:26:29.308 Looks like this, um. 256 00:26:29.308 --> 00:26:34.019 Right. 257 00:26:34.019 --> 00:26:38.159 Um. 258 00:26:38.159 --> 00:26:42.028 And this will be a 10 here and this can be. 259 00:26:42.028 --> 00:26:46.828 Okay, um. 260 00:26:46.828 --> 00:26:50.729 Well. 261 00:26:55.048 --> 00:26:58.409 Hand up to here. 262 00:26:58.409 --> 00:27:03.209 Hello. 263 00:27:05.189 --> 00:27:09.689 So. 264 00:27:09.689 --> 00:27:14.999 The probability that the waiting time I think with a half is 50. so. 265 00:27:19.439 --> 00:27:27.838 Probabilty sorry accessing to 0. 266 00:27:35.278 --> 00:27:40.618 So the probability X is less than 0 equals. 0. 267 00:27:40.618 --> 00:27:45.778 Probability of X equals. 0 exactly. Is 1 half. 268 00:27:45.778 --> 00:27:51.959 And the probability that 0, less or equal to X. 269 00:27:51.959 --> 00:27:55.499 I think we can. 270 00:27:55.499 --> 00:27:58.739 Is going to be. 271 00:27:58.739 --> 00:28:01.769 Yeah. 272 00:28:01.769 --> 00:28:05.098 Divided by 20. 273 00:28:05.098 --> 00:28:09.088 Probably, that's greater than 10 equals. 1. 274 00:28:11.098 --> 00:28:19.288 So that's the CDF for this mixed thing. So there's a point X equals 0 that point has the probability of 1 half. 275 00:28:19.288 --> 00:28:24.538 And then there is a continuous thing from there on the 10. 276 00:28:24.538 --> 00:28:28.709 So that is the, um. 277 00:28:28.709 --> 00:28:31.709 Um, because. 278 00:28:31.709 --> 00:28:38.278 If you're waiting for the Uber and it's not there already, it could have 5 at any time in the next 10 minutes. 279 00:28:38.278 --> 00:28:48.028 Uniformly, so this is to make it work out. So effects as 0, this comes to 1, half affects is 10. 280 00:28:48.028 --> 00:28:55.769 This is 1020. S and a half and a half, which is 1 and it's linearly in between. So yeah, I wait my hand and took the short term. 281 00:28:56.909 --> 00:28:59.939 Anything else okay. So. 282 00:28:59.939 --> 00:29:03.659 So that is the, um. 283 00:29:03.659 --> 00:29:10.138 To do okay. 284 00:29:10.138 --> 00:29:17.398 That's the able to distribution function for this for this mix the screen continuous. 285 00:29:18.628 --> 00:29:26.519 Um, now I can find the density function. 286 00:29:26.519 --> 00:29:29.788 So that's going to be. 287 00:29:31.499 --> 00:29:34.888 And so this thing here, this will be. 288 00:29:34.888 --> 00:29:38.128 Call it big. Yeah. So this. 289 00:29:38.128 --> 00:29:41.788 Oh, that whole thing there big kind of correct. Okay. 290 00:29:42.808 --> 00:29:49.199 Okay, so. 291 00:29:51.659 --> 00:29:56.519 The density function. 292 00:30:03.598 --> 00:30:09.898 The big half of X X a little Equifax. 293 00:30:09.898 --> 00:30:14.219 Hello okay so but that will be. 294 00:30:16.378 --> 00:30:23.459 Is it's 0, X is less than 0. 295 00:30:24.989 --> 00:30:28.048 And it's. 296 00:30:28.048 --> 00:30:32.398 Sort of undefined. 297 00:30:32.398 --> 00:30:39.419 Effects exactly. Equal. 0. 298 00:30:39.419 --> 00:30:43.229 And it's, um. 299 00:30:44.608 --> 00:30:48.209 X over, um. 300 00:30:50.219 --> 00:30:56.429 And it is, let me scroll back a page here. 301 00:30:56.429 --> 00:31:06.028 So affects, so if I look at this thing here, the function takes a jump, the killer just takes a jump at 0. so there's no derivative. 302 00:31:06.028 --> 00:31:09.509 And here, the derivative is going to be. 303 00:31:09.509 --> 00:31:16.979 On 20th okay, because if we go over here. 304 00:31:16.979 --> 00:31:20.098 Um. 305 00:31:23.009 --> 00:31:28.318 So, if we go over here by 10 to go up by 1 and half. 306 00:31:28.318 --> 00:31:33.239 There's a slope there is 120th. 307 00:31:33.239 --> 00:31:37.769 Okay, so, in this interval there the, um. 308 00:31:39.598 --> 00:31:45.179 It's 120th and if we go over here to the right it's flat. 309 00:31:45.179 --> 00:31:50.249 Slope is 0 and right at X equals tenants undefined again. 310 00:31:51.449 --> 00:31:59.669 So. 311 00:31:59.669 --> 00:32:04.078 Black. 312 00:32:05.759 --> 00:32:11.009 Hello. 313 00:32:11.009 --> 00:32:15.358 Hello. 314 00:32:15.358 --> 00:32:21.628 And define. 315 00:32:21.628 --> 00:32:25.919 Effects equals 1 and 0. 