WEBVTT 1 00:00:17.759 --> 00:00:24.839 So, good afternoon class probability class. 2 00:00:24.839 --> 00:00:29.760 16, my universal question is. 3 00:00:29.760 --> 00:00:34.140 Can you hear me. 4 00:00:37.829 --> 00:00:42.420 Thank you. Okay, so. 5 00:00:42.420 --> 00:00:48.750 The reason the delay I was trying to get my iPad working. My pad is not connecting to the. 6 00:00:49.765 --> 00:01:02.994 Not connecting to my laptop to share the screen. So what I'm going to have to do today is I'll show you points from the show, you points from the textbook and we'll try and get it working for Monday. I don't know what is going on with any case. 7 00:01:02.994 --> 00:01:04.915 Thanks, Thomas again. 8 00:01:05.189 --> 00:01:09.030 So. 9 00:01:09.030 --> 00:01:13.260 Sharing the screen. 10 00:01:13.260 --> 00:01:18.930 So, again, okay, there is no class on. 11 00:01:18.930 --> 00:01:23.670 Thursday contend president Jackson's town meeting or take a break. 12 00:01:24.144 --> 00:01:38.064 And I'll do some of this review next time. Mathematica is let me just show you a quick introduction to Mathematica. It works with. 13 00:01:38.879 --> 00:01:47.579 Just show it works with mathematics, it works with algebra so I'll just give you a little teaser off it actually. 14 00:01:51.989 --> 00:01:56.189 It would. 15 00:02:00.299 --> 00:02:05.700 Silence. 16 00:02:09.210 --> 00:02:12.389 Silence. 17 00:02:12.389 --> 00:02:17.639 So, mathematics was written by a physicist called steep fall from and. 18 00:02:17.639 --> 00:02:23.639 This is a little too small to read, but we'll start a new document and we'll make it bigger here. 19 00:02:23.639 --> 00:02:30.360 And, okay, so you can see this here. 20 00:02:30.360 --> 00:02:38.129 And just see what you can do is you can do things like. 21 00:02:38.129 --> 00:02:42.599 Said, variable sex equals 2. 22 00:02:42.599 --> 00:02:50.430 And then whatever, but now we can do things like, why equals? I don't know. 23 00:02:50.430 --> 00:02:58.409 Like, okay, that's sort of like mad lab, but here's what we can do is. 24 00:02:59.729 --> 00:03:06.389 We can do things like integration the differentiations. Like, I can do something like integrate. 25 00:03:08.789 --> 00:03:13.650 I can say something like integrate. 26 00:03:13.650 --> 00:03:17.069 Squared with respect to X. 27 00:03:17.069 --> 00:03:24.479 It says invalid expression I going to make. It goes. 28 00:03:24.479 --> 00:03:29.400 Silence. 29 00:03:29.400 --> 00:03:35.520 And I'm trying to do a demo. 30 00:03:36.960 --> 00:03:40.169 X2 X. 31 00:03:41.819 --> 00:03:45.389 Be doing the integrate. 32 00:03:45.389 --> 00:03:49.319 Try the other 1. 33 00:03:53.430 --> 00:04:03.780 Silence. 34 00:04:07.229 --> 00:04:12.780 I tried to do a demo. It worked earlier. It's not working now so. 35 00:04:21.959 --> 00:04:25.350 Silence. 36 00:04:25.350 --> 00:04:28.499 Oh, okay. We'll do the demo next week. 37 00:04:28.499 --> 00:04:31.858 Okay, we're back to chapter. I'm 5 now. 38 00:04:31.858 --> 00:04:35.338 Chapter 5 is talking about. 39 00:04:36.988 --> 00:04:49.079 I'm talking about multiple variable probability cases and just to review the thing is that you're doing a, um. 40 00:04:49.079 --> 00:05:00.298 You're doing an experiment and the experiment may have of outcomes, which produce more than 1, random variable. So I toss a. 41 00:05:00.298 --> 00:05:09.658 On charge of the law under dart dartboard and I look at the X Y position and that's 2 random variables coming out of the experiment. 42 00:05:09.658 --> 00:05:21.389 And and these random variables, they may be independent of each other, or they may be correlated or haven't defined correlation. 43 00:05:22.468 --> 00:05:36.713 Precisely, yeah, I'll get to it later, but we started looking at an example this example here and we have a packet switch and a packet switch as input ports and it is output ports. And the idea is that it routes across bar router. 44 00:05:36.713 --> 00:05:42.384 Let's say it routes a packet which comes in and around the packet to 1 of the output ports. 45 00:05:42.689 --> 00:05:57.569 And and the experiment is, will packets come randomly now so a packet comes in randomly and it make to either input port. 46 00:05:57.569 --> 00:06:02.009 And then when it comes, it may be switched. 47 00:06:02.009 --> 00:06:07.108 Randomly to 1 of the to 1 of the output ports. 48 00:06:07.108 --> 00:06:11.459 And. 49 00:06:11.459 --> 00:06:21.449 Okay, what we want to look at is well, what's happening? The probability distribution for things coming. 50 00:06:21.449 --> 00:06:30.899 Coming things on the output ports. Now what dissertation here is in Greek ladder that's. 51 00:06:30.899 --> 00:06:36.509 That's the random experiment, and with facilitation here says like a 1. 52 00:06:38.098 --> 00:06:41.098 Well, the 1st is an ordered pair that the 1st thing. 53 00:06:41.098 --> 00:06:47.728 Is it's a packet came in. 54 00:06:48.293 --> 00:06:57.233 On the 1st input port and it wants to go to the 1st output port. So the 1st component here is what's coming into the 1st input part. 55 00:06:57.233 --> 00:07:08.783 The 2nd component is what's coming into the 2nd input Ford and all the packets are labeled a, and we look at the number after it and that's the output port that that packet. 56 00:07:09.059 --> 00:07:13.079 Wants to go to so this is the experiment here so. 57 00:07:13.079 --> 00:07:21.358 At the ending thing here a, to a 2 says each input port got a packet and each packet wants to go to output port 2. 58 00:07:21.358 --> 00:07:26.579 So, the random variables, the X Y, random variables are. 59 00:07:26.579 --> 00:07:29.579 The number of packets on each output port. 60 00:07:29.579 --> 00:07:33.028 And an interesting case here, say. 61 00:07:33.028 --> 00:07:39.178 So, look at the last thing you too. So, 2 packets came in and. 62 00:07:40.619 --> 00:07:51.689 No, Pat, both of them want to go the output port number 2. so X is a number of packets going to 1. that's 0 wise the number of packets going to output for 2 and that is 2. 63 00:07:53.274 --> 00:08:07.553 And we want to look at the probability mass function of things now. Okay. So we have probabilities on the experiment here and each input port gets a packet with probability 1, half and the 2 input. Ports are independent. 64 00:08:07.889 --> 00:08:13.168 So, court at the time, no input port gets a pack and a half the time. 65 00:08:15.713 --> 00:08:28.884 1 input each input for gets a packet and record of the time before it gets back court at the time to 1st, we have a question is the test on April 5th and yes. That was announced last time the 2nd, term test will be on April. 66 00:08:29.363 --> 00:08:30.803 So you'll have. 67 00:08:31.379 --> 00:08:37.408 2 weeks, so let me write that down. 68 00:08:37.408 --> 00:08:41.759 Yes, okay. 69 00:08:41.759 --> 00:08:46.259 And then we want to start this. 70 00:08:48.149 --> 00:09:01.019 When I said here, then we want to start calculating probability, mass functions and for something as simple as this, we can just start crafting things. You can look at all the various things that might happen. 71 00:09:01.019 --> 00:09:11.399 We get no packets either output port. If don't packets came to either input port, that's a half a half. So it's a quarter insulin and we can work through all of the, all of the examples here. 72 00:09:12.448 --> 00:09:16.918 And and different ways, we can graph it and that sort of thing. 73 00:09:16.918 --> 00:09:20.578 I mentioned this thing last time, so I won't review it. 74 00:09:20.578 --> 00:09:31.739 That talks about that thing there. I mentioned the marginal thing. Last time you sum up 1 of the margins. 75 00:09:32.543 --> 00:09:40.163 Okay, we can go nothing interesting. There, the joint CDF I mentioned that quickly last time review it quickly. 76 00:09:40.163 --> 00:09:53.724 So, it's a joint shift to distribution function of X and Y, and this would be if this is our graph here, X and Y, for them mass functions is, it's the probability of something happening in that quarter plane below and below. 77 00:09:53.783 --> 00:09:57.053 And less than that point is the definition up there. 78 00:09:57.833 --> 00:10:12.413 I may make a touch bigger for you there. This is the definition of the top, right? Big app the CDF X and Y, axis less than that value. And also why, as I said, there should be a column in there actually typesetting issue. 79 00:10:14.038 --> 00:10:19.769 Okay, there's a definition right there. Okay. And some. 80 00:10:19.769 --> 00:10:23.278 And we saw a quick example of that. So that's. 81 00:10:23.278 --> 00:10:26.698 Okay. 82 00:10:28.288 --> 00:10:38.278 And from the joint CDF, we can subtract and get the the margin. Well, we can get the marginal CDF. 83 00:10:38.278 --> 00:10:43.649 Okay, just for 1 variable so. 84 00:10:44.759 --> 00:10:50.249 And I sketched out something like this last time when my, um. 85 00:10:50.249 --> 00:10:53.249 When my screen sharing thing was working. 