WEBVTT 1 00:05:01.949 --> 00:05:05.788 Now, can you hear me now that any better. 2 00:05:05.788 --> 00:05:14.129 Thank you don. Okay, what I did is I log out of my computer and I logged in again and that reset something. 3 00:05:14.129 --> 00:05:21.598 And in my end, it was showing everything worked, but okay. 4 00:05:21.598 --> 00:05:29.999 So, now, the theory is, you can see the screen, but again, we have. 5 00:05:29.999 --> 00:05:36.869 And we have practice, so. 6 00:05:39.569 --> 00:05:42.928 Okay. 7 00:05:45.119 --> 00:05:54.869 Sorry about that mess. Okay. So the theory is this is engineering profitability class, 19, April, 12 2021. 8 00:05:54.869 --> 00:06:01.439 And 1st, any opinions about the exam last week. 9 00:06:01.439 --> 00:06:04.918 Questions. 10 00:06:12.778 --> 00:06:16.499 No probability to sebatian of us. 11 00:06:16.499 --> 00:06:23.488 Being able to see the screen. It's a very good question. You tell me. 12 00:06:25.949 --> 00:06:33.119 Can you see the screen. 13 00:06:39.983 --> 00:06:48.863 Thank you. Okay we are moving on and profitability. We saw probability for 1, random variable. 14 00:06:48.863 --> 00:06:55.524 We saw probability for 2 random variables and now we are seeing probability. 15 00:06:55.889 --> 00:07:00.418 For a vector of random variables. This is in chapter 6. 16 00:07:00.418 --> 00:07:06.329 And I got some summary points on the blog, but I'll walk through here. 17 00:07:06.329 --> 00:07:13.528 Where would you get a vector of random variables? Well, a signal and audio signal for perhaps you are. 18 00:07:14.754 --> 00:07:27.774 Sampling at 40,000 times a 2nd, because you want to capture frequencies up to 20 kilohertz. Perhaps. And each voltage is a random variable. You have a vector of those random variables would be 1 example. 19 00:07:28.678 --> 00:07:36.329 Or you're trying to video signal each pixels are beach pixel 3, random variables. 20 00:07:36.329 --> 00:07:49.199 Now, why this is useful remember we had this running example of transmitting a goal to translating a signal 1st of it was plus or minus 1 then it was a continuous a real number. 21 00:07:49.199 --> 00:07:52.559 And we wanted and the transmission was error prone. 22 00:07:53.934 --> 00:07:57.624 And we wanted to reconstruct what the most likely transmitted signal was. 23 00:07:57.954 --> 00:07:58.194 Well, 24 00:07:58.194 --> 00:07:59.514 here we have, 25 00:07:59.543 --> 00:08:14.213 maybe you're transmitting a video signal and it's noisy and you want to then reconstruct the signal or you might also the next level is you want to decide what's the best way to transmit it now that's getting the image 26 00:08:14.244 --> 00:08:15.593 processing courses. 27 00:08:16.223 --> 00:08:19.494 So, that's getting really deep, but at least give you a superficial idea. 28 00:08:19.798 --> 00:08:26.519 What the issue is any case we have us a vector X1 to. 29 00:08:26.519 --> 00:08:32.519 Real folks in a computer and that vectors a random variable. 30 00:08:32.519 --> 00:08:36.839 Okay, good now. 31 00:08:38.249 --> 00:08:43.589 And we did means it variances we did correlations before. 32 00:08:43.589 --> 00:08:54.149 And plan to do them again. Example, 61 here, we have packets arriving at a package switch and. 33 00:08:54.149 --> 00:08:59.458 In this case, the switches 3 inputs and. 34 00:08:59.458 --> 00:09:10.109 3 outputs, let's say, and it's like a cross bar switch or something. I, here's the thing. Not use example 61. so, is that each of the 3 inputs. 35 00:09:10.109 --> 00:09:14.278 Has stuff coming from a different place and each input. 36 00:09:14.278 --> 00:09:17.969 Of the 3, which is a 50, 50 chance of having a packet there. 37 00:09:17.969 --> 00:09:28.619 So total number of packets, it did switch. It could be 0 that's going to be like, probably 18. it could be 3 probability 18. it could be somewhere in between. This is just, you. 38 00:09:28.619 --> 00:09:32.519 But, you know, whatever you bind on your probabilities. 39 00:09:32.519 --> 00:09:41.519 In any case you got to factor X1 X2 X3, which is the number random variables for the number of inputs. 40 00:09:41.519 --> 00:09:45.869 Package and input X1 X2 X3. 41 00:09:47.759 --> 00:09:55.948 On counts example, 62, we've got our chip and the chips been divided into regions. 42 00:09:55.948 --> 00:10:05.158 And each region as might be a plus on random variable describing how many packets, how many failures in that region. 43 00:10:05.158 --> 00:10:19.558 For example, they made it a little more complicated here where you got the number of defects and then the coordinates the defects. And so on example, 63 is the audio signal I talked about. 44 00:10:19.558 --> 00:10:25.229 You sampling at 20,000 times a 2nd, maybe and each. 45 00:10:25.229 --> 00:10:31.408 Sample that's a random variable. They're highly correlated. Of course, but that's what makes it interesting. 46 00:10:31.408 --> 00:10:42.028 Okay now see you random variables, the sector with some and scalers, let's say. 47 00:10:42.028 --> 00:10:45.719 And skipping through some details of this. 48 00:10:45.719 --> 00:10:50.759 Well, you have a region probability that you're in that region or something. 49 00:10:50.759 --> 00:11:04.019 Now, how do we okay. Start getting a little more quantitative you got a cumulative distribution function for the vector just as you had for a scale or anything you had. 50 00:11:04.019 --> 00:11:14.639 To to a scale or error random variables here. Random variables. The factor of scalers so the cumulative distribution function. 51 00:11:14.639 --> 00:11:25.499 Big a** CDF, that lower case big accepts the name of the random variable. And the argument is a particular value for the random variable. 52 00:11:25.499 --> 00:11:35.519 And the CDF is the probability that each component of the random variable is less than, or equal to the cut off. That's the argument here. 53 00:11:35.519 --> 00:11:40.019 Is quite a simple extension from the 2 variable. 54 00:11:40.019 --> 00:11:45.899 And the nice thing with this, this definition is, it works for discrete. It works for continuous. It works for mixed. 55 00:11:45.899 --> 00:11:51.509 Okay, and then you can. 56 00:11:52.708 --> 00:12:07.019 You can get they talk about here marginal things where you drop 1 of the random variables for example, for the CDF, the CDF for you drop the last round of variables. You just choose the original CDF and put an incentive. 57 00:12:09.058 --> 00:12:18.958 Okay example, where you have a back to, you might want to do the max. We talked about that a little before. I was just. 58 00:12:18.958 --> 00:12:29.068 Week and a half ago now Here's a case radio trends is talking sample 6 for the radio transmitter. 59 00:12:29.068 --> 00:12:36.538 Okay, it's trying to transmit the signal 3 times, because on 3 different paths, because some pass might be. 60 00:12:36.538 --> 00:12:46.109 Have noise in it or something, and so the receiver, it receives the 3 signals and each have a different strain. 61 00:12:47.339 --> 00:13:00.328 Now, maybe the thing is, if the signal, it's a message is to get through, then the, at least 1 of these 3 cases, it has to get through with a strength of 5 or more. 62 00:13:01.313 --> 00:13:12.413 So oh, more than 5 add to these good fiber less is bad. So we're interested in the case where none of those 3 signals get through strong enough. 63 00:13:12.864 --> 00:13:21.774 So, X1 less equal to 5 X to less equal to 5 X3 also less than equal to 5. in other words, we want to know the probability that the max of X1 X2 X3 is equal to 5. 