WEBVTT 1 00:00:11.964 --> 00:00:18.925 Hey, good afternoon people I don't know if things are working, but my universal question. 2 00:00:29.789 --> 00:00:33.570 We can even professor. 3 00:00:37.799 --> 00:00:45.990 Can you hear me anyone. 4 00:00:47.070 --> 00:00:51.840 We can even professor. 5 00:00:53.009 --> 00:01:03.630 No, yes, thank you. Chris. Okay, the reason I'm checking is every so often people can't hear me. Okay so. 6 00:01:03.630 --> 00:01:12.900 Interior watching the chat window, and it just talk to my line of sight slightly. But so this is probability. 7 00:01:12.900 --> 00:01:17.909 Whoops, 1 more thing to get going. 8 00:01:17.909 --> 00:01:21.420 A, 2nd, here. 9 00:01:42.775 --> 00:01:43.435 Right. 10 00:01:49.950 --> 00:01:54.719 Too many tools and. 11 00:01:57.030 --> 00:02:04.409 Everything is sort of, in theory. 12 00:02:04.409 --> 00:02:08.939 Working that this theory, and there is practice, of course. 13 00:02:10.919 --> 00:02:20.370 So, like to do today, I will show you a touch more of Matlab and do some more stuff from chapter 4. 14 00:02:20.370 --> 00:02:32.400 Single variable probability things and the 1st thing is exam 2 will be April 5th same rules as before. I posted asked for opinions on Webex last night. So. 15 00:02:32.400 --> 00:02:40.259 Okay, Matlab we have a Matlab window over here. 16 00:02:40.259 --> 00:02:44.460 And working today. 17 00:02:44.460 --> 00:02:50.340 So, and again, just some things about Matlab, you can. 18 00:02:51.389 --> 00:03:02.729 You can create a sequence of numbers with call in here 1 to 10. if you want to go every to something like that we could assign at Tom. 19 00:03:03.840 --> 00:03:10.259 For example, and you can do I make it smaller to. 20 00:03:10.259 --> 00:03:19.560 To make it easy, you can work with factors as I mentioned before times, 10 multiplies everything by 10. 21 00:03:19.560 --> 00:03:34.319 So, I'm a does stuff like that and excuse me I'm talking too much today. Let me show you a few more things. I listed them in the class for yesterday. Actually, yesterday. 22 00:03:34.319 --> 00:03:37.860 Um. 23 00:03:37.860 --> 00:03:48.840 So, we can do things like what to show them quite simply probability distribution function for something like. 24 00:03:48.840 --> 00:03:53.189 Binomial or something, if this works. 25 00:03:56.189 --> 00:04:02.580 Anyway, give it some parameters. I don't know. 26 00:04:02.580 --> 00:04:09.689 Okay, what did I do here? 27 00:04:11.159 --> 00:04:17.610 Oh, right. You can go back and edit something here. 28 00:04:25.615 --> 00:04:33.115 Good so, the PDF does the probability density function. The 1st arguments. Depend on what distribution you're doing. 29 00:04:33.444 --> 00:04:42.264 The 1st argument is the vector of values that you want to evaluate it at and then we have the parameters for by normally it would be that. 30 00:04:42.264 --> 00:04:51.115 And that, for example, that you might recognize our force, you find out and if you want for something big, suppose I want it for. 31 00:04:51.809 --> 00:04:59.639 May be a 100. don't do a fair guy is always. 32 00:04:59.639 --> 00:05:04.559 That is always fair coin, but maybe I just wanted for the values perhaps. 33 00:05:04.559 --> 00:05:17.879 Around the middle or something, we could do something like that and sell the value. You can do something like that and we could some, we could do something like them and go back. 34 00:05:20.249 --> 00:05:25.319 And we could sum something like this, and it better add up to 1. 35 00:05:26.939 --> 00:05:31.588 Sums up to 1 if we don't want the. 36 00:05:31.588 --> 00:05:41.759 You know, we could also get say the CDF go back to my favorite example here and that would be the continuous the cumulative. So. 37 00:05:41.759 --> 00:05:50.399 Adding up. Okay, normal. I give examples here. Um, binomial um. 38 00:05:50.399 --> 00:05:58.228 We can do a bar chart of something like this. It's really nice being able to add it to previous line to a bar chart of that. 39 00:05:58.228 --> 00:06:02.879 Come on our chart. 40 00:06:02.879 --> 00:06:08.519 Good, it's up there at the top here. 41 00:06:08.519 --> 00:06:11.728 Now, I can if I. 42 00:06:15.088 --> 00:06:18.658 Oh, either quite other considerations. 43 00:06:20.548 --> 00:06:29.428 Well, I asked for comments and somebody, I only got a couple of comments. The 1 person said they got the 5th said they could not do the 5th. So. 44 00:06:30.959 --> 00:06:42.329 That was a consideration or comments that people gave. Okay so we got bar chart. We could go back and re, edit the previous thing and do PDF and so on. 45 00:06:42.329 --> 00:06:49.649 Sort of thing like that. Okay, you could do a normal distribution. 46 00:06:49.649 --> 00:06:57.209 Um. 47 00:06:57.209 --> 00:07:07.079 And down here, the parameters are different, it's the mean and standard deviation. 48 00:07:07.079 --> 00:07:10.889 And let me make if say, from. 49 00:07:13.978 --> 00:07:24.538 Minus 3 may be in steps of point 5 or something to 3 and with a mean of 0 and Saturday deviation of 1 perhaps. 50 00:07:24.538 --> 00:07:30.869 So this is, why do we get. 51 00:07:30.869 --> 00:07:34.949 So, I have examples like that here, so. 52 00:07:37.858 --> 00:07:41.069 What else to have as examples here? 53 00:07:41.069 --> 00:07:48.389 Farm disagreed. Okay. You can also, um. 54 00:07:48.389 --> 00:07:53.939 Again, each particular 1 has argument Hassan, if we. 55 00:07:53.939 --> 00:08:01.858 So, you notice the plot, it has 1 plot up at a time well, you can have it do more than 1 plot and when you do a new product, replaces the old. 56 00:08:01.858 --> 00:08:08.399 Um, we could try and. 57 00:08:11.488 --> 00:08:18.209 See. 58 00:08:22.769 --> 00:08:29.879 And, okay, so, you know, is for to say 0, to. 59 00:08:29.879 --> 00:08:33.989 10 or something well, argument could be a. 60 00:08:33.989 --> 00:08:39.178 Just probably able to get your number of things and a parameter might be. 61 00:08:39.178 --> 00:08:46.229 See, if this works here, I'm making this up that I'm going along bang. It worked. 62 00:08:46.229 --> 00:08:52.139 So, the mean is 2 and from 0 to 10. so it peaks it 2 and. 63 00:08:52.139 --> 00:09:01.889 If we did a plus on, let's say, and this could be a fraction. So it could be say, 1, it's each 1 shifted to the left. 64 00:09:01.889 --> 00:09:11.158 I could make it say point 5 really shifted to the left. Um, and then I could say. 65 00:09:12.504 --> 00:09:27.234 5, to me is 5 shifted more does it? Right? And so she already with the mean of 5 it's looking pretty close to a normal because with plus on the standard deviation is a square root of the mean. So, the main is 5 Saturn. 66 00:09:27.234 --> 00:09:39.293 Deviation is 2.2. and so 0 is and so this is not completely symmetric, but it's not so enormously far off. And if I do a plus on with the mean of 10. 67 00:09:40.558 --> 00:09:46.528 Um, I mean, I could bigger numbers here say the 20 and mean of 10. 68 00:09:46.528 --> 00:09:55.109 It's looking really close to a normal so it's fairly symmetric. Not completely, but it's pretty good. So. 69 00:09:55.109 --> 00:09:59.188 And. 70 00:09:59.188 --> 00:10:09.208 Okay, so also this would be is suppose you wanted suppose you had to do a croissant. 71 00:10:09.208 --> 00:10:13.948 Suppose you want to find, say from 5 up for the sign here. 72 00:10:13.948 --> 00:10:24.958 20 say, you could say, it's always, you want to find the right tail tails to say 0 to 5. 