86 00:10:56.099 --> 00:11:00.749 Just checking again still not working. Okay. 87 00:11:02.698 --> 00:11:06.389 Silence. 88 00:11:08.519 --> 00:11:13.288 And another example, I can look at doing that there. 89 00:11:13.288 --> 00:11:17.188 Oh, we can work out what they're doing here. 90 00:11:22.014 --> 00:11:36.594 Well, the marginal for f is we got the joint, the marginal CDF just on X, or we've integrated out. Why? Basically to convert the 2 variable to the 1 variable CDF. 91 00:11:36.594 --> 00:11:38.724 We just left the other variable equal infinity. 92 00:11:39.359 --> 00:11:45.359 And so if we have this for X, and Y, we want the margin, the CDF just on X. 93 00:11:45.359 --> 00:11:59.818 If where we've converted from integrating out, why if we're looking at the density function, we said why it goes to infinity and this makes it as a minus beta Y is 0 and we will get that for the PDF for X. 94 00:11:59.818 --> 00:12:10.859 Okay, and did for why the 1 variable marginal density function for Y, is we set X sequel to infinity in the. 95 00:12:10.859 --> 00:12:14.068 In the 2 variable joint CDF. 96 00:12:14.068 --> 00:12:17.698 Okay. 97 00:12:17.698 --> 00:12:22.229 Similar sort of thing happening. 98 00:12:26.759 --> 00:12:31.168 Yeah, okay here. 99 00:12:36.058 --> 00:12:39.359 I'm just going to try something on my. 100 00:12:47.609 --> 00:12:53.729 It's trying to. 101 00:12:53.729 --> 00:12:57.899 Silence. 102 00:13:04.708 --> 00:13:08.578 Trying to turn off and on the iPad, but. 103 00:13:08.578 --> 00:13:13.918 Still cannot connect. Okay now. 104 00:13:15.149 --> 00:13:19.558 Okay, what's the new thing that's happening here is. 105 00:13:19.558 --> 00:13:25.859 So so we have random variables we had then this was causing the trouble of course, on that. 106 00:13:25.859 --> 00:13:38.308 1st test we have a discreet random variables where the number of outcomes is finite or countably infinite and we have the continuous ones where it's an uncomfortably infinite number of cases. 107 00:13:39.509 --> 00:13:42.538 Now, we may have a joint situation. 108 00:13:42.538 --> 00:13:47.788 Where 1 is discreet and the other is continuous and. 109 00:13:47.788 --> 00:13:55.979 Here's a natural example of it. It's not artificial at all. It's it's a natural example. Excuse me and. 110 00:13:55.979 --> 00:14:05.369 And because the input communications channel, that's noisy. And the input is discreet plus or minus 1 it's translating 1 bit. 111 00:14:05.369 --> 00:14:14.428 Morse code, for example, Morse called Telegraph 1st, Telegraph cable under the Atlantic Ocean ended up in. 112 00:14:14.428 --> 00:14:20.129 Telegraph and at heart's content Newfoundland. 113 00:14:20.129 --> 00:14:24.328 You know, they transmit voltage are not a voltage and it'd be 1. 114 00:14:24.328 --> 00:14:35.818 For a certain time that would be 1 bit. Okay so so, X is random variable. It's the input to our channel. It's plus or minus 1 fault and say 50, 50 keep life. Simple here. 115 00:14:35.818 --> 00:14:38.969 But the problem is that the channel adds noise to the. 116 00:14:38.969 --> 00:14:45.808 Bit so, and the noise is, we're going to say, make it uniformity to distribute for minus to. 117 00:14:45.808 --> 00:14:56.339 To 2, so, X is discrete and is the noise as uniform and then the signal is X plus and so. 118 00:14:56.339 --> 00:15:00.328 Why the received signal is now. 119 00:15:00.328 --> 00:15:03.538 Continuous. 120 00:15:04.708 --> 00:15:19.109 Now, we want to start playing games, just having fun with it. Ultimately what you want to do is you receive a certain voltage you want to figure out, what was the most likely signal to have been transmitted, given that you. 121 00:15:19.109 --> 00:15:33.418 Received a certain voltage and what makes it interesting and a more general thing we'll see later is the transmitted bits are not equal probability. Like, we're transmitting a scan of the page as more white pixels and black pixels. For example. 122 00:15:33.418 --> 00:15:41.818 Okay, so any so, in this case, we want to find a joint probability here that. 123 00:15:41.818 --> 00:15:51.149 X is plus 1 and also why is actually equal to 0 Why exactly equals 0 probably less than equal lesson doesn't matter. So. 124 00:15:53.729 --> 00:16:00.538 So, this is actually using stuff from chapter 4. really? Um, so. 125 00:16:01.828 --> 00:16:06.359 Wants a probability that X is 1 wildlife center equals some. 126 00:16:06.894 --> 00:16:21.563 Y, Lowercase why upper case, why is the name of the random variable Lowercase wise that particular value? So we're saying, so big why? That's a little, why is the it was talking about the random variable why sites? Equal to some bag. 127 00:16:21.803 --> 00:16:25.464 Okay, so we got that combo thing so it's X equals 1 and. 128 00:16:25.739 --> 00:16:33.989 Why next equal to Y well, we have stuff from the from earlier and that's conditional probability. 129 00:16:33.989 --> 00:16:39.599 Why why give an exodus 1 times the probability that axis 1. 130 00:16:39.599 --> 00:16:43.918 Now, before I go to the next page, you can look at this now, you can start. 131 00:16:43.918 --> 00:16:53.609 Working with that. Well, probably X is 1. that's 50%. That one's easy. Look at the probability. Why this signal why given access? 1. 132 00:16:53.609 --> 00:16:59.729 Well, big wise X, plus N, so big why? It's actually a little why that means. 133 00:16:59.729 --> 00:17:03.869 X plus, and last week, the little Y. 134 00:17:03.869 --> 00:17:09.749 Lex is 1, so, 1, plus and less a little wiser and is equal to. 135 00:17:09.749 --> 00:17:15.778 Little Y, minus 1 and we know the distribution of. 136 00:17:15.778 --> 00:17:21.358 And it's uniform, so you see, now, we can, so before I go to the next page. 137 00:17:21.358 --> 00:17:24.868 Given this 1st, step here, you can start thinking about what you would do. 138 00:17:24.868 --> 00:17:32.219 And so the probability that. 139 00:17:32.219 --> 00:17:36.449 Big wide I said, why give next equals 1 it's going to be. 140 00:17:37.888 --> 00:17:42.449 Well, this is they just compressed the steps that I just told you so. 141 00:17:42.449 --> 00:17:53.398 Basically, and is uniform on minus 2 to 2. so, and plus 1 X is 1 in this case uniform and minus 1 to 3. 142 00:17:53.398 --> 00:17:59.489 So, the probability wise less and something is how far it is into that interval for minus 1 to 3. 143 00:17:59.489 --> 00:18:02.999 And that will get that number here white plus 1 over 4. 144 00:18:05.068 --> 00:18:08.189 So then we picked some particular. 145 00:18:08.189 --> 00:18:22.858 Big, why was aerospace? 1 quarter, and we multiply by the production so I would get 18. so, this is a probability that the input signal is 1 and simultaneously output say, don't is negative. 146 00:18:22.858 --> 00:18:31.949 Is 18, and if you think about it, this is a probability of 9th that we will. 147 00:18:31.949 --> 00:18:35.909 Not correctly decode an input signal of 1. 148 00:18:35.909 --> 00:18:43.798 You know, the input was positive. Noises. Uniform output was negative. That's a bad case. Cause the output is negative. You're going to. 149 00:18:43.798 --> 00:18:51.449 I think that more likely the input is negative too, and you'll be correct, but he'll be wrong in each of the time. 150 00:18:52.769 --> 00:19:03.659 And Ditto affects is minus 1, the output is going to be positive an 8 of the time. So, what this is saying with this level of noise, it's going to corrupt the input a quarter of the time. 151 00:19:03.659 --> 00:19:11.848 So, okay, joint PDF of 2 continuous, random variables. 152 00:19:11.848 --> 00:19:19.078 Continuous intuitive thing I'll skip through that. 153 00:19:19.078 --> 00:19:23.608 So, you got so probably access in that. 154 00:19:24.534 --> 00:19:30.144 X with a bold faces is the vector X, which is the scalar X and the scale, or Y, together as a pair. 155 00:19:30.294 --> 00:19:40.134 So the probability that random variable vector is in some region, just integral to the density over the region that makes sense integrating over the whole. 156 00:19:40.439 --> 00:19:44.759 Region guess 1, that's required and. 157 00:19:45.838 --> 00:19:51.058 Against it's a requirement for a density function. If this is not 1, that's not a legal density function. 158 00:19:51.058 --> 00:20:00.328 Okay, and you can go from the CDF for the continuous case to the PDF by doing it. 159 00:20:01.558 --> 00:20:11.249 That double derivative and so on and again, so the probability that they're in a certain regions to integrate the density function over the region. 