64 00:13:25.229 --> 00:13:39.778 And that we can do is well, if the Max is actually defy, it's a quote, just saying all 3 are each separate because I used to define that probability is down here. 65 00:13:39.778 --> 00:13:43.739 That CDF on 5 5 5. okay. 66 00:13:46.469 --> 00:13:51.749 Now, if this is a discreet case. 67 00:13:51.749 --> 00:14:06.298 Each of the random variables I discreet. We can talk about a point probability called the joint probability, mass function and it's joint because it's got the end things and that's this here equation 65. the probability that. 68 00:14:06.298 --> 00:14:10.109 Each of the random variables is exactly equal to that value. 69 00:14:11.759 --> 00:14:21.389 And if you've got a region here of several points, the probability that your vectors in that region is some over all the points in the region. That's what we have here. 70 00:14:21.389 --> 00:14:27.688 Oops, sorry. 71 00:14:35.004 --> 00:14:46.014 Okay, now you get marginal things, which is a problem where you just want to probabilities for a couple of the random variables, having the given values and the others you don't care about. 72 00:14:46.349 --> 00:14:52.469 So, what you can do is get a some overall possible values for the variables you don't care about. So. 73 00:14:53.729 --> 00:15:07.528 Here equation 67, we are interested in the value of and we don't care about the others. So we some overall possible values of all of the other, the end minus 1 other ones that we don't sum over. 74 00:15:08.609 --> 00:15:20.009 So are summing over oops sorry about the 2nd thing scrolls, but I don't want it to you. See, we saw over X1 X J, minus 100 J plus 1 to check. And that gives us. 75 00:15:20.009 --> 00:15:23.999 No probability of the next J has this precise value. 76 00:15:26.729 --> 00:15:41.668 You can do it over, so that's where to send about and the probability for 1 specific value margin. You do marginal where you care about 2 variables and you don't care about the N minus to see some overall values to the end minus. So you don't care about. 77 00:15:41.668 --> 00:15:47.219 That's joint now, the next thing is you can talk about a conditional. 78 00:15:47.219 --> 00:15:55.139 And it's the same definition as we had. I don't know where it was chapter 2 or something. 79 00:15:55.139 --> 00:16:06.568 So, we have 6 sign a, we're interested and what can we say, but the probability of different values of X, and if we know. 80 00:16:06.568 --> 00:16:12.089 The other end minus 1 values and this is a definition basically. 81 00:16:12.089 --> 00:16:15.749 It's the probability of. 82 00:16:15.749 --> 00:16:19.769 Where are we fix all and random variables divided by. 83 00:16:19.769 --> 00:16:23.879 The marginal where we fix the N minus 1 that are the. 84 00:16:23.879 --> 00:16:29.609 He might on the right side as a bar here. So this is why we need the marginal things. 85 00:16:29.609 --> 00:16:33.538 And you get a chain thing, like, in 6, 9, be. 86 00:16:34.828 --> 00:16:49.163 Where it's another way of giving the point functions, the probability of X1 to accent of that specific point and it's a product of a lot of different conditional things. Here. We look out of starting working from the. Right? The right. 87 00:16:49.163 --> 00:16:58.014 We've got the probability of X1 is a particular value left times. The probability of X2 is given value given X1 had that value. We know it has. 88 00:16:58.014 --> 00:17:08.304 And then all our way left in the middle dotted out would be the probably the X3 has the desired value, given index 1 and 2 as the values that we know they have and multiply through extending this. 89 00:17:13.588 --> 00:17:18.538 6, 5. 90 00:17:21.328 --> 00:17:30.419 Again, we got this package switch and the package switch has 3 inputs and 3 outputs. What it does is it move things around, like a railway switch yard or something. 91 00:17:32.578 --> 00:17:44.038 And each input the, what's happening in each input is independent of the other 2 inputs. Each input has a probability of 50, 51, half of. 92 00:17:44.038 --> 00:17:47.159 A packet arriving and. 93 00:17:47.159 --> 00:17:55.259 And then it's a total number of packets that arrived at all 3 inputs combined. So it's from 0 to 3. 94 00:17:55.259 --> 00:17:59.398 And this is the probability is just by no meal. 95 00:18:01.854 --> 00:18:02.453 Now, 96 00:18:08.574 --> 00:18:09.683 we're interested, 97 00:18:09.983 --> 00:18:15.294 so the probability of thing of the packets at each output port, 98 00:18:15.683 --> 00:18:18.144 we saw the thing a chapter 2 chapters that goes. 99 00:18:20.578 --> 00:18:24.298 It's, it's using a multi nomial thing that. 100 00:18:24.298 --> 00:18:35.578 If we, it's the derivation assumes that each input packet can go independently to any 1 of the 3 output ports. That's the underlying. 101 00:18:35.578 --> 00:18:47.818 The base underlying this formula and what we want to know is we don't care which packet goes to, which output we care about the total number of packets that went to each output. 102 00:18:49.048 --> 00:18:52.949 And the thing is, so if we, if an packets. 103 00:18:52.949 --> 00:19:01.709 Came in then, and so I went to output 1. J1 to 2. K went to out what 3 where I checked for scape was then. 104 00:19:01.709 --> 00:19:07.048 Pakistan banners, and they were also not created in this switch. 105 00:19:07.048 --> 00:19:17.848 Then this see, here is the probability that I went to the 1st output J to the 2nd and Kate as the 3rd, given that we had and total. Okay. 106 00:19:20.939 --> 00:19:30.838 But this thing here, it's a conditional. This is a probability distribution for each for packet pages the outputs given that we had and packaged total. 107 00:19:30.838 --> 00:19:36.269 But now we want the unconditional probability of. 108 00:19:36.269 --> 00:19:40.979 The outputs getting J and K package, regardless of the value of. 109 00:19:42.058 --> 00:19:50.338 Where is this sign here? I'm trying to circle. The handover is conditional on their being end package total so we want it regardless of and. 110 00:19:52.558 --> 00:20:07.138 And the way it works is this unconditional probability is equal to the conditional probability times. The probability distribution for an. 111 00:20:09.148 --> 00:20:17.759 As a previous page here so T X I. J. K. given in times of probability, they're being and package and that set 3 choose and 1 over 8. 112 00:20:19.019 --> 00:20:24.868 Okay, for any given value of by J and K. 113 00:20:26.909 --> 00:20:36.659 And you can start working it out here. I'm plugging in different numbers. This would be a case, use something like mathematical perhaps. So, maybe I'll show it to you. Thursday. 114 00:20:37.949 --> 00:20:44.999 Okay. 115 00:20:44.999 --> 00:20:53.999 So, the probability of each output getting 0, that's the easy thing, because that happens only if each input gets 0, that happens in 8. so, the time. 116 00:20:55.499 --> 00:21:00.538 Probability of output 1, getting 1 packet. 117 00:21:00.538 --> 00:21:04.229 And you're getting 0 packets well. 118 00:21:04.644 --> 00:21:17.394 If there is 1 packet total, then there's a 4th chance it goes to output 1 and that's the 3rd here. Now we've got to multiply that by the probability. Is there is 1 packet total. That's 38. 119 00:21:17.394 --> 00:21:22.044 so, 324 is the probability that the 1st output gets. 1, the other 2 outputs gets 0. 