73 00:10:24.958 --> 00:10:30.359 That's bang there and now you want to find what's the sum. 74 00:10:34.943 --> 00:10:45.173 Before I go for something, so what you can do with the so you can use math lab to compute numbers where there's no simple form for a partial some of the bullets on. 75 00:10:46.649 --> 00:10:51.568 You know, we could try 0 to 10. 76 00:10:51.568 --> 00:10:55.438 We could try 1000. 77 00:10:56.063 --> 00:11:10.374 1, pretty close. Okay, so you could find that probably density function sums, plotted to see what it looks like. And so on stuff like that. And I have the code here. You can do. 78 00:11:13.349 --> 00:11:22.019 Yeah, okay. Um, also some stuff we can do random numbers. Um. 79 00:11:22.019 --> 00:11:36.928 Whatever you notice how it pops up help for you. That's nice. So, what this is doing is just 3 by 3 matrix of this random number as, as I mentioned other things here. No. 80 00:11:36.928 --> 00:11:48.749 There's a 1, so random generate trend of numbers from 0 to 1, but you can produce a matrix of them or something and. 81 00:11:50.009 --> 00:11:58.889 Oops, and every time you do what you get different, random numbers, and if you don't want them scaled. 82 00:11:58.889 --> 00:12:08.249 You know, and you could scale them. Obviously, I could take this round a matrix modified by 10 thing. And so now they're each element is uniform from, um. 83 00:12:09.568 --> 00:12:13.589 Yeah, from 0 to 10. 84 00:12:15.239 --> 00:12:22.019 Make it something just Charlie. Okay. Now. 85 00:12:22.019 --> 00:12:25.889 Actually say 5, here. 86 00:12:25.889 --> 00:12:32.788 Okay, now, and again, every time you get your numbers and you could, for example. 87 00:12:32.788 --> 00:12:37.499 I introduce you say to large large I suppose I find the mean of that. 88 00:12:39.778 --> 00:12:53.698 Risk predict mean expect. Okay. It's I mean, every time I do it, I get a different number. It's on the average it's a half, but only 5 of them, it's not a half, but suppose instead of. 89 00:12:53.698 --> 00:12:59.308 5 numbers I do a 100 numbers, and the mean is going to be fairly close to point 5. 90 00:12:59.308 --> 00:13:03.599 And we'll quantify what that means. If I do a 1000 numbers. 91 00:13:03.599 --> 00:13:08.578 It's much closer if I do 10,000 numbers closer, it's still. 92 00:13:08.578 --> 00:13:14.129 Okay, uniform numbers we can get. 93 00:13:16.739 --> 00:13:20.068 Just into jurors, that's all. 94 00:13:22.769 --> 00:13:26.428 Actually, the numbers for energy. 95 00:13:28.283 --> 00:13:39.683 The 1st argument actually is the range because it's 0 to 1, does it mean for an injured? So I can get rounded yours in a certain range here. So I could have done it with the random floats. 96 00:13:39.894 --> 00:13:45.594 And Donna truncation on or multiplied up and done application or something but okay. 97 00:13:46.739 --> 00:13:51.538 Probably also. 98 00:13:51.538 --> 00:13:55.979 Yeah, just a vector of them um, you can also get, um. 99 00:13:55.979 --> 00:13:59.399 Normal random numbers and. 100 00:14:01.979 --> 00:14:05.249 So, we get normal something like. 101 00:14:05.249 --> 00:14:08.519 See, what this does. 102 00:14:08.519 --> 00:14:11.519 Yeah. 103 00:14:12.839 --> 00:14:18.269 Well. 104 00:14:18.269 --> 00:14:28.139 Yeah, there's a major so these are normal number is the calcium, normal mean 0 center. Deviation 1 see. 105 00:14:28.139 --> 00:14:33.389 Get them in a matrix or a vector or something and I have a good example. Silver. 106 00:14:33.389 --> 00:14:40.979 Okay show you how I can plot stuff. 107 00:14:43.798 --> 00:14:49.288 Let me go back to PDF, so we do a PDF say, binomial. 108 00:14:49.288 --> 00:14:53.129 Make a normal normal. 109 00:14:53.129 --> 00:14:57.028 And, um. 110 00:15:01.469 --> 00:15:05.308 And, um. 111 00:15:08.308 --> 00:15:13.708 Okay, wondering I do here. 112 00:15:22.168 --> 00:15:26.999 The minus 3 to 3. 113 00:15:26.999 --> 00:15:34.139 Center deviation. 114 00:15:34.139 --> 00:15:38.849 Okay. 115 00:15:38.849 --> 00:15:43.739 Now, if I want to plot it, I could also say. 116 00:15:50.129 --> 00:15:58.948 Let's say here, what is copying what I got there and I'd say X equals I give it this way to do just the stream a thing saying. 117 00:16:01.318 --> 00:16:10.259 Text and then I could say something like that X, I could say Y, equals on PDF. 118 00:16:10.259 --> 00:16:15.119 Normal. 119 00:16:15.119 --> 00:16:19.859 And, um. 120 00:16:19.859 --> 00:16:23.278 Okay, so now I could. 121 00:16:23.278 --> 00:16:27.298 Let's see what happens here. 122 00:16:30.418 --> 00:16:36.568 That's my clock. You go party planning here. Okay. 123 00:16:38.938 --> 00:16:43.798 And you can scroll it up if your. 124 00:16:46.139 --> 00:16:50.818 Okay, and I show here how we could bought 2 things. Um. 125 00:16:52.198 --> 00:16:55.408 We got why here? 126 00:16:55.408 --> 00:17:03.359 Um, it's made quite too normal to so I could say plot. 127 00:17:03.359 --> 00:17:11.068 X Y2 recorded good here too. I now I could plot the 2 of them together. 128 00:17:11.068 --> 00:17:17.159 And. 129 00:17:19.949 --> 00:17:25.979 Okay. 130 00:17:25.979 --> 00:17:33.358 Did I go wrong here? 131 00:17:39.088 --> 00:17:48.028 Oh, of course, it's why not why 1. 132 00:17:49.949 --> 00:17:55.348 Time lucky. 133 00:17:55.348 --> 00:17:58.499 So, I just wanted to things like that. 134 00:18:01.884 --> 00:18:11.513 So that if you do a 3 D plot, you can actually grab the plot and chain move it around and so on. Okay. 135 00:18:11.513 --> 00:18:19.584 So, quick blurb on how my Matlab can we use if we're doing some of this stuff and I got the things that I did here. 136 00:18:19.858 --> 00:18:24.628 Oh, okay. And again, my opinion of it, I've mentioned it last time. 137 00:18:24.628 --> 00:18:32.519 It's okay. Okay. 138 00:18:32.519 --> 00:18:40.348 Now, I got a little blurb here in probability density because it can get. 139 00:18:40.348 --> 00:18:43.618 Touch confusing. 140 00:18:43.618 --> 00:18:52.138 Suppress this for the moment. Oh, does anyone have anyone anything else you'd like me to do? Put into mathematics into math lab. 141 00:18:52.138 --> 00:18:57.749 Yeah, okay. So, um. 142 00:18:58.949 --> 00:19:07.199 So, I've got a blurb here on probability density, as I said. 143 00:19:08.878 --> 00:19:21.509 It can get confusing. So I thought this example with units would help you and this also ties into functions of a random variable. 144 00:19:21.509 --> 00:19:25.019 Also, vaguely, which is about where we are in chapter 4. 145 00:19:25.019 --> 00:19:28.318 So, the thing is this that. 146 00:19:28.318 --> 00:19:32.729 Um, I suppose I got English versus metric. 147 00:19:34.288 --> 00:19:42.808 And, oh, by the way there's 2 different definitions and U. S laws on the conversion to English and metric. 148 00:19:42.808 --> 00:19:47.848 For some purposes, and in both cases the, um. 149 00:19:47.848 --> 00:19:54.058 Metric unit is the primary 1 and there's laws defining how the English system. 150 00:19:54.058 --> 00:20:01.739 Relates to the metric system. 1 law says that an inch is exactly 2.54 centimeters. 151 00:20:01.739 --> 00:20:05.939 Another law says that. 152 00:20:05.939 --> 00:20:11.608 And then is a meter divided by 39.37. exactly. 153 00:20:11.608 --> 00:20:19.288 For 2, different applications of the inch now, these 2 different definitions are not exactly data call. 154 00:20:19.288 --> 00:20:25.138 They are they differ in, like, 1 part and kind of the 7th or something. So. 155 00:20:25.138 --> 00:20:29.578 No, 1 accused laws and necessarily being. 