WEBVTT 1 00:01:11.879 --> 00:01:20.129 Okay, good afternoon class if anyone can hear me. 2 00:01:20.605 --> 00:01:22.644 This is probably class 15, 3 00:01:22.674 --> 00:01:23.004 March, 4 00:01:23.034 --> 00:01:23.424 18th, 5 00:01:23.424 --> 00:01:23.545 2021, 6 00:01:24.355 --> 00:01:30.564 so a universal question and you hear me because again, 7 00:01:30.564 --> 00:01:35.665 I have no good feedback about whether thanks Daniel. 8 00:01:35.665 --> 00:01:38.484 So, what is happening today? 9 00:01:40.469 --> 00:01:47.640 1st, um, next Thursday, president Jackson has a spring town meeting so that. 10 00:01:47.640 --> 00:01:59.340 And so I'm going to cancel class Thursday that also compatible with what I said to start that. Occasionally we'd have a day off because there are no official long weekends. 11 00:01:59.340 --> 00:02:06.599 After the start of the semester since February, so this will be 1. so next Thursday you do what you want. 12 00:02:06.599 --> 00:02:15.659 Attend the virtual town meeting sleep like yeah, I know it's 3. P. M. going to hike whatever you like. Okay. And no homework do then also. 13 00:02:15.659 --> 00:02:25.650 So, today we're just doing a few pieces from chapter 4 and then if we get time. 14 00:02:25.650 --> 00:02:39.594 Going to chapter 5, which is 2 random variables. Yeah try and example or 2 page c2nd page 169. 15 00:02:47.879 --> 00:02:51.389 Okay. 16 00:03:01.259 --> 00:03:04.680 Trying to keep everything overlapped reasonably here. 17 00:03:04.680 --> 00:03:10.050 And again, what we have at the top here is a. 18 00:03:11.069 --> 00:03:22.560 Is a useful thing. This is the right tale for the normal Gaussian distribution. So you might want to put a note on page. 169 is useful. 19 00:03:22.560 --> 00:03:27.930 Just to show you what it is, just to remind you. 20 00:03:29.159 --> 00:03:38.099 Let's see here. 21 00:03:44.759 --> 00:03:49.169 Okay, so again we have the normal distribution here. 22 00:03:50.729 --> 00:03:56.939 And so Q is the right tail. So that is actually fairly. 23 00:03:56.939 --> 00:04:01.259 0. 24 00:04:03.719 --> 00:04:07.860 And so on, so. 25 00:04:14.069 --> 00:04:19.110 That it's going to be. 26 00:04:19.110 --> 00:04:23.430 And so on. 27 00:04:23.430 --> 00:04:26.759 Sorry, that's okay. 28 00:04:26.759 --> 00:04:30.749 I see yes. Would that be a question? 29 00:04:30.749 --> 00:04:35.189 Greater than 0, that would be company that is 0. 30 00:04:35.189 --> 00:04:42.928 That would be the probability that X is greater than or equal to a. 31 00:04:42.928 --> 00:04:47.038 Somewhere this is a here, let's say. 32 00:04:48.238 --> 00:04:57.389 So, in more detail on me, draw it out more neat plays in. So, for Q0, that would be. 33 00:04:59.158 --> 00:05:03.149 Huh. 34 00:05:03.149 --> 00:05:06.658 There okay here. 35 00:05:06.658 --> 00:05:12.149 Okay, and Q1 would be. 36 00:05:12.149 --> 00:05:23.098 Again, we try and be 1, let's say 0 and 11 would be this. 37 00:05:23.098 --> 00:05:29.579 Okay, um, queue of minus 1 would be. 38 00:05:32.038 --> 00:05:35.819 And so on. 39 00:05:35.819 --> 00:05:40.588 And then 2 of minus 1 would be. 40 00:05:40.588 --> 00:05:47.488 That's approximately point 85 give or take. Okay. So what does the queue mean? 41 00:05:48.899 --> 00:05:57.418 It's a probability that its argument is greater than or equal to, of the random variable, being greater than equal to that value. 42 00:05:57.418 --> 00:06:01.348 So, um. 43 00:06:01.348 --> 00:06:05.249 Let me write that out. 44 00:06:05.249 --> 00:06:08.908 Black and works better if you're. 45 00:06:08.908 --> 00:06:15.119 Hello. 46 00:06:20.009 --> 00:06:27.059 And Gary, you called X and where X is a go see in. 47 00:06:28.288 --> 00:06:32.459 Variable okay. 48 00:06:35.009 --> 00:06:42.838 If we scroll back up in the book here, they have a defined here so. 49 00:06:42.838 --> 00:06:52.528 Fi is the left tail? Why just electrical engineers for summaries and standardized on? Q. 50 00:06:54.209 --> 00:06:57.329 Okay. 51 00:06:57.329 --> 00:07:02.189 So, if we're looking, at example of 422 here. 52 00:07:04.829 --> 00:07:10.649 Um, so. 53 00:07:15.353 --> 00:07:23.363 If we look at this, what's happening here? So it's a noisy communication system. We're going to see examples like this all semester. 54 00:07:23.783 --> 00:07:37.733 We have an input voltage V and it gets scaled by the scale factor alpha and then a calcium random noise gets added to it. So alpha 1, and it goes in random variable has. 55 00:07:38.249 --> 00:07:41.968 Um, means 0 and Sigma 2. 56 00:07:41.968 --> 00:07:49.528 So so, let me just stop right there and show you how to show how you can use. 57 00:07:49.528 --> 00:07:53.129 How you can use queue. 58 00:07:54.809 --> 00:07:58.889 If we think about, so, in here. 59 00:08:02.548 --> 00:08:11.668 With some, it's Kelsey, and with me equals 0 and equals 2. so basically. 60 00:08:11.668 --> 00:08:20.759 So, what's the probability that, um. 61 00:08:20.759 --> 00:08:23.999 The, and. 62 00:08:23.999 --> 00:08:27.838 Say. 63 00:08:27.838 --> 00:08:34.708 Is greater than equal to 2. okay. 64 00:08:36.328 --> 00:08:41.489 So that will be queue of 1. 65 00:08:49.139 --> 00:08:52.678 Because and has. 66 00:08:52.678 --> 00:09:00.719 Is Sigma too? I mean, how would I do that? I mean, I could I could define some. 67 00:09:00.719 --> 00:09:03.928 Some new random variable, let's say. 68 00:09:03.928 --> 00:09:10.229 Call it what letters haven't we used yet? 69 00:09:10.229 --> 00:09:13.678 Call it Tom. 70 00:09:13.678 --> 00:09:22.259 Called Adobe or something, we're going to say W equals and over 2. okay. Just find a new random variable. 71 00:09:23.999 --> 00:09:30.119 So, sorry, I have to scroll up by a whole page, but so just to review. 72 00:09:30.119 --> 00:09:43.649 And is Gaussian with equals 0 same equals 2. W, equals, um, and divided by 2. so now. 73 00:09:43.649 --> 00:09:53.339 What are W, some math statistics W, will have immediately closed 0 and Sigma equals 1 for them W under here or something. 74 00:09:53.339 --> 00:10:02.759 Yeah, that's just common sense. Okay. So now the probability the N is greater than anything. 75 00:10:02.759 --> 00:10:13.438 You know, let's call it alpha or whatever that's going to be the probability. The W. 76 00:10:13.438 --> 00:10:17.099 Is greater than alpha divided by 2. 77 00:10:17.099 --> 00:10:20.578 Okay. 78 00:10:20.578 --> 00:10:32.639 So let's say so, if we want so probability to end greater than 2, that's the probability that w's greater than 1 and that's going to be Q1. 79 00:10:33.899 --> 00:10:40.438 And we look into the table, and that is point 1 6. 80 00:10:40.438 --> 00:10:44.908 Okay, so you see how we can do things like that. 81 00:10:46.469 --> 00:10:50.938 So now, professor, yes. 82 00:10:50.938 --> 00:10:54.599 Why did you do that on the table? Like, how do you how did you use the table? 