160 00:20:12.538 --> 00:20:16.828 And marginal we talked about it before we integrate out 1 of the variables. 161 00:20:18.959 --> 00:20:23.848 Okay, now an easy case. 162 00:20:24.959 --> 00:20:34.348 For random pair would be it's uniform in some Square, like a 2 to form in the square 0 to 1. so. 163 00:20:34.348 --> 00:20:46.019 So the density is going to be 1 inside the square and 0 outside you integrate over the square you get 1 it's always positive or it's always non negatives. This is a legal density function. 164 00:20:47.128 --> 00:20:54.088 Now, and you integrate and you get the CDF. I did something like that. Now. 165 00:20:55.469 --> 00:21:08.213 The nasty thing with the CDF for these 2 variable things is that you tend to get a lot of special cases with a density function is easy. It's 1 inside in 0 outside. 166 00:21:08.723 --> 00:21:11.874 But now you start doing integral on this. 167 00:21:13.439 --> 00:21:19.169 I showed you this last time, you get several different cases. 168 00:21:19.169 --> 00:21:22.499 And there's actually 5 of them. 169 00:21:22.499 --> 00:21:27.568 If you're to the lower or below, or left the square, it's arrow. 170 00:21:27.568 --> 00:21:31.558 Inside that Square, it's X Y, if you, um. 171 00:21:31.558 --> 00:21:40.108 If you're Bob's a square, it's actually to the right of the square. It's why and point 5 is on the next page. If you're above into the right. 172 00:21:40.108 --> 00:21:44.189 It's going to be 1, so there's 5 different cases here. 173 00:21:44.189 --> 00:21:53.368 Now, I'll try to show you some mathematical examples lately. 1, nice thing about Mathematica is it handles things with a number of different special cases and so. 174 00:21:56.459 --> 00:22:00.598 This example, here with. 175 00:22:03.959 --> 00:22:08.009 With the 2 variable, um. 176 00:22:08.604 --> 00:22:22.523 Density function, Lowercase accidents. So you want to get the constant, or you'd get the constant because the thing is, if you integrate X and Y minus that, if any to infinity, it has to give 1. so you get the integral and then you can find. C. 177 00:22:22.523 --> 00:22:24.324 so that part you see there. 178 00:22:26.128 --> 00:22:34.709 Okay, you may want to find some more complicated things or is the probability that. 179 00:22:36.209 --> 00:22:44.068 Say some function of the random variables of something like X plus Y, equals 1 X and Y are the 2 random variables in this experiment. 180 00:22:44.068 --> 00:22:49.949 Well, basically. 181 00:22:49.949 --> 00:22:53.249 It would be the probably the. 182 00:22:53.249 --> 00:22:58.949 You'd integrate the density function over the triangle, which goes up to X plus Y, equals 1 1. 183 00:22:58.949 --> 00:23:02.278 But it's still after this line. 184 00:23:02.278 --> 00:23:08.578 With salt minus 1 and integrate density function over that and you'll get the probabilities. So. 185 00:23:11.669 --> 00:23:17.578 Now, what they're doing here is just doing medical over the regions where the density function. It's not 0. 186 00:23:17.578 --> 00:23:24.479 It's just easier because integrating minus symphony to infinity, you just integrate over the years. What's not. 187 00:23:24.479 --> 00:23:31.709 Okay, now here is something new and important and slightly complicated. 188 00:23:31.709 --> 00:23:35.098 We saw the calcium the normal distribution for. 189 00:23:35.098 --> 00:23:35.578 1, 190 00:23:35.574 --> 00:23:38.453 variable now, 191 00:23:38.453 --> 00:23:42.864 remembering the importance of the calcium thing is that any reasonable, 192 00:23:43.044 --> 00:23:51.384 random variable if you let and get big starts looking like a Gaussian distribution and happens fairly fast in most cases. 193 00:23:51.384 --> 00:23:51.834 So. 194 00:23:52.078 --> 00:24:03.749 You know, if you go looking hard, you'll find counter examples of the couch. I showed you last time is exempt because she doesn't have any moments you've tried to integrate. 195 00:24:03.749 --> 00:24:08.788 Find the mean expected value for the calcium the integral diverges. 196 00:24:08.788 --> 00:24:14.038 It does not exist. Well, if the other girls converge and. 197 00:24:14.038 --> 00:24:19.769 For the moments things start looking like a 1000 for 1 barrier, but Here's a case of a Gaussian. 198 00:24:19.769 --> 00:24:27.239 With 22 on 2, random variables X and Y. 199 00:24:27.239 --> 00:24:35.669 And to make life easy, we're assuming that the standard deviation on each of them is 1. 200 00:24:35.669 --> 00:24:40.558 But we've added 1 more parameter and it's called role. 201 00:24:40.558 --> 00:24:46.739 And. 202 00:24:46.739 --> 00:24:54.628 That's going to be a correlation coefficient and that's going to capture how X and Y, depend on each other. 203 00:24:54.628 --> 00:25:04.138 So, if role was 0, X and Y, are independent, knowing X does not tell you anything about why. 204 00:25:04.138 --> 00:25:09.598 The other extreme if role was very close to to 1. 205 00:25:09.598 --> 00:25:21.358 Then, if you knew X, you Y, would basically be equal to accept as a slight little noise added and the limit as row equal to 1. then. 206 00:25:21.358 --> 00:25:25.439 Why would equal X, but the thing is, we're getting. 207 00:25:25.439 --> 00:25:33.179 0, in the denominator new way, we'd have to use some sort of floppy tiles rule or something to figure out what that means in that case. 208 00:25:33.179 --> 00:25:36.449 And if role was minus 1. 209 00:25:36.449 --> 00:25:47.909 Then what as row approaches minus 1, then why becomes equal to minus X? So we call role correlation coefficient and 0. 210 00:25:47.909 --> 00:25:51.058 Is no relation? No correlation between them. 211 00:25:51.058 --> 00:25:54.298 And plus or minus 1. 212 00:25:54.298 --> 00:25:57.628 They're tied to each other and real cannot be bigger than 1. 213 00:25:57.628 --> 00:26:06.449 Now, there's other ways you could have 2 random variables that are each separately. Gaussian but this is a special case. That's what the footnote says. 214 00:26:06.449 --> 00:26:15.148 So that's for the 2, random variable case continuous, random variables. If we want the marginal. 215 00:26:15.148 --> 00:26:21.328 Density function on just actually integrate out why? And they show it down here. 216 00:26:21.328 --> 00:26:27.028 If you're doing this X is a constant inside here, so you can pull backs out. He's a mindset squared out. 217 00:26:27.028 --> 00:26:30.358 And I won't hit every step, but. 218 00:26:34.288 --> 00:26:38.669 You see the concept as you integrate this out and. 219 00:26:38.669 --> 00:26:51.088 Well, this is what it looks like here when X and Y, kind of be related to each other, it's a spell shaped curve, but it's long and narrow for the 2 variables. So. 220 00:26:51.088 --> 00:26:57.298 Any case we integrate things out and we will get something like. 221 00:26:57.298 --> 00:27:02.128 The marginal PDF for X is this and that's. 222 00:27:02.128 --> 00:27:05.939 But it is, that's the 1 variable. 223 00:27:05.939 --> 00:27:12.568 Yeah, for Gaussian so we have this form of the joint Gaussian. If I can scroll back up here. 224 00:27:12.568 --> 00:27:18.298 Than, in fact, the marginal PDF for either variables separately is here. 225 00:27:18.298 --> 00:27:22.048 1, variable calcium. Good. 226 00:27:22.048 --> 00:27:25.618 Okay, now. 227 00:27:25.618 --> 00:27:34.318 Independence and so I talk informally about independence of 2, random variables. Again, we toss the dart dartboard. 228 00:27:34.318 --> 00:27:37.469 X and Y, and probably independent of each other. 229 00:27:37.469 --> 00:27:41.669 You do your idea experiment so. 230 00:27:41.669 --> 00:27:44.818 This is a formal definition, so. 231 00:27:44.818 --> 00:27:48.148 This is how we choose to define independence. 232 00:27:48.148 --> 00:28:01.919 And what's the same thing? Just 1 variables we just carried this definition over. So probably that access some value or group of values in some area. And why does. 233 00:28:01.919 --> 00:28:07.739 X and Y, are independent if we just take the 2 single variables separately in the problem is multiply. 234 00:28:10.378 --> 00:28:16.798 So, we already saw this basically earlier, but this is formally for 2 variables coming out of an experiment. 235 00:28:19.499 --> 00:28:22.919 Okay. 236 00:28:24.179 --> 00:28:28.858 And given that definition, it means density functions just multiply. 237 00:28:33.778 --> 00:28:41.038 Okay, now, if I go back to this, okay, now on page. 238 00:28:41.038 --> 00:28:45.538 55, if I go back a number of pages to. 239 00:28:47.578 --> 00:28:53.608 I need to do to try not to get seasick. 240 00:28:55.259 --> 00:28:59.038 Silence. 241 00:29:00.358 --> 00:29:04.648 Here we go. 242 00:29:04.648 --> 00:29:07.679 Figure 5, 6 now. 243 00:29:08.939 --> 00:29:13.528 Again, so this was our, we're tossing to dice. 244 00:29:13.528 --> 00:29:18.449 And each dice separately is fair with each dice separately. 