156 00:20:29.578 --> 00:20:37.138 Consistent with each other, so, in any case. So Here's an example to try to help you understand. 157 00:20:38.788 --> 00:20:47.009 I kept it up some of the some of the issues with functions of random variables. 158 00:20:47.009 --> 00:20:50.249 So, get a uniform random variable on 0 1. so it's. 159 00:20:50.249 --> 00:21:01.048 1, and maybe it's feet, let's say axis intuitively feet. Okay. So I can even try sketching it out. 160 00:21:01.048 --> 00:21:06.808 Where is my stream window? 161 00:21:10.979 --> 00:21:16.138 Insist on having a black boundary. That's all. I have to do this. 162 00:21:16.138 --> 00:21:26.669 Oh, if you're wondering why I don't use a microphone headset while lecturing now. Well, 1st, my laptop. 163 00:21:26.669 --> 00:21:34.709 Has some better than average microphones as laptops goal and the 2nd thing is. 164 00:21:34.709 --> 00:21:41.189 I'm writing onto the iPad with an eye pencil and the pencils connected the iPad with. 165 00:21:41.189 --> 00:21:45.838 And so I. 166 00:21:45.838 --> 00:22:00.773 Um, so if I Bluetooth on, then with the iPad, and also with my laptop, the laptop will connect to the iPad. And then that effect is it uses up all the blue to slots and doesn't make it easy to use a Bluetooth headset. 167 00:22:01.078 --> 00:22:04.919 Any case so. 168 00:22:04.919 --> 00:22:11.189 Come on here. 169 00:22:15.148 --> 00:22:18.868 Yeah, so the simple thing. 170 00:22:18.868 --> 00:22:24.659 So f, here and it's going to be like this. Okay. 171 00:22:25.919 --> 00:22:32.878 Okay, and so we get the probability of something in here. 172 00:22:32.878 --> 00:22:38.489 Say point 5, 2.51.01. 173 00:22:38.489 --> 00:22:42.028 Now, if we change to centimeters. 174 00:22:42.028 --> 00:22:46.858 Okay, what this is going to look like, is going to be. 175 00:22:46.858 --> 00:22:50.638 This because there's some. 176 00:22:52.679 --> 00:22:56.429 So this here okay, so this is fee. 177 00:22:56.429 --> 00:23:00.479 And this here is centimeters, so it's going to be. 178 00:23:03.868 --> 00:23:10.828 So, this, this, this height here it's going to be 130 is because it has to integrate out to 1. 179 00:23:10.828 --> 00:23:17.429 So so X equals 1 in the interval here. 180 00:23:17.429 --> 00:23:24.298 Half of Y equals 130. yes. And the reason is the integral to half of Y. 181 00:23:25.469 --> 00:23:29.489 Throughout a 30 has to be 1, just as with X. 182 00:23:31.348 --> 00:23:36.689 Because the girls 0 to 1, half of X DX has to be 1. okay. 183 00:23:36.689 --> 00:23:41.818 So so how we would convert from X to Y. 184 00:23:42.989 --> 00:23:46.528 Is the thing any value of why. 185 00:23:46.528 --> 00:23:53.578 Could be 30 X. okay. So if X was a 10th of a foot, why would be 3. 186 00:23:53.578 --> 00:24:04.019 3 centimeters. Okay. And what this means. So this is a conversion and what the conversion for f. of why. 187 00:24:06.328 --> 00:24:11.608 And it's going to be 130. okay. 188 00:24:11.608 --> 00:24:14.699 So, it's the reverse thing. 189 00:24:14.699 --> 00:24:22.888 And from here you get over, so we go in the inverse here. So. 190 00:24:22.888 --> 00:24:28.679 F Y, equals 1 over di why by DX? 191 00:24:28.679 --> 00:24:32.759 Okay, okay. 192 00:24:32.759 --> 00:24:40.949 And I talk about it up there and this relates to functions of a random variable. 193 00:24:42.449 --> 00:24:47.308 Okay, some more stuff actually. 194 00:24:47.308 --> 00:24:54.328 Yeah, that's a reasonable point. And may come back to back later some examples but okay. 195 00:24:57.358 --> 00:25:05.459 Functions of random variables. Let me hit some of that. Let's see page 175 and so on. 196 00:25:05.459 --> 00:25:08.608 Excuse me. 197 00:25:11.278 --> 00:25:16.709 We go here. 198 00:25:29.368 --> 00:25:34.888 Okay, we saw some of these things before here. Something they're fairly simple. 199 00:25:40.439 --> 00:25:52.739 Let's look at for 2009 for a minute or 2, because again, it's a useful example. The on Garcia, the price of the book is horrible, but at least they do have. 200 00:25:52.739 --> 00:25:56.489 A lot of nice examples and if I look here. 201 00:25:58.469 --> 00:26:10.648 Again, we got the case where you're I joke you specter for spectrum, communications, whatever, but any of these. 202 00:26:10.648 --> 00:26:21.239 You know, servers that are providing capacity, they provision only a finite amount, affects a certain amount of capacity. That's not enough to handle everyone at the same time. 203 00:26:21.239 --> 00:26:33.929 Like, highways, you're driving on, or people who live up in Clifton park in this area, and you're going home at the end of the business day. You're driving over the twin bridges, which is. 204 00:26:33.929 --> 00:26:39.838 And there's a backup because the bridge was not designed for the busiest hour of the 24 hour day. 205 00:26:39.838 --> 00:26:49.138 So that's so we'd like to compute the probability or the average delay for that again. Your. 206 00:26:49.138 --> 00:26:55.828 You're trying to make a phone call is a fixed number of channels there, not enough channels for everyone to talk at the same time. 207 00:26:55.828 --> 00:26:58.949 So. 208 00:26:58.949 --> 00:27:02.519 So, what do we do. 209 00:27:02.519 --> 00:27:13.499 So, we wanted we want to throw some math at it. Okay. And this is another take at this. So we're assuming we get end independent speakers. People want to make a phone call, whatever. 210 00:27:13.499 --> 00:27:24.479 And each speaker, which is fairly small, and we're assuming the speakers are independent. So the number of people who want to talk at a given time is binomial. 211 00:27:24.479 --> 00:27:32.368 And so there's end people that might want to talk, but on the average, only end times fee will want to talk, but it could be more. 212 00:27:32.368 --> 00:27:41.368 So, and we've got in and the supplier basically built and channel M, channels are M, is less than in. 213 00:27:42.023 --> 00:27:54.413 And so we want to start and so the actual number that want to talk is the random variable X. so, every time we run this experiment, X is a different number. 214 00:27:54.413 --> 00:27:58.374 The number of people that want to talk is different every time. Every time we do this. 215 00:27:59.124 --> 00:28:11.723 So, now we want to know the number of people that could not talk signals, got to start got started if X is less than equal to N, y0 effects is bigger than and then Y, is X minus M. 216 00:28:12.719 --> 00:28:18.598 And the, the notation, they have a little, Super plus sign there, which is. 217 00:28:18.598 --> 00:28:21.719 X minus M, but not less than 0. 218 00:28:21.719 --> 00:28:28.679 Okay, so and it's from 0 and why it's here to a minus and minus and. 219 00:28:28.679 --> 00:28:38.128 Okay, so, you know, you invent notation whenever it's useful and it'll Super. Plus is a useful thing to have. So we have it. 220 00:28:38.128 --> 00:28:42.449 Okay, so now we want to know. 221 00:28:42.449 --> 00:28:50.189 Well, the 1st thing we want to know is what's the probability that every customer is happy? So, what's the probability that no, 1 is dropped? 222 00:28:50.189 --> 00:28:54.028 So that's the probability that why is 0. 223 00:28:54.028 --> 00:29:05.669 Which is the probability that XS 0 to M. so that's that some there that partial some here piece of buy from 0 to M. so Here's a place you might want to. 224 00:29:07.078 --> 00:29:11.128 You might want to go Matlab or something. 225 00:29:11.128 --> 00:29:15.929 Let me know let me just demo it perhaps. 