83 00:10:54.599 --> 00:11:00.448 So the table has queues, so I wanted Q1. It'll be right here. 84 00:11:00.448 --> 00:11:04.229 1.559 minus 1. 85 00:11:04.229 --> 00:11:07.948 Which is 1.5 so that is. 86 00:11:07.948 --> 00:11:12.178 Point 1 5, 9 so okay. 87 00:11:14.124 --> 00:11:14.693 Okay, 88 00:11:14.903 --> 00:11:21.774 so still an example for 22 so we can see here where I wrote down, 89 00:11:21.774 --> 00:11:29.994 you can use the table to find tail probabilities for normal variables whose standard deviation is not. 90 00:11:30.778 --> 00:11:36.509 I mean, let me do 1 more example, just for fun. 91 00:11:36.509 --> 00:11:40.078 Now, there's another example. 92 00:11:43.708 --> 00:11:49.918 I don't know imagine something else a. 93 00:11:49.918 --> 00:12:01.288 Running out of letters, they go around very well with mean 1 and. 94 00:12:01.288 --> 00:12:05.219 And Sigma 3. okay. 95 00:12:06.629 --> 00:12:10.499 What's the problem? 96 00:12:10.499 --> 00:12:16.828 That K is positive. 97 00:12:16.828 --> 00:12:20.908 Okay, how would we do that? 98 00:12:20.908 --> 00:12:26.158 Well, I could standardize the thing 1st solution. 99 00:12:26.158 --> 00:12:30.028 It's to find a new random variable. 100 00:12:30.028 --> 00:12:37.918 Call it L and we're going to make it a minus 1 over 3. so now for L. 101 00:12:37.918 --> 00:12:42.568 Little case down here, so the mean of, um. 102 00:12:42.568 --> 00:12:49.019 So, for this, the mean of Al is, it's going to be 0. 103 00:12:49.019 --> 00:12:57.089 And the Sigma Val is going to be won this house mean and standard deviation transform. So. 104 00:12:57.089 --> 00:13:00.899 So, um. 105 00:13:00.899 --> 00:13:04.499 Yes, that'd be 2 thirds where the Sigma. 106 00:13:06.839 --> 00:13:11.458 Um, I don't think so. Um. 107 00:13:13.259 --> 00:13:20.038 Because a translation does not change the Sigma and a scale. 108 00:13:20.038 --> 00:13:25.078 Changes the Sigma linearly. So, um. 109 00:13:26.188 --> 00:13:30.359 I mean, okay, so there's a problem with that. Let me do an aside here. 110 00:13:32.219 --> 00:13:35.729 So, and it will come back to that. So. 111 00:13:37.979 --> 00:13:48.028 So so I have so we have a random variable X, and we have some sort of expected value and some sort of. 112 00:13:48.028 --> 00:13:51.479 The page didn't change. 113 00:13:51.479 --> 00:13:55.528 Come on there, um. 114 00:13:55.528 --> 00:14:04.379 And and some sort of standard deviation. Okay now, let's say to fine. 115 00:14:05.153 --> 00:14:10.043 Why new random variable, because we're going to functions of random variables. That's the section in this chapter. 116 00:14:10.313 --> 00:14:20.994 So we can, let's define why is a X plus B, so what happens is expected value of why is going to be a times the expected value. 117 00:14:21.298 --> 00:14:25.918 Plus be. 118 00:14:25.918 --> 00:14:29.099 And a standard deviation of why. 119 00:14:29.099 --> 00:14:43.198 It is going to be a time to center deviation of X and the variance will be the square. So the variants of Y will be a squared variants of X. 120 00:14:45.323 --> 00:14:59.033 So, they do the obvious thing, um, expected value shifts and scales and then the variance just scales deviation of variance to scale and it has to be a, and a squared because of do dimensional analysis. 121 00:14:59.668 --> 00:15:08.969 Okay, so that's it. So I'll be the end of the aside here. 122 00:15:10.649 --> 00:15:16.499 Okay, so if I come back to the previous thing here, then. 123 00:15:16.499 --> 00:15:23.009 So, if I to find L is K minus 1 over 3. 124 00:15:23.009 --> 00:15:29.908 Oh, I see. Oh, you mean a 3rd for the shift. 125 00:15:29.908 --> 00:15:37.438 Let's see now. 126 00:15:39.599 --> 00:15:42.989 Okay. 127 00:15:48.359 --> 00:15:55.739 Oh, I see what you're talking about my mistake. Whoever said the 3rd email. Me. That's what's a point? 128 00:15:55.739 --> 00:16:01.168 What I would really want to say to clean this up is. 129 00:16:06.448 --> 00:16:20.249 No, I'm getting confused. 130 00:16:20.249 --> 00:16:21.714 I got myself confused. 131 00:16:21.744 --> 00:16:36.503 I'm just plugging. 132 00:16:37.288 --> 00:16:41.788 The previous I prefer out. Yeah, you did. 133 00:16:41.788 --> 00:16:47.818 Ask Katie 3 minus 1 over 3. 134 00:16:47.818 --> 00:16:51.958 Yeah, so. 135 00:16:53.188 --> 00:17:02.158 That right, yeah, I think actually, what I might want to do is. 136 00:17:12.239 --> 00:17:22.798 Okay, let's try something like that then and see. 137 00:17:22.798 --> 00:17:26.608 Manage getting myself confused, but. 138 00:17:26.608 --> 00:17:30.659 Okay, let's think that's more likely to be. Right so. 139 00:17:32.249 --> 00:17:39.959 Okay, so now, what I want again is I'm wanting this this here, um, the probability that. 140 00:17:39.959 --> 00:17:44.909 Okay, it's greater than equal to 0. okay. So. 141 00:17:46.709 --> 00:17:53.759 So, okay, great and equal to 0 is 0, Alice minus 1. so. 142 00:17:53.759 --> 00:17:56.759 So that here, um. 143 00:17:56.759 --> 00:18:03.328 Okay, greater than or equal just because it equals the probability with alligators and minus 1. 144 00:18:03.328 --> 00:18:10.048 And that's going to be queue of minus 1 and that's going to be about point 85 give or take. 145 00:18:10.048 --> 00:18:15.838 So, professor yes. So how did you get that view with Alan the Sigma about. 146 00:18:17.489 --> 00:18:31.378 Using this rule from the aside thing up here, where, if we scale the, if we scale around and variable. 147 00:18:32.398 --> 00:18:46.019 If we got around 0 X, if we for any distribution doesn't have to be normal, then if we've Y, equals X, plus speed and the expected value of why is a times expected value of X plus B. 148 00:18:46.019 --> 00:18:57.088 That's going to be, let's see if I can scroll back in the page. So we're on page 160 and I let me go back to where we see something like that. 149 00:19:10.769 --> 00:19:17.669 Well. 150 00:19:21.239 --> 00:19:25.769 Um. 151 00:19:42.088 --> 00:19:48.868 Silence. 152 00:19:50.249 --> 00:19:54.689 Silence. 153 00:20:13.138 --> 00:20:17.489 Silence. 154 00:20:19.169 --> 00:20:23.398 Silence. 155 00:20:38.788 --> 00:20:51.088 Eva. 156 00:20:52.979 --> 00:20:58.648 Well, here's the thing about shifting etc. I've X plus these expected value of X. 157 00:20:58.648 --> 00:21:02.249 A C and. 158 00:21:09.358 --> 00:21:13.288 The variant scales here um. 159 00:21:21.538 --> 00:21:29.308 Well, you can just work it out from 1st principles. Let's just do that here. 160 00:21:29.308 --> 00:21:44.098 Okay, so you've got random variables X and Y, why it was a plus B. 161 00:21:48.148 --> 00:21:51.868 Silence. 162 00:22:02.818 --> 00:22:07.919 Okay, now, so we have to. 163 00:22:13.588 --> 00:22:20.338 Okay, so now we have to convert from D. Y. D. X and. 164 00:22:27.479 --> 00:22:33.058 We have the thing that I did, um. 165 00:22:33.058 --> 00:22:36.239 Last time where. 