316 00:32:25.919 --> 00:32:30.898 Greater than 1, so. 317 00:32:30.898 --> 00:32:35.159 So, the density function is the. 318 00:32:35.159 --> 00:32:40.318 Flow of the cumulative distribution function if whatever it has a slope. 319 00:32:40.318 --> 00:32:43.798 And if it jumps and fans, there's no smoking. 320 00:32:43.798 --> 00:32:50.519 Point so. 321 00:32:50.519 --> 00:32:53.939 Okay. 322 00:32:56.338 --> 00:33:00.118 Okay. 323 00:33:00.118 --> 00:33:08.219 And so if we want say, a question say. 324 00:33:10.858 --> 00:33:16.019 What's the probability. 325 00:33:16.019 --> 00:33:23.939 I wait probably say 5 to 7 minutes. 326 00:33:23.939 --> 00:33:29.278 So, there's 2 ways I can do it. Um. 327 00:33:30.989 --> 00:33:34.318 And let me scroll up. 328 00:33:37.259 --> 00:33:42.148 So answer 1. 329 00:33:42.148 --> 00:33:46.558 It would be half of 7-f 5. 330 00:33:46.558 --> 00:33:50.009 And. 331 00:33:50.009 --> 00:33:57.538 And after effects for these things, and that the 7, that will be 1 app. 332 00:33:57.538 --> 00:34:01.588 +720 years. 333 00:34:01.588 --> 00:34:05.219 Which is. 334 00:34:06.959 --> 00:34:11.518 720 s, in this here. 1 half . 335 00:34:11.518 --> 00:34:16.289 5 point here 1520 years. 336 00:34:16.289 --> 00:34:19.498 It was 2, 5 years. 337 00:34:19.498 --> 00:34:26.759 10 answer to is we integrate the density function. 338 00:34:28.079 --> 00:34:36.719 5 to 7, and in this interval, it's 120th equals undergo 5 to 720th. 339 00:34:36.719 --> 00:34:41.009 Yes, and that is to. 340 00:34:42.148 --> 00:34:48.509 So, we integrate the density function over in interval. 341 00:34:50.548 --> 00:34:57.088 And then, sometimes to show the function is more convenient sometimes the density function is more convenient. 342 00:34:57.088 --> 00:35:01.648 You pick. 343 00:35:01.648 --> 00:35:08.789 So, I showed you some examples here that able to function just for a uniform, random variable and for a mixed random variables. 344 00:35:08.789 --> 00:35:12.719 Which is a combination of a discreet and a, and a. 345 00:35:12.719 --> 00:35:16.708 Can you okay? 346 00:35:21.568 --> 00:35:26.338 I want to show you some more examples of this. Um. 347 00:35:26.338 --> 00:35:30.599 There is a page, um. 348 00:35:31.798 --> 00:35:38.938 We could do an exponential random variable and say. 349 00:35:38.938 --> 00:35:49.438 And this is define for. 350 00:35:50.998 --> 00:35:56.068 0, and the probability that X is greater than. 351 00:35:57.179 --> 00:36:02.159 Some value a would be. 352 00:36:04.289 --> 00:36:07.289 Minus Labor day. 353 00:36:07.289 --> 00:36:12.778 So, let's just do a check here. Um. 354 00:36:12.778 --> 00:36:17.728 Probability X grading through 0. 355 00:36:17.728 --> 00:36:23.429 11 is a parameter. -0 equals 1. 356 00:36:23.429 --> 00:36:26.759 Good probably I'll do the X. 357 00:36:28.259 --> 00:36:36.119 Equal to something really, really big affinity or something. It's either a minus lab infinity. 358 00:36:36.119 --> 00:36:39.929 Stay close to the minus infinity equals 0. 359 00:36:39.929 --> 00:36:47.009 Okay, so the problem is X, Ray, a, is exponentially decreasing. 360 00:36:47.009 --> 00:36:56.789 So, the CD app equals the probability of the random girls less unable to that. 361 00:36:56.789 --> 00:37:00.628 And that's going to be 1-either the last 1. 362 00:37:02.338 --> 00:37:06.599 Application of this is while our. 363 00:37:06.599 --> 00:37:12.389 For something well, we had the discrete thing, Greg and so this is a continuous analog. 364 00:37:12.389 --> 00:37:16.858 We have the discreet geometric thing. This is a continuous version of it. Um. 365 00:37:20.668 --> 00:37:26.009 Exponential random variable is the continuous. 366 00:37:28.018 --> 00:37:33.748 Version of the discreet geometric, random variable. 367 00:37:39.358 --> 00:37:44.699 Okay, that's great. We call geometric if it's continuously call it exponential. 368 00:37:44.699 --> 00:37:49.768 So, and, um. 369 00:37:52.108 --> 00:37:58.108 And so what it will be used for is the probability that this uranium Adam. 370 00:37:58.108 --> 00:38:01.469 Has not decayed at a certain time. 371 00:38:01.469 --> 00:38:05.969 So this will be the probability. 