226 00:29:15.929 --> 00:29:20.608 So, we're doing so what I'm doing on page. 227 00:29:21.959 --> 00:29:25.169 175 example. 228 00:29:25.169 --> 00:29:29.699 429, and let's say, here's Matlab. 229 00:29:30.749 --> 00:29:37.888 And we're going to use Matlab, we can make it a little bigger. So, let's suppose I have. 230 00:29:37.888 --> 00:29:45.838 I'm making this up as I'm going along, makes life more exciting. So I'm doing a demo who knows? It'll break, but. 231 00:29:45.838 --> 00:29:49.739 We'll see. So, let's suppose I've got a 1000 customers. 232 00:29:49.739 --> 00:29:56.969 And let's suppose a probability that a customer wants to speak at any given time, is. 233 00:29:56.969 --> 00:30:03.509 Oh, I don't know it suppose. 234 00:30:03.509 --> 00:30:09.269 10% or something. Okay, so end times is 100. 235 00:30:11.669 --> 00:30:15.959 So, let's suppose that the. 236 00:30:17.699 --> 00:30:22.679 That's suppose that the supplier builds. 237 00:30:22.679 --> 00:30:26.068 120 channels. Okay. 238 00:30:26.068 --> 00:30:36.598 So now, what's the probability that everyone is happy? Okay so, on the average, 100 channels. 239 00:30:36.598 --> 00:30:42.388 Are are needed sometimes less sometimes more. So the. 240 00:30:44.278 --> 00:30:48.989 You know, so the supplier says I'm going to build 120. 241 00:30:48.989 --> 00:30:55.798 So now, what's the probability probability? 242 00:30:58.169 --> 00:31:07.858 Everyone happy. Okay. Well, so we're going to go. Oh, sorry. I was. 243 00:31:07.858 --> 00:31:11.669 Well, let me pull up Matt lab and let us see. 244 00:31:11.669 --> 00:31:16.499 My Matlab here, you'd be. 245 00:31:16.499 --> 00:31:20.459 Okay, so I'm going to have. 246 00:31:20.459 --> 00:31:23.578 Pdf. 247 00:31:23.578 --> 00:31:27.749 Oh, no meal. 248 00:31:27.749 --> 00:31:32.308 And I'm going to some I'm. 249 00:31:33.898 --> 00:31:45.298 I'm going to come back to that 1st argument, but it's and is 100.1 that argument back in there. 250 00:31:45.298 --> 00:31:48.419 I'm going to want, um. 251 00:31:51.989 --> 00:31:55.828 I want to probability up to 120. okay. 252 00:31:57.058 --> 00:32:10.739 So this is the, if that makes sense, I got all the binomial probability. The probability that there's 0, people want to talk 1 people up to 120 people want to talk. Okay. 253 00:32:10.739 --> 00:32:18.689 And so some of that is going to be the probability that everyone is happy summit. 254 00:32:27.239 --> 00:32:33.689 Oh, wow. That is pretty good. 255 00:32:33.689 --> 00:32:42.598 Basically, if I provide 20%, more than the mean, the probability that anyone is bigger than the. 256 00:32:43.949 --> 00:32:51.358 That is more than a 120 is effectively 0. that's a surprise at chili. Let me try. 257 00:32:52.648 --> 00:32:56.219 Just 11010% more. 258 00:32:56.219 --> 00:33:04.588 I am just enough. 259 00:33:04.588 --> 00:33:18.719 That's the thing I did something wrong. Oh, too late 9,000 day and it's a 1000. that's better. 260 00:33:19.979 --> 00:33:23.729 Okay, so. 261 00:33:24.773 --> 00:33:39.473 Anyone went awake, you had gotten a point. Okay so I have a 1000 people, 10% chance. Any person wants to talk me. Number of people want to talk is a 1000 times point. 1. that's 100. okay. So specter provides 120 channels. 262 00:33:41.189 --> 00:33:46.679 20% more than the mean, it was a 98% chance. That's enough. 263 00:33:46.679 --> 00:33:57.269 It's surprising, but this is, it's because the mean is 100 the actual numbers going to be costing fairly closely around 100. 264 00:33:57.269 --> 00:34:07.169 I mean, let's suppose there's only they want to save a little more money. Let's suppose they have only 10% more than you need. 265 00:34:07.169 --> 00:34:12.719 So, 87% suppose they have no more than you need. 266 00:34:12.719 --> 00:36:12.719 Silence. 267 00:40:15.893 --> 00:40:32.934 Eva. 268 00:40:33.030 --> 00:40:39.000 Silence. 269 00:40:39.000 --> 00:40:44.309 Okay, can you hear me now? Is this any better. 270 00:40:44.309 --> 00:40:51.210 Thank you. Okay, so my 1st laptop crashed. I'm in my 2nd laptop so. 271 00:40:52.289 --> 00:40:56.280 Let's see, continue on from the textbook here. 272 00:40:57.570 --> 00:41:00.840 And. 273 00:41:00.840 --> 00:41:04.139 Silence. 274 00:41:10.889 --> 00:41:14.460 Silence. 275 00:41:14.460 --> 00:41:18.780 I don't know what happened with that. 276 00:41:18.780 --> 00:41:23.250 Silence. 277 00:41:31.559 --> 00:41:35.219 3rd here. 278 00:41:36.539 --> 00:41:40.559 Silence. 279 00:41:40.559 --> 00:41:45.059 Okay. 280 00:41:46.110 --> 00:41:49.860 Silence. 281 00:41:52.920 --> 00:41:57.719 Silence. 282 00:41:59.190 --> 00:42:02.579 Silence. 283 00:42:02.579 --> 00:42:11.309 Okay, I'm sure I think I'm sharing my screen and. 284 00:42:12.869 --> 00:42:17.969 Or in the old laptop. 285 00:42:18.989 --> 00:42:22.139 Silence. 286 00:42:22.139 --> 00:42:28.230 Oh, okay. 287 00:42:28.230 --> 00:42:31.710 Any of your functions where you were saying, okay. 288 00:42:31.710 --> 00:42:37.530 We were seeing so many functions and so on. 289 00:42:37.530 --> 00:42:44.969 Okay, where some relatively new stuff here. 290 00:42:44.969 --> 00:42:49.920 Okay, so. 291 00:42:54.449 --> 00:43:00.329 Okay, still good. 292 00:43:00.329 --> 00:43:04.230 Silence. 293 00:43:04.230 --> 00:43:09.150 So thank you, thank you. Okay, so. 294 00:43:09.150 --> 00:43:15.480 Well, we're doing now in section 4.6 on my backup laptop. I also. 295 00:43:15.480 --> 00:43:20.190 Don't want to take the time to get my iPad connected, but. 296 00:43:20.190 --> 00:43:33.630 So these are these qualities give you estimates for some for some random variable? The thing is that. 297 00:43:33.630 --> 00:43:45.570 There may be a random variable where, you know, what's mean and it's variance, but you don't know. But that's all, you know, you don't know the exact distribution, but, you know, the mean and variance. 298 00:43:45.570 --> 00:43:51.570 Or if we're talking to a ticket really simple, just, I mean, suppose we're talking students at our Pi. 299 00:43:51.570 --> 00:43:59.760 And let's suppose that, you know, well, you know, that the height is now negative. We don't have students with negative height. 300 00:43:59.760 --> 00:44:06.179 And let's suppose you've looked at the students and the main height is. 301 00:44:06.179 --> 00:44:11.250 I don't know 55 and a half feet. Okay. 302 00:44:12.480 --> 00:44:18.510 So, now you want to know what's the probability that a student is more than. 303 00:44:18.510 --> 00:44:24.989 7 feet high or something, it's going to be small, but if you know. 304 00:44:24.989 --> 00:44:30.570 Nothing but the mean height of a student, you don't know how the stool tied to distributed. 305 00:44:30.570 --> 00:44:41.159 Then you can still say something about the probability that as students height is greater than some number. 306 00:44:41.159 --> 00:44:44.400 And that's called the mark of any quality. 