166 00:22:40.288 --> 00:22:48.538 It's going to be value why actually. 167 00:22:48.538 --> 00:22:51.778 So. 168 00:22:51.778 --> 00:22:56.489 Um, because. 169 00:22:57.778 --> 00:23:01.858 Again, effects was feet and why is meters. 170 00:23:03.209 --> 00:23:07.078 Then it's going. Okay, so now we're going to have, um. 171 00:23:07.078 --> 00:23:15.568 And we're going to throw in so here X's Y minus B a. so now what we're going to have. 172 00:23:32.784 --> 00:23:33.953 Sorry, what am I doing? 173 00:23:34.378 --> 00:23:41.638 Like, for why is, um. 174 00:23:41.638 --> 00:23:46.019 Next must be 1 or a. 175 00:23:47.729 --> 00:23:52.679 Yeah. 176 00:23:52.679 --> 00:23:58.528 He hacks okay now so that's going to be. 177 00:24:28.858 --> 00:24:38.548 Confusing myself right now. So that is the effects. 178 00:24:45.778 --> 00:24:50.219 It is 1. 179 00:24:51.358 --> 00:24:57.058 And I've managed to I've managed to confuse myself actually. 180 00:24:57.058 --> 00:25:05.128 So, and continue on Monday. 181 00:25:05.128 --> 00:25:12.358 So so you raised a valid point. Um, let me think about that. So. 182 00:25:12.358 --> 00:25:16.318 Yeah, I'll continue that whole example on Mondays and. 183 00:25:16.318 --> 00:25:20.009 I don't want to waste the class this time while I am confused myself. 184 00:25:20.009 --> 00:25:30.719 In any case so I'll continue that. Exactly. But let's look at some new stuff that and remind me don't let me forget. 185 00:25:33.959 --> 00:25:38.068 Some new random variables here. 186 00:25:38.068 --> 00:25:43.709 Okay, you 422 there's a gamma random variable and, um. 187 00:25:46.588 --> 00:25:57.929 It's got 2 parameters and it's a continuous thing on 0 to infinity and it's nice because by setting the. 188 00:25:57.929 --> 00:26:04.528 2 parameters, which are alpha in Lambda we get a number of special cases now. 189 00:26:04.528 --> 00:26:16.588 The government and it's got a generalization of the factorial code upper case cam and uppercase cam fan. Plus 1 is factorial. So it's a factorial thing where it's argument can be a fraction. 190 00:26:17.604 --> 00:26:32.394 And 2, special cases they mention here the Chi square, random variable issues for statistical tests. What a statistical test is that. 191 00:26:32.699 --> 00:26:41.548 You've got some population and you think it follows a certain distribution and the high scar test will tell you. 192 00:26:41.548 --> 00:26:46.499 How likely your guess is so. 193 00:26:46.499 --> 00:26:51.689 What would be an example of Chi square test? 194 00:27:28.229 --> 00:27:33.118 Silence. 195 00:27:36.659 --> 00:27:45.358 Silence. 196 00:27:53.429 --> 00:27:59.338 And this is the note, Eva. 197 00:28:00.358 --> 00:28:05.429 And you get numbers like this, let's say. 198 00:28:17.848 --> 00:28:23.278 Okay, so. 199 00:28:23.278 --> 00:28:30.898 Get these numbers get these hit number of hits. 200 00:28:30.898 --> 00:28:34.949 Ok, so. 201 00:28:34.949 --> 00:28:38.128 If, um. 202 00:28:40.229 --> 00:28:46.858 Form. 203 00:28:46.858 --> 00:28:50.009 We're hitting each square. 204 00:28:59.519 --> 00:29:02.788 What's the probability. 205 00:29:27.749 --> 00:29:33.659 What's a probability you'd see something this even in the Chi square. 206 00:29:33.659 --> 00:29:37.019 As used here. 207 00:29:38.638 --> 00:29:42.749 Okay, oh. 208 00:29:44.034 --> 00:29:56.634 Statistics is a compliment to, on probability probability we have distributions and we compute the probability of seeing things. Like, we toss a coin 10 times and we see 6 heads or tails. That's probability. 209 00:29:57.683 --> 00:30:03.653 Statistics is we observe something. And then what's the probability that. 210 00:30:03.989 --> 00:30:12.239 That the distribution fit, so Chi square be useful for something like that. We'll see that later. That's 1 application of the. 211 00:30:12.239 --> 00:30:20.159 Of the gam, another application is this is the Erlang random variable and they talk about it. 212 00:30:21.449 --> 00:30:31.739 Here it's the sum is a sum of exponential random variables will be an airline memorandum variable. 213 00:30:31.739 --> 00:30:35.788 So the example they have 424 here. 214 00:30:35.788 --> 00:30:45.929 Is we got your system and you got a component and 2 spares and when once you. 215 00:30:45.929 --> 00:31:00.324 Turn a component on it fall it's lifetime falls and exponential is an exponential random variable. So you'd like to know something about what's the probability that your factory is still working at time key? 216 00:31:00.564 --> 00:31:02.453 She wants to probably distribution. 217 00:31:02.759 --> 00:31:16.469 That at least 1 of your that you all 3 of your parts haven't prime and the 2 spirits haven't died by time team. Well, each 1 as a distribution exponential. 218 00:31:16.469 --> 00:31:22.348 See, 1 to random variable, which the sum of the lifetime of the 3. 219 00:31:22.348 --> 00:31:31.199 Components, which would be the some, you want to do random variable to some of 3 exponential, random variables and that's going to be Erlang and. 220 00:31:31.199 --> 00:31:39.058 So that's useful there so we'll throw some numbers at the thing later, but that's an executive summary of it. 221 00:31:39.058 --> 00:31:50.034 So is the gamma random variable so it's argument to 0 to infinity and it's useful because it has 2 parameters and by picking particular values for the 2 parameters, 222 00:31:50.034 --> 00:31:51.534 you get these other useful things, 223 00:31:51.923 --> 00:31:55.854 Chi square and stuff like that. 224 00:31:56.189 --> 00:32:09.838 Another 1 of these generic probability distribution Jenna is called beta at random variable and it's defined for an argument from 0 to 1 and it's also got 2 parameters. 225 00:32:09.838 --> 00:32:16.709 And be there, and again by setting particular values. 226 00:32:16.709 --> 00:32:21.538 For the 2 parameters you get interesting special cases. 227 00:32:21.538 --> 00:32:27.509 Well, a simple 1 is if a equals 1 and vehicles 1, it's a uniform random variable 0 to 1. 228 00:32:27.509 --> 00:32:41.038 And we may crunch some numbers added later. I'm giving you ideas today. Another 3rd. Interesting. Random variable is the koshi random variable. 229 00:32:41.038 --> 00:32:46.259 And I'll show you how you would get it physically how it could happen. 230 00:32:46.259 --> 00:32:56.159 It's been a pointer. 231 00:32:58.019 --> 00:33:03.148 So, got some sort of point here here. 232 00:33:05.159 --> 00:33:08.939 And the RV is the point. 233 00:33:08.939 --> 00:33:14.699 On the X axis, the pointer points to. 234 00:33:17.429 --> 00:33:21.088 Okay, and that. 235 00:33:22.138 --> 00:33:30.808 That is koshi. Okay so it's got physical applications now. 236 00:33:30.808 --> 00:33:43.169 The thing was the cow, she is, it has no moment if you try to compute the mean of it. Oh, well, what does it looks like the distribution it's sort of like. 237 00:33:44.848 --> 00:33:50.338 It's sort of like the normal, but the tails are patter. It doesn't go to 0 as quickly. 