372 00:38:07.048 --> 00:38:10.349 Hey. 373 00:38:10.349 --> 00:38:15.389 You know. 374 00:38:15.389 --> 00:38:18.840 Hello. 375 00:38:20.309 --> 00:38:26.579 This is not. 376 00:38:26.579 --> 00:38:31.139 Say by time or something. 377 00:38:32.250 --> 00:38:37.650 That's an application of it, so. 378 00:38:39.329 --> 00:38:45.900 So. 379 00:38:48.570 --> 00:38:53.639 Okay, so again this has the. 380 00:38:53.639 --> 00:38:57.900 Uh, filter function. 381 00:38:57.900 --> 00:39:01.230 Hello. 382 00:39:01.230 --> 00:39:06.840 The density function, the probability density function, not related to. 383 00:39:06.840 --> 00:39:15.929 Portable document format . 384 00:39:15.929 --> 00:39:19.619 You know, the minus land X, minus land. 385 00:39:21.210 --> 00:39:27.059 To actually call. Okay. 386 00:39:29.070 --> 00:39:32.369 So, it starts off large and it gets small. So you don't want. 387 00:39:32.369 --> 00:39:37.019 Okay. 388 00:39:42.179 --> 00:39:45.659 Okay, so this is a very widely used, um. 389 00:39:45.659 --> 00:39:51.869 Function so, X application is the 1 application is. 390 00:39:51.869 --> 00:39:56.610 Well, the lifetime of the radioactive Adam. 391 00:39:56.610 --> 00:39:59.909 Probability has indicated at a certain point. 392 00:39:59.909 --> 00:40:06.510 Another big application of this is something called inter arrival times. 393 00:40:07.650 --> 00:40:13.230 There. 394 00:40:19.769 --> 00:40:25.260 So this is the time between. 395 00:40:25.260 --> 00:40:32.519 Too radioactive or 2 calls for a call center. So. 396 00:40:34.110 --> 00:40:38.190 But here are the random variable is the time. 397 00:40:38.190 --> 00:40:42.090 Okay. 398 00:40:42.090 --> 00:40:46.559 Between calls to the call center. 399 00:40:46.559 --> 00:40:52.679 For. 400 00:40:53.820 --> 00:40:59.130 To Adam's. 401 00:40:59.130 --> 00:41:05.639 Etc so it complementary to crosstalk. 402 00:41:15.570 --> 00:41:21.809 So, we talked about the call center thing. There's like a 1Million people may call the call center, but. 403 00:41:21.809 --> 00:41:27.960 The chance is 1 in 100,000 anyone in the wheels on an average, say gate candidate will call. 404 00:41:27.960 --> 00:41:33.570 So, the number that are going to call in the day or the hour, that's a plus on random variable. 405 00:41:33.570 --> 00:41:39.840 The time between adjacent calls is the geometric is an exponential random variable. 406 00:41:39.840 --> 00:41:42.960 In this case, so the compliments the crosstalk. 407 00:41:45.000 --> 00:41:48.000 A relevance thing for call centers. 408 00:41:48.000 --> 00:41:54.059 Is if the time it takes for you to process a call is greater. 409 00:41:54.059 --> 00:41:57.750 Then the expected time between calls. 410 00:41:57.750 --> 00:42:01.500 Then calls are going to follow, so. 411 00:42:03.059 --> 00:42:06.059 Um. 412 00:42:06.059 --> 00:42:09.119 I'll write that down. 413 00:42:09.119 --> 00:42:13.440 Okay. 414 00:42:13.440 --> 00:42:18.090 So. 415 00:42:18.090 --> 00:42:23.550 Is a call to the call Senator. 416 00:42:37.139 --> 00:42:40.980 Oh, I'm sorry. Okay. 417 00:42:46.889 --> 00:42:54.570 Then you have a problem. Okay. 418 00:42:54.570 --> 00:43:02.280 And you can get the expected value at the time because. 419 00:43:02.280 --> 00:43:07.320 Expected value of X, plus the integral of x time. 420 00:43:12.329 --> 00:43:16.230 And so you can get, you can work that out. So. 421 00:43:16.230 --> 00:43:25.349 Okay, so we've seen we're seeing continuous and mix basically, mostly continuous, random variables. 422 00:43:25.349 --> 00:43:30.420 We see in the uniform on different intervals. We've seen the exponential. 423 00:43:30.420 --> 00:43:41.909 Hello. 424 00:43:45.929 --> 00:43:51.030 While you have a problem, I mean, calls are going to file up faster than you can handle them. 425 00:43:54.059 --> 00:43:58.409 Let me show you a, another, um. 426 00:43:58.409 --> 00:44:04.320 Continuous brand new variable. This 1 is the most important 1 so. 427 00:44:04.320 --> 00:44:07.559 Next I'm going to show you. 428 00:44:11.489 --> 00:44:24.780 It's called the. 429 00:44:26.760 --> 00:44:34.320 Also known as the normal. Okay so. 430 00:44:37.500 --> 00:44:49.619 And the reason it's important is when you get a lot of observations, a lot of occurrences, almost every other distribution starts looking like this. 431 00:44:49.619 --> 00:44:53.190 So, why it's important. 