307 00:44:44.400 --> 00:44:48.389 It's not a close bound, but, um. 308 00:44:49.949 --> 00:44:57.119 But it's but it's something at least it's better than nothing. So. 309 00:44:57.119 --> 00:45:06.929 Let me check, let me say the main height since was 5 feet, just to make the numbers easy. So, the probability that a student is more than 5 feet high. 310 00:45:10.405 --> 00:45:24.025 Is less critical to 1, or that makes sense. The probability the student would be more than 10 feet. High is less than a half. What you think it's going to be something like 0, actually, but at least it gives you some bound. The probability the student is more than. 311 00:45:24.719 --> 00:45:28.650 5.5 feet high is let. 312 00:45:28.650 --> 00:45:33.389 Is no more than 5 over 5.5 and so on. 313 00:45:33.389 --> 00:45:40.380 So, the America, any quality, all it knows is the mean of a distribution and it. 314 00:45:40.380 --> 00:45:45.960 Gives you the probable it puts a bound on the probability that. 315 00:45:45.960 --> 00:45:50.610 Of getting an outlier, and they worked to the derivation here. 316 00:45:50.610 --> 00:46:03.840 It's not a tight bound, but there are distributions. Well, there are distributions that will hit that bound. Exactly. They're weird ones, but they will. Okay. So now rule. 317 00:46:03.840 --> 00:46:09.809 In science engine is if you have more information, you can make more inferences. So suppose. 318 00:46:09.809 --> 00:46:16.320 We also have the Chevy chef we also have the variance of the height in addition to having. 319 00:46:16.320 --> 00:46:29.400 The mean, so now we can do the Chevy chef and equality and what this is we'll give a double tailed probability. The probability that you're more than a certain amount away from. 320 00:46:29.400 --> 00:46:40.349 The mean, so we know the mean and the standard deviation or the variance is the square of the 2nd deviation but nothing else. So it could be a totally weird non distribution. 321 00:46:40.349 --> 00:46:46.139 Gotta be legal, but kind of negative probabilities probably always have to integrate to 1 and so on. 322 00:46:47.280 --> 00:46:59.730 So, we've got this thing here, so if we have students in the mean, height is 5 feet and the standard deviation is 1 foot. So, the variance is 1 foot also 1 squared then the probability that. 323 00:46:59.730 --> 00:47:04.170 Particular student is more than a foot off the mean is no more. 324 00:47:04.170 --> 00:47:09.539 Well, just thinking because 1 over 1, too loose, it's say, no more than 2 feet off the mean. 325 00:47:09.539 --> 00:47:13.679 Is cannot be more than a quarter. 326 00:47:13.679 --> 00:47:26.010 And probably going to be much less. So, in any case, if, you know, only the mean, you use a mark off, if you use a mean, plus if, you know the mean and variance against the Chevy ship, any quality. 327 00:47:26.010 --> 00:47:36.300 And in doing an example here, and if, you know, like, you know, it's a normal distribution, which is true for. 328 00:47:36.300 --> 00:47:40.079 In many cases, I still give an example here. 329 00:47:40.079 --> 00:47:48.929 For response times, if, you know, I mean, if he measured the mean and the standard deviation 15 and 3, the probability is more than 5 seconds away. 330 00:47:48.929 --> 00:47:53.219 Has to be less than a 3rd 36 actually. Okay. 331 00:47:56.130 --> 00:48:07.139 Yeah, and if you know more, if, you know, it's a calcium, then you get much tighter. You got an example here for 39. 332 00:48:07.139 --> 00:48:12.539 Let's assume it's a normal distribution and so the probability. 333 00:48:12.539 --> 00:48:16.469 That you're more than K segments away from the mean. 334 00:48:16.469 --> 00:48:24.719 By the Chevy chef is less than 1 over K squared. Whereas from if you notice that Gaussian is very much less and using example here cables to. 335 00:48:24.719 --> 00:48:39.389 So, the galaxy and the probability within 2 segments of the mean is 95% probably more than 2 segments away. Is 5% or is it Chevy shared with gave up bound of 25%? 336 00:48:39.389 --> 00:48:43.710 But fair 40, they're showing you an example where the Chevy shove bound. 337 00:48:43.710 --> 00:48:47.400 Is tight, but their weird non monotonically distributions. 338 00:48:47.400 --> 00:48:57.119 So right, and the point here, the more information you have, the better the balance you can get, that just makes sense. Okay. 339 00:48:57.894 --> 00:49:12.324 Okay, transform methods. I'm skipping somewhat characteristic function. I'm skipping you get them in some other courses and I can't if I do the whole book page by page, then we won't get we won't get as much done. 340 00:49:12.324 --> 00:49:14.485 So generating functions. 341 00:49:15.210 --> 00:49:18.300 Again, I'm skipping some like. 342 00:49:18.300 --> 00:49:23.699 Skipping okay class and so on. Okay. 343 00:49:23.699 --> 00:49:30.510 Oh, okay. Here's a new topic now. oopsee reliability. So. 344 00:49:32.159 --> 00:49:36.869 It's calculating failures, reliability, stuff like that. 345 00:49:36.869 --> 00:49:42.510 For liability, I put a little blurb. Let me pull up the. 346 00:49:42.510 --> 00:49:48.150 My website. 347 00:49:52.260 --> 00:49:56.400 Silence. 348 00:49:56.400 --> 00:49:59.550 Silence. 349 00:49:59.550 --> 00:50:12.030 Okay, so I got a blurb here about the inequality. Send me do I'm motivated here mark off and Chevy chef. 350 00:50:13.650 --> 00:50:23.400 And a good point about assumption so, the more the more, you know about the distribution, the more you can compute, but your assumptions had better be correct. 351 00:50:23.400 --> 00:50:29.969 And have an example here, real world examples. People make assumptions that are false. 352 00:50:31.650 --> 00:50:35.940 Talking too much today. 353 00:50:35.940 --> 00:50:47.340 And so they make assumptions about the distribution about how they say the stock market operates and they produce models, which usually work. 354 00:50:47.340 --> 00:50:52.380 It can't prove they don't work until they stop working and I got a link here to. 355 00:50:52.380 --> 00:50:56.610 A hedge fund called long term capital management some years ago that. 356 00:50:56.610 --> 00:51:10.889 They, they came tolerably with, inside of bringing down the U. S. economy. They were very highly leveraged and they made some assumptions they were run by to Nobel laureates. Unpack doesn't mean Nobel laureates are always bright or right so. 357 00:51:10.889 --> 00:51:18.030 Okay, and somebody who knows something about it the header J. P. Morgan gave me diamond. He's a famous person in the finance world. 358 00:51:18.030 --> 00:51:28.469 Observe the markets wings more widely than it's statistically reasonable if your assumptions are so the point he's making you statistical assumptions can be wrong. So. 359 00:51:28.469 --> 00:51:35.010 And so reliability you want to get the fundamentals, right? 360 00:51:35.010 --> 00:51:39.809 So, just, um. 361 00:51:41.489 --> 00:51:47.550 And the motivate reliable, they've got a blur guy like putting in real world stops. So. 362 00:51:47.550 --> 00:51:55.019 40, so 44 years ago today, the green island bridge collapsed. 363 00:51:55.019 --> 00:52:02.760 It didn't look like it did today. It guys it was undermined by a spring flood. 364 00:52:04.199 --> 00:52:11.820 That's the current 1 and see history 1, the original 1 it burnt down at 1 point. 365 00:52:11.820 --> 00:52:17.940 In any case, it collapsed at 1 point March. 1577. okay. 366 00:52:17.940 --> 00:52:22.710 So, it wasn't excuse me. 367 00:52:22.710 --> 00:52:28.440 I'm alone in the room you're not going to get. Okay. Okay. So it wasn't reliable. 368 00:52:28.440 --> 00:52:33.210 I have a quick quick bridge. 90 collapsed. 369 00:52:33.210 --> 00:52:36.690 10 years later i1 killed 10 people. 370 00:52:36.690 --> 00:52:39.750 Is a video of it going down actually so. 371 00:52:40.224 --> 00:52:51.864 And so, again, getting into reliability. So famous alumnus of our Pi was roebling. Jr. built the Brooklyn Bridge took 15 years to build it for the 1st, couple of years. 