120 00:21:23.338 --> 00:21:31.469 All right, 0 or the other 2 cases, the probability that the 1st, 2 outputs get 1 and the 3rd 1 gets there. So that's. 121 00:21:31.469 --> 00:21:35.729 Um, so that probably this happening, given that there are 2 packets. 122 00:21:35.729 --> 00:21:42.838 It's this thing here, it's. 123 00:21:45.953 --> 00:21:59.513 2 nights according to this here, then we got the private. Got ahead. The probability is there are in fact 2 packets and that's free to get this year, which you could reduce. If you wanted, you can knock out 112. exactly. 124 00:21:59.544 --> 00:22:07.703 Housing is 1 day. Then you've got the probability that again, 200 you do it's the same way. 012 and so on. So. 125 00:22:09.479 --> 00:22:15.479 So, you get all the different cases here for distributions of packets on the output lines. 126 00:22:18.239 --> 00:22:27.598 And then if you want to find something say that probability that the 1st output got more than the 3rd out, but you just add up the relevant cases here. 127 00:22:27.598 --> 00:22:33.328 So that was a nice example of discreet of a discreet that. 128 00:22:34.378 --> 00:22:43.558 Now, we're going to look at the random variable is a continuous factor. Each of its end components is continuous. 129 00:22:43.558 --> 00:22:48.419 Before you to the components was discreet, you could also have a mixed thing. 130 00:22:50.249 --> 00:23:03.598 And the probability that the vectors in a certain region, so now you've got the density function, little left. So the probability of the vectors in summary Janae just integrate over that read, integrate little app over that region. 131 00:23:03.598 --> 00:23:16.979 And then the way to get little from big app, and so on it, just derivatives here. So, again, you've got your cumulative. And that's the probability that. 132 00:23:16.979 --> 00:23:21.269 For the region being lower left below that vector point. 133 00:23:21.269 --> 00:23:25.739 Which is integrated each, each scale or component from minus affinity to that value. 134 00:23:25.739 --> 00:23:33.209 Of the density, and then the density is just the derivative of the partial derivative above the. 135 00:23:33.209 --> 00:23:38.699 Cdf and again, just like we did with the discrete you could. 136 00:23:38.699 --> 00:23:53.249 Integrate out to get the marginal density for any 1 of the random variables. So, for any 2 or for any subset, you want it. And what you do is, you integrate out the variables. You do not want you to integrate amount for entity. 137 00:23:53.249 --> 00:24:07.979 And you got conditional ones again, you want the conditional density of X ban, given, you know, you've fixed the other end minus ones and it's just the. 138 00:24:10.169 --> 00:24:15.568 It's the density for all of them divided by the density for the N minus 1 that are fixed. 139 00:24:15.568 --> 00:24:25.348 Okay, and you get this product thing here, the conditional density index and given, you know, the other end minus 1. 140 00:24:26.939 --> 00:24:40.439 Or the unconditional 1 up here of X is a conditional back, send give, you know, the other X minus 1 times the conditional X minus X2 and minus 1 giving you the only other and minus 2, blah, blah, blah. Okay. Now. 141 00:24:42.509 --> 00:24:54.628 So, our, basically, our most common, continuous, random variable is the calcium, the normal and that's because the other. 142 00:24:55.648 --> 00:25:01.588 Random variables start looking like a Gaussian very quickly. Almost every case. 143 00:25:02.878 --> 00:25:07.558 Okay, so here question example, 6, 6, we have a. 144 00:25:08.818 --> 00:25:17.699 It's an example here, it's a galaxy it's not a most general joint, but it's it's 1 joint calcium. 145 00:25:17.699 --> 00:25:26.278 And so what we have here is so this little f, it's a density. So if we look over here, what's happening. 146 00:25:26.278 --> 00:25:40.199 It's either the minus and each of the things here, it's X1 square it's X2 square and so an X3 square X3 square it is a half in front of it because it's got a different standard deviation the next 1 next to. 147 00:25:40.794 --> 00:25:52.104 And there's this product term here X1 X2 and it's minus. It's because X1 of next 2 are negatively correlated in this formula example of a particular 3 variables. 148 00:25:52.104 --> 00:25:58.193 And at the bottom, the 2 pie square root of minus pies, they're to normalize it. 149 00:25:59.459 --> 00:26:03.479 So, you integrate and you want to get overall few variable it has to be 1. 150 00:26:04.769 --> 00:26:10.888 Okay, so this is our 3 variable density. What do we want to do with it? 151 00:26:10.888 --> 00:26:25.769 Conditional on our marginal on X1 X2 and conditional next to say, give an X1 X3 or X1 X. Ray. Okay. If you want the marginal density for X1 X3, what you do is you integrate out X2. 152 00:26:25.769 --> 00:26:37.439 And that's what they're showing down here and they pulled out either the minus 6, sub, 3 squared over 2 because it's a constant dependent next 1 and next to. 153 00:26:37.439 --> 00:26:40.528 So, he just pulls it out. 154 00:26:42.058 --> 00:26:47.788 Okay, so we had this thing here, we integrate over to. 155 00:26:49.019 --> 00:26:54.298 And you could call up Mathematica, or you could have fun doing it by and. 156 00:26:54.298 --> 00:27:00.959 And if you integrate out, you'll get that thing down here. 157 00:27:02.429 --> 00:27:08.999 So, if we stop to look at this, so this is the marginal for X1 and X free. 158 00:27:08.999 --> 00:27:20.578 Regardless of the value of X to integrated over the whole X2. And so what this is in this particular case is X1 X3 are separately 1 variable Gaussian. Okay. 159 00:27:20.578 --> 00:27:28.648 With variance. 160 00:27:28.648 --> 00:27:33.538 It doesn't have to happen that way, but in this particular case. 161 00:27:33.538 --> 00:27:41.878 X1 and X3, they're not correlated with each other. So the form in this case will separate out. And so these are 2 separate. 162 00:27:41.878 --> 00:27:46.078 1 variable cow scans, they're independent. 163 00:27:46.078 --> 00:27:59.999 The mean is 0, and the variance this 1 now, we want to do something else say, the conditional next to give a next 1 and next 3 the definition is the 3 variable density. 164 00:27:59.999 --> 00:28:06.959 Which is here divided by the conditional up there divided by the. 165 00:28:06.959 --> 00:28:17.909 Density for X1 X2, regardless X1 X3, regardless of 2. so we divide through by the thing up here and we get that. So this is the. 166 00:28:19.499 --> 00:28:24.388 Conditional value X to give the next 1 and 3. 167 00:28:27.269 --> 00:28:31.739 And if we think about that. 168 00:28:31.739 --> 00:28:35.729 What it is. 169 00:28:38.519 --> 00:28:49.199 Sit and think what that is. So this is the density of X to given that we know 1 and 3. 170 00:28:49.199 --> 00:28:57.749 Well, it depends on 1 doesn't depend on 3 because X2 X year, independent of each other, but it does depend on 1. 171 00:28:57.749 --> 00:29:03.449 So, this is a Gaussian, but it's a house in where the mean of X to. 172 00:29:03.449 --> 00:29:07.019 Depends on X1 so. 173 00:29:08.219 --> 00:29:12.298 So, next 1 is different. The next 2 will be different also. 174 00:29:12.298 --> 00:29:16.138 And that makes sense because they're correlated with each other. So. 175 00:29:16.138 --> 00:29:19.348 And here they throw in some numbers, so. 176 00:29:19.348 --> 00:29:23.848 X, the mean value of X to is X1 over square to 2. so. 177 00:29:24.929 --> 00:29:33.598 It positively correlated. Okay. 178 00:29:35.459 --> 00:29:40.169 67 shows us some more things that can happen. 