245 00:29:18.449 --> 00:29:28.048 Uh, 16, Jan to each 1 of the 2 faces of 6 faces, coming up. However, for some unknown reason, the 2 dice try to are tracking each other. 246 00:29:30.148 --> 00:29:34.229 So the probabilities say that both faces come up a 2. 247 00:29:34.229 --> 00:29:38.338 Which should be 136 is now 240 seconds, which is. 248 00:29:38.338 --> 00:29:48.479 Not 136, so given that definition are these 2 dice independent and the answer is, no, they are not independent because. 249 00:29:48.479 --> 00:29:53.669 I look at this particular thing here, saved probability of both faces coming up a 2. 250 00:29:55.078 --> 00:30:05.334 If they were independent would be 136 each 2 separately is 166636butinfact according this table will see the pair of twos to 40 seconds of the time, which is bigger than 106. so, by that definition, these. 251 00:30:11.189 --> 00:30:14.189 2 dice are dependent on there and. 252 00:30:14.189 --> 00:30:20.009 They are not independent. Okay because that fails that definition that the probability of these. 253 00:30:20.009 --> 00:30:27.419 Joint thing, X and Y, together, having this value to and 2 is not equal to the product of X having that value. 254 00:30:27.419 --> 00:30:31.138 For any why times are probably of why having that value. 255 00:30:31.138 --> 00:30:36.929 For any X. 256 00:30:38.638 --> 00:30:46.888 Okay, so this is 519 here. It's dependent. 257 00:30:48.058 --> 00:30:55.169 I'm all going to do this example later. What is happening here is we're transmitting a long. 258 00:30:55.169 --> 00:31:01.828 Message on the net or store it on a file, we're cutting it up into tactic sized packets. 259 00:31:01.828 --> 00:31:08.128 Plus a fraction, and then this long experiment we are. 260 00:31:08.128 --> 00:31:14.459 Cues the random well, the link to the original message is some exponentially distributed random variable. 261 00:31:14.459 --> 00:31:20.909 And cues the number of full packets and our is the size of the fractional last packet. 262 00:31:20.909 --> 00:31:24.568 And up to, for quotes and an, are for remainder. 263 00:31:24.568 --> 00:31:29.368 And this is computing RQ and are independent of each other. 264 00:31:29.368 --> 00:31:33.239 And I'll work this out when. 265 00:31:33.239 --> 00:31:36.568 My pad gets working again. 266 00:31:36.568 --> 00:31:46.469 Turns out, they're independent, but okay, so this is doing experiments on our 2 things independent or not. 267 00:31:46.469 --> 00:31:50.489 Okay. 268 00:31:50.489 --> 00:31:55.378 I could feel an exam question on if I were psychic, I could imagine in 2 weeks. 269 00:31:55.378 --> 00:32:02.999 Okay, now, for 1, random variable, and I still have to work out that other thing was expected to. 270 00:32:04.048 --> 00:32:08.729 Why next and so on, but leave that till I get my iPad going. 271 00:32:08.729 --> 00:32:14.519 But we saw moments, like, a moment is a mean Central. 272 00:32:14.519 --> 00:32:18.118 2nd central moment would be the variance and so on. 273 00:32:18.118 --> 00:32:29.608 If you can hire order moments, they start characterizing and find or detail what the distribution looks like. And if you've got all the moments, it's like having expansion, tailor, expansion function you have the function. 274 00:32:29.608 --> 00:32:33.808 Ignoring weird paranoid cases. 275 00:32:33.808 --> 00:32:42.568 Okay, that is 1 right now for 2 random variables we would want some expect joint expected values and so on. 276 00:32:45.118 --> 00:32:52.949 We can compute, they're going to have a function of them so we can so we got to join Tom density function f of X. Y. 277 00:32:52.949 --> 00:32:57.868 And we might have for the brand, and we might have some function of the random variable and. 278 00:32:57.868 --> 00:33:04.078 And 1, to find the expected value of G, X, and Y, we had the density of X and Y. 279 00:33:04.078 --> 00:33:14.519 That would be the definition and some of the example here. 280 00:33:14.519 --> 00:33:23.189 We've got 2 random variables and what's the expected value? What's the distribution of the sum? What's expected value of the sum? So. 281 00:33:25.078 --> 00:33:30.239 And again, this starts off simple, but it starts getting a little more complicated. 282 00:33:30.239 --> 00:33:36.598 So, the random variable might be tell us a coin. Is it heads or tails do that twice? That's a special number of heads. 283 00:33:36.598 --> 00:33:42.509 But looking at it that way, it's very simple. Okay. But but we're getting something interesting here. 284 00:33:44.638 --> 00:33:50.098 What the what this says is that the expected value of the sum of 2, random variables. 285 00:33:50.098 --> 00:33:54.838 Is the expected value is a sum of the expected values now. 286 00:33:56.878 --> 00:34:11.429 Well, you might say that's not surprising, but here's the interesting thing is that the does not depend on anything about the random variables X and Y, this equation. 526. it's true. Regardless. 287 00:34:11.429 --> 00:34:16.139 Of our X and Y, independent dependent correlated or whatever. 288 00:34:16.139 --> 00:34:26.938 This is always true. Work out how it is this the definition so, the X Prime, plus swipe prime presidents integrated over X prime why? Prime minus a fee to infinity. 289 00:34:26.938 --> 00:34:31.438 Well, you separate the underground somewhat 200 girls. 290 00:34:33.509 --> 00:34:39.028 1, on the left side here. 291 00:34:40.559 --> 00:34:45.838 And what we're doing is we're integrating out why on. 292 00:34:45.838 --> 00:34:53.818 Joint density well, that's the definition of the marginal density of X. so we see here the D. why that's. 293 00:34:53.818 --> 00:34:57.748 Marginal density annex at lower case x X. 294 00:34:57.748 --> 00:35:09.329 And did on the why it's sitting on the left is expected value of X on the watch secondary. And why so, this is surprising fact here that if you add to random variables, then the expectations at. 295 00:35:09.329 --> 00:35:13.409 Regardless of anything else, I think, correlate it or not. 296 00:35:13.409 --> 00:35:21.958 Dependent or not X and Y, need not be independent and this extends to. 297 00:35:23.039 --> 00:35:27.239 Some of them, obviously by induction. 298 00:35:30.088 --> 00:35:33.478 Now, that's true for the sum. 299 00:35:33.478 --> 00:35:38.099 It's not too for any other combination. It has the product. 300 00:35:38.099 --> 00:35:41.369 Of 2, random variables then. 301 00:35:42.929 --> 00:35:47.579 It's true only if they are. 302 00:35:49.349 --> 00:36:03.418 Independent of each other. So, if we've got some joint function at some function of X and Y, some function of x time, some function of why it decomposers. 303 00:36:03.418 --> 00:36:07.378 Then the expected value of. 304 00:36:07.378 --> 00:36:16.378 Expectation that the product of the expected value, so that function on next and other function on why this here happens only if they're independent. 305 00:36:17.458 --> 00:36:22.528 Okay um, so we got. 306 00:36:22.528 --> 00:36:30.719 Now, we can start talking about moments, so the expected value of joint moments. So actually the J. Y. the K that's. 307 00:36:30.719 --> 00:36:37.528 Vacate joint moment, it's not the central central omit is we'd subtract out. The means. This is just the moment. 308 00:36:37.528 --> 00:36:40.768 An obvious definition here. Okay. 309 00:36:43.079 --> 00:36:51.179 And again, we kind of the marginal moments on X suggest Y, okay. 310 00:36:53.608 --> 00:37:05.518 And we'll work to get with examples next class and if you subtract the mean out from X, and Y, and we had expected value of that to the J to the, that is your. 311 00:37:06.599 --> 00:37:11.039 Central moment of the J order and acts in the case order and why. 312 00:37:13.798 --> 00:37:18.208 Now, if, um. 313 00:37:18.208 --> 00:37:25.469 For Jane caving 1, then it says the 1st order central moment in X and Y. 314 00:37:25.469 --> 00:37:34.798 Expected value of X minus the mean times. Why minus? I mean, that's got a definition and we call it the CO variance of X and Y. 315 00:37:34.798 --> 00:37:44.219 If we just acts, we'd call it the variance just why the variance of Y, variance of actually be X Y, and Z X squared. 316 00:37:44.219 --> 00:37:48.539 They check it out you that so, but X and Y, we got to pair here. 317 00:37:49.768 --> 00:37:53.969 And we give it a new name, and we call it the call variance. 318 00:37:55.559 --> 00:38:03.208 And you can play with it and multiply things out and get some other ways to work with it. And so on, depending. 319 00:38:03.208 --> 00:38:11.190 Now, what's interesting about the CO variant? Is it, um. 320 00:38:11.190 --> 00:38:17.579 It captures how X and Y, relate to each other. 321 00:38:17.579 --> 00:38:21.900 So, if they track each other, the CO will be larger. 