372 00:52:51.864 --> 00:53:02.005 He's spending money, but there's not obvious what he's spending money on, because he's spending the money on the foundations. They're still there. There would. Some of them are still they're prepped embedded 1870, give or take. 373 00:53:03.000 --> 00:53:06.420 A 150 years later, it's still there. 374 00:53:06.420 --> 00:53:09.869 He had a lot of redundancy in so on. So. 375 00:53:09.869 --> 00:53:13.650 Not a big spreadsheet ever collapsed. Okay. So. 376 00:53:13.650 --> 00:53:17.969 The guy used intuition and so on. 377 00:53:17.969 --> 00:53:30.239 And let me go back to the definition. I'll come back to my examples here so I'll go to my blog and then I'll go through the book. So, this is my summary here. Reliability. 378 00:53:30.239 --> 00:53:38.250 So, it's a probability that item is still functioning. So, if I buy an industry, say an incandescent light bulb. 379 00:53:38.250 --> 00:53:46.110 You know, they're fairly obsolete now, but there's still some of them around. So incandescent light bulbs were standardized to have a 1000 hour. 380 00:53:46.110 --> 00:53:49.260 Life expectancy it was an industry. 381 00:53:49.260 --> 00:54:03.474 Standards group that establish that only a 1000 that could make them on. So so what's the probability? It's still functioning at after 100 hours? 500 hours a 3000 hours. So it's at 0 hours. It's 1. 382 00:54:03.474 --> 00:54:06.324 it starts off functioning. We assume it's not dead on. 383 00:54:07.739 --> 00:54:16.769 It's not born dead, so we got there live. Probably it's still functioning and that's big. And it's just the inverse of the cumulative distribution function. 384 00:54:16.769 --> 00:54:29.159 Give an example here for the exponential CDF is 1 minus heated. The lamb 2. T. so the reliability is 8 of the minus lamb to T the probability that's still alive and time T. 385 00:54:29.159 --> 00:54:34.650 And lamb does just a parameter like you mean, sort of thing. Okay. So now. 386 00:54:34.650 --> 00:54:37.889 What you want is the meantime to failure. 387 00:54:37.889 --> 00:54:41.610 Okay, and that's just well, you've got the. 388 00:54:41.610 --> 00:54:45.420 If you got the liability, then. 389 00:54:46.710 --> 00:54:51.059 Well, it's just turns out it's probably density function. 390 00:54:51.059 --> 00:54:55.710 It works, it works into this and so the. 391 00:54:55.710 --> 00:55:01.559 You can get the meantime to failure and. 392 00:55:01.559 --> 00:55:04.889 That's important. So how long. 393 00:55:04.889 --> 00:55:11.909 So, we got sticky example automobiles. You can't go forward the can you just see variable. 394 00:55:11.909 --> 00:55:17.519 Electronic transmissions are pretty bad so what's the meantime to. 395 00:55:17.519 --> 00:55:21.000 To failure of the transmission. 396 00:55:21.000 --> 00:55:26.880 It's a 5,000 dollar part when it fails, how do I know okay. 397 00:55:26.880 --> 00:55:34.440 So, and then you've got the failure, so, giving you a new ideas here, then you got the failure rate. 398 00:55:34.440 --> 00:55:37.590 Um, and the failure rate is. 399 00:55:39.150 --> 00:55:45.119 Is that if it's still alive, that probably it's going to fail the next Delta. T. so. 400 00:55:45.119 --> 00:55:48.269 So, if we have a lightbulb, let's say. 401 00:55:48.269 --> 00:55:54.119 And called incandescent ones, because they don't last forever safaris. We know. 402 00:55:54.119 --> 00:55:59.429 If you don't abuse some, so so if we have. 403 00:55:59.429 --> 00:56:05.309 Say, I'd like Bob that's already a 1000 hours old. 404 00:56:05.309 --> 00:56:19.920 And it's still alive so, what's the probability of it is going to fail in the next 2nd so That'll be the failure rate. Well, you normalized of course, you know, it's like definition of the density function. So you got the failure rate is. 405 00:56:19.920 --> 00:56:24.570 You know, it's like it's sort of like a reliability given that hasn't died yet. 406 00:56:25.855 --> 00:56:39.414 Okay, and so I'm going to work this into current events. Okay so here's fun. History of unreliable stuff. robin's great. Just never fell down. You've all seen the video that Nicole was narrow bridge swinging itself to death. 407 00:56:39.775 --> 00:56:46.945 So, did you call our bridge? Designers were smarter than their old legs they knew they can analyze. They're smart enough. They didn't need all this redundancy. 408 00:56:47.125 --> 00:56:58.704 The Brooklyn Bridge is like 3 ways of diagonal bracing because the rumblings they said, hey, the spreads is 1500 feet long between the towers. It's going to swinging the wind when things swing in the wind. 409 00:56:59.579 --> 00:57:11.429 Complicated things happen. So senior probably said, no, I don't want my bridge swinging in the wind. So, grace, grace, grace, Grace every which way? And that's why his bridge is still there. 410 00:57:11.429 --> 00:57:17.309 Okay. 411 00:57:17.309 --> 00:57:27.389 Example another way to look at reliability. It's a little morbid, but since, you know, death rates with corporate 19 with the virus are in the news now. 412 00:57:27.389 --> 00:57:33.300 We have things while reliability is like, life expectancy. So. 413 00:57:33.300 --> 00:57:42.389 If you're born alive, which is a definition for born alive, I can't remember what it is. You got to survive at least a day or something. 414 00:57:42.389 --> 00:57:54.269 And if you're dying to come out, it's not born live necessarily so got the probability. So the probability you live to a given age, and these are actual numbers, if you're. 415 00:57:54.269 --> 00:58:01.829 Born alive basically 99% chance. You'll get to age 20. we don't have much child mortality anymore. 416 00:58:01.829 --> 00:58:05.250 So, and. 417 00:58:05.250 --> 00:58:08.849 You know, 96% chance, you'll get to age 40. so this is like. 418 00:58:08.849 --> 00:58:12.269 Your reliability for people. Okay. 419 00:58:12.269 --> 00:58:15.809 88% chance to get to age 60, that sort of thing. 420 00:58:15.809 --> 00:58:20.400 So, and then for people again. 421 00:58:20.400 --> 00:58:26.969 The meantime to failure would be to ask is your life expectancy so. 422 00:58:26.969 --> 00:58:35.550 Life expectancy at the US at versus say, 78. so that would be the meantime to failure for a baby. 423 00:58:37.019 --> 00:58:42.480 Then we can get to failure rate for people. 424 00:58:42.480 --> 00:58:51.719 So fail your rates, the probability that you're going to die next year, and that will in the next Delta key to normalize for Delta T. 425 00:58:51.719 --> 00:58:54.719 So, 20 year old. 426 00:58:55.889 --> 00:58:59.969 Basic point 1% probability you'll die if you're a male. 427 00:58:59.969 --> 00:59:06.360 You'll die this year or point 0, half of that. If you're a female or a 3rd of that. If you're a female. 428 00:59:06.360 --> 00:59:20.429 Using the statistical terminology, females have a higher reliability than males and a life expectancy sense in a mortality sense and these numbers, like double, every 8 years or something. So if a year old. 429 00:59:20.429 --> 00:59:28.650 7% chance, he'll die in the next year. If you're a male 5% if you're a female so so this is failure rate. 430 00:59:28.650 --> 00:59:39.570 Okay, so, for people you've got reliability life expectancy. It probably is still alive meantime to failures the probability. 431 00:59:39.570 --> 00:59:44.730 Is your life expectancy and you failure rate probability you're going to die in the next year. 432 00:59:45.900 --> 00:59:52.559 Relevant the current events is what covert has done is, it's not a full year of life expectancy. 