179 00:29:41.699 --> 00:29:45.358 We got 3 a cascade of 3. 180 00:29:45.358 --> 00:29:57.088 Random variables. Excellent. It's uniform. 01 X2 has to be less the next 1. this is we're defining in our problem. X2 is uniform and 0. X1. 181 00:29:57.088 --> 00:30:06.118 That's the next 1, so if we plot X1 and next 2 together, the non 0 region is a triangle. 182 00:30:07.769 --> 00:30:17.459 Now, it's 3 is uniform and 0 X2. So we put we plot the 3 of them together. It's going to be this tetrahedron. 183 00:30:20.064 --> 00:30:34.374 Okay, so now we want the joint density for X and the marginal of 3 X3 is clearly going to be clustered towards the left side. 184 00:30:36.028 --> 00:30:44.159 Because it has to be less next to which has to be less than X1, which is uniforms. X3 is more likely to be small than big. But let's. 185 00:30:46.439 --> 00:30:51.028 Quantify this? Okay so. 186 00:30:52.558 --> 00:31:00.749 It gets boring calling you mix 1 X2 X3 all the time. So, for some of the time is going to call them X. Y, and Z. 187 00:31:01.769 --> 00:31:13.019 Consistency is fun. Okay. So, here's the thing. We got our joint density here so the definition showed a few things back to the joint density. X1 X2 X3. 188 00:31:13.019 --> 00:31:17.489 It's the, it's the. 189 00:31:17.489 --> 00:31:27.358 Conditional Z, given X and Y, Z X3 times a conditional of why give an X times the conditional times the unconditional effects. Now. 190 00:31:30.929 --> 00:31:37.288 Now, we can work now, we started the right the unconditional density of X. it's just 1. 191 00:31:37.288 --> 00:31:46.259 Okay, the conditional on why given acts since why has to be less than X it's uniform in the region. 0, 2 X. 192 00:31:46.259 --> 00:31:57.388 And outside that, it's 0, so it's uniform and the regions here direct, and it has to be normalize integrates out to 1. so the conditional density of why give an X1 over X. 193 00:31:58.648 --> 00:32:05.548 The conditional density is Z given X and Y, it's uniform in a region where it's less than Y. 194 00:32:05.548 --> 00:32:08.878 And outside that 0. 195 00:32:08.878 --> 00:32:12.868 So the conditional density of Z is 1 over. Why. 196 00:32:14.219 --> 00:32:17.548 If Z is less and Y, otherwise they're all. 197 00:32:19.199 --> 00:32:26.338 So the so the unconditional density of the vector is 1 over X Y, provided. 198 00:32:26.338 --> 00:32:31.048 Axes and the regions 0 to 1 wise and their agents 0 to acts. 199 00:32:31.048 --> 00:32:37.048 And it does not depend on Z is in the region 0, to wide up here in this. 200 00:32:37.048 --> 00:32:47.189 Fruitful inequality and this is the joint density function. It does not depend on Z provided Z is from 0 to Y. 201 00:32:47.189 --> 00:32:51.719 Oh, okay. 202 00:32:54.088 --> 00:33:04.019 Now, if we want the just the density for Y and Z, we don't care what access we integrate out. X. 203 00:33:04.019 --> 00:33:13.108 And dissenter girl, why is a constant here integrating 1 over axes on natural log? 204 00:33:13.108 --> 00:33:19.469 And we're going to get that, and we want the joint on. 205 00:33:20.788 --> 00:33:28.348 X3 alone for any integrating out X2. So we integrate this thing here. 206 00:33:28.348 --> 00:33:32.999 And we get that so. 207 00:33:32.999 --> 00:33:37.739 They agree with me that spies to the left. 208 00:33:40.318 --> 00:33:48.778 Okay, we can cascade these conditionals. 209 00:33:48.778 --> 00:33:55.229 Is the next question is we're interested in independence now. 210 00:33:55.229 --> 00:33:58.769 There's a definition that will just gave you. 211 00:34:02.368 --> 00:34:07.259 The sector is. 212 00:34:07.259 --> 00:34:11.548 The sector random variables is independent. 213 00:34:11.548 --> 00:34:21.659 So probably is a component at this joint probability here that each component is in a certain region as the product of the problem of these separate probabilities. 214 00:34:21.659 --> 00:34:35.398 We had the same thing with 2 factors. Now that's equivalent to saying that the cumulative thing can be decomposed like that. 215 00:34:35.398 --> 00:34:48.509 And if it's discreet, it's equivalent of saying the math function can be composed or if it's continuous, a density function can be decomposed. So, this is a definition for. 216 00:34:48.509 --> 00:34:53.398 The components of the vector, random variables, independent of each other. 217 00:34:58.679 --> 00:35:07.679 An example here we have an infected, and for some noise means 0 standard deviation 1. 218 00:35:07.679 --> 00:35:12.148 This would be the, this is a formula for the joint. 219 00:35:12.148 --> 00:35:15.628 Garcia, and and there's no correlation. 220 00:35:15.628 --> 00:35:25.858 So, you can take this thing here, you can split it apart and do end separate pieces to the end components. So this, in this case, there is. 221 00:35:25.858 --> 00:35:35.699 They're independent, if in the exponent up here, there were cross components of X1 times X2 and so on. Then they would not be independent anymore. 222 00:35:38.219 --> 00:35:44.699 Okay, so our vector of random variables, we want to do things with it. 223 00:35:44.699 --> 00:35:51.898 We want to compute functions of it, like the me and then and sample variance for something. 224 00:35:54.148 --> 00:36:02.009 Okay, if it's 1 function. 225 00:36:02.009 --> 00:36:10.498 G of the random variable then you just integrate out. This is like, what's a pair thing to get the. 226 00:36:11.728 --> 00:36:22.108 Say the cumulative for Z or something it's a little abstract. Let's get a specific example here. 227 00:36:22.108 --> 00:36:27.568 Min, and Max, we've got our end. 228 00:36:27.568 --> 00:36:32.699 Random variables. They're independent, they're the same distribution. 229 00:36:32.699 --> 00:36:37.469 Doesn't have to be calcium could be anything what we're saying. They're independent at the moment. 230 00:36:37.469 --> 00:36:47.099 I got 2 functions here. Dolby will be the max of the N, and Z will be the men of them. So we've got 2 functions of our vector random variable. 231 00:36:47.099 --> 00:36:52.679 And what can we know about W, and Z? What is their distribution. 232 00:36:55.619 --> 00:37:03.298 Okay, now we can guess that is going to be caution to the right and is going to be closer to the left. 233 00:37:03.298 --> 00:37:08.998 But, you know, we can formalize at a touch. So W, is the max. 234 00:37:08.998 --> 00:37:14.728 Let's look at the the CDF a cumulative on so. 235 00:37:14.728 --> 00:37:22.768 Down here so big f Dolby you, that's a probability at the random variables lesson given value. W, well. 236 00:37:22.768 --> 00:37:37.228 This is, this is equal since W, so, Max, if the random variable big Dolby is less and little Dolby, you are equal, then each of the end components has to be less than or equal right up here. So the max. 237 00:37:37.228 --> 00:37:40.409 Of all of the buy has to be less painful to you. 238 00:37:40.409 --> 00:37:48.358 Well, since I said that the X's were independent, the same factors and this comes out to the probability. 239 00:37:48.358 --> 00:37:52.438 Then each of the ex survive that single job you all multiplied together. 240 00:37:53.938 --> 00:37:58.528 Well, each of these pieces is the cumulative on X. 241 00:37:58.528 --> 00:38:05.909 So, this thing here, this is a cumulative on where the argument is stop you to the end and this is the. 242 00:38:08.278 --> 00:38:20.309 This is the density. This is the CDF on the Max for the men. I'm not going to. You just subtract 1 calculated complemented calculate and complement again. 