WEBVTT

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Hi.

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Okay, good.

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Can you see my screen Thank you? Yeah, I had to log out and then restart it.

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No, no screen yet.

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Oh, my name's I'm sharing my screen.

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Eva.

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Stop.

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Start now.

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Looks like, I should be sharing it now.

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Oh, um.

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Starting.

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It's so crazy. Yeah, I can.

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Hello.

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Okay, I stopped and restarted sharing after said logging notes and logging in again.

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So now, okay.

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Okay, if you can still.

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See things then.

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Yeah, for now, you're right what I did is I stopped sharing and I restarted sharing again, so.

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Similar to hate computers after a while. Okay. We're back to example. 536. okay. What's happening here? The part of the example to show this conditional expectation we got our chip. It's getting random defects. It's Paul saw.

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Random variable, and there happened to be K defects on the whole chip.

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Okay, cosmic race now we're interested in a little piece on the chip. For example, there may be several processors on the 1 chip and 1 process, or may be bad.

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But any particular processor is good if it has say.

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So many defects. Okay so the primary random variable X, the total number defects on the chip, but this 1 region of the chip.

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If the total chip is Kay.

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Defects this 1 region, the number of defects on it is a separate, random, variable, new, random variable. Why? So defects on say this corner this quarter of the chip. That's why X's defects on the total trip. Okay. Now.

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The random variable, why for defects on this quarter of the chip it's conditional on how many defects there are all together.

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Reasonable thing, if it's a quarter of the chip, it would have a quarter as many effects on the average and it would be a separate croissant would be a reasonable thing, or you could actually use a binomial. Well, we'll talk soon with some facade. Okay. So, what we have here now.

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This left estimation thing probability is the total number defects is K, then we've got the conditional probability that the defects in this quarter. That's why given that the total is K.

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so we have the expected value here for the number of defects in this quarter.

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Given the total is K, How's your? And then we, some over K and that gives us the expected value.

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For the number of defects in this quarter.

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In general, regardless of the of the number of total number detects, we've averaged, we've set out to the equation now. Okay so that's expected value of this.

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You know, 2nd, random variable. Why? So that's what's going on here and then just the right half of the line here.

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Works it's true so the total number of defects that exits croissants. So its mean is alpha that's the parameter. And so alpha here and the expected number of defects.

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In the quarter given that the total is K, then that's going to be.

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A probability of a defect.

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Being in.

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Being in that quarter, that would be P, if there's a defect probably to the quarter, that speed and then we're doing an expectation. So we multiply by K and we saw it out. And in this thing here.

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He's a constant you pull it out and the sum of all the K of X equals. K. well, that's the mean for the plus side. So this is an example here for the expected value defects in this quarter.

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Of the whole ship. Okay the binary communications channel.

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Yeah, we've worked through that into arrival.

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Talk a little about that before. Now. What's new here?

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Is that okay?

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So, number of customer arrivals in a particular time was croissant. It's the customers who are independent of each other. Okay. So, now just working through some of that there.

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And if it's on then with parameter beta T. it'd be beta equals 1. then. The thing was on. If you double the time interval, the expected number of hits.

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Cosmic raise whatever customers doubled salt. So so if you double the interval, then the number of customers contact, raise, whatever, that larger interval at the scales upgrade energy with t that makes sense and forth on parameters.

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Here that beta for people's want to just total beta team. And the next line here, this is from dismiss compete in a long time ago for the parts on. Okay, well, the parameters and the.

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The variance for is.

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Day to day also and so.

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The severity and such a square of the deviation off the mean. So the expected value for and squared. They basically just.

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I'll set it again, so.

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Effective value and squared minus beta T would be beta teen and just add it back in.

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I'm leaving out some details. Okay. And then.

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So this expected number of customers for a given case, then we integrate T out here.

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Okay, functions are 2 random variables.

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Okay, what's an application of here?

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Maybe motivate why this is useful.

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Let's talk video transmission, let's say.

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You may think of a color signal is red, green, blue.