432 00:44:57.090 --> 00:45:06.690 And then grows to infinity almost every other. 433 00:45:13.050 --> 00:45:20.340 That's looking like the. 434 00:45:23.489 --> 00:45:26.670 This is a law of large numbers. 435 00:45:28.019 --> 00:45:31.380 Very, you know, but there's lots of walls and large numbers. 436 00:45:34.349 --> 00:45:41.760 So, it will drop, but just, it happens. You take binomial. 437 00:45:41.760 --> 00:45:45.420 And you have more and more and song. 438 00:45:45.420 --> 00:45:49.860 But the means not too small other things. 439 00:45:49.860 --> 00:45:54.090 In most cases, and this happens quite fast actually. 440 00:45:54.090 --> 00:45:57.960 So, on the head of blog. 441 00:45:59.070 --> 00:46:04.170 So, what I'm showing you now, is, was the 3rd example continuous and variable. 442 00:46:04.170 --> 00:46:07.679 It's called the calcium it's also called the normal. 443 00:46:07.679 --> 00:46:11.579 The word normal is picked because it's so normal and. 444 00:46:11.579 --> 00:46:15.150 Because other distribution start, they all start looking like this. 445 00:46:15.150 --> 00:46:21.000 Almost, and this is the famous bell curve. 446 00:46:24.179 --> 00:46:28.469 Thank you. Sorry. Okay. 447 00:46:28.469 --> 00:46:36.119 It goes to minus symphony to infinity. 448 00:46:36.119 --> 00:46:40.860 And this is X and ActiveX, and it looks like this. 449 00:46:43.079 --> 00:46:47.670 Okay. 450 00:46:47.670 --> 00:46:52.739 And it is f. 451 00:46:59.280 --> 00:47:08.340 Okay, and so it's the metric and the mean is 0, standard deviation is 1. 452 00:47:10.619 --> 00:47:17.099 So, it's it's a mess, it's got this exponential my, and so on and the. 453 00:47:17.099 --> 00:47:22.079 Kill divert function does not have a simple form. There's no closed for. 454 00:47:22.079 --> 00:47:27.150 Though there are approximations. 455 00:47:27.150 --> 00:47:31.440 Good. 456 00:47:32.670 --> 00:47:38.340 And this for Halloween, I'd be wondering what's happening in the attic, but. 457 00:47:38.340 --> 00:47:42.690 Okay, so. 458 00:47:42.690 --> 00:47:47.130 Um. 459 00:47:47.130 --> 00:47:50.610 Show you a little about this, um. 460 00:47:50.610 --> 00:47:59.400 Hmm. 461 00:47:59.400 --> 00:48:04.199 Okay. 462 00:48:04.199 --> 00:48:09.300 After the lock up, people don't pay the tuition. Know I don't know. Um. 463 00:48:09.300 --> 00:48:12.329 Faculty don't produce, I don't know. Um. 464 00:48:12.329 --> 00:48:18.030 Okay, so you can also have with some mean. 465 00:48:18.030 --> 00:48:25.260 So the general mean and oh, I'll look it out later but in any case. 466 00:48:25.260 --> 00:48:29.400 The, the. 467 00:48:29.400 --> 00:48:35.849 Okay, cause he had a girl's say from Infinity to. 468 00:48:35.849 --> 00:48:45.300 X0 Square 1 2. 469 00:48:45.300 --> 00:48:48.480 Yes, there or something for the newsletter. 470 00:48:48.480 --> 00:48:52.920 And there's no closed for. 471 00:48:55.380 --> 00:48:58.590 Okay, um, you use. 472 00:48:58.590 --> 00:49:07.860 Use the calculator, or in the past they would use tables. 473 00:49:10.230 --> 00:49:15.210 So, for an exam, I might give you a set of tables. Okay. Um. 474 00:49:16.530 --> 00:49:20.130 And then there's a, for general mean, and. 475 00:49:22.559 --> 00:49:26.369 Saying, well, what do we have. 476 00:49:26.369 --> 00:49:30.090 I don't know. 477 00:49:33.840 --> 00:49:38.340 Hello. 478 00:49:38.340 --> 00:49:42.539 Yeah. 479 00:49:42.539 --> 00:49:45.960 Um. 480 00:49:45.960 --> 00:49:49.679 So, for example, s, a T scores or something, um. 481 00:49:50.820 --> 00:49:54.869 Or something, he might have me, we have 500. 482 00:49:54.869 --> 00:49:59.820 My being 100. 483 00:49:59.820 --> 00:50:03.599 If this will do it that way, I keep changing. 484 00:50:03.599 --> 00:50:10.019 Um, I'll give you some rough. 485 00:50:10.019 --> 00:50:14.699 Numbers here. 486 00:50:14.699 --> 00:50:20.489 Um, Dave on the. 487 00:50:22.230 --> 00:50:26.670 Here okay. Yeah, let me write it better. 