243 00:38:21.360 --> 00:38:27.329 Now, I do an example. 244 00:38:27.329 --> 00:38:31.559 The problem is, my iPad was not connecting again. 245 00:38:31.559 --> 00:38:36.659 Before I give it 1 more, try to see. 246 00:38:36.659 --> 00:38:39.900 Let's see what we do here. 247 00:38:49.530 --> 00:39:02.190 Okay, it's still not connecting. 248 00:39:07.530 --> 00:39:11.429 Oh, okay. Otherwise I would have plugged into a specific thing here. 249 00:39:26.909 --> 00:39:30.960 Yeah, I'll keep walking through the books and I don't know what it is. 250 00:39:34.980 --> 00:39:46.199 Another example, this nice Singley on Garcia, we have examples, which are. 251 00:39:46.199 --> 00:39:50.579 To reality more often than not. 252 00:39:56.219 --> 00:40:07.650 Let me see if this yeah, it's not working. Okay. 253 00:40:15.119 --> 00:40:23.670 So, we've got a server and it's it's a web server, let's say, and Scott requests is and possible clients. 254 00:40:23.670 --> 00:40:27.989 Could generate request for the web page. 255 00:40:27.989 --> 00:40:39.269 And they're different actually, some of the clients are request a lot of pages and some requests only a few. 256 00:40:39.269 --> 00:40:48.869 So, the Jace client, the Jace possible source is generating service requests that are exposed to distributed. 257 00:40:48.869 --> 00:40:52.199 This is in her arrival time of. 258 00:40:52.199 --> 00:40:57.449 So, your memory exponential again is a compliment of the time so. 259 00:40:57.449 --> 00:41:09.150 So the number of requests in a given time, interval would be a plus on the time. The arrival time between consecutive requests is exponential. 260 00:41:10.289 --> 00:41:19.409 And if you can figure out, if you forget the formula for exponential, you Haitian might have your little cheat sheet, which has the page number that big table on it. 261 00:41:19.409 --> 00:41:31.440 Okay, so we want to know the arrival times between consecutive requests. And the reason is, is if the arrival time is smaller. 262 00:41:31.440 --> 00:41:36.539 Then the time to process the request, and we're going to get a backlog. 263 00:41:36.539 --> 00:41:46.980 Okay, well, so again, so we got the server here. There's N, possibles clients sources. 264 00:41:46.980 --> 00:41:51.420 Each of them is a different inter arrival time. 265 00:41:51.420 --> 00:41:58.829 Which is an exponential random variable and so the server starts getting backlogged. 266 00:41:58.829 --> 00:42:04.710 If any of these possible sources makes a request. 267 00:42:04.710 --> 00:42:09.869 Okay and so we're interested. 268 00:42:09.869 --> 00:42:14.969 In the random variable, which is the random variable to the next. 269 00:42:14.969 --> 00:42:19.980 Request from any of the sources. 270 00:42:19.980 --> 00:42:24.750 So, we're going to sit and the, and the minimum. 271 00:42:24.750 --> 00:42:32.489 Time over the end sources. So I knew random variables. Z is the minimum of the times to each of the end sources. 272 00:42:32.489 --> 00:42:37.530 Okay, and. 273 00:42:38.909 --> 00:42:44.699 So, I did the Max before, just so the minimum 2 compliments. 274 00:42:44.699 --> 00:42:50.159 And we're assuming things are not correlated size, waving my hands a little. 275 00:42:50.159 --> 00:42:53.789 We're getting the product of the. 276 00:42:53.789 --> 00:42:59.250 1, minus CBS and for exponential than this, and the. 277 00:42:59.250 --> 00:43:05.130 Cdf is this minus? I'm going to see ballpark together. We get that. 278 00:43:05.130 --> 00:43:12.630 And so, now again, this is random variable for the time until the 1st request from anyone arrive. 279 00:43:12.630 --> 00:43:16.710 And that's that turns out to be exponential. 280 00:43:16.710 --> 00:43:21.329 With the parameters, the some of the lamb just over the separate sources so. 281 00:43:23.340 --> 00:43:33.030 So, if the sources rate lamb to I, then all the sources together exponential with the rate of the sum of all of the rates. 282 00:43:35.130 --> 00:43:41.340 That was an example with men 611 is an example with Max. 283 00:43:41.340 --> 00:43:46.199 We got. 284 00:43:46.199 --> 00:43:49.469 And redundant systems in our cluster. 285 00:43:51.090 --> 00:44:00.150 And as well, and each of the subsystems, its lifetime is exponential. 286 00:44:00.150 --> 00:44:04.170 With parameter lamb, but they're all the same this time. Make your life easier. 287 00:44:04.170 --> 00:44:09.150 And again, so it was this excellent of distribution, just to remind you it's memory list. 288 00:44:09.150 --> 00:44:16.380 So, the probability of it failing does not depend on how old it is. So maybe it fails if a cosmic re, it. 289 00:44:16.380 --> 00:44:27.480 Okay, so now we've got these end subsystems each as parameter Lambda the whole cluster. It will be running as long as at least 1. 290 00:44:27.480 --> 00:44:34.920 Of these subsystems is running, so and we want to. 291 00:44:34.920 --> 00:44:41.699 Attach a formula to that so, each of the sub systems that. 292 00:44:41.699 --> 00:44:49.170 It's lifetimes around invariable, exponential parameter. We want to know what is the lifetime as a whole cluster. 293 00:44:49.170 --> 00:44:53.489 And that's going to be the max of the lifetime to the end components. 294 00:44:54.809 --> 00:45:05.429 He said components right? Exponential random variable. The cluster is random variables. The max of those things it's a function of random variables. It's a function of a random vector. Okay. 295 00:45:05.429 --> 00:45:12.179 Vector is all the access. So what is this thing here for you? Well, we did it 2 pages ago. 296 00:45:12.179 --> 00:45:19.500 And this will be the density, the cumulative distribution function for the max. 297 00:45:19.500 --> 00:45:29.639 And that's here, this does not have such a simple expression. You can expand it out and so on things not always simplify. 298 00:45:32.730 --> 00:45:35.760 Transformations. 299 00:45:37.829 --> 00:45:46.860 We've seen getting 1 function, you could get end functions of the N, random variables and you could do joints and stuff like that. So. 300 00:45:51.750 --> 00:46:03.090 An example here where we've got a random vector X and we've got end linear transformations of it. 301 00:46:03.090 --> 00:46:06.329 And each of them is a. 302 00:46:06.329 --> 00:46:10.349 X plus B. okay. 303 00:46:11.519 --> 00:46:26.489 And well, sorry, each of them, it's a plus B so what we've done is we've transformed 1 random vector X to a 2nd, random variable Z. 304 00:46:26.489 --> 00:46:30.630 And we would like to know, what can we say about Z. 305 00:46:33.239 --> 00:46:37.889 Andrew to do, and just a summary as we have it down here. 306 00:46:39.030 --> 00:46:45.659 I'll skip over, you can work out the detail. I'll give you what the meaning of the summary is. 307 00:46:45.659 --> 00:46:49.440 Well, the density of. 308 00:46:49.440 --> 00:46:53.940 Disease it's density for access where you transform all the. 309 00:46:53.940 --> 00:47:02.429 These 2 axes here is, and at the front of it is a scale thing to make. Excuse me. 310 00:47:02.429 --> 00:47:08.400 Yeah, I didn't work on this lecture is the goal is to make this thing integrate out to 1. 311 00:47:08.400 --> 00:47:19.260 So, what we're seeing here in chapter 6, are some vector random variable, these components around new variable and to do some stuff with the min and Max and transform. 312 00:47:19.260 --> 00:47:26.219 I'll skip the start thing. 313 00:47:28.920 --> 00:47:36.750 Do some fun how far didn't start thing. 