238 00:33:50.338 --> 00:34:02.368 Well, you can see here, you can look at the denominator, but 1 over X squared it's going to 0 as actually gets big, but not as fast as normal, which has either the minus X squared. Okay. So. 239 00:34:02.368 --> 00:34:12.148 The thing was the cow, she is, it's got no moments if you try to intergrade excess effects and divergence. So if you do something like this integral of X f, effects. 240 00:34:12.148 --> 00:34:17.489 Yes, it diverges. 241 00:34:17.489 --> 00:34:21.599 It can't be computed so. 242 00:34:21.599 --> 00:34:25.739 So, it's an example of a. 243 00:34:25.739 --> 00:34:30.389 For well defined, probably distribution that has no moments. 244 00:34:30.389 --> 00:34:37.588 For hotels, other some of these ones and so on, but just get some examples. 245 00:34:40.079 --> 00:34:44.068 Yeah, so but okay, um. 246 00:34:45.179 --> 00:34:51.298 So, I mentioned some of them here, some problems, um. 247 00:34:52.528 --> 00:35:04.498 Let me, I'm confused myself and then I'll do with these other over the weekend and I'll do these problems on Monday. I'll do some new stuff at the moment in chapter 52 random variables. 248 00:35:04.498 --> 00:35:12.958 Give you ideas and executive summaries and we'll do some, um. 249 00:35:21.208 --> 00:35:31.978 Do some to. 250 00:35:37.978 --> 00:35:43.528 That's why I write to page numbers often on things, but. 251 00:35:58.108 --> 00:36:01.798 Okay. 252 00:36:02.514 --> 00:36:12.204 So chapter 5, we're talking about pairs of random variables. So if I toss a dart at a dartboard, for example, look at where it hits, we've got X. 253 00:36:12.204 --> 00:36:25.793 and Y, if your ID introduction to engineering design, and you have your catapult, you fire, and you record where it lands X and Y, I've done sections of ID for a number of years. 254 00:36:25.793 --> 00:36:32.153 You've got 2 random variables. So the 1 experiment E are firing your catapult. 255 00:36:32.699 --> 00:36:38.009 And where it lands that 1 experiment generates 2 random variables. 256 00:36:38.009 --> 00:36:41.579 Pair of random variables and. 257 00:36:41.579 --> 00:36:47.969 They're resulting from the same experiments so we want to perhaps work with. 258 00:36:47.969 --> 00:36:52.768 Means and mass functions density functions and so on with them as a pair. 259 00:36:53.784 --> 00:37:02.603 And we get reasonable extensions for defining probably mass function if it's discreet, the density function PDF, if it's continuous. 260 00:37:03.023 --> 00:37:10.974 And in either case the cumulative distribution function works for discrete continuous and mix the mastosis mess here. 261 00:37:11.278 --> 00:37:17.219 Now, the thing is this is that the 2 random variables. 262 00:37:17.219 --> 00:37:28.798 They may be depended on each other, which we might call correlated or not. And in your ID experiment, they're probably not correlated that, you know, you. 263 00:37:28.798 --> 00:37:33.599 You know, the Marshall model that you fire, it's off to the right or the left. 264 00:37:33.599 --> 00:37:42.148 Call that excellent say it's off above and below far and near call that why X and Y are probably not related to each other. 265 00:37:43.103 --> 00:37:56.454 Okay, so but it took random variables example. I use Monday. I'm looking out the window right now. I'm just stuck 10 mile smart Pi, looking out, and it's raining. I could look at the weather today. 266 00:37:56.603 --> 00:38:04.824 That's experiment is the weather on some date and my 2 random variables could be the amount of sunshine and the amount of rain. 267 00:38:05.188 --> 00:38:13.380 And I can quantify the amount of sunshine, of course, because I've got my solar panels and I've got. 268 00:38:13.380 --> 00:38:23.219 And I've got an app on my phone, which shows me how much electricity my solar panels generate. And that could be used to quantify the sun. 269 00:38:23.219 --> 00:38:29.789 You know, the sign on really nice day a couple of days ago I got over 40 kilowatt hours. 270 00:38:29.789 --> 00:38:34.619 Of sun, um, today, I'm probably going to get a couple of kilowatt hours. 271 00:38:34.619 --> 00:38:46.679 So, and so the experiment is the weather on is on a, is pick a day and so the sun and the rain are going to be. 272 00:38:46.679 --> 00:39:00.329 They're going to track opposite of each other. They're going to be dependent on each other and there's a number we can compute for how to pet linearly dependence called a correlation coefficient. It goes to minus 1 to 1 from. 273 00:39:00.329 --> 00:39:10.170 Minus 1 means the 1 variable tracks in the opposite direction to a variable. Plus 1 means attracts in the same direction. 274 00:39:11.219 --> 00:39:16.440 Okay, and so I got some points here, um. 275 00:39:18.750 --> 00:39:25.139 And I'll just hit you some of the points in the chapter, then we'll do examples. 276 00:39:26.969 --> 00:39:31.110 We got some outcome toss point or something. 277 00:39:31.110 --> 00:39:39.960 Okay, I got an example 5.1 here we look at students at height and weight. So the random experiment is we pick a student. 278 00:39:39.960 --> 00:39:44.969 And the outcome is still the height and weight are going to be positively correlated. 279 00:39:44.969 --> 00:39:48.449 So, okay. 280 00:39:48.449 --> 00:39:51.690 Um. 281 00:39:51.690 --> 00:39:57.449 Example to. 282 00:39:57.449 --> 00:40:00.659 Might be again, you got to the user. 283 00:40:00.659 --> 00:40:05.309 Can I look at a web ad, or skip the web ad? 284 00:40:06.659 --> 00:40:13.289 And well, and then 2, random variables with a number of times. 285 00:40:13.289 --> 00:40:19.440 He said it looks at the edge versus the number of times. They go to the next page or something. 286 00:40:19.440 --> 00:40:28.139 Okay, Here's an interesting 153 communications. 287 00:40:28.139 --> 00:40:41.969 So, you're, let's suppose you're starting a large file on your computer and let's suppose the disk has blocks that are, let's say a modern assets is the actual block size of, say, 4 kilobytes. 288 00:40:41.969 --> 00:40:46.260 And so, let's suppose you've got a 10,000 bite file. 289 00:40:46.260 --> 00:40:56.489 That you're storing, so it's going to take 2 full blocks of 4 kilobytes each and then a final fraction of a block. That'll be a little less than half a block. 290 00:40:56.489 --> 00:41:00.719 So you have a random experiment. 291 00:41:00.719 --> 00:41:06.510 The you pick a random file and then the. 292 00:41:06.510 --> 00:41:18.630 Random variables you got to measure are the number of full bites, full blocks that it would take to store and then that's the 1st strategy very well. And the 2nd, round new variable is the amount of a fractional block. 