488 00:50:26.670 --> 00:50:30.570 Okay. 489 00:50:33.000 --> 00:50:37.920 Huh. 490 00:50:49.349 --> 00:50:54.090 Hello. 491 00:51:01.019 --> 00:51:04.380 Better. 492 00:51:09.599 --> 00:51:13.199 And scores would be an example. So. 493 00:51:17.670 --> 00:51:22.289 So, again, you don't compute these things by hand. 494 00:51:24.420 --> 00:51:30.090 That's like the most of the useful formulas you have to use computers for. 495 00:51:31.710 --> 00:51:46.320 The way it is, um, but there are some rules of thumb you can use to play with this thing. 496 00:51:46.320 --> 00:51:50.940 Hello. 497 00:51:50.940 --> 00:52:00.150 Looks like this 1-1. 498 00:52:00.150 --> 00:52:03.659 To, um. 499 00:52:05.519 --> 00:52:09.570 Give or take, it's like a 6 and a 6 thing here. Um. 500 00:52:09.570 --> 00:52:13.980 So the cumulative thing f, access will be. 501 00:52:13.980 --> 00:52:19.800 Point 5 is the point. 83 is would be about point 1. 502 00:52:19.800 --> 00:52:23.099 7 or something, um. 503 00:52:23.099 --> 00:52:28.559 For 2, it's 98%, I think, um. 504 00:52:29.699 --> 00:52:36.570 I'm doing this structure. 505 00:52:36.570 --> 00:52:40.739 That maybe correct myself for Monday. 506 00:52:40.739 --> 00:52:51.150 And so on, so this is the thing with the normal distribution 2 thirds of a time you're within 1 standard deviation of the mean. 507 00:52:51.150 --> 00:52:54.989 So, if it's a test with the mean of 500. 508 00:52:54.989 --> 00:52:58.440 And the Sigma standard deviation of 100. 509 00:52:58.440 --> 00:53:02.219 2 thirds of the time the scores are from 400 to 600. 510 00:53:02.219 --> 00:53:08.190 So, the 98. 511 00:53:08.190 --> 00:53:13.650 Like I said, I'm doing this the memory I've got 9,598. Oh, correct. 512 00:53:15.090 --> 00:53:19.829 You can do a rule of thumb things. 513 00:53:19.829 --> 00:53:25.559 And there. 514 00:53:28.860 --> 00:53:35.670 Thank you that's big enough. You deserve a point to email them. 515 00:53:42.929 --> 00:53:49.110 Yeah, 2 thirds of the time here within a segment of this. 516 00:53:50.219 --> 00:54:02.760 And the probability within 2 segments of the mean is like, 98% 3 segments. It's like 99.8% or so. 517 00:54:02.760 --> 00:54:06.179 I think it's going down exponentially more than exponentially fast. 518 00:54:07.289 --> 00:54:12.659 Now, the way you use it as an approximation for something like binomial, you match. 519 00:54:12.659 --> 00:54:17.489 The means the abbreviation, so. 520 00:54:22.650 --> 00:54:26.730 Okay. 521 00:54:28.050 --> 00:54:32.940 Hello. 522 00:54:35.639 --> 00:54:39.719 You match the meeting and segment. 523 00:54:39.719 --> 00:54:46.409 So, provide oatmeal, so binomial the mean and P. 524 00:54:46.409 --> 00:54:50.670 And the Sigma square root event. 525 00:54:51.929 --> 00:54:55.079 Square root of actually. 526 00:54:55.079 --> 00:54:58.619 I may do people that have here. 527 00:54:58.619 --> 00:55:01.829 Meaningful and over 2. 528 00:55:01.829 --> 00:55:04.920 Sigma is the square root of that over to. 529 00:55:04.920 --> 00:55:10.320 So then you can match find day Garcia. 530 00:55:10.320 --> 00:55:14.519 That, um. 531 00:55:14.519 --> 00:55:21.480 I got her 1st, like, it. 532 00:55:21.480 --> 00:55:30.869 So, you can match it and the, it's a, it's a good approximation fast for surprisingly small and. 533 00:55:30.869 --> 00:55:36.389 5 or so okay. 534 00:55:36.389 --> 00:55:40.289 Um. 535 00:55:45.090 --> 00:55:50.099 Good. 536 00:55:51.869 --> 00:55:55.469 For small already. 537 00:55:58.619 --> 00:56:01.619 Say, 5 or something. 538 00:56:02.909 --> 00:56:06.150 To approximate a plus on. 539 00:56:11.519 --> 00:56:20.280 You want alpha big enough. 540 00:56:22.920 --> 00:56:27.090 That there is a left. 541 00:56:32.880 --> 00:56:40.559 If we do focus on here, um, you know, if any, if Alpha is 0. 542 00:56:40.559 --> 00:56:44.550 You know, we're going to get something like this. I'll equals 0. okay. 543 00:56:44.550 --> 00:56:49.230 But if we do something like alpha equals 1. 544 00:56:49.230 --> 00:56:52.650 It's going to be something like that, or whatever. 