314 00:47:41.309 --> 00:47:47.010 3rd time I'd come back to it, but we're running at a time in this semester. So. 315 00:47:49.409 --> 00:47:58.650 Students Steve is just a fun thing. I mentioned the story before we're going back to 1900 timeframe again. Us brewery in Ireland. 316 00:48:00.954 --> 00:48:10.284 Different batches of beer, different alcohol concentrations and so they would sample them and measure them and sampling would take time and cost money. 317 00:48:10.284 --> 00:48:19.795 So they hired a mathematician called Boston to try and calculate how much testing what they have to do as a beer to. So they would know what the. 318 00:48:20.099 --> 00:48:26.820 Which gives us, atch was and he developed something things called students to distribution and so on. 319 00:48:26.820 --> 00:48:37.590 He may he probably send her a pseudonym dentist didn't want him to publish under his name because this is a trade secret. Maybe shouldn't publish at all. I don't know how to use his real name. 320 00:48:37.590 --> 00:48:46.769 And students, I'm getting ahead of statistics is a way to tell of 2 different batches have really, really have the same mean. 321 00:48:48.389 --> 00:48:54.809 So always my hands and skip over that. 322 00:48:58.590 --> 00:49:02.400 Some, I may skip over for now and. 323 00:49:04.409 --> 00:49:14.880 Accepted values, so you want to know what the mean of the mean of a function of the random variables are. 324 00:49:14.880 --> 00:49:19.289 So, he got the vector actually got a function g of X. 325 00:49:19.289 --> 00:49:28.679 And we want to know what the expected value of g access and we just integrate out here, integrate all alibi. So g. 326 00:49:28.679 --> 00:49:37.710 Kinds of density if it's continuous or some over the point probability it's discrete. Nothing really new there. 625. so. 327 00:49:39.389 --> 00:49:43.739 1 example, for g would be the sum of the components. 328 00:49:43.739 --> 00:49:49.050 And it turns out the expected value for the summit is to some of the expected guys. We saw that before. 329 00:49:50.369 --> 00:49:53.670 And the product works, only if you're independent. 330 00:49:53.670 --> 00:50:02.670 So, in any case, we can now start thinking about how the components. 331 00:50:02.670 --> 00:50:06.150 Of the vector are. 332 00:50:06.150 --> 00:50:11.219 Restrict our controlled by each other, we're going to get into his correlations. 333 00:50:11.219 --> 00:50:17.130 Work her way up to it. Well, you can add the vector mean just called. 334 00:50:17.130 --> 00:50:25.079 Full pay Sam's full face, exercise, already expected value and that's the vector of the expected value of the separate components. 335 00:50:26.519 --> 00:50:29.820 Whether or not, they're correlated that doesn't matter here. 336 00:50:33.329 --> 00:50:43.590 Definition of okay. 337 00:50:45.659 --> 00:50:52.500 Let's drop something. Okay you can have a correlation matrix. 338 00:50:52.500 --> 00:50:55.769 And it captures how. 339 00:50:55.769 --> 00:51:05.309 It's expected value of well, 1st, the X squared, and also for any care, except 2 times X3 and so on. So. 340 00:51:08.130 --> 00:51:16.349 So, the correlation matrix, which is different from the correlation coefficient, the terminologies annoying. 341 00:51:16.349 --> 00:51:20.309 It's a matrix of all these 2nd order expected values here. 342 00:51:21.864 --> 00:51:34.945 And then you can, you can work that and the next thing is a variance matrix where we centralize to the random variables. We take the expected value, not as X by but of X or by minus. 343 00:51:36.505 --> 00:51:36.925 So. 344 00:51:37.230 --> 00:51:46.679 It's the correlations about the mean of each 1 you might say this is a CO variance Matrix, which we and Garcia calls big K. 345 00:51:46.679 --> 00:51:57.960 Now, here's the thing about this 1, is that if X X1, and next up to are independent of each other than this joint expectations, going to be 0. 346 00:52:00.030 --> 00:52:06.210 Any case we look at K, diagonal component is, are the variances. 347 00:52:06.210 --> 00:52:10.199 Take the square root, you get the standard deviation. So. 348 00:52:11.760 --> 00:52:17.250 And the off diagonal elements are how much they're different pairs. 349 00:52:17.250 --> 00:52:25.590 Of the components are correlated if a particular pair of the X buyer not correlated at, but it will be 0. 350 00:52:25.590 --> 00:52:33.150 If they're strongly correlated, it will be a positive number. If they're strongly inversely. 351 00:52:33.150 --> 00:52:42.840 Correlated it will be a negative number, so, and they do different examples of it here. So. 352 00:52:45.389 --> 00:52:56.159 What this particular case? 66 was it was this 3 variable Gaussian where this is a term up here. We're X1 and next 2 had this. 353 00:52:56.159 --> 00:53:02.909 Negative correlation, because this is minus X1 X2 pair and the exponents here. 354 00:53:02.909 --> 00:53:06.570 But X3 was independent of X1 and next 2. 355 00:53:06.570 --> 00:53:09.929 So this was a density up here. 356 00:53:12.690 --> 00:53:18.869 And now you could work through and find the K and say, find expected values. 357 00:53:20.730 --> 00:53:27.000 And so you can they actually calculate what's a correlation coefficient of X1 and next 2 is. 358 00:53:27.000 --> 00:53:30.960 It's negative. I know we can find the. 359 00:53:30.960 --> 00:53:39.329 Correlation, what do they call it up here? The variance Matrix has the correlation coefficient in it and. 360 00:53:39.329 --> 00:53:47.400 Variances so actually, this has a CO variances in variances and so. 361 00:53:47.400 --> 00:53:58.739 And we'll get something like that. So next 1, next year independent, next to next year, independence far, we've got zeros and X1 X2 are negatively correlated to the negative. 362 00:54:02.159 --> 00:54:07.349 And they got more compact expressions for this here. 363 00:54:07.349 --> 00:54:20.909 Okay, now you've got linear systems where you got a random vector modified by him and by end Matrix, say to give a random vector, why. 364 00:54:20.909 --> 00:54:33.239 And now what they're computing here is how the expected value gets transformed, you take the vector of means for X and multiplied by aid. You get some extra means for why. 365 00:54:33.239 --> 00:54:40.170 Occasionally things are simple called maryanne's matrix. 366 00:54:42.719 --> 00:54:47.219 And put an a transpose around it so on. 367 00:54:52.050 --> 00:54:59.039 Okay, if it's uncorrelated things a little easier. So. 368 00:55:00.690 --> 00:55:06.840 What they're talking about here is a touch. Interesting. 369 00:55:06.840 --> 00:55:11.429 The suppose paragraph. 370 00:55:11.429 --> 00:55:18.750 Okay, what's happening here you go to random vector, but the components are correlated. 371 00:55:18.750 --> 00:55:26.550 What you would like to do is model pipe by a matrix. So the new random vectors components are not. 372 00:55:26.550 --> 00:55:33.960 Correlated give an example, say, color television. 373 00:55:33.960 --> 00:55:37.619 Yeah, it's a 3 vector red green, blue. 374 00:55:37.619 --> 00:55:43.079 Now, in practice with natural scenes, the red green and blue. 375 00:55:43.079 --> 00:55:51.719 Component they're correlated that you have a bright pixel for R. G. and B are all 3 high dark pixels are all 3 low. 376 00:55:51.719 --> 00:55:56.789 The saturated pixels will red as high and green and blue are low. 377 00:55:56.789 --> 00:56:02.670 Yeah, they happen, but they're not as common. What's more common with the color. 