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3 dimensional vector and maybe somewhat random.

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You know, you look at different pixels in your image, or in your movie, but it's actually.

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They don't actually use the transmit red, green and blue separately. What's done? Inside? The video processor is thinking of RGB as a 3 vector. They rotate it and scale it to a new basis. And 1 basis is total brightness of the scene. Call.

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That why let's say for why access another basis would be the.

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Perhaps see,

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and the total brightness green is the strongest component of it,

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another thing might be component might be sort of the red how red the scene is and that'd be approximately red minus green 3rd component would be how blue the scene is that'd be approximately minus screen.

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Everything's from scale to be normal and so okay, so.

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The input is a red green blue signal, that's to say what the sensor would send, but this thing gets rotated and scaled to a new 3 factor. And this is a 3 vector now, which is compressed and transmitted. So we'd want to work with properties of this.

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So this is a function.

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This is a function of the random variables, initial random for a random pixel and the same but we'd actually were processing this function as a figure out and unbearable.

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And the reason we do that, if things work out better, if we do, that does correlate some of the noise and so on. Okay. So this is a motivation for why we would want to have a function of 2, random variables. Perhaps.

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Or an example, got random points in the plane. It may be X. Y, and for some reason you want to.

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Convert to raw data.

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Well, here's the, you know, maybe you're tossing a dart and row captures directly. How accurately the dart hit and data captures which direction? It's often initial random variables. You've got a function of them and that's.

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What's that? That's what we're talking about here in this section. So, you know, it's a reasonable thing to do.

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In any case, you got your input random variables X and Y, and you find some function Z or maybe even several things. Okay.

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Now, convenient way to work with this is work with the.

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Cumulative distribution function, so you got the probability that.

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The random variable, especially in particular value, lower Casey CDF on Z and so you'd integrate over X and Y, and it would be integral over.

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The regions for which.

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G X and Y are less and C. okay.

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So, to use an example, we're just doing the sum up through random variables, connects in line. We defined a new random variables.

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And so we know about X and Y, we know their PDF but can we say about the.

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Okay.

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Well, if we look at the CDM, so we've got X and Y, here on the graph X. plus Y, so the CDF for some Z means that X plus Y, has to be less than say. So.

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X plus Y, being a content itself straight line slope minus 1.

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So the so the, that's the integrating the joint density function, X and Y, over this gray region That'll give us the probability that the.

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Big Z is less than say okay.

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And that'd be this, we integrate X minus syncing to infinity and we integrate Y from minus Anthony disease minus.

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X, and we're integrating the joint density function for there and then the, that's the cumulative. The density is, you take the derivative here with respect to Z.

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now, the only place the andrew's end to here is that upper bound on the inner integral.

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So that so the derivative is just actually the.

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Marginal value that it comes out to that right there.

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So the density on the being of particular values, we integrate after X and Y, or.

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Over X. Y Z. okay.

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Is a convolution okay right down here. Oh, okay.

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Up here, now, 2554that'SA general thing for general X and Y, if they're related to each other, if it happens that they're independent of each other, then that joint density function, it affects unwind.

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So you split it, you separate out into 2 single variable density functions. That's like definition independence naturally and we get this thing, 555 the density function for the sum of X and Y, effect to the wire independent.

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Okay, now let's.

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Do an example here by 40 so this is.

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To calcium are important, but we're going to assume that they're independent. That's too easy. We're going to assume that they to joint Gaussian and they've got a correlation coefficient ROI equals minus a half.

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So, in that case.

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There's a formula here that we did, it was a definition actually, for what the density function for joint calcium was particular role is is defined several pages back. And that's actually that middle thing down here. I'm trying to circulate with the curser. If you can see it.

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Ignore the integral and that's that's that right there. So what we got, we got in front well, it's 2 variables, so it's 1 over 2 pies.

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It was 1 grab 1 over square root 2 by 2 variables 1 over 2 Pi and we've got this thing in here square root of 1, minus row squared and that's necessary.

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To make the whole density integrate up to 1. so, inside here we have, so.