322 00:38:21.900 --> 00:38:26.099 If they're independent of each other, it will tend to be smaller. 323 00:38:27.150 --> 00:38:31.320 If wide track sects of the negative direction. 324 00:38:31.320 --> 00:38:39.030 You know, if why tends to be low when X is high and vice versa, the CO variants will tend to be negative. 325 00:38:40.440 --> 00:38:44.190 So Co, variance code captures how X and Y. 326 00:38:44.190 --> 00:38:51.960 Track each other and how widely spread out there if they're really tightly clustered, it's going to be small. 327 00:38:51.960 --> 00:38:58.079 So, if we're thinking of your ID experiment, or you're firing a marshmallow. 328 00:38:58.079 --> 00:39:03.150 And you're rated on how tightly clustered your marshmallow hits? 329 00:39:03.150 --> 00:39:06.809 And you'll get a larger score. 330 00:39:06.809 --> 00:39:12.599 If your Co variants is smaller closer to 0 and absolutely smaller. 331 00:39:12.599 --> 00:39:19.019 Okay, now if they're independent. 332 00:39:19.019 --> 00:39:27.690 Again, I'm giving you a sort of just high level, intro management level. I put this from the previous thing. If they're independent. 333 00:39:27.690 --> 00:39:33.090 And the formal national definition of independence, it turns out the coherence will be 0. 334 00:39:35.429 --> 00:39:42.659 So so independence implies the variances 0. 335 00:39:44.215 --> 00:39:57.985 The reverse doesn't necessarily happen and now the thing with the CO variance is that if you do dimensional analysis of it, 336 00:39:57.985 --> 00:39:59.275 it's got units of. 337 00:39:59.909 --> 00:40:09.989 Lanes squared, because access units of length and X squared as units of light squared X Y, well, of units of lane square and also so. 338 00:40:11.550 --> 00:40:18.449 Correlation coefficient is units of lengths, the coal variants as units of length squared. Now. 339 00:40:19.590 --> 00:40:25.889 So, the sanitation is like to normalize it to remove that dependence on the unit. 340 00:40:25.889 --> 00:40:29.340 And so we make it scale and variant. 341 00:40:29.340 --> 00:40:37.139 I take the variance and dividing by the 7 deviations of X and Y, the standard deviation also as units of length. 342 00:40:37.139 --> 00:40:49.920 So, we've done here, we get something called row of X and Y, that's the correlation coefficient that you 1st saw in the equation for the 2 variable calcium distribution. 343 00:40:51.510 --> 00:40:56.880 And so we do this, we've got the thing there, and that will be the correlation coefficient. 344 00:40:56.880 --> 00:41:04.320 Of X, and Y, now that will capture how X and Y track each other it linearly. 345 00:41:04.320 --> 00:41:13.769 So, if they're independent, it will be 0 If Y equals X. exactly. It will be 1. if Y, is a. 346 00:41:13.769 --> 00:41:17.610 Just a scale of X of Y, equals 10 X. 347 00:41:17.610 --> 00:41:26.610 It will also be 1 if why so is a translation of X Y, equals 10 X plus 3 let's say roll will also be 1. 348 00:41:26.610 --> 00:41:29.909 If wise negative, 10 X. 349 00:41:29.909 --> 00:41:36.599 Role will be minus 1 and will capture that. Why still track sex but in the negative way. 350 00:41:36.599 --> 00:41:41.159 Correlate correlation coefficient so. 351 00:41:41.159 --> 00:41:47.519 Now, those of you that get the statistics, which a touch on the end of the course. 352 00:41:47.519 --> 00:41:54.119 This gets talked about a lot in sociology and whatever you got the correlation between 2 variables. 353 00:41:54.119 --> 00:42:01.199 And is it significant like, is it meaningful could have happened just by chance or something? 354 00:42:02.699 --> 00:42:08.820 Okay, they do some stuff with it here. 355 00:42:11.429 --> 00:42:19.440 So and okay, so the thing independent was, you can multiply the separate probabilities to get the joint profitability. 356 00:42:19.440 --> 00:42:25.860 That's a general concept. Correlation captures a linear dependence and now example. 357 00:42:25.860 --> 00:42:30.090 527 here it teaches something important. 358 00:42:30.090 --> 00:42:33.840 That variables could be dependent on each other. 359 00:42:33.840 --> 00:42:41.550 But not linearly correlate correlation is a linear concept. If this is a more complicated relation than. 360 00:42:41.550 --> 00:42:44.940 Things may depend on each other, but not be correlated. 361 00:42:44.940 --> 00:42:48.570 And that is shown in this thing here 527. 362 00:42:51.090 --> 00:42:57.900 So, what we have is we have a unit circle unit means that is a radius of 1. 363 00:42:57.900 --> 00:43:02.519 And the random our experiment is, we pick a random angle. 364 00:43:02.519 --> 00:43:06.539 Data, and we look at the point on the circle. 365 00:43:06.539 --> 00:43:11.639 So, that experiment is in the spinner. 366 00:43:11.639 --> 00:43:17.699 And look at where it hits on the circle and from that experiment, we get 2 random variables X and Y. 367 00:43:19.320 --> 00:43:24.179 Now, we know they're dependent on each other because. 368 00:43:24.179 --> 00:43:28.889 Give an X and Y, is plus or minus the square root of 1 minus X squared. 369 00:43:28.889 --> 00:43:34.320 So, for any value of X, is that most 2 values a lie so yeah. 370 00:43:34.320 --> 00:43:37.800 It's, um, is a relation they're dependent. 371 00:43:37.800 --> 00:43:42.809 And if we were sticked ourselves, say, the top half circle for any value of axis. 372 00:43:42.809 --> 00:43:55.050 1 value why that's dependent. Yeah, but they, if we throw in our definition of correlated, they're not correlated. So they're absolutely dependent on each other, but they're not. 373 00:43:55.050 --> 00:44:02.880 Correlated and this is a lesson here. Correlation is only a linear idea. Defendant has more general idea. 374 00:44:02.880 --> 00:44:08.489 And giving the example here and computes what the correlation is. It's 0. 375 00:44:11.159 --> 00:44:18.690 Now, here is like our joint exponential distribution and. 376 00:44:20.340 --> 00:44:26.400 And we want to find any of these things, like, expected valuable variance and raw expected value. 377 00:44:26.400 --> 00:44:30.239 This is our PDF in here the 2 of you to the minus actually in the minus why. 378 00:44:30.239 --> 00:44:38.190 Integrated out, non negative integrate out. You get 1. so the 2 is correct the expectation you. 379 00:44:38.190 --> 00:44:43.860 Put X Y, to start is the expectation for X times. Y. 380 00:44:43.860 --> 00:44:50.369 Not to and why so that's the function you want. The explanation of is X Y, and you integrate it out. 381 00:44:50.369 --> 00:44:58.739 And, um, you get you get, um. 382 00:44:58.739 --> 00:45:02.099 You get 1. 383 00:45:02.099 --> 00:45:09.030 We're going to take a minutes if we expect to see if I get mathematical going again. This is annoying. 384 00:45:10.800 --> 00:45:16.980 Well, I know my problem. 385 00:45:18.119 --> 00:45:22.650 Like, to find something for X. 386 00:45:22.650 --> 00:45:28.079 Sorry about that. 387 00:45:28.079 --> 00:45:32.820 Okay, it'll the old 1. 388 00:45:34.469 --> 00:45:37.679 Don't say, okay. 389 00:45:39.449 --> 00:45:43.829 And the great. 390 00:45:43.829 --> 00:45:47.429 Squared with X. 391 00:45:53.309 --> 00:45:58.530 Still is 2 a, and B. 392 00:46:01.769 --> 00:46:14.400 Bingo finally integrate a squared. I get a cube over 3. 393 00:46:14.400 --> 00:46:17.550 Differentiate I integrate sign. 394 00:46:21.210 --> 00:46:29.519 Oh, minus calls. Good differentiate big day. Differentiate. 395 00:46:29.519 --> 00:46:35.820 Signed. 396 00:46:35.820 --> 00:46:44.039 Close good integrate that thing. 397 00:46:45.059 --> 00:46:50.550 And again, instead of X and Y, I'm going to use a and B. 398 00:46:50.550 --> 00:46:56.820 And a great. 399 00:46:58.590 --> 00:47:04.920 And you do to a Times B, times to. 400 00:47:06.179 --> 00:47:09.570 Exponential minus. 401 00:47:09.570 --> 00:47:13.949 Exponential. 402 00:47:13.949 --> 00:47:19.619 To be. 403 00:47:19.619 --> 00:47:25.230 We from. 404 00:47:25.230 --> 00:47:31.110 0, to a. 405 00:47:31.110 --> 00:47:36.210 Thing go, and that's what we have down here. Oh, okay. 406 00:47:37.860 --> 00:47:45.449 If you factor it out, it worked and I'd say, integrate that with respect to a, which is X. 407 00:47:47.280 --> 00:47:52.050 Um. 408 00:47:52.050 --> 00:47:56.820 Hey. 409 00:47:56.820 --> 00:48:03.389 Darrow to infinity. 410 00:48:04.650 --> 00:48:07.800 1, okay. 411 00:48:09.929 --> 00:48:13.679 It was faster and Mathematica than not doing this thing. 412 00:48:13.679 --> 00:48:16.829 By hand okay. 413 00:48:16.829 --> 00:48:24.179 Motivation to learn Mathematica did the girls your friend shells and so on. 414 00:48:25.349 --> 00:48:29.639 Integrate something like the normal thing. 415 00:48:32.880 --> 00:48:39.119 And you do to do exponential. 