293 00:41:18.630 --> 00:41:29.309 Or if you're setting packages over the net, it would get broke the long message. We could broken up into packets. So that it takes so many full packets. And then a fractional packet. 294 00:41:29.309 --> 00:41:36.030 And so you 2 random variables with the number of full packets and the size of a fractional packet. 295 00:41:36.030 --> 00:41:39.719 And we want to look at distributions of that, for example. 296 00:41:40.949 --> 00:41:52.590 That sort of thing no, you can do combos of random variables and you can look at combos of them. So if you got. 297 00:41:52.590 --> 00:41:58.500 2, random variables, X and Y, coordinates in a plane then here a. 298 00:41:59.094 --> 00:42:13.494 The event that X, plus Y, is less than equal to 10 the event to your point is in that gray region, below into the left of that line with slope minus 1, for example, and now we might want to compute the probability of something like that happening. 299 00:42:13.800 --> 00:42:19.559 Your value shapes, so you can have so if you have input to random variables X and Y. 300 00:42:19.559 --> 00:42:24.869 You can find define events. 301 00:42:24.869 --> 00:42:32.219 Which are shapes on the plane, and then you can start asking for probability of these events happening and so on. 302 00:42:32.219 --> 00:42:35.940 Um. 303 00:42:35.940 --> 00:42:45.630 And if you got discreet, random variables, give any 2 discrete, random variables, X and Y, you can make a pair of them. Now, you've got a 2 random variables system. 304 00:42:45.630 --> 00:42:54.119 And and they talk about and we'll talk about some examples here. 305 00:42:55.260 --> 00:43:00.449 Well, let me just write down some examples. Actually then we'll go back to the book. 306 00:43:03.719 --> 00:43:09.119 I mean, it could be independent. Um, okay. 307 00:43:22.079 --> 00:43:28.199 And so. 308 00:43:30.570 --> 00:43:35.429 Equals 1 say. 309 00:43:35.429 --> 00:43:42.150 There's coin heads Y, equals 3rd coin just a 2nd here. 310 00:43:42.150 --> 00:43:49.019 Why it costs 1 that's the 2nd. 311 00:43:50.969 --> 00:43:57.030 Into something like that. Okay. So now we can start talking about the probability, you know. 312 00:43:57.030 --> 00:44:04.440 Actually goes on and close 1, it's going to be 1 quarter and that sort of thing. 313 00:44:04.440 --> 00:44:13.559 So that's a 2 random variable example. These things are clearly independent, but it starts giving you the ideas, and we can talk about. 314 00:44:13.559 --> 00:44:24.239 So, we can do a probably we can do a probably mass function here or something like that. 315 00:44:24.239 --> 00:44:28.500 X y0. 316 00:44:28.500 --> 00:44:33.929 1 terra fine and the values are going to be like. 317 00:44:33.929 --> 00:44:45.030 1 corridor that corridor, something like that. So the probability mass function on 2 random variables and they are. 318 00:44:45.030 --> 00:44:49.500 They're, they're independent of each other. 319 00:44:49.500 --> 00:44:54.090 For 2, random variable, continuous case. 320 00:44:54.090 --> 00:44:57.150 Let me start a new page here. 321 00:44:58.829 --> 00:45:03.449 Silence. 322 00:45:06.059 --> 00:45:12.539 So, we'll pick pick a point uniformly. 323 00:45:12.539 --> 00:45:16.619 In the square. 324 00:45:16.619 --> 00:45:22.710 Okay. 325 00:45:22.710 --> 00:45:29.699 And then we're going to get things like, I, for some particular value of X and Y. L. B1. 326 00:45:30.809 --> 00:45:39.090 Okay, so that's getting, you know, that sounds so simple. It's like, why am I spending time on it? 327 00:45:39.090 --> 00:45:45.059 Try to fill the time somehow. Now, let me make the touch harder. Now. Let me say. 328 00:45:45.059 --> 00:45:57.059 This is a line X plus Y, equals 1 and this is X and this is why 0 and 1. so now the new experiment is pick a point. 329 00:45:59.039 --> 00:46:02.400 Uniformly. 330 00:46:03.510 --> 00:46:10.619 In the triangle okay now. 331 00:46:13.260 --> 00:46:25.739 Okay, so so now it might get a little interesting because. 332 00:46:25.739 --> 00:46:35.309 Maybe, you know, now there's a relation between X and Y. 333 00:46:44.489 --> 00:46:49.619 X and Y. 334 00:46:49.619 --> 00:46:54.719 Um. 335 00:46:54.719 --> 00:47:05.130 So, because if because here's the thing. 336 00:47:06.570 --> 00:47:09.659 It's X plus Y, last name the 1, sir. 337 00:47:09.659 --> 00:47:12.719 X. 338 00:47:15.269 --> 00:47:19.320 Why okay. 339 00:47:19.320 --> 00:47:25.679 Okay. 340 00:47:26.699 --> 00:47:31.889 And correlate so. 341 00:47:33.719 --> 00:47:41.880 They're correlated so if, um. 342 00:47:41.880 --> 00:47:50.880 And we could actually, that's we could actually do some stuff on this. Let me draw this figure again here. 343 00:47:52.530 --> 00:47:57.570 So, we're in this region in here. Okay. So. 344 00:47:57.570 --> 00:48:02.039 And this is a density function, so after it. 345 00:48:03.809 --> 00:48:07.860 It's going to be some constant Tom. See. 346 00:48:07.860 --> 00:48:16.260 If 0 less seem to X less than 1 is, they're listed for why less vehicle to 1 minus X. 347 00:48:16.260 --> 00:48:20.969 And 0, otherwise else. Okay. 348 00:48:23.130 --> 00:48:26.280 Um, so the question is, what is C. 349 00:48:26.280 --> 00:48:31.409 Well. 350 00:48:33.780 --> 00:48:38.550 0, to 1 see. 351 00:48:41.219 --> 00:48:46.739 Okay, that's going to be the inter girl. Um. 352 00:48:49.619 --> 00:48:56.130 And that's going to be, um. 353 00:48:58.679 --> 00:49:06.360 Well, that's going to be the same as the under girl. Actually see the undergraduate x0 to 1. 354 00:49:06.360 --> 00:49:20.190 Which is X Square, which you see over to. 355 00:49:20.190 --> 00:49:25.139 Okay, so see will be 2. okay so. 356 00:49:26.460 --> 00:49:31.380 If I can scroll up, if I'm scrolling too quickly, then let me know. 357 00:49:32.849 --> 00:49:40.260 Okay, so actually, so C equals 2. so f of X and Y equals. 358 00:49:40.260 --> 00:49:46.079 2 axis from 0 to 1 and Y. 359 00:49:46.079 --> 00:49:50.190 Is 70 to 1 minus X and again, it's this thing here. 360 00:49:50.190 --> 00:49:56.730 Okay, now we can get marginal distributions marginal. 361 00:49:59.219 --> 00:50:04.409 Marginal PDF is that a great integrate out of variable? 362 00:50:07.079 --> 00:50:11.400 Great out of variable. 363 00:50:11.400 --> 00:50:21.329 So, FX of X is going to be the integral effects and Y, X comma. Why integrating out why. 364 00:50:21.329 --> 00:50:26.309 Okay, well it's going to be. 365 00:50:34.710 --> 00:50:41.639 But this is defined all, but this is positive only for Y, from 0 to 1 minus. 