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If it was 1 very, but it would be to the minus X squared over 2. if it was 2 things that were independent of each other, it would be either the minus X squared minus Y squared over 2.

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but they are correlated with row and again, roast correlation coefficient. If they track each other perfectly row is 1 dimension list. If they track each other inversely, it's minus 1.

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if they're independent of each other, it's 0 and any role that's less than.

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Certainly, it's always less than a half the X and Y, basically.

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There for practical purposes, fairly independent of each other.

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So, okay, so an exponent here, if we have a non 0 row X squared minus to roll.

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X times or Z minus X, X and Z here plus.

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Well, call, that's why minus 2 row X Y, plus Y squared and the whole exponent divided by 21 minus row Square. It's sort of to normalize things. Okay. So, in any case, that's the joint Gaussian.

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For X and Y, being correlated with correlation correlation a row. Okay. Now.

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We want again, we're looking at the sum of the X and Y.

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And this is a density for the song we integrate.

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The the 2 variable density for action.

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X plus Y, equals see, that's an integrate out, integrate over here for all the X Doo, Doo, Doo, Doo, Doo and.

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We get down to the thing at the bottom here.

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Well, we had a specific role here of minus a half and to plug it all in. We get this and this.

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So, we get the answer down here and so the sum of it's with correlated.

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Calcium, and that is a calcium.

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With you in a variance.

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Okay, so some of calcium is a calcium well correlated according to our.

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Type of correlation, or it's a variable joint. Okay.

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That's nice here. So.

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What's happening next to is that.

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We assume we've got a system with despair, and I've got a computer here if the computer fails to have another computer thing beside me often.

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I had to use it. Okay, so.

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And so we want to.

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Figure out the density functioning for the lifetime of this redundant system.

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I'll give you another example at home. I have to test the power walls in my garage. It's a total of about.

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Oh, 27 kilowatt hours. And for all I know I may be the only person in the Albany area that has the test of power walls.

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I'm not positive, but I really think so. So, in any case, if Niagara Mohawk National grid fails, I could run my house for a day or so day or 2 on my on my power wall even ignoring my solar panels. Okay.

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Which, at the moment are producing about a 3rd, not today right now, but this time of year, producing about a 3rd, more electricity than I used on the average.

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So, a whole month of March, they produce about 20% more electricity than I use and of course, it's getting sunnier. So you thought it was system with redundancy here I've got National grid and then I've got my.

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Battery such a system with redundancy in any case.

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So, what's happening here in this example? 541.

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Nice realistic example, twos, you know why we like the buckets prices.

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Enormous, but it has nice examples and a lot of them.

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Okay, so you got the main system, we assume it's exponentially distributed lifetime and again, the exponential distribution remember its memory list. So the probability of it failing today.

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Does not depend on how old it is, so.

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Okay, and then when the 1st system fails, then we power up the backup, which has again fails with a.

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So, back to initial distribution, what's a different parameter? Let's say.

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Because perhaps we're using different technologies for the main 1 in the stand by.

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And we want to now know what's our lifetime of the system with the back up.

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Okay, so key is a lifetime of the main.

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Key to want teachers the lifetime of the backup they're both random and the lifetime of the.

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Big combined thing is T1 plus 2, it's a random variable to some other random variables. Okay. And so.

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So, this here is the exponential distribution and X's non negative.

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And he had a and landed at the front is just to make the thing integrate out to 1 and that's for 1. and then for the 2nd 1. well.

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Why is Eva and put that in here.

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And for the.

155
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Okay, now there is something here. What we've done is the 2nd system. We don't turn it on and it's all time Max versus some X being random. So this is exponential.

156
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But you notice that C minus X, because we're counting the lifetime of the original system backup from time 0, even though we don't turn it on until time X that's why I, Betsy minus X here. And this is valid only.

157
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For Z, bigger than x X and this is 0 because.

158
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Well, just hasn't been turned on you. Okay now. So we want the lifetime density on Z and it's the convolution of these 2.