378 00:56:02.670 --> 00:56:09.750 Image is, it's very bright, or it's very dark. It's less common that it'd be very saturated. 379 00:56:09.750 --> 00:56:23.429 Okay, now well, we want to do an RGB. It's a vector. What we want to do is rotate it to a different 3 dimensional vector space or the new rotated. 380 00:56:24.510 --> 00:56:28.289 Back there, the components are not correlated. 381 00:56:28.289 --> 00:56:34.829 And the reason we want to do this is we're going to compress that. 382 00:56:34.829 --> 00:56:37.949 Comparison or transmit. 383 00:56:38.574 --> 00:56:52.914 And it's easier to work with this components. If they're not correlated with each other, we can compress them separately for example and we'll get a nice compression technique or we transmit them separately. Well, it all transmitted together. 384 00:56:52.914 --> 00:57:05.155 But using a technique, which transmits each voltage separately. Now, because it turns out and works nicer with the compression or transmission, if the 3 components are not correlated. 385 00:57:05.460 --> 00:57:14.250 And in fact, this is what they did in the history of color television, it took the RGB, and they rotated that. 386 00:57:14.250 --> 00:57:25.500 And 1 component was called Y, and it was for brightness, total brightness. It turned out to be point 6, red plus point 6 screen plus point 3, red plus point 1 blue. 387 00:57:25.500 --> 00:57:30.449 And so the 1st of all was told brightness the next was. 388 00:57:30.449 --> 00:57:36.210 Red minus green, and the 3rd was blue, minus green and the 3 that were not correlated with each other. 389 00:57:36.210 --> 00:57:44.280 Very much then they compressed and transmitted the 3 separately. Okay, so that's the motivation for this. 390 00:57:45.929 --> 00:57:50.579 And so we're going to talk about here is if we have the. 391 00:57:51.659 --> 00:58:03.900 Pairwise correlations Co, variances whatever. Then it helps us figure what should a, the matrix, a B that we're going to multiply X by. So the results are not correlated. 392 00:58:07.889 --> 00:58:17.400 And they talk about it there. I skipped the, I'm skipping characteristic function just because I've got to skip some things too much in the book. 393 00:58:18.989 --> 00:58:27.510 Characteristic functions. I'm ignoring anything that's start. 394 00:58:27.510 --> 00:58:31.739 Probability ignore it. Okay. 395 00:58:31.739 --> 00:58:35.460 Jointly Gaussian. 396 00:58:35.460 --> 00:58:40.349 Okay. 397 00:58:40.349 --> 00:58:54.690 Okay, random Victor X, the excise Gaussian, but they're correlated as correlations and so on. So this is actually the definition for what we're going to call the joint ghost in this case here. So. 398 00:58:54.690 --> 00:59:02.940 The exponential and we got to make this case and then we subtract out and so on. 399 00:59:05.820 --> 00:59:15.239 And cases Co, variance Matrix, the diagonal is the variance and the, the off tag and also the variance. 400 00:59:17.369 --> 00:59:30.750 And so we're going to have here a roll vector X, minus transposed kind of the inverse of the variance matrix column vector X and. 401 00:59:30.750 --> 00:59:38.010 So this nets out to a scaler here at 61, and then we divide here by described as a determinant. 402 00:59:38.010 --> 00:59:45.059 Of the coherence that just to scale the thing out. Oh, okay. That's how we can do. And. 403 00:59:49.440 --> 00:59:56.219 Okay, I just have an example here. We might perhaps work out next time. 404 00:59:59.849 --> 01:00:11.519 And just given all our definitions. It's very if have certain pairwise variances in this joint calcium is 0, then those 2 components are also independent. 405 01:00:11.519 --> 01:00:17.670 So, that's how they show that down here. So, and surprise conditionals. 406 01:00:17.670 --> 01:00:21.750 Up here. Oops. 407 01:00:24.389 --> 01:00:29.400 So, the end components is to join calcium and. 408 01:00:29.400 --> 01:00:35.340 Now, we want the conditional, the ends 1, given the X1 2 X and minus 1. 409 01:00:36.750 --> 01:00:39.989 Away from my hands, a little skip some details. 410 01:00:39.989 --> 01:00:43.800 But we can. 411 01:00:43.800 --> 01:00:50.280 Mark it out here, I think Thursday I might do lots of examples here. I'm giving you executive summary. So. 412 01:00:52.530 --> 01:00:56.880 Sorry to get a little messy ignore stars. 413 01:01:03.210 --> 01:01:07.889 Estimation okay. Um. 414 01:01:09.059 --> 01:01:16.860 What is happening here? Think of our communications channel. 415 01:01:17.335 --> 01:01:26.034 Signal sources acts we add our noise and X. plus N is why that's our received signal. We don't know. X. 416 01:01:26.034 --> 01:01:35.335 we do know why we want to know is what is the probability of the different values of X given the why that we saw. 417 01:01:35.639 --> 01:01:40.170 And, in fact, we want to maximize something. 418 01:01:41.429 --> 01:01:47.099 And that is down here. So again, if you can see what I. 419 01:01:47.099 --> 01:01:56.039 Circling sir, so we want so you had the probabilities of any given transmitted signal given. 420 01:01:56.039 --> 01:02:01.349 The receipts signal, and we want the transmitted signal, which has the. 421 01:02:01.349 --> 01:02:06.119 Rage is probability given this received signal. 422 01:02:06.119 --> 01:02:16.320 We want the value of X so what this is down here what we want is the value of acts that maximize. 423 01:02:16.320 --> 01:02:27.420 The probability of that value of X, given why? And that's called a maximum a Estimator and may be. 424 01:02:27.420 --> 01:02:30.690 A Latin, and after. 425 01:02:31.889 --> 01:02:46.679 Well, so, this conditional probability here I've given, why is we bring in base again and to probably do this why give an extensive probably of X divided by the probability of why now to find the denominator then we got to find. 426 01:02:46.679 --> 01:02:50.550 Some all the different cases and so on but okay. 427 01:02:51.630 --> 01:03:01.199 So, there's some way effort in tying that denominator probably to why give an extra part of the definition of the problem. Probably X. the denominator takes a touch of work. 428 01:03:01.199 --> 01:03:10.199 Okay, in any case we take this formula here and we want to find the value of X that maximizes that. 429 01:03:10.199 --> 01:03:18.630 Okay, so now. 430 01:03:18.630 --> 01:03:25.949 There's a problem with this method, this map estimator. 431 01:03:27.389 --> 01:03:35.550 And the problem is this that it requires that we know the preorder probabilities of X. 432 01:03:38.155 --> 01:03:52.164 Totally sensible thing we've seen this before some examples that if the noise is very small, then then why is going to be close to X. okay. So the most probable value of access is going to be pretty close to the most probable value. Why? 433 01:03:52.164 --> 01:03:53.965 If if the noise is small. 434 01:03:54.269 --> 01:04:00.630 If the noise is large, then it's going to swamp the ax a lot. 435 01:04:00.630 --> 01:04:10.590 And whenever X Y, says how to affect so much, the probability is a lie and the case. But the thing is doing this optimally requires. So, you know, the, a priority for X but. 436 01:04:10.590 --> 01:04:14.250 Maybe, we don't know. 437 01:04:14.250 --> 01:04:17.489 You don't always know everything we want to know. 438 01:04:17.489 --> 01:04:20.550 So, maybe we. 439 01:04:20.550 --> 01:04:25.079 Know, what the noise formula is, and we know. 440 01:04:25.079 --> 01:04:33.750 When we, when we see what we're seeing with the why but we don't know the for the access, but we'd still like to do something. 