366 00:50:41.639 --> 00:50:45.000 So, that is going to be. 367 00:50:50.369 --> 00:50:59.039 So, um, so this is going to be defined as, um. 368 00:51:00.630 --> 00:51:04.889 It's actually going to look like it's going to look like the undergo that. So so the. 369 00:51:04.889 --> 00:51:08.969 Pdf for X is going to be. 370 00:51:12.985 --> 00:51:27.715 Which makes sense, because this is the 2 variable thing up here X and Y, you integrate down and you're going to get the density density of X and so on. So you have a 2 variable random variable. You can integrate out. 371 00:51:27.989 --> 00:51:36.780 X and Y, get something like that can get. I know we could do that for why also we get the same thing. So. 372 00:51:36.780 --> 00:51:42.179 So that's 1 of the points. You have 2 random variables here. 373 00:51:42.179 --> 00:51:46.860 That is the, um. 374 00:51:46.860 --> 00:51:51.119 That's the PDF. The CDF. 375 00:51:51.119 --> 00:51:56.670 Page page, so the CDF. 376 00:51:56.670 --> 00:52:06.780 Big f, it was a probability that. 377 00:52:09.780 --> 00:52:13.710 And why center you go to why. 378 00:52:13.710 --> 00:52:20.460 So, if I have my examples of Triangle thing here. 379 00:52:22.530 --> 00:52:27.420 X and Y, so the PDF for this point here. 380 00:52:28.980 --> 00:52:33.900 Is a probability there the PDF of another point here. 381 00:52:33.900 --> 00:52:45.114 Let's say will be the probability that this come on shift shift shift. There you go. Okay. So the CDF is the lower left. 382 00:52:45.114 --> 00:52:57.985 So, the CWS accumulate density function for a point is the probability that the random variable is below and to the left of that point, including the, including the boundary lines that's less than or equal there. 383 00:52:58.375 --> 00:53:00.355 And this means that it will handle. 384 00:53:01.110 --> 00:53:12.210 Discrete as well as continuous. So if I use say to demonstrate the CDF with my coin example, um. 385 00:53:12.210 --> 00:53:19.739 So, what are the 2 coins. 386 00:53:22.019 --> 00:53:28.980 So, again, what the, what the mass function was again, we had a quarter. 387 00:53:28.980 --> 00:53:32.190 1, quarter quarter. 388 00:53:32.190 --> 00:53:43.949 In 1 quarter, then this is for the 1st coin being tails or heads and the 2nd coin being tails their heads. Okay. 389 00:53:43.949 --> 00:53:51.119 So, for example, the CDF here. 390 00:53:53.190 --> 00:53:57.960 Just 1 quarter the, um, CDF here. 391 00:53:59.010 --> 00:54:03.090 That's 1 I need more colors. 392 00:54:05.969 --> 00:54:09.659 The CDF, let's get fancier. 393 00:54:10.800 --> 00:54:15.449 The CDF here is 1 half. 394 00:54:17.400 --> 00:54:22.710 The CDF here is 1 quarter. 395 00:54:22.710 --> 00:54:26.039 The CDF. 396 00:54:26.039 --> 00:54:30.269 Here there's 1. 397 00:54:30.269 --> 00:54:34.829 The CDF here. 398 00:54:37.380 --> 00:54:45.269 Is 1 half and so on each 1 of these cases, what I mean is for the CDF for that point there. 399 00:54:45.269 --> 00:54:53.610 So, for here, for example, would be to CDN for that point. 400 00:54:53.610 --> 00:55:08.550 So, the cubital distribution function in this probability, mass function examples, the probability that the outcome was to the lower and to the left. So then the light colored green. 401 00:55:09.385 --> 00:55:22.465 Value up here. Well, everything that can happen is below to the left. So the idea for that is 1 and so on the orange point here it might be X equals 2 Y, equals a half. Perhaps. 402 00:55:23.304 --> 00:55:29.244 2 possible outcomes are inside that quarter plane. So, the CDF for that, because 1 half. 403 00:55:29.550 --> 00:55:34.500 And so on, so we can do Canada distribution function for. 404 00:55:34.500 --> 00:55:39.000 Discrete to a variable. 405 00:55:39.000 --> 00:55:43.019 Random variables who could also do it for continuous. 406 00:55:43.019 --> 00:55:47.579 The triangle things that touch complicated. Let me. 407 00:55:47.579 --> 00:55:55.889 Do. 408 00:55:55.889 --> 00:55:59.730 Say you in a form. 409 00:55:59.730 --> 00:56:08.519 Square or something so let's suppose we've got something like that. 02121. 410 00:56:08.519 --> 00:56:12.750 So, your random experiment is pick a point. 411 00:56:12.750 --> 00:56:18.929 Okay, and just the density function. 412 00:56:20.789 --> 00:56:30.900 That's going to be 10 X1 and realistic or Y, listening for 1 and 0 otherwise. 413 00:56:30.900 --> 00:56:34.170 Now, the CDF. 414 00:56:34.170 --> 00:56:39.869 If I pick a point here, CDF. 415 00:56:42.000 --> 00:56:46.320 Big after the CDF some scripts to say. 416 00:56:48.300 --> 00:56:55.619 So, the easy case is X and Y, are in 0 and 1 that's going to be X and Y. 417 00:56:55.619 --> 00:57:00.659 Yes. 418 00:57:00.659 --> 00:57:04.860 But we got lots of other cases here. Okay. 419 00:57:04.860 --> 00:57:11.429 Um, we might have something over here. Okay. 420 00:57:11.429 --> 00:57:18.389 And effects is bigger than 1 and why is some 0 to 1? Then it's going to be Y. 421 00:57:22.260 --> 00:57:25.530 And. 422 00:57:25.530 --> 00:57:31.409 It's going to be. 423 00:57:31.409 --> 00:57:35.039 Colors see. 424 00:57:35.039 --> 00:57:42.269 Green, let's suppose we got something up here. 425 00:57:42.269 --> 00:57:51.929 Going to be 1, if X cratering greater than equal to 1 and Y, great equal to 1 it's going to be. 426 00:57:54.510 --> 00:58:01.050 Somewhere down here 0, if. 427 00:58:01.050 --> 00:58:04.139 X less ankle 0 or. 428 00:58:04.139 --> 00:58:07.530 Why did I say close 0 and so on. 429 00:58:07.530 --> 00:58:14.880 And other special cases so so the thing with these discreet. 430 00:58:14.880 --> 00:58:19.320 Well, these 2 random variable things. 431 00:58:19.320 --> 00:58:24.750 Here you end up with more special cases. I mean, I didn't even list all the special cases here, so. 432 00:58:26.670 --> 00:58:31.590 But it gives you an idea so we could actually. 433 00:58:31.590 --> 00:58:35.250 Labels them out. 434 00:58:35.250 --> 00:58:39.059 I could label them something like this at Chile. 435 00:58:44.969 --> 00:58:48.719 C. F, why. 436 00:58:48.719 --> 00:58:53.760 Well, if we're in here, it's X times. Y. 437 00:58:53.760 --> 00:58:59.190 If we're here to 0, if we're out to the right. 438 00:58:59.190 --> 00:59:04.380 If we're out to the right here, it's going to be why. 439 00:59:05.789 --> 00:59:11.880 Wrap above it, it's going to be X if in the top, right? It's going to be 1. 440 00:59:11.880 --> 00:59:19.289 So, 12345, special cases, I guess, for the CDF. 441 00:59:19.289 --> 00:59:30.510 Okay, and you can do things like this, you can have expected value in this case it's really easy. 442 00:59:30.510 --> 00:59:30.894 Of course, 443 00:59:30.894 --> 00:59:32.304 expected value of Y, 444 00:59:33.414 --> 00:59:36.985 and you can good things like the expected value of X squared, 445 00:59:36.985 --> 00:59:51.474 what should be caps in here because that's a random variable value of X squared and that's going to be the integral of X squared DX 0 to 1 execute over 3 equals 4th and 446 00:59:51.835 --> 00:59:54.295 2nd value of Y squared equals 1. 