433 00:59:52.559 --> 00:59:57.570 1, full year, so okay. 434 00:59:57.570 --> 01:00:03.989 Okay, and then before we get back to the book, some. 435 01:00:03.989 --> 01:00:18.599 Some other terminology, so you want to go redundant now to improve reliability. You've got multiple systems and you get you a big system is several sub systems. If they're all necessary. 436 01:00:18.599 --> 01:00:22.320 Then the reliability is multiply and center liabilities fraction. 437 01:00:22.320 --> 01:00:36.030 It gets get smaller, so if I take my car, let's say all 4 tires have to work if I'm going to drive. So, the reliability of my car would be the product of their liability of all 4 tires. 438 01:00:36.030 --> 01:00:40.289 And it's smaller. 439 01:00:41.340 --> 01:00:47.369 So, as long redundancy, if it's if they're in parallel, if I, if I've got redundancy, then. 440 01:00:47.369 --> 01:00:53.849 Then, basically the compliments of their liabilities multiply and that number gets larger. 441 01:00:53.849 --> 01:00:57.780 So. 442 01:00:57.780 --> 01:01:02.639 See, again, talking too much here. 443 01:01:02.639 --> 01:01:06.840 So, let's suppose I have to drive to our Pi. 444 01:01:08.099 --> 01:01:14.219 I can take my car and my wife's car and assuming she doesn't doesn't want it. Then. 445 01:01:14.219 --> 01:01:18.659 You know, if either car works that I can get to our Pi. 446 01:01:18.659 --> 01:01:29.159 And so their liability of the 2 car system is depends only 1 car functioning so that we've modified the compliments of them. 447 01:01:29.159 --> 01:01:32.280 Um. 448 01:01:32.280 --> 01:01:40.170 An example here before I get back to the book, I just want to put in what I've typed in, because it's a nice summary. 449 01:01:40.170 --> 01:01:49.860 For disks, well, especially with hard drives, you know, they, they would fail without warning at maybe after a few years. 450 01:01:49.860 --> 01:01:57.360 That stuff, so, people embedded redundant systems of disks and. 451 01:01:57.360 --> 01:02:05.039 So you might, and so you got checksums and error correction and that sort of stopped. So you might perhaps have. 452 01:02:05.039 --> 01:02:08.789 3 disks, but only 2 of them need to be functioning. 453 01:02:08.789 --> 01:02:14.699 So so, basically have the 3 digits 1 of them could fail. 454 01:02:15.780 --> 01:02:23.309 And so this will be called a redundant array of inexpensive. Well, the guy who invented it, David Patterson, he's a. 455 01:02:23.309 --> 01:02:33.420 Famous computer hardware person, a lot of good work at Berkeley, and he invented the idea, and he called it redundant array of inexpensive disks. 456 01:02:33.894 --> 01:02:44.155 And because the idea is, you could bunch up a pile of cheap disks and get a system that was more reliable than the individual cheap this. So you could use cheap tests and get away with it, save money. 457 01:02:44.635 --> 01:02:52.465 So, he commented that when his idea was adopted by industry, they redefined rate to mean, redundant array of independent desks. 458 01:02:52.739 --> 01:02:58.199 So any case, so, here we get the reliability of. 459 01:02:59.429 --> 01:03:02.670 If you got redundancy, the reliability, the system goes up. 460 01:03:02.670 --> 01:03:14.789 But there are ways you can use this array of disks where you Stripe each block over several desks, and you get increase speed. But there, there's no redundancy. Reliability goes down. 461 01:03:14.789 --> 01:03:23.460 See, your raid, your disc arrangement called a raid and can illustrate increased reliability or decreased liability, depending how you do it. 462 01:03:23.460 --> 01:03:26.880 Okay, so that's. 463 01:03:28.860 --> 01:03:34.530 Yes, now I can go back to the book and show you a little of this. 464 01:03:35.909 --> 01:03:39.539 Let's see here, I don't need that. 465 01:03:39.539 --> 01:03:47.159 And anything happening no, no messages. 466 01:03:47.159 --> 01:03:50.219 Okay. 467 01:03:50.219 --> 01:03:54.449 Okay, so. 468 01:03:54.449 --> 01:04:02.130 Okay, so that was my summary now, look at what the book says about reliability again you got the failure rate. 469 01:04:02.130 --> 01:04:08.760 I'll do examples on Thursday I meantime to failure, I've mentioned. 470 01:04:08.760 --> 01:04:12.690 Um, so. 471 01:04:12.690 --> 01:04:21.389 Okay, so now getting Excel details, he got the failure rate probably going to die in the next Delta T given. You're still alive at the start. Okay. 472 01:04:21.389 --> 01:04:29.579 And for people, for example, the failure rate rises, the older you are, the more likely you are to die tomorrow. Okay. 473 01:04:29.579 --> 01:04:34.170 So, but, um. 474 01:04:35.034 --> 01:04:46.105 Perhaps the failure rate, and maybe talk about, say radioactive Adams the probability of a decaying does not depend on how old the admin system is no memory. We talk about this. 475 01:04:46.704 --> 01:04:52.224 And so if it's a radioactive Adam, it's failure rate is constant. 476 01:04:54.000 --> 01:05:02.070 And so, now here, we're going backwards you're saying if you have a problem distribution whose failure rate is constant. 477 01:05:02.070 --> 01:05:13.320 Then, what can we, what do we know about the distribution and what this is proving here is that if the failure rate is constant, it is an exponential distribution. 478 01:05:13.320 --> 01:05:20.699 It must be and this works through the proof of that. So, the exponential distributions the only distribution whose failure rate is constant. 479 01:05:22.440 --> 01:05:26.789 So, the probability of dying does not depend on how old it is. 480 01:05:28.500 --> 01:05:42.960 Unlike electronic components, so that's where most things look like this they start off, they've got this initial mortality problems and then they mature they have a low failure, right? And then it goes up as they get old age. So. 481 01:05:42.960 --> 01:05:50.909 2 for people, also, the probability of very young babies dying is higher as they get a little older that probably goes down because there are these. 482 01:05:50.909 --> 01:05:54.719 Nate neonatal issues. Okay. 483 01:05:54.719 --> 01:06:01.590 So, failure rate, reliability of systems when they're redundant. 484 01:06:03.239 --> 01:06:08.820 Now, this is actual and is a function called why Bill I may talk about because into some of this. 485 01:06:08.820 --> 01:06:18.119 Okay, another distribution has got parameters again here you could have systems that are sequential. So the system is less reliable and its components. 486 01:06:18.119 --> 01:06:25.440 Or they're in parallel, the system is more alive. Well, then its components, perhaps if any 1 of these is required. Okay. 487 01:06:27.144 --> 01:06:40.405 And we're doing some math on it here, give me an executive summary of this part of the chapter. And we actually throw math at you. If each 1 is exponential with some parameter, then what happens? Okay. More complicated things. 488 01:06:40.824 --> 01:06:43.195 Computer mentioned generating random variables. 489 01:06:45.329 --> 01:06:49.380 You want to be careful if you do it yourself, because you'll probably do it badly. 