159
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That's that thing down here and you.

160
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It actually simplifies nicely. Well.

161
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Sort of, I guess, more or less, and we get the density functioning. We can see Atlanta squared the. So this sum here is not exponential anymore.

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It's something called the airline distribution with M equals 2. so you add.

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Hey, of these exponential random variables you get an airline K distribution. So this is an application number of life.

164
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Oh, okay.

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Conditional and again, so the conditional density.

166
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I'll say why given.

167
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A Z given why I'm sorry?

168
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Um, you just integrate over the given, it's like a softball and you said, Assam, it's the conditional thing that we integrate out by there.

169
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Because you've got the conditional density of why I was given a particular value of why we multiply that by the density of that particular value apply and we integrate outline.

170
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So, what we're going to do is an example here of quotient.

171
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So, we have X and Y.

172
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They're exponential and we want to know what is the quotient of those 2.

173
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So, I'm too tired to think of an example right now, but it actually has an application. We're not just making up an example just.

174
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Because we're sadistic. Well, maybe we're sadistic. That's a separate question but this is not evidence of it. Okay.

175
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And, okay, so X and Y, you're exponential and we want to know.

176
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What about X divided by? Why.

177
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So, the density of the density of the given, why.

178
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A particular value of why these just X scaled up by 1 over why.

179
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That's what I'm saying here. So, so the entity to see for particular value of why is.

180
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The density of access, half of vacs and, um.

181
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And then this scaled by Y, here.

182
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Well, why X? Y, Z. okay. That's how you get the Y Z and they're given why.

183
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And why they put absolute value around why here? I don't actually know.

184
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Because we know that why is some exponential? It's.

185
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Negative so, these vertical bars here absolute value not conditional.

186
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I don't know what they're doing there. In any case. We now we integrate over why we integrate out.

187
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Why.

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Integrate wine and.

189
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I don't know, I didn't mind infinity because if why is less than 0, then the density of why is 0 okay in any case integrate out here.

190
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And, um.

191
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We get this here. Well, we have the conditional here in this definition up additional over there.

192
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They're independence and now we plug in definitions.

193
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And we get and waving my hands a little we get the density for.

194
00:37:25.528 --> 00:37:33.329
Being that, so if we the ratio 2, exponentials, this is the PDF of the results.

195
00:37:34.733 --> 00:37:44.963
Okay, transformations again you transform random variables that gave you some reasons such as video processing.

196
00:37:45.204 --> 00:37:51.384
You rotate from the RTV vector space into something like it's called the Y, vector space.

197
00:37:52.048 --> 00:38:02.759
History of colored television, they rich and get those sorts of rotations with discrete electrical components like resisters and so on.

198
00:38:03.778 --> 00:38:14.940
And it's amazing and of course, it was also very inaccurate because you discreet components drift a lot. And that would change the color of the signal.

199
00:38:16.199 --> 00:38:23.400
So early colored TV said knobs, you would turn to adjust the caller to make it the least worst. Okay. Any case.

200
00:38:25.050 --> 00:38:29.550
You can transform and get a joint.

201
00:38:31.260 --> 00:38:36.719
A joint entity function, or the serious Expedia advocated a bunch of right here.

202
00:38:36.719 --> 00:38:39.869
For some transformation of them.

203
00:38:39.869 --> 00:38:45.150
Webex, and why that's still an example of it. It makes it easy or here.

204
00:38:45.150 --> 00:38:59.519
But 2, random variables, X and Y, and want to know what about the min and Max. So the 2 new random variables as the men of X Y, and the Max, and we would like to know the joint entity function of X and Y.

205
00:38:59.519 --> 00:39:06.659
Help you and Z together, because they're going to be correlated with each other. Obviously, for example, W, cannot be bigger and see.

206
00:39:07.344 --> 00:39:21.625
Okay, so the joint CDF of Bellevue and Z well, from the definition of joint, that's a problem lower left quadrant probability.