447 00:59:54.295 --> 01:00:06.355 and also now, what's no, you can now start talking about the expected value of X times Y, and stuff like that. And this will capture the relation between X and Y. 448 01:00:08.280 --> 01:00:13.590 So the expected value of X Y, and Louisiana girl. 449 01:00:13.590 --> 01:00:17.880 Have X Y. 450 01:00:17.880 --> 01:00:25.650 X y1. Okay. And in this particular case. 451 01:00:28.980 --> 01:00:33.510 It's going to be 1, I think. 452 01:00:36.989 --> 01:00:45.750 No, no, no, I have to put X Y, in here, of course, because that's my expected value. Okay. 453 01:00:47.280 --> 01:00:51.389 So, we can check what the expected value of X Y, is, um. 454 01:01:05.610 --> 01:01:11.909 Ok, and we can separate it out. 455 01:01:15.780 --> 01:01:21.179 Okay, I'm screwing things up again. 456 01:01:28.230 --> 01:01:34.440 So the expected value of X Y. 457 01:01:34.440 --> 01:01:40.349 A quarter. Okay. Um. 458 01:01:41.940 --> 01:01:47.730 So, and this will capture the relation between the 2 of them if X and Y. 459 01:01:47.730 --> 01:01:58.710 Are varying to gather the expected, this will be higher and then the expect, and then it's the opposite way. It will then be lower. 460 01:01:58.710 --> 01:02:02.579 And we can start doing things like. 461 01:02:02.579 --> 01:02:08.070 We can then compute this by. We can do things like. 462 01:02:13.079 --> 01:02:27.780 That's like a cool variance. In this particular case, it's going to be 1 quarter, minus a half times 1, half equals 0 and that means basically, no, no correlation between X and Y. 463 01:02:34.349 --> 01:02:44.489 If it were positive, there'd be a positive relationship between the whose negatives it'd be a negative relationship between them. 464 01:02:44.489 --> 01:02:47.760 So this is an introduction to some simple. 465 01:02:47.760 --> 01:02:58.409 To some simple 2, variable problems that we can have, we've got the joint probability density or mass function. We, the lower left. 466 01:02:58.409 --> 01:03:05.849 Integral the lower left square that will give us a cumulative distribution function on the 2 variables and we can find. 467 01:03:05.849 --> 01:03:15.030 We can integrate out the PDF for the company mass function to get a 1 variable thing. Just for 1 variable. We could do means we can do. 468 01:03:15.030 --> 01:03:19.139 Joint expectations, like the expected value of X times Y. 469 01:03:19.139 --> 01:03:23.730 And stuff like that, so. 470 01:03:23.730 --> 01:03:32.639 So, let's see some more examples ideas. So the book, I'll hit the book in more detail, but this is giving you. 471 01:03:34.710 --> 01:03:38.130 And intuitive feeling here are some of this. 472 01:03:41.670 --> 01:03:47.099 Well, let's look, actually, for example, 55 nice example here. 473 01:03:47.099 --> 01:03:51.780 What's happening is you're going to package switch. 474 01:03:51.780 --> 01:03:56.550 Let me draw, let's say scroll. 475 01:04:01.710 --> 01:04:09.389 Okay. 476 01:04:11.550 --> 01:04:21.539 So you got you got 2 input ports here and a packet arrives at each fort was probabilty a half. 477 01:04:21.539 --> 01:04:28.559 But it's independently, so we could get 0 packets 1 Packard or 2 packets. So. 478 01:04:31.139 --> 01:04:35.670 Each, and it might. 479 01:04:35.670 --> 01:04:41.489 Independently. 480 01:04:41.489 --> 01:04:45.869 Have a packet. 481 01:04:45.869 --> 01:04:54.900 Okay, it was probabilty 1 half. Okay. And then the packets. 482 01:04:59.130 --> 01:05:02.369 And then. 483 01:05:02.369 --> 01:05:08.789 And each input packet goes. 484 01:05:08.789 --> 01:05:11.969 To either. 485 01:05:13.110 --> 01:05:17.460 Output is probably well, see, 1 half. Okay. 486 01:05:19.559 --> 01:05:27.630 So, now we want to look at so now that's the so, this is the experiment. 487 01:05:30.449 --> 01:05:36.539 We let we let it go what until some input packets arrive. 488 01:05:36.539 --> 01:05:40.739 And then, um. 489 01:05:40.739 --> 01:05:49.619 And we look at the number of number of outputs going to port 1 or port 2 here. 490 01:05:49.619 --> 01:05:58.349 And, and basically our, we use the Greek letter. 491 01:05:58.349 --> 01:06:04.619 Is there for the actual I'm experiment and we show we show the 2 outputs. 492 01:06:04.619 --> 01:06:08.969 Um. 493 01:06:08.969 --> 01:06:16.500 And then X and Y, are the number of outputs going to the I'm going to the 2 packets. So. 494 01:06:16.500 --> 01:06:20.670 And then we want to look at X and Y, and how they. 495 01:06:20.670 --> 01:06:31.139 And how they relate to each other. So, so actually would be the number of packets and I would put. 496 01:06:31.139 --> 01:06:36.090 Part 1, and why it would be the number of packets and output for too. So. 497 01:06:41.364 --> 01:06:53.784 And, okay, and what this shows here, the 1st line is the packet of the 1st, port design output board 1 or 2, and a packet of the 2nd port designed for 1 and 2. 498 01:06:53.815 --> 01:06:57.744 so a 2 way 1 means the 1st input got a packet for output. 499 01:06:58.019 --> 01:07:03.690 2 and the 2nd input kind of packet for 1. so each output gets 1 and 1. so on. 500 01:07:03.690 --> 01:07:11.309 Okay, so we might start wanting to compute then probability mass function. So those are the. 501 01:07:11.309 --> 01:07:14.699 So the inputs is either port. 502 01:07:14.699 --> 01:07:18.030 Input port either did not get or did get. 503 01:07:18.030 --> 01:07:21.449 I did get a packet, let's say. 504 01:07:21.449 --> 01:07:26.400 So so the total number of input packages, there are 1 or 2. 505 01:07:26.400 --> 01:07:36.449 And the outputs, some output thing. Okay. So we want to start doing probability mass function with a tabular thing here, showing various. 506 01:07:36.449 --> 01:07:39.719 The random variables, X and Y, from the experiment. 507 01:07:40.735 --> 01:07:53.965 And then we can start looking at the probabilities, the probability that neither output port got a packet means was 00 means. There were no packets coming in at all, which is a quarter because each input, 50, 50 out of pocket, or did not. 508 01:07:53.965 --> 01:07:56.875 So, and so we can compute these various things here. 509 01:07:57.510 --> 01:08:02.789 Find a way to put this as a table, perhaps table a here. 510 01:08:02.789 --> 01:08:05.880 And, um. 511 01:08:07.050 --> 01:08:21.270 And then we can start doing marginal probabilities some down the margins, and get the probably distribution mass function for acts on its own or probably mass function for apply on its own, for example, by summing down or across. 512 01:08:22.529 --> 01:08:33.119 So, if we come across is probably the best functions for why Y, equals 0916tothe time 1616tothe time and 2116ofthe time. So. 513 01:08:35.369 --> 01:08:38.579 So, we can start doing stuff like that. 514 01:08:40.199 --> 01:08:43.470 Okay. 515 01:08:43.470 --> 01:08:47.520 What's happening and figure 5 6. 516 01:08:49.289 --> 01:08:52.590 We're tossing. 