490 01:06:49.380 --> 01:06:53.340 Um, and. 491 01:06:55.105 --> 01:07:09.414 So, you called Matlab or your favorite package? I'd like to write and C plus plus you preferred Python, they all have packages to do it. Well, there's something called Emerson twister. If you want to point to a good. 492 01:07:09.690 --> 01:07:13.889 I don't know rejoin Adam or twister is good. 493 01:07:13.889 --> 01:07:18.539 It's quite fine. So any case you can do it in Matt lab here so. 494 01:07:20.034 --> 01:07:32.664 You to do what we're talking about here is if suppose you want a random normal Gaussian distribution variable, you can get her company uniform, random variable by doing a transformation with. 495 01:07:34.829 --> 01:07:38.369 And, okay. 496 01:07:38.369 --> 01:07:45.150 Other things here, so generating random variables gamma. 497 01:07:45.150 --> 01:07:55.409 I haven't showed you really what gamma is. Gab has got several parameters for particular values of the parameters. You can generate other random variables. 498 01:07:56.699 --> 01:08:01.590 Functions I mentioned mixtures. 499 01:08:01.590 --> 01:08:05.550 Yeah. 500 01:08:05.550 --> 01:08:09.510 Excuse me. 501 01:08:09.510 --> 01:08:13.349 Silence. 502 01:08:14.670 --> 01:08:18.630 Don't worry about entropy. I'm not going to worry about. 503 01:08:18.630 --> 01:08:29.699 Is a starred thing, and I want to spend time a little later in the book, and not just hit every chapter in the book. So okay. 504 01:08:29.699 --> 01:08:36.600 So, I'm going to skip and Trippy skip plan. I may use it later. 505 01:08:36.600 --> 01:08:46.050 So, that was chapter 4 basically, I'll come back and hit it in more depth on Thursday but. 506 01:08:47.520 --> 01:08:57.989 You know, let me some executive summary of what we saw here. We saw the cumulus distribution function, the left tail of the distribution. 507 01:08:57.989 --> 01:09:11.250 And the CDF is nice, because it works, it's valid for discreet, random variables and it's also valid for continuous. We can use the CDF in either case. 508 01:09:11.250 --> 01:09:16.470 So that's nice. Um, you've got a continuous random variable. 509 01:09:16.470 --> 01:09:31.465 We have the density function, so discreet, random variables. We call it the probably mass function continuously called the density function, and we integrated over a little interval and we find out there's a probability that the event happened in that interval. 510 01:09:31.739 --> 01:09:37.829 So just have to be a little involved in any size interval. Okay. 511 01:09:37.829 --> 01:09:47.970 We can transform a random variable. I showed you a simple example, which with centimeters to feed may show you a more complicated. 1. 512 01:09:47.970 --> 01:09:51.869 Then we got. 513 01:09:51.869 --> 01:10:04.350 Functions got statistics of the random variable there ways to summarize the Raspberry. You got the mean and the variance and other moments I'm not much talking about the variance is a central 2nd moment. 514 01:10:04.350 --> 01:10:10.770 Conditional lines that we know some other knowledge it's by conditionals in chapter 3 nothing new. 515 01:10:10.770 --> 01:10:23.640 And then if we know only some statistics, so the random variable, but not the details. If we don't only the mean, we can do a mark off inequality and it will tell us something. 516 01:10:23.640 --> 01:10:32.640 About the probability of an outlier if we also know the variance, we can then use the Chevy shaped quality, which tells you a little more about the probability of an outlier. 517 01:10:32.640 --> 01:10:38.579 If we also know what distribution it really is, like, normal, then we can tell a lot. 518 01:10:39.659 --> 01:10:43.770 So, and that was, um. 519 01:10:44.335 --> 01:10:59.154 6 or, I mean, I can count. Okay, I'll do some examples on Thursday. Well, I also showed you Matlab, which is nice, because you can work with real numbers with Matt lab and being a 1000 I showed you could have end being a 1Million even. 520 01:10:59.909 --> 01:11:09.659 Okay, so I'll show you mathematical also at some point that I'll run into some examples on Thursday, and I may show you Mathematica, which works with. 521 01:11:09.659 --> 01:11:18.090 Expressions algebra. Matlab does numbers. Mathematica does algebra so cool. 522 01:11:18.090 --> 01:11:24.060 I like it at least you're an engineer, you got to like, math and to someone. 523 01:11:24.060 --> 01:11:31.439 I mean, I'm judgmental if you don't like math, you're going to have your life as an engineer is going to be difficult. 524 01:11:31.439 --> 01:11:35.279 Okay, so that has that. 525 01:11:35.279 --> 01:11:46.949 Let me just give you a teaser about what's going to happen in chapter 5 is you've got pairs of random variables. Like, I'm tossing a dart at a dark dark board. It hits at a position, got X and Y. 526 01:11:46.949 --> 01:11:54.510 2 random variables, let's say, and if we are tossing a dart at a dartboard. 527 01:11:54.510 --> 01:12:08.850 Then the X and the Y, it may be not correlated with. They don't depend on each other. If I know the X value or the dark dark but doesn't tell you tell me anything about the Y value. 528 01:12:08.850 --> 01:12:12.329 But maybe the pairs are dependent, like. 529 01:12:12.329 --> 01:12:20.760 I since I have the solar panels on my roof, I'm interested in how much sunlight we get each day. You know, I can check my phone and. 530 01:12:20.760 --> 01:12:25.920 At us to see the 1st, 2 weeks of March. 531 01:12:32.244 --> 01:12:39.715 The 1st, 2 weeks of March, I have my house use the other 93 kilowatt hours, but my solar panels have generated 488 kilowatt hours. 532 01:12:41.550 --> 01:12:48.539 Now, today it's going to use a little more this evening, but you get the idea. So I may just sit in St, amount of sunlight. 533 01:12:49.619 --> 01:12:54.149 Maybe Molson sit in the amount of rain because they got a lot and. 534 01:12:54.149 --> 01:13:01.829 You know, is it going to be brown or green to rain? So I got 2 random variables for the day sunlight and rain. 535 01:13:01.829 --> 01:13:11.340 They're going to be related. Okay because if I have a lot of rain, I've got less sunlight. So I got a so my experiment is pick a day. 536 01:13:11.340 --> 01:13:21.750 My random 2, random variables from that experiment of picking a day as sunlight that day. And how a train that they, those 2 random variables are related to each other. 537 01:13:21.750 --> 01:13:26.819 And we're going to say they're negatively correlated with each other high sun means low rain. 538 01:13:26.819 --> 01:13:33.239 So, we'll be seeing that sort of thing with pairs. So that's the sort of thing we'll be seeing. 539 01:13:33.239 --> 01:13:38.520 In in chapter 5, so. 540 01:13:39.265 --> 01:13:53.725 So, a reasonable point to stop, as I said, we're going to chapter 4 more. I showed you my 1st, I showed you my blurb that I typed on transformation of a random variable, reduce the fee, and then ran to chapter 4 more. 541 01:13:54.145 --> 01:13:57.204 And let me see, where is my blog again? 542 01:13:57.479 --> 01:14:01.560 Here we go and. 543 01:14:03.510 --> 01:14:14.310 And 2, random variables, like I said, I'll work, this will be Thursday and Monday, and just by popular demand. 544 01:14:14.310 --> 01:14:20.550 Random variables. So okay. 545 01:14:20.550 --> 01:14:29.399 A few decades ago, there was a numbers game in Arizona where they never, they never generated a 9. actually, it was a. 546 01:14:29.399 --> 01:14:33.119 Like, a month before it was noticed, so okay. 547 01:14:33.119 --> 01:14:39.539 So, you Thursday. 548 01:14:39.539 --> 01:14:46.439 Let me just say questions. 549 01:14:46.439 --> 01:14:51.869 Have fun and maybe we'll see what hardware fails Thursday and locks up. 550 01:14:53.039 --> 01:14:57.060 Okay fun. 551 01:14:57.060 --> 01:15:01.500 Questions. 552 01:15:01.500 --> 01:15:06.180 Bye bye.