207
00:39:21.864 --> 00:39:30.835
So, here so the big f. W. so this is a joint CDF W and Z. and that should be a comma there. W. com, let's see.

208
00:39:30.925 --> 00:39:41.664
Typo cycles that's the probability that W center equals Z slash. Dolby is the man and the max. So we have this thing here.

209
00:39:41.940 --> 00:39:53.309
That's the joint, and probably a minute and Bob you and also the Max is less than or equal. Okay.

210
00:39:56.190 --> 00:40:05.099
So so if we look here.

211
00:40:08.130 --> 00:40:11.610
So, we've got X and Y, here.

212
00:40:14.070 --> 00:40:18.750
And that out to be this region.

213
00:40:27.840 --> 00:40:36.179
Oh, that's confused myself, man. Less. Yeah.

214
00:40:39.030 --> 00:40:50.070
And have to be it's sort of a region like that. Yeah. Okay.

215
00:40:50.070 --> 00:40:55.230
Yeah, okay so that's.

216
00:40:56.730 --> 00:40:59.730
And an example here.

217
00:41:01.079 --> 00:41:07.500
So, the joint the, and sees a man and the Max, that's the CDF.

218
00:41:07.500 --> 00:41:11.579
And then just, it works out to be this so.

219
00:41:13.320 --> 00:41:23.039
Okay, Here's I'm converting to polar notation and.

220
00:41:23.039 --> 00:41:28.469
So, X and Y is case, they're independent calcium and.

221
00:41:28.469 --> 00:41:32.280
Means 0, Sigma.

222
00:41:32.280 --> 00:41:41.550
Now we convert to to polar so R, squared of X Y, squared and status are can apply over X.

223
00:41:43.679 --> 00:41:50.639
This case we have to work out simply well, it's 1 reason we're doing it so the joint CDF on our data.

224
00:41:50.639 --> 00:41:57.179
Is the probability are the data listen data and that is the.

225
00:41:59.219 --> 00:42:03.809
Well, this here is comes out to be our here and.

226
00:42:03.809 --> 00:42:08.010
And they're integrating well data.

227
00:42:08.010 --> 00:42:19.469
Doesn't affect it, so I'll hit this more detail next time. Basically, the summary sorts of thing the CDF is the probability that are.

228
00:42:19.469 --> 00:42:26.550
It's less than this arc and fade is less than this line and that's that shaded region there. And that's the.

229
00:42:26.550 --> 00:42:32.400
Cws on our data the question is what's the probability of that happening?

230
00:42:33.599 --> 00:42:39.989
And our net, so to be Kelsey and and.

231
00:42:39.989 --> 00:42:43.110
Is uniform it turns out so.

232
00:42:47.130 --> 00:42:52.559
We can do a more general thing, linear transformation of X and Y.

233
00:42:52.559 --> 00:42:57.780
Say to be the in Dolby you matrix qualification.

234
00:42:58.949 --> 00:43:08.940
And you go back and forth, we assume it's convertible. Otherwise the 2 variable thing collapses down to a 1 variable thing.

235
00:43:10.885 --> 00:43:25.074
And so we want to so we have the density on X and Y, we'd like to know what's the density? Well, the probabilty so we construct a little square or a little rectangle. I'm sorry little parallel or gram even.

236
00:43:25.135 --> 00:43:31.945
But so the probability that we're at some X and Y, and that's the corner of the rectangle.

237
00:43:32.250 --> 00:43:40.349
With the CX is tied to see why the activity wire small. So the thing on the left, that's the probability that our X Y, fall in that little.

238
00:43:40.349 --> 00:43:46.349
Box the density times, the size of the box, the Y, so.

239
00:43:46.349 --> 00:43:53.130
If we transform that little box in the X, Y, coordinate system to the V. W, coordinate system.

240
00:43:53.130 --> 00:44:01.409
Then the probabilities have to be the same and so that's the density on the Dolby you times.

241
00:44:01.409 --> 00:44:12.300
The area of that little box that's written here is says areas of Pamela gram, and that's going to be like the J Colby. It's going to be the.