545 00:56:52.650 --> 00:56:58.349 You know, if you do something like, I don't know, Al equals 5 or something who knows though? 546 00:57:01.440 --> 00:57:04.800 That's okay, so now, um. 547 00:57:04.800 --> 00:57:07.889 Because the thing for Hassan. 548 00:57:09.659 --> 00:57:14.699 The mean is alpha and assign deviation to square root of alpha. 549 00:57:14.699 --> 00:57:22.710 So you want again, she wants to go to develop a notice will be less no. So I don't think we'll 5 for redevelopment. 2.2. 550 00:57:22.710 --> 00:57:27.030 Then it's going to be okay, so. 551 00:57:27.030 --> 00:57:31.110 So, the normal distribution becomes a useful approximation quite quickly. 552 00:57:31.110 --> 00:57:35.369 So, any calculator is any good. 553 00:57:36.869 --> 00:57:41.550 As it built into it, not your core function calculators, but. 554 00:57:41.550 --> 00:57:44.969 Okay, so. 555 00:57:46.739 --> 00:57:52.650 I've shown you a uniform, I showed you exponential. I showed you. 556 00:57:52.650 --> 00:57:56.340 Calcium also known as normal. 557 00:58:00.389 --> 00:58:03.690 And then I'll show you some more later, but. 558 00:58:05.940 --> 00:58:09.599 No, you might be wondering. 559 00:58:10.920 --> 00:58:18.449 You know, applications of the kilter distribution function. It's another thing to learn. Well, what I'll show you now. 560 00:58:18.449 --> 00:58:22.889 Is a place we're using it makes solving a problem faster. 561 00:58:22.889 --> 00:58:29.880 Um. 562 00:58:34.980 --> 00:58:42.030 Hello. 563 00:58:45.900 --> 00:58:49.889 Okay, here's my random experiment. 564 00:58:51.090 --> 00:58:56.159 Pick 2 uniform. 565 00:58:58.170 --> 00:59:01.559 R. V. in the interval. 566 00:59:04.769 --> 00:59:09.780 And 1 and report the smaller. 567 00:59:12.690 --> 00:59:22.409 So, I've got 2 athletes running a race, random times and I want to know what's the winning time. 568 00:59:22.409 --> 00:59:27.750 Okay, so that's the RV the random variable. 569 00:59:27.750 --> 00:59:32.940 Is the smaller is a smaller 1 okay. 570 00:59:34.590 --> 00:59:42.869 What is distribution. 571 00:59:44.130 --> 00:59:51.030 So. 572 00:59:51.030 --> 00:59:54.449 Hey, do this so how am I going to do that? 573 00:59:57.420 --> 01:00:02.250 That you be the 1st, random variable. 574 01:00:02.250 --> 01:00:05.369 The, the, the 2nd. 575 01:00:05.369 --> 01:00:08.550 And Dolby is a minimum of the tube. 576 01:00:08.550 --> 01:00:11.909 Okay. 577 01:00:11.909 --> 01:00:16.230 I want the distribution function for W. 578 01:00:17.849 --> 01:00:22.079 And now for you. 579 01:00:22.079 --> 01:00:25.440 The CDF looks like that. Okay. 580 01:00:25.440 --> 01:00:30.989 And the PDF looks like that. Okay. 581 01:00:30.989 --> 01:00:35.639 They're only 1. okay and V. it's the same. Okay. 582 01:00:40.110 --> 01:00:44.250 The airport. 583 01:00:46.050 --> 01:00:50.940 So, um. 584 01:00:50.940 --> 01:00:55.380 Hello. 585 01:00:56.699 --> 01:01:05.070 So, let's so basically random variables. 586 01:01:05.070 --> 01:01:10.079 Actually, put a minimum of the. 587 01:01:11.159 --> 01:01:17.880 Then, um. 588 01:01:17.880 --> 01:01:21.389 Excellent I seem to some value. 589 01:01:22.920 --> 01:01:26.099 If both random variables. 590 01:01:26.099 --> 01:01:29.969 Um. 591 01:01:29.969 --> 01:01:42.780 Let me do Max, essentially, if you're not going to scream at me too much for editing stuff. 592 01:01:47.099 --> 01:01:55.769 Then. 593 01:02:03.570 --> 01:02:06.960 Okay. 594 01:02:06.960 --> 01:02:12.960 So, the rather the maximum is less than X. 595 01:02:12.960 --> 01:02:18.179 If the random variable you is less able to X and. 596 01:02:18.179 --> 01:02:21.599 Hey, listen to X. okay. 597 01:02:23.039 --> 01:02:29.340 So. 598 01:02:31.530 --> 01:02:39.059 Can you hear it? So the probability that, uh. 599 01:02:43.949 --> 01:02:47.159 Half of access access the probability. 600 01:02:49.139 --> 01:02:54.599 So this is the, um, let me write that. Some words, I think. 601 01:02:56.099 --> 01:02:59.969 So the probability. 602 01:02:59.969 --> 01:03:04.469 That this it was a probability. 603 01:03:04.469 --> 01:03:14.940 That time is the probability. 