416 00:48:39.119 --> 00:48:44.400 Minus X minus squared. 417 00:48:46.679 --> 00:48:53.429 On you to write 2. 418 00:48:53.429 --> 00:48:57.360 With respect to a. 419 00:49:00.179 --> 00:49:07.980 And or for the it's a com. 420 00:49:07.980 --> 00:49:19.019 It's the par central for the normal folks, they call it or if there's so many different standard we saw 2 different names for BQ. And so, and if I put in a value for a. 421 00:49:19.019 --> 00:49:23.519 A say from 0 to infinity. 422 00:49:26.489 --> 00:49:32.070 Scared to fly over too. Yeah. Um. 423 00:49:33.239 --> 00:49:41.130 It's the 2 is on the other below instead of above, because, you know, from 0 to infinity instead of my. 424 00:49:41.130 --> 00:49:44.610 Okay. 425 00:49:44.610 --> 00:49:49.019 Mathematica can be useful, um. 426 00:49:52.440 --> 00:49:55.469 I wanted that is a number. 427 00:49:57.329 --> 00:50:09.150 Silence. 428 00:50:09.150 --> 00:50:15.869 If I wanted it as a number to a 100 digits or something, I could also do it. 429 00:50:15.869 --> 00:50:20.369 Okay. 430 00:50:20.369 --> 00:50:23.460 Telling this thing here. 431 00:50:24.630 --> 00:50:28.889 Yeah, and correlation coefficient. 432 00:50:28.889 --> 00:50:35.369 Back up here again, and I can work it out. I wanted to. 433 00:50:38.639 --> 00:50:43.170 Okay, so conditional probability, it's come back to this for a few minutes here. 434 00:50:43.170 --> 00:50:48.840 I got a motivation ticket of the communications channel now. 435 00:50:48.840 --> 00:50:54.570 Transmitting a signal X or Y, X, which is a plus or minus 1. 436 00:50:54.570 --> 00:51:00.750 Noise gets added to it and you had a received signal why. 437 00:51:00.750 --> 00:51:07.769 And you look at why, and you'd like to find out what do you know about X? 438 00:51:09.269 --> 00:51:13.889 And maybe the probability of different types of access knocked. 439 00:51:13.889 --> 00:51:21.090 Equal and this is a simple thing for this course. And the real world, the noise might be not so simple. 440 00:51:21.090 --> 00:51:25.139 As a normal distribution in the real world. 441 00:51:25.139 --> 00:51:31.500 Success of you don't just have 1 act you got a whole vector of access and they. 442 00:51:31.500 --> 00:51:35.460 May be correlated with each other, so. 443 00:51:35.460 --> 00:51:40.170 Example, if I look at this page of the textbook, and I'm transmitting it. 444 00:51:40.170 --> 00:51:48.150 I mean, black is a lot less common than white, but at both black and white, they tend to occur and runs actually. 445 00:51:48.150 --> 00:51:52.289 And if you know, the previous pixel, you know, the next pixel. 446 00:51:52.289 --> 00:51:56.159 A higher probability, they tend to be bunched up in runs. 447 00:51:56.159 --> 00:52:02.519 Now, you know, that, then you might like to work that into your calculations of conditional probability. 448 00:52:02.519 --> 00:52:09.690 Or if you're transmitting letters and a textbook in a book, let's say, and, you know, they're a letter a C. 449 00:52:09.690 --> 00:52:15.929 Face who's had if you're from the United Kingdom and or her colonies, then. 450 00:52:15.929 --> 00:52:25.980 You would there's correlations and so if the letters are getting mangled in transmission, like, maybe it's more code and there's some errors. 451 00:52:25.980 --> 00:52:32.969 Then you'd want to use relations if you see a queue most of the time, the next sliders. Are you? Not always, but generally. 452 00:52:32.969 --> 00:52:37.440 Take a conditional probability conditional expectation. Okay. 453 00:52:40.230 --> 00:52:51.960 This goes way back to chapter 2 probability of why some, why give an access of joint probability divide that by that probability of X we saw on this mentioned. 454 00:52:51.960 --> 00:53:00.570 Okay or discreet, random variables, use the mass function and you divide. 455 00:53:00.570 --> 00:53:03.719 Okay. 456 00:53:05.130 --> 00:53:20.010 So this is an example so, this is our dice experiment. Remember we were tossing to dice and each dice independently looks fair. 457 00:53:20.010 --> 00:53:26.550 But the problem is that they track each other for some unknown reason. If you like physics, you could probably. 458 00:53:26.550 --> 00:53:30.179 Think of something to described is a class how this is likely to happen. 459 00:53:31.469 --> 00:53:39.150 But so if you see if the 1st dye is a 5 and cold dices die, it's the 1st dies of 5. 460 00:53:39.150 --> 00:53:44.519 Well, then the 2nd dye is more likely than. 461 00:53:44.519 --> 00:53:50.250 More than 16 probability that it will also be a 5 because because the dissect tracking each other. 462 00:53:50.250 --> 00:53:58.349 So, we want well, we want to throw numbers added there, probably that probably distribution what the mass function on why. 463 00:53:58.349 --> 00:54:04.019 If the 2nd dye is the 1st day was a Y, and that is. 464 00:54:04.019 --> 00:54:07.440 By definition is the. 465 00:54:07.440 --> 00:54:13.019 Probability of 5 and wide divided by the marginal probabilities in the 1st 1 is 5. 466 00:54:15.420 --> 00:54:24.510 And so the probability, if the why being a 5 give the 5 is 27, because it's. 467 00:54:24.510 --> 00:54:27.750 240 seconds divided by a 6. 468 00:54:27.750 --> 00:54:31.199 The probability for the 1st time to be anything other than 5. 469 00:54:31.199 --> 00:54:36.179 1234 or 6 is again, we saw in that formula, but. 470 00:54:36.179 --> 00:54:48.780 The top the numerator is 140 seconds and divided by 617. so this is a way we can find conditional probability of 1, random variable given the other 1. 471 00:54:48.780 --> 00:54:55.800 Or the conditional probability of any function of the given any other function. 472 00:54:57.300 --> 00:55:04.710 Okay, so I'll just give you all again. I'm giving you a high level. 473 00:55:04.710 --> 00:55:12.090 Stuff here, but, um. 474 00:55:15.780 --> 00:55:18.809 We have a chip integrated circuit. 475 00:55:20.190 --> 00:55:24.809 And we're assuming the defects happen randomly. 476 00:55:27.599 --> 00:55:32.250 Independent of each other maybe they're caused by. 477 00:55:32.250 --> 00:55:35.760 Particles calls Smith, grace, hitting something. 478 00:55:35.760 --> 00:55:42.300 And so a reasonable model is that the number of defects is a croissant. 479 00:55:42.300 --> 00:55:45.449 Brand new variable with some parameter mean alpha. 480 00:55:47.190 --> 00:56:00.389 Okay, now we maybe the chip has several sub regions. Like, if you're Nvidia and you're manufacturing and graphics processing unit. 481 00:56:00.389 --> 00:56:05.760 It has on it made might have on it may be 16 streaming multi processors. 482 00:56:05.760 --> 00:56:09.659 Which is basically independent of each other so. 483 00:56:11.130 --> 00:56:19.139 So too many defects happen and 1 streaming multi process. It just got note but the other ones are still functioning. 484 00:56:19.139 --> 00:56:32.760 So, video will do is then going to bend the output. They're going to see how many of the steering multi processes work and the more that work, the higher the price they'll charge for it. 485 00:56:32.760 --> 00:56:37.409 Videos everyone does the same thing. It's a reasonable thing to do. 486 00:56:37.409 --> 00:56:43.619 But now we want to estimate how many chips of varying qualities are going to get out of this. 487 00:56:43.619 --> 00:56:50.820 So so we've got to reach so the total chip had. 488 00:56:50.820 --> 00:56:54.510 Alfa defects. 489 00:56:56.909 --> 00:57:00.840 You know, on average, but it's false on. Distributor could be more could be less. 490 00:57:00.840 --> 00:57:06.210 But we got a region on the chip it's 1, streaming, multi processor colon. 491 00:57:07.889 --> 00:57:15.929 And we're interested is that in this particular region, is it going to be good after the thing? 492 00:57:15.929 --> 00:57:23.519 So we want to know how many defects this region is going to have, or we want to know the random. 493 00:57:23.519 --> 00:57:28.590 Variable in the grad distribution so, um, so any. 494 00:57:28.590 --> 00:57:31.800 Ray zapping the chip. 495 00:57:31.800 --> 00:57:37.349 Has a probability of finding in this regions region is 1 processor on the chip. Let's say. 496 00:57:38.519 --> 00:57:41.550 So total chip costs are distributed. 497 00:57:41.550 --> 00:57:44.940 I mean, is alpha. 498 00:57:44.940 --> 00:57:48.090 This particular region on the chip. 499 00:57:48.090 --> 00:57:53.460 Probability of a defect that somewhere on the chip of hitting here is P. 500 00:57:53.460 --> 00:57:57.960 And we want to learn something about what's happening to this region are now. 501 00:57:57.960 --> 00:58:04.559 Intuitively it might say, look, I mean, of alpha defects totally P of any 1. 