242
00:44:12.300 --> 00:44:17.460
The determinant of the matrix. A. B. C. D. and Tom.

243
00:44:21.000 --> 00:44:27.480
Dp over that, that that out to be the determined dividing by the determinant. So.

244
00:44:30.659 --> 00:44:36.059
Down here. See. Okay so not so to.

245
00:44:36.059 --> 00:44:46.739
Here basically, so, and this side, this is a tutorial that I did a classic 2 ago. You remember so, this is it's doing and just.

246
00:44:46.739 --> 00:44:55.739
For a more general thing where we're transforming the variables with this linear transformation was this matrix. A. B. C. D. so.

247
00:44:55.739 --> 00:45:05.699
So, Z is a time tact and X is a minus 1 disease. So the density function epilepsy. That's what we want. It's a density function of that.

248
00:45:05.699 --> 00:45:09.809
Is the extensive function divided by the determinant?

249
00:45:09.809 --> 00:45:13.500
So, and.

250
00:45:14.760 --> 00:45:17.909
How we can transform density functions.

251
00:45:17.909 --> 00:45:21.599
So, the initial what's a rectangle will become this parabola gram.

252
00:45:21.599 --> 00:45:26.429
Okay, um.

253
00:45:26.429 --> 00:45:30.030
That was in general now, if we do it for joint.

254
00:45:32.130 --> 00:45:41.250
Density joint calcium. Okay. So got X and Y, and would say we're doing a rotation let's say.

255
00:45:41.250 --> 00:45:45.750
Oh, actually, this is doing.

256
00:45:45.750 --> 00:45:50.159
Yeah, it's a rotation here. Okay.

257
00:45:52.679 --> 00:45:57.480
45 degrees 1 way or the other. Okay. And.

258
00:45:59.340 --> 00:46:04.440
So and the determinant.

259
00:46:05.639 --> 00:46:09.030
It's 1, well, it's.

260
00:46:10.679 --> 00:46:14.730
No, sorry no, just square would have to start to. Okay.

261
00:46:19.650 --> 00:46:28.739
So the density works out to a it is 1. I'm.

262
00:46:33.269 --> 00:46:37.380
I got to think about just there may be a typo any case.

263
00:46:37.380 --> 00:46:43.829
So, we can get the density of some of this rotation of the joint calcium.

264
00:46:47.039 --> 00:46:52.079
Now, what we're doing here is we're assuming that they may be correlate. You've got this row in here.

265
00:46:52.079 --> 00:46:58.110
Okay, so if we rotate at.

266
00:46:58.110 --> 00:47:08.159
Okay, okay now.

267
00:47:10.375 --> 00:47:25.164
So before we had our joint thing, we assumed they had, they were means 0 and Sigma 1 but there was that 1 parameter role to the correlation coefficient that was then now, what's happening here in 59 is that we're generalizing this.

268
00:47:25.164 --> 00:47:38.275
So, we sold the 2 and they're related by some correlation coefficient but now we allow the 2 calcium to a different means and segments. So X.

269
00:47:39.539 --> 00:47:42.840
As mean, and 1 in Sigma signal 1.

270
00:47:42.840 --> 00:47:47.460
And why has mean M2 segments segment 2.

271
00:47:47.460 --> 00:47:53.340
And they have a correlation coefficient role X and Y, now.

272
00:47:53.340 --> 00:48:00.750
And this is a definition here as well. This is the joint density function of X and Y, know.

273
00:48:00.750 --> 00:48:10.050
It's actually a definition, it's not a conclusion because we could combine 2 galaxy in any way. We like, you know.

274
00:48:10.050 --> 00:48:20.579
We could say that the, you know, but this is the way we choose to combine them and we get this thing here and it's actually conceptually.

275
00:48:20.579 --> 00:48:28.409
Really? No there, because we look at what we're doing here X, minus 1 over Sigma 1. so this is just normalizing X. okay.