604 01:03:18.840 --> 01:03:23.610 So this is the 2 independent variables the probability. 605 01:03:23.610 --> 01:03:27.360 I mean, if X is the greater than then. 606 01:03:27.360 --> 01:03:34.530 Both random variables you and we have to be less they will do X. so, the probability of that happening is the product of the to. 607 01:03:34.530 --> 01:03:37.559 So. 608 01:03:40.320 --> 01:03:47.039 So, we're going to get this and so the probability. 609 01:03:48.719 --> 01:03:52.590 This is the product of the probabilities. 610 01:03:53.760 --> 01:03:59.789 Because they're independent. 611 01:04:01.440 --> 01:04:08.070 So you and V are independent. 612 01:04:09.449 --> 01:04:19.710 So, X is the greater because the probability that the greater the 2 is less than some values, the probability that the 1st. 613 01:04:19.710 --> 01:04:24.420 And the guy was less than that value times. The problem the 2nd, 1 is less than that bags. 614 01:04:24.420 --> 01:04:31.679 Independence, so so what this means is f. 615 01:04:33.150 --> 01:04:39.570 So. 616 01:04:40.800 --> 01:04:47.190 So, for the maximum of 2, random variables, I multiply the CDs I multiply it. 617 01:04:47.190 --> 01:04:50.880 Distribution functions. Oh. 618 01:04:53.340 --> 01:04:56.610 Hello. 619 01:05:01.349 --> 01:05:04.800 So. 620 01:05:07.289 --> 01:05:11.820 But half of you that equals 1 for that. 621 01:05:11.820 --> 01:05:15.090 Thing is the same thing. 622 01:05:16.920 --> 01:05:20.429 So of the maximum of 2. 623 01:05:22.949 --> 01:05:26.130 I do here. 624 01:05:28.409 --> 01:05:32.460 I don't know. 625 01:05:34.500 --> 01:05:38.820 Of course, because it looks like that. 626 01:05:38.820 --> 01:05:43.349 Squared. 627 01:05:43.349 --> 01:05:50.639 So, for the max of the 2 uniform, random variables, this is the CDM. Um. 628 01:05:50.639 --> 01:05:54.960 Square 0 x0 1. 629 01:05:54.960 --> 01:06:06.179 If that is greater than equal to 1, and in general, the max of K uniform random variable. 630 01:06:08.880 --> 01:06:12.989 The okay here. 631 01:06:12.989 --> 01:06:16.170 It goes to the. 632 01:06:16.170 --> 01:06:20.639 0, 1. 633 01:06:20.639 --> 01:06:28.829 And to do this, using the density function, it has to do a native row and it would guide get the same answer. 634 01:06:28.829 --> 01:06:34.500 But it would be mess here. 635 01:06:40.019 --> 01:06:44.489 So, if I plot it, um. 636 01:06:45.869 --> 01:06:50.760 That. 637 01:06:56.369 --> 01:07:04.800 Anyway. 638 01:07:14.849 --> 01:07:19.710 So, if I plotted. 639 01:07:19.710 --> 01:07:24.030 And, of course too, so the. 640 01:07:24.030 --> 01:07:30.329 Squared and then the density function. 641 01:07:33.900 --> 01:07:37.170 So, if I plot it. 642 01:07:37.170 --> 01:07:43.829 And it looks like this put all that looks like that. 643 01:07:45.389 --> 01:07:53.159 I'm sorry, it looks like that. 644 01:07:54.449 --> 01:08:02.789 What this means is it, it's bias to the right. Okay. It. 645 01:08:05.159 --> 01:08:08.639 Okay. 646 01:08:08.639 --> 01:08:15.059 I E, the maximum of 2 uniform random variables is more likely to be high that to be low. 647 01:08:26.369 --> 01:08:29.850 Hi. 648 01:08:29.850 --> 01:08:34.739 You know, okay makes sense. 649 01:08:34.739 --> 01:08:38.609 So so we can find. 650 01:08:38.609 --> 01:08:44.250 Go back to a little bit. 651 01:08:44.250 --> 01:08:48.539 We could do this for the minimum also. Um. 652 01:08:50.460 --> 01:08:56.369 I do the minimum now. Um. 653 01:09:01.109 --> 01:09:09.329 Well, it's just a compliment in this case 1-big, half of X. 654 01:09:09.329 --> 01:09:12.750 Well, they call it a 2 here for the 2. 655 01:09:17.010 --> 01:09:21.899 And this here will be 1-X Square. So the big 1. 656 01:09:23.489 --> 01:09:29.819 Is. 657 01:09:32.909 --> 01:09:37.859 Um. 658 01:09:37.859 --> 01:09:42.810 0, to 1, if I get this. Right? Um. 659 01:09:44.039 --> 01:09:47.189 And. 660 01:09:49.050 --> 01:09:56.159 Sorry to 0 It ends up in 1. 661 01:09:58.470 --> 01:10:03.779 And kind of do this. 662 01:10:04.949 --> 01:10:10.020 Hello, and the. 663 01:10:10.020 --> 01:10:15.270 The density function is a derivative of those 2. 