502 00:58:04.559 --> 00:58:13.289 Given 1 happening, and our than the mean number of defects, setting region is going to be alpha times P. you'll, you're going to be right actually, but let's work it out. 503 00:58:13.289 --> 00:58:17.670 So well. 504 00:58:17.670 --> 00:58:22.739 We could say this is a nice chance to be perhaps. So if. 505 00:58:22.739 --> 00:58:31.199 Kay defects total and the probability of any particular defect hitting this region is P. 506 00:58:31.199 --> 00:58:39.570 Then the random variable for their being, say, J, defects on this region, it's going to be a. 507 00:58:39.570 --> 00:58:42.570 By Dolby random variable. 508 00:58:42.570 --> 00:58:47.639 Um, total being K, and we want to know Jay defects out of K. 509 00:58:47.639 --> 00:58:59.849 And probably anyone is Pete, and again, we only need this in the total number defects in this region and of the total number of defects hitting the chip. K. we don't care which J. of them. 510 00:58:59.849 --> 00:59:07.469 Are in this region, we just care about how many and we assume everything's independent of everything else. So that's going to be this here. Okay. 511 00:59:07.469 --> 00:59:14.610 The probability of there being J defects in this interesting region, given this K defects total. 512 00:59:14.610 --> 00:59:23.460 And if Jay is bigger than K, it's terrible, because it can't happen. Oh, okay. So now we want to know this. 513 00:59:25.409 --> 00:59:31.320 So, how are we going to do this. 514 00:59:31.320 --> 00:59:34.980 We need to know probably so probably J given K. 515 00:59:36.179 --> 00:59:39.239 And we know the probability of K, that's. 516 00:59:39.239 --> 00:59:42.599 Probably a J given Kay that's. 517 00:59:42.599 --> 00:59:46.559 Binomial probability for Kay. That's croissant. 518 00:59:46.559 --> 00:59:56.250 We want the marginal on Jay on integrate out. We don't care what K is. Don't care what the total number defects the only care how many defects there are in this region. 519 00:59:56.250 --> 01:00:00.510 So we want to integrate okay, well, some out, because it's discrete. We've got this thing here. 520 01:00:00.510 --> 01:00:09.809 So the 1st thing is the conditional probability of having J, defects in the region, given a 2nd part is a probability of K, defects total. 521 01:00:09.809 --> 01:00:13.679 We some over K and we get this here. 522 01:00:16.320 --> 01:00:20.429 Play a little algebra. algebra's fun. I'm not joking. It is fun. 523 01:00:20.429 --> 01:00:25.829 And due to and it comes down to here and this is on. 524 01:00:25.829 --> 01:00:33.300 And as facade with a parameter of alpha P. so the number of defects hitting this interesting region, it's croissant. 525 01:00:33.300 --> 01:00:38.070 With a mean of alpha Pete, which is what you might have expected from common sense, then. 526 01:00:38.070 --> 01:00:43.769 You'd be right sometimes commonsense is right. Okay. 527 01:00:48.510 --> 01:00:53.340 So, in particular, so let's suppose here that you're. 528 01:00:53.340 --> 01:00:59.969 That interesting that that interesting sub chip is good if it has no more than 2 defects, and you can calculate the problem. So. 529 01:00:59.969 --> 01:01:04.170 Okay, that was great. 530 01:01:04.170 --> 01:01:09.539 Continue with it's the same idea, but no miss here. So you got conditional. 531 01:01:09.539 --> 01:01:15.599 Cdf cumulative. 532 01:01:17.670 --> 01:01:24.269 So, the cumulative this, this is going to be new. This is the cumulative. 533 01:01:24.269 --> 01:01:31.260 Distribution function on why for some value of accident. 534 01:01:31.260 --> 01:01:35.909 2, random pharaoh's X and Y, that's your location on the left. 535 01:01:37.110 --> 01:01:43.079 So, it's a probability and it makes sense. It's the joint probability that why is equal to why. 536 01:01:43.079 --> 01:01:51.719 For and access this value divided, probably access. So this point 539 it just follows on what we had in chapter 2. so. 537 01:01:54.000 --> 01:02:00.449 And then given the thing on the left, the cumulative, if you can do the derivative. 538 01:02:00.449 --> 01:02:04.889 If it exists and you'll get the. 539 01:02:04.889 --> 01:02:11.550 The conditional density function on why given some value of facts it's a derivative of. 540 01:02:12.960 --> 01:02:18.840 That up there. Okay. 541 01:02:21.780 --> 01:02:27.150 And another way to look at it, we just integrate the conditional. 542 01:02:27.150 --> 01:02:30.750 Density function for all of the whys and the interesting region. 543 01:02:30.750 --> 01:02:36.780 Okay, so back to our big example. 544 01:02:36.780 --> 01:02:41.280 Binary communication system we're transmitting a bit. 545 01:02:41.280 --> 01:02:46.739 Plus or minus Juan here, they're not equal probability. 546 01:02:46.739 --> 01:02:51.840 Um, the plus 14th and the minus 1, 2 thirds. 547 01:02:51.840 --> 01:03:02.460 That's the 1st way we're messing with you. The 2nd way we're messing with you is that the noise is not just uniform over an interval. Now, the noise is calcium. 548 01:03:02.460 --> 01:03:07.500 And but we're going to keep it simple standard deviation being 1 for the moment. 549 01:03:07.500 --> 01:03:16.199 And why is our binary X plus our Gaussian and so let's this mix discreet continuously to natural thing that can happen. 550 01:03:17.789 --> 01:03:23.280 Okay, so we want to find conditional outputs given either input. 551 01:03:23.280 --> 01:03:30.150 And then the 3rd thing, this is something that's, we're actually getting useful. 552 01:03:32.940 --> 01:03:39.210 If the received signal was positive, what's the probability of the transmitted signals was positive you know, that's. 553 01:03:40.440 --> 01:03:44.159 What we want to know, and this is actually the probability that. 554 01:03:44.159 --> 01:03:49.619 We could decode the signal correctly. Okay. There was saved then. 555 01:03:49.619 --> 01:03:55.590 You know, the negative of that, the Congress that's going to be an error. 556 01:03:55.590 --> 01:04:00.570 So, how do we do this is what we want to know if the received signals positive. 557 01:04:00.570 --> 01:04:06.690 What's a probability transmitted signals policy? It felt much more than I have, but let's see. 558 01:04:06.690 --> 01:04:10.320 Okay, um. 559 01:04:12.239 --> 01:04:20.670 Well, we're going to take this through and staff, so this time for fun, we're going to use a cumulative dense distribution function on why. 560 01:04:20.670 --> 01:04:25.469 Given X is positive. 561 01:04:26.969 --> 01:04:38.429 Well, that's a problem by definition this the definition here, probably a wise lesson. Why give an excess plus 1 now why is X? Plus? N. okay. So the thing on the left is that is. 562 01:04:38.429 --> 01:04:47.070 And we assume axis 1, that's part of the condition. So why why means X means X plus and less regular why. 563 01:04:47.070 --> 01:04:54.570 Which means, and plus 1 is, I should have white because this 1 here. So this is a probability that N plus 1 less grateful to why. 564 01:04:54.570 --> 01:04:57.809 So, what we've done here is. 565 01:04:57.809 --> 01:05:01.800 It's out no. Okay. It's this is just an expression on why. 566 01:05:01.800 --> 01:05:07.980 Well, then endless plus 1, why means? And less echo Y, minus 1. 567 01:05:07.980 --> 01:05:13.800 But N is normal. So now this is just that partial and a goal. 568 01:05:13.800 --> 01:05:22.530 Or a Gaussian here that now this we can, we can integrate it in a closed form. However. 569 01:05:24.239 --> 01:05:30.690 We can look into a table for any particular value of why we can look into a cable and see what it is. 570 01:05:33.179 --> 01:05:39.510 Like, if Y is 1 and Y, minus 1 is 0, and just hit her goal is going to be 1 half. 571 01:05:40.650 --> 01:05:50.760 For example, okay, so I guess some examples here effects is 1. 572 01:05:51.960 --> 01:06:01.349 And it says, send a girl here that's queue of minus 1 actually queue being engineering the right tale. The right tail probability. 573 01:06:01.349 --> 01:06:05.429 My day 10 affects is minus 1 point. 574 01:06:05.429 --> 01:06:10.380 The opposite 16 so what this is saying. 575 01:06:10.380 --> 01:06:14.250 Is that if the transmitted signal is 1. 576 01:06:14.250 --> 01:06:20.070 56 of the time the received signal is positive and 16 of the time it's negative. 577 01:06:22.050 --> 01:06:26.519 Okay, so 16 of the time, the noise is so large that it. 578 01:06:26.519 --> 01:06:29.789 It flipped the transmitted bit basically. 579 01:06:29.789 --> 01:06:36.719 Okay, good so far. So this is the probability is a bit came to correctly. 5 6. 580 01:06:36.719 --> 01:06:41.340 But member the transmitted bits are not equally probable. 581 01:06:42.449 --> 01:06:45.570 So, it's basically. 582 01:06:45.570 --> 01:06:52.679 2 thirds of the time the transmit had been as minus not plus. And now we want to these are called prior probabilities. 