441 01:04:33.750 --> 01:04:41.789 You can't do as much as you did before, but do you do something maybe and this will be another type of Estimator here. So. 442 01:04:43.320 --> 01:04:46.980 What this other type down here is it. 443 01:04:48.300 --> 01:04:51.960 In a sense assumes that all the axes are equally probable. 444 01:04:53.094 --> 01:05:06.054 A little sloppy, but it's not far off if we know the probabilities of acts and we put them into our formula to get the best. Guess at ex, given why? 445 01:05:06.414 --> 01:05:11.905 If we don't know the probabilities for accidents to say they're equal or something and. 446 01:05:12.480 --> 01:05:15.690 You know, maybe we'll get something useful out of it. 447 01:05:15.690 --> 01:05:29.639 That's what you do so why we spend time on it. So that's this bottom thing on the page here. This is what it nets out to. So we look at each value of X. we get the probability of seeing that. Why. 448 01:05:31.380 --> 01:05:38.519 And now what we do is we find the X which gives the highest value thing that why. 449 01:05:38.519 --> 01:05:43.530 So, we're maximizing this. 450 01:05:43.530 --> 01:05:48.539 Probability of why give an extra find the value of X and maximizes that. 451 01:05:48.539 --> 01:05:53.550 Yeah, it's sort of like the previous 1, assuming all the parties for acts of the same. 452 01:05:55.170 --> 01:06:01.380 Okay, and so this is a different type of Estimator where we don't know the price for. 453 01:06:01.380 --> 01:06:05.789 And that's got a different name maximum likelihood estimator. So. 454 01:06:05.789 --> 01:06:14.190 Okay, and. 455 01:06:15.449 --> 01:06:22.110 This nets out to the is we want to maximize the density. 456 01:06:22.110 --> 01:06:25.769 Value of access, maximize the density effects forgiven X. 457 01:06:25.769 --> 01:06:34.349 We're actually gonna do some given via given why and that's where we know the priors on. 458 01:06:34.349 --> 01:06:41.489 And if we don't, we'll have the other, they're complementary things. 459 01:06:41.489 --> 01:06:45.269 Of why give an X? The other 1 is X and Y, so. 460 01:06:46.469 --> 01:06:55.260 There's the map Estimator at the and the estimator, the maximum like. 461 01:06:56.670 --> 01:06:59.789 We're going to give different answers, but. 462 01:07:01.380 --> 01:07:08.280 Sometimes or we're seeing here are some examples. 463 01:07:11.699 --> 01:07:15.389 And I'll look at joint Kelsey and might say. 464 01:07:17.519 --> 01:07:22.590 This is a horrible mess. 465 01:07:26.550 --> 01:07:30.150 All the derivation looks at the answer here. 466 01:07:33.690 --> 01:07:43.260 Okay, so X and Y are joint and they have a correlation coefficient row and each have their standard deviation segment. 467 01:07:43.260 --> 01:07:47.340 And there means am and. 468 01:07:47.340 --> 01:07:51.510 This is a conditional density on ex given why. 469 01:07:51.510 --> 01:07:56.340 And the maximum is this here. 470 01:08:00.030 --> 01:08:09.269 We just look at them and what this thing means. Well, you so you got the EMS you correct for the means not being 0 So you can ignore that. 471 01:08:11.159 --> 01:08:23.369 The segments are correcting for X and Y, having different standard deviations. Okay. So that means were 0 and the segments were 1. 472 01:08:23.369 --> 01:08:26.489 The next map would be row times. Y. 473 01:08:29.460 --> 01:08:32.789 What this means is that. 474 01:08:34.560 --> 01:08:46.770 Effects, and why track each other quite closely, then the best guess for axis about why? But if they don't track each other very well. 475 01:08:46.770 --> 01:08:53.640 Correlation is low and just because why is make doesn't mean it should be big. 476 01:08:54.324 --> 01:09:08.274 And that's why we got road times. Why there? So, let's suppose ROAS point 01, then the map for access point. 01 time is why so, what this is saying is if X and Y are not well correlated, then just because why is big does not mean that should be very big. 477 01:09:09.750 --> 01:09:14.520 It's going to drag ex along a little, but not very much. That's what's happening there. 478 01:09:18.300 --> 01:09:26.670 The maximum likelihood estimate, or where we don't know the prior on, that turns out to be a different formula. 479 01:09:30.149 --> 01:09:34.859 And, um. 480 01:09:34.859 --> 01:09:48.060 And we get here, or we don't know anything about the prior and so here, the value of access maximizes the likelihood thing. It's a different formula in here. 481 01:09:50.819 --> 01:09:56.489 Okay, and. 482 01:09:56.489 --> 01:10:03.390 Now, we're going to start getting fancier and so on hit that you tell her and just giving me highlights. 483 01:10:04.590 --> 01:10:11.189 Other types of estimators estimators are important. Okay because we have the noisy. 484 01:10:11.189 --> 01:10:17.520 Channel we see what we see, we want a good Estimator on what was so. 485 01:10:17.520 --> 01:10:22.800 And they're comparing different types of estimators that are oh, okay. 486 01:10:22.800 --> 01:10:26.310 It's a good point to stop. I think. 487 01:10:27.630 --> 01:10:41.250 And, yeah, some more estimators and so on happening, we'll stop, give or take around here. So, let me review just a number of new things today. 488 01:10:41.250 --> 01:10:46.439 What we're doing today was just I get back to the start here. 489 01:10:46.439 --> 01:10:52.439 We're doing this is chapter 6 with vectors getting work. It's 6. 490 01:10:56.159 --> 01:11:03.930 Okay, big chapter. Okay. 491 01:11:06.569 --> 01:11:14.789 So, talking about vector random variables so we have end random variables. They form a vector and then. 492 01:11:14.789 --> 01:11:24.329 Lots of examples here and talked about the joint cable to function for the vector. It's just probability that the points. 493 01:11:24.329 --> 01:11:39.175 Below and left the given value and then you could mass function to discrete density functions. It continues. We could start having finding probabilities here by changing together. Conditional things, get integrate things out. 494 01:11:39.175 --> 01:11:40.225 Good marginals. 495 01:11:40.529 --> 01:11:51.390 Not some nice examples here. The independence, the components is a vector independent if he's probabilities can separate out. 496 01:11:51.390 --> 01:11:58.140 Functions of random variables we saw like min and Max are some nice examples. 497 01:11:58.140 --> 01:12:08.579 And then we started to start section and we started looking at. 498 01:12:08.579 --> 01:12:20.430 Some functions of random variables, woman and maxes and functions and so transformation thing, then we start looking at expected values and. 499 01:12:20.430 --> 01:12:28.770 Correlations pairwise correlations between the components of the vector, given the things involving may receive. 500 01:12:28.770 --> 01:12:38.579 And some translating your transformations difficult thing is you transform it because you want to correlate the components. 501 01:12:38.579 --> 01:12:46.979 And again, okay, so that's. 502 01:12:46.979 --> 01:13:00.390 A quick summary of the non start section. So, calcium is your big example big type of random variable to be vector with components calcium, which might, or might not be pairwise correlated. 503 01:13:00.390 --> 01:13:11.039 So, Thursday, we'll continue on maybe do some, bring it down to earth do some examples. If I can get it working, I might even show you some examples and Mathematica. 504 01:13:11.039 --> 01:13:16.680 And so have fun and I'll see you. 505 01:13:16.680 --> 01:13:18.720 Thursday.