664 01:10:16.409 --> 01:10:20.399 Which does that. 665 01:10:20.399 --> 01:10:29.579 So, the minimum of to function, it's more likely to be to the left the low value that makes a 2 record makes total sense. 666 01:10:29.579 --> 01:10:33.840 And we can do this easily using the distribution functions. 667 01:10:33.840 --> 01:10:37.439 Hello. 668 01:10:37.439 --> 01:10:42.210 Um. 669 01:10:42.210 --> 01:10:48.600 You could find also the 2nd of 3 if you want to do stuff like that and so on. 670 01:10:48.600 --> 01:11:01.050 You find orders statistics, so that's a nice application or you could do it with the density functions, but it's easier to use the terms of distribution function. 671 01:11:03.270 --> 01:11:11.130 That's a nice point to solve, um. 672 01:11:11.130 --> 01:11:15.720 Review what I was doing, so we're into chapter 4. 673 01:11:15.720 --> 01:11:19.979 We're just talking about mostly about continuous distribution functions. 674 01:11:19.979 --> 01:11:25.859 Although we did start off by saying that we've seen continuous distribution. 675 01:11:25.859 --> 01:11:30.390 We did start by saying the cumulative distribution function. 676 01:11:30.390 --> 01:11:36.000 Which is the probability of the random variables up to a specific value. 677 01:11:36.000 --> 01:11:41.100 And it is nice because it handles both discreet, random variables. 678 01:11:41.100 --> 01:11:44.369 And continue with random variables, and even the mixed thing. So. 679 01:11:44.369 --> 01:11:49.020 You sometimes get a random variable, which is mixed. So. 680 01:11:49.020 --> 01:11:55.560 Maybe, it sits on a specific number or maybe it's over a range so nice example Uber. 681 01:11:55.560 --> 01:12:01.439 Is a good probability over there now if not, it's smeared out over a continuous range. 682 01:12:01.439 --> 01:12:05.340 And we saw a uniform which you've seen before. 683 01:12:05.340 --> 01:12:10.800 We've seen I showed you the exponential random variables, decaying exponentially. 684 01:12:10.800 --> 01:12:14.489 And it's the probability that. 685 01:12:14.489 --> 01:12:21.449 Your radioactive your Adam is not yet perhaps or a customer has not yet come to your server. 686 01:12:21.449 --> 01:12:29.670 And then we saw the normal also called the calcium distribution, which is the most important continuous distribution. 687 01:12:29.670 --> 01:12:37.260 Because most of the others start looking like that as then gets figures as Bell Curve distribution. 688 01:12:37.260 --> 01:12:42.779 You cannot work it by hand you have to use a calculator, but that's the like. 689 01:12:42.779 --> 01:12:48.659 But really quickly like, binomial pen equals 5 or 10 already. calcium's quite good. 690 01:12:48.659 --> 01:12:55.770 Hassan, and equals 5, certainly equals 10. you wouldn't even notice the difference and then equals 5 already. 691 01:12:55.770 --> 01:12:59.069 It's quite good. So. 692 01:13:00.090 --> 01:13:04.800 So these are the big things I'll show you more distributions on Monday. 693 01:13:04.800 --> 01:13:14.250 And also I showed an example of the continuous. 694 01:13:14.250 --> 01:13:20.550 The chemical function where you solving a problem is easier if you use it. 695 01:13:20.550 --> 01:13:25.800 That was a new random variables, the maximum or the minimum of several others. 696 01:13:25.800 --> 01:13:31.140 And I want the distribution for this fax for men, and it's easier to do with the. 697 01:13:32.220 --> 01:13:39.390 See, what else? We're late getting home or 2 online. I'm sorry, but I gave you a week after I put it online. 698 01:13:39.390 --> 01:13:42.630 And what I'll put a 3 online also fairly soon. 699 01:13:42.630 --> 01:13:47.789 We got, we got to start you on this stuff and again, if you'd like to read more. 700 01:13:47.789 --> 01:13:51.899 I put I typed some notes into my blog and also. 701 01:13:51.899 --> 01:14:00.630 Textbook is actually well written. Well, I'll stay down here. If you have questions other than that have a good weekend. 702 01:14:00.630 --> 01:14:04.079 Hello. 703 01:14:04.079 --> 01:14:08.250 Right. 704 01:14:13.770 --> 01:14:16.979 Actually. 705 01:14:20.789 --> 01:14:25.979 Hm. 706 01:14:25.979 --> 01:14:29.292 Oh.