583 01:06:52.679 --> 01:07:04.949 And after the transmission, those are the scarier probabilities before and after before the transmission, the prior probabilities after it at the scarier probabilities. 584 01:07:04.949 --> 01:07:09.750 So, the so the probability of next is 1 is 4th. 585 01:07:10.800 --> 01:07:15.809 So, now we want the probability that they received signal why. 586 01:07:15.809 --> 01:07:20.099 Was positives who wants a posterior probability basically. Well. 587 01:07:21.570 --> 01:07:35.579 If the transmit and singles pauses with saved signals positive 56 of the time but the transparency was positive. 4th of the times modify that. Then we've got 16 of the time that the received signal got flip 2 thirds of the time to signal. It's minus. 588 01:07:35.579 --> 01:07:43.619 And this nets out to 3839% of the time, the received signal was positive. So. 589 01:07:43.619 --> 01:07:49.800 So the transmitted signal is positive. 4th of the time 33%. 590 01:07:49.800 --> 01:07:55.920 The receipt signal is positive of 39% of the time, because it always smeared things out. Okay. 591 01:07:55.920 --> 01:08:00.329 It's, it's the smear this. Okay so. 592 01:08:02.429 --> 01:08:08.130 Now, so this is the problem. So, the receive signals Charles, native 3839% of the time. 593 01:08:09.780 --> 01:08:18.720 Now, what we want to do is the final step that if the received signal is positive, what's the probability that the transmitted signal was polished? And so. 594 01:08:18.720 --> 01:08:26.939 So, what this is saying is if the receipts and it was positive, what's the probability that came through? Right? Okay. It didn't get smeared so much. It got wrong. 595 01:08:26.939 --> 01:08:32.699 Okay, so that's a probability. The transit index, those 1, given the receipt signal was positive. 596 01:08:32.699 --> 01:08:45.359 Space, so we have the conditional probability to receive those positive transcenters positive probability. The transmitted was positive. Divided by the probability. The received is positive crunch, crunch, crunch, crunch, 73%. 597 01:08:47.010 --> 01:08:56.220 So, if the received signal was positive, 73% of the time, the transmitted signal was was positive. So, 73% of the time. 598 01:08:56.220 --> 01:09:00.659 And it's more likely plus and minus. That's what we should infer. 599 01:09:00.659 --> 01:09:05.250 Okay, but if we infer this, we'll be right to 73% of the time. 600 01:09:07.739 --> 01:09:10.859 And 27% of the time will be wrong. 601 01:09:10.859 --> 01:09:19.829 Now, if you think I'm going to slow for you, if you want to start thinking things you might say well, how does this depend on the noise. 602 01:09:19.829 --> 01:09:30.329 I mean, if the noise was 0, then this is going to be 1, that's annoyed is really small. This is going to be 1. okay. It always is really, really large. This is going to be 50, 50. 603 01:09:30.329 --> 01:09:37.470 You won't be able to see the transmitted signal so you can, you know, you could get quantitative with that. You could graphic. 604 01:09:37.470 --> 01:09:43.979 Another thing, so you might look at the effective noise. 605 01:09:43.979 --> 01:09:51.539 Now, if you look at the effective noise, you know, maybe in an economic situation, you could have smaller noise, but it's going to cost you money. 606 01:09:51.539 --> 01:09:56.279 And by even an exponential relationship that. 607 01:09:56.279 --> 01:09:59.340 Having the noise called, doubles the cost or something. 608 01:09:59.340 --> 01:10:02.880 So, now you might actually start thinking here. 609 01:10:02.880 --> 01:10:06.810 About you have. 610 01:10:06.810 --> 01:10:12.180 And again, the relation of quality of the signal versus the cost. 611 01:10:13.680 --> 01:10:20.159 You might also look at the look at things. 612 01:10:20.159 --> 01:10:23.460 I'm correct correction. 613 01:10:23.460 --> 01:10:31.739 Error correction systems I mean, many of you seem to error correction code. You talked about it very briefly back in chapter 2. 614 01:10:31.739 --> 01:10:46.590 Simple thing, he transmit 3 times involved and so there's a thing where you could, and we looked out how much that reduces the noise. So right here you could say, okay, if we triple the transmission cost transmit everything. 3 times. 615 01:10:46.590 --> 01:10:57.779 Then, what effect does that have on the note? You could start playing games like that but of course, 3 times and doing a vote is a really primitive way of doing it. You could. 616 01:10:57.779 --> 01:11:01.170 Read Solomon just watch more sophisticated methods. 617 01:11:01.170 --> 01:11:04.649 So, you might look at effects of error correction on this. 618 01:11:04.649 --> 01:11:09.569 You would also look at the effects so, here it was 2 to 1. 619 01:11:09.569 --> 01:11:14.340 Minus the plus now you might look at the effects of different types of. 620 01:11:14.340 --> 01:11:19.109 Of force information, like, you might do something like a, a. 621 01:11:19.109 --> 01:11:23.130 Often code on the source and this 1. 622 01:11:23.130 --> 01:11:26.250 Compress the code and make it 50, 50. 623 01:11:26.250 --> 01:11:32.279 Plus or minus 1, and look at the effects of that. So you see. 624 01:11:32.279 --> 01:11:37.170 You could start making this as a building block, interviewed a full complicated system. 625 01:11:38.609 --> 01:11:43.350 Okay, and that is a reasonable point to stop. Now. 626 01:11:43.350 --> 01:11:55.500 Okay, so, and so the end of page 264, so we did today, is we saw my iPad not connecting to my laptop again grumble grumble and. 627 01:11:57.204 --> 01:12:05.515 We saw more 2, variable stuff. We saw concepts. Like Cole variance is a big idea and that's a joint variance. 628 01:12:05.515 --> 01:12:18.625 And it's 0, we saw the concept of independence of the joint probability is a project to the 2 separate probabilities. This Co variance. It's like a variance with 2 variables and with, and the cove if the. 629 01:12:19.380 --> 01:12:23.489 Variables are independent. The CO variance is 0. 630 01:12:24.510 --> 01:12:28.409 Now, the coal variances units of length squared so we can't. 631 01:12:28.409 --> 01:12:33.869 Factor out the units and make a dimension list number called a correlation coefficient. 632 01:12:33.869 --> 01:12:37.199 Also minus 1 to 1 and. 633 01:12:37.199 --> 01:12:44.250 1 means that why next track each other it couldn't be proportional why? It could be 5 x plus 3 and it's still going to row is still 1. 634 01:12:44.250 --> 01:12:48.659 We saw a simple version of a 2 variable Gaussian. 635 01:12:48.659 --> 01:12:52.560 Where the correlation coefficient worked its way into the equation. 636 01:12:52.560 --> 01:13:01.859 And there's more general 1st for is the 2 variable accounts and that's a nice simple 1. the 1 we saw the 2nd deviation were both 1. but there was that row in there. 637 01:13:03.180 --> 01:13:06.989 And the density function is like a hail, but it's scratched. 638 01:13:06.989 --> 01:13:10.739 Like, a rich and Pennsylvania or something that got scrunch and. 639 01:13:10.739 --> 01:13:14.039 Top down or different from left and right. So we saw that. 640 01:13:14.039 --> 01:13:19.229 And we saw coherence, correlation, coefficient and some joint. 641 01:13:19.229 --> 01:13:23.670 Some joint stuff here. Okay. 642 01:13:23.670 --> 01:13:37.890 Okay, so Thursday, no class Monday I'll continue on and do more example the other new thing I showed you was Mathematica Mathematica works with algebra. It can do integrations differentiations. 643 01:13:37.890 --> 01:13:45.210 And I'll show you more examples of that. If you're curious what went wrong for me is i1st set X equals too. 644 01:13:45.210 --> 01:13:52.380 And so actually, it's no longer just an abstract variable. It was a constant and you try to integrate to you get. 645 01:13:52.380 --> 01:13:56.939 To X or something, and that was causing the problems here. 646 01:13:56.939 --> 01:14:10.824 Well, at the X1 X was 2 D2 and mathematically felt like this. And once I switched to using a and B, instead of X and Y, everything worked and even if I started the new window, it still carried over the old values. I could have cleared X. 647 01:14:10.824 --> 01:14:14.395 and so on and made an abstract variable again, but that would. 648 01:14:16.529 --> 01:14:21.659 Yeah, I would have taken some time because I can't remember the syntax or that at the moment. 649 01:14:21.659 --> 01:14:32.279 Anything big like, Mathematica, you build a private cheat sheet and that's how you work it. Okay. So, Monday, next Monday, we'll all do some more of these examples expectations. So. 650 01:14:32.279 --> 01:14:37.020 Functions and stuff and so here are just walking your way through the book. 651 01:14:37.020 --> 01:14:40.739 Of do piles of examples, perhaps on Monday so. 652 01:14:40.739 --> 01:14:45.149 I have a good week and see you then.