517 01:08:53.609 --> 01:08:58.319 We're tossing to dice. 518 01:08:58.319 --> 01:09:01.920 But they have a problem, they're not random. 519 01:09:04.050 --> 01:09:18.210 But they're non random in a correlated sort of way if they were random than the probability of anything half of any pair of numbers would be 136. but what's happening here is the 2 dice. 520 01:09:18.210 --> 01:09:21.989 Are more likely than chance to have the same value. 521 01:09:22.645 --> 01:09:33.895 How that would happen I don't know. But the probability of both dice showing a 1, let's say 240 seconds where, if it was random, it would be 132nd. 522 01:09:33.895 --> 01:09:40.704 It would be 136 to 40 seconds is 101 and that's more than 136. and the problem is everything else is scaled down proportion. 523 01:09:44.430 --> 01:09:51.449 Okay, so get a weird sort of thing there and then we can do marginal things. Now, the funny thing. 524 01:09:51.449 --> 01:09:55.199 With figure 5.6, we're tossing the 2 dice that are. 525 01:09:55.555 --> 01:10:09.925 Related to each other, is that the marginal probabilities are all 16 so the probability the 1st stop showing a 1 is 16. you add up 540 seconds into 40 seconds. We get 740 seconds. That's a 6. so we look at each die separately. It's fair. 526 01:10:09.925 --> 01:10:17.545 It's only a combination of the 2 dice that is unfair. They track each other to a small. 527 01:10:22.800 --> 01:10:29.010 Yeah, okay so they talk about it here the 2 loaded dice. 528 01:10:29.010 --> 01:10:41.399 The marginal thing I mentioned, you summed down some out 1 variable integrate out 1 variable. Um, if I go back to the dicing up here. 529 01:10:41.965 --> 01:10:52.614 So you want to find the marginal probability that the 1st dye shows a 1 it is the 1st type being a 1 and the 2nd di, being everything, all 136 and you sum down that column. 530 01:10:52.614 --> 01:11:01.645 You get a 6 that's the marginal on the 1st dye showing up 1 the marginal for the 2nd di, showing a threes at some across row 3 and you'd get 1 6. 531 01:11:02.640 --> 01:11:11.250 So that's March, it'll probably mass function to sum up 1 of the variables. Nothing fancy there. 532 01:11:11.250 --> 01:11:20.670 And for the for that support switcher experiment, you could get marginal I showed you the plot a while back. 533 01:11:20.670 --> 01:11:25.260 Okay. 534 01:11:27.359 --> 01:11:33.569 59, I'll do in detail Monday. Maybe this is the. 535 01:11:33.569 --> 01:11:48.114 Thing where we're transmitting along message, that's been broken up into packets and maybe packets that are a 1024 bites a number of them plus a fraction leftover. So, the experiment is around a message. 536 01:11:50.305 --> 01:12:00.114 Which has some distribution geometric distribution for the total link to the message and from that experiment we get 2 random variables in our Q. 537 01:12:00.114 --> 01:12:08.545 is the full number of full packets and our is the link to the last fractional packet. So, 2, random variables from the 1 experiment. 538 01:12:08.880 --> 01:12:16.649 And now we might want to learn something about you and are, like, what are their means and so on. 539 01:12:16.649 --> 01:12:23.310 And what are, what are they, what are their probabilities? And this works out that here. 540 01:12:25.079 --> 01:12:30.569 Then the joint joint community distribution functions, and so on. 541 01:12:30.569 --> 01:12:36.180 Okay, work out some math later, so. 542 01:12:36.180 --> 01:12:42.390 Yeah okay so that's a reasonable place to stop. I'm. 543 01:12:43.854 --> 01:12:57.204 Well, here this is 511 is talking about the uniform square thing. So what we saw today was we saw some more 1 variable things, and I got myself confused. So I'm totally simple thing. 544 01:12:57.234 --> 01:13:03.055 I just don't want to take your classes time. Why? I'm confused myself. All worked out so we saw some. 545 01:13:04.170 --> 01:13:11.340 Exercises with 1, random variable I'll do some more and we also saw some. 546 01:13:11.340 --> 01:13:25.710 A couple of more common probability distributions, gamma and beta Egypt, 2 parameters and by picking particular values for the parameters, you can make some important special cases. 547 01:13:25.710 --> 01:13:33.689 Gamma the tie square distribution is helps. You do a test of. 548 01:13:33.689 --> 01:13:42.899 A statistical tests that assume distribution is in fact reasonable for some experiments. So. 549 01:13:42.899 --> 01:13:47.159 You toss well, the simple thing you tell us here. 550 01:13:48.324 --> 01:13:59.694 Coin 10 times and get 7 heads and 3 tails you could use Chi square to find the probability of that happening. It's the coin was fair now for something. Simple. Like that is easier ways to do that. That would be an example. 551 01:13:59.694 --> 01:14:11.274 The Chi squared the gamma also can be used a special cases, the Erlang distribution and that's probably distribution for the some of them exponential random variables. So. 552 01:14:13.045 --> 01:14:25.015 Typical thing you're bringing here, you've got a pile of spares, and as each part wears out, you fire up you power up the next spare and each pairs distribution for its lifetime is exponential. 553 01:14:25.314 --> 01:14:33.715 Then the Erlang would give you the probably distribution that this machine is still functioning. I. E, that you haven't yet used up all EMS fares. 554 01:14:34.470 --> 01:14:42.149 And as a special purpose, 1 was the beta distribution, which you make special values it gets interesting special cases. 555 01:14:42.149 --> 01:14:56.460 And then we moved on to the 2 variable thing, this chapter here chapter 5 and you've got 2 random variables that come out of the same experiment to experiment as Utah, the dark and 2, random variables, the X and Y, from where the dark hits. 556 01:14:56.460 --> 01:15:09.329 And then each random variables separately, of course, is mean and variance, you can start doing cumulative, you can do a joint distribution or mass function, joint. 557 01:15:09.329 --> 01:15:14.939 Cumulative distribution function, which is the probability for the lower left quarter playing. 558 01:15:14.939 --> 01:15:19.380 Below the point, and then you can, we will start seeing how. 559 01:15:19.380 --> 01:15:22.920 Getting measuring relations between the 2 random variables. 560 01:15:24.060 --> 01:15:29.880 So that's a reasonable. 561 01:15:29.880 --> 01:15:35.579 Place to stop so have a good weekend and maybe the weather clears up. 562 01:15:35.579 --> 01:15:41.789 And I'll stay around for a minute in case it's any questions. 563 01:15:41.789 --> 01:15:49.739 Other than that see, you virtually Monday and again, next Thursday a week from today no class. 564 01:15:49.739 --> 01:15:55.170 Take a day off, go to the spring town meeting whatever you like. 565 01:15:55.170 --> 01:15:58.170 I know homework due next Thursday either. 566 01:16:05.039 --> 01:16:09.689 No questions.