276
00:48:28.409 --> 00:48:41.250
And why minus him to every signature? That's normalizing why? So we've got to use the general case where X and Y means in segments but what does he quite? What does formula? Does it just normalize the 2 of them? Okay.

277
00:48:42.750 --> 00:48:47.639
And then it's got the row in here that we saw before.

278
00:48:47.639 --> 00:48:53.550
We got this cross term in here, minus 2 row times X. Y.

279
00:48:53.550 --> 00:49:01.230
And then we normalize inside the exponent with this thing here and we normalize the whole thing down there at the bottom.

280
00:49:02.309 --> 00:49:06.510
Notice we got the Sigma 1 Sigma 2 in the square root of 1 minus rose Square.

281
00:49:06.510 --> 00:49:11.760
Okay, and if you integrated X and Y, out, you would get 1.

282
00:49:13.230 --> 00:49:22.230
Okay, and it's a bell shape thing that squash and the greater the absolute value of row is the, it is.

283
00:49:23.550 --> 00:49:32.429
Okay, this is just looking at the counter line so.

284
00:49:33.510 --> 00:49:43.650
Middling value of the role if we're always 0, these would be circled as row approach plus or minus 1. this would almost be a straight line.

285
00:49:45.300 --> 00:49:56.909
It's showing it here again just nicely blooded thing on the right is an absolutely larger well, we're always minus, but it's absolutely larger than the thing on the left.

286
00:50:03.000 --> 00:50:08.400
And they're just showing that to them.

287
00:50:09.840 --> 00:50:14.730
The Sigma expresses how stretched out the thing is in general.

288
00:50:14.730 --> 00:50:28.320
Access segment is larger than Y, segment. I'd be Sigma 1 Sigma to stretch that in the X direction. Why? Sigma is bigger stretch stone in the wind direction and the center is where the M1 end.

289
00:50:30.869 --> 00:50:41.250
Okay, and that we had the joint density there, we can integrate out why get the marginal density on X in its own.

290
00:50:41.250 --> 00:50:47.250
Get something like this here.

291
00:50:47.250 --> 00:50:51.809
This is what you'd expect. Okay. It's got to me in a segment.

292
00:50:51.809 --> 00:51:00.059
Why the same now this starts getting very interesting here. The conditional.

293
00:51:00.894 --> 00:51:09.565
So, X and Y, it's a joint distribution there's a correlation coefficient. So they're tracking each other to some extent. Not perfectly.

294
00:51:10.164 --> 00:51:16.045
So, if you know why it tells you something about X and if, you know, X, it tells you something about why.

295
00:51:16.349 --> 00:51:19.530
Not everything, but something. Okay.

296
00:51:19.530 --> 00:51:29.969
So, we didn't do conditionals here so your conditional density. So if we know why what's an additional density on X given some value for why.

297
00:51:29.969 --> 00:51:37.980
And that's we defined this way, way back. And this is a definition for the conditional. It's the joint density divided by the density at that. Why.

298
00:51:37.980 --> 00:51:42.059
So, we plug in and we plug in and we get this mess here.

299
00:51:43.110 --> 00:51:51.000
By 64. okay. Um.

300
00:51:55.739 --> 00:52:03.269
Now, it's a bit of a mess, but sometimes life is missing.

301
00:52:03.269 --> 00:52:09.869
And so it's not as bad as yeah, it's a bit of a mess.

302
00:52:09.869 --> 00:52:13.050
But as I said, sometimes life is missing.

303
00:52:13.050 --> 00:52:23.550
Okay, so for particular values of why then X is.

304
00:52:24.719 --> 00:52:28.800
It's the calcium okay. He has a minus so.

305
00:52:30.059 --> 00:52:33.269
And so on for some means Sigma.

306
00:52:37.320 --> 00:52:41.519
And here's the conditional mean and the variants here.

307
00:52:43.019 --> 00:52:47.280
Let's look at what this means. Let's look at the variance. 1st, so.

308
00:52:47.280 --> 00:52:57.719
What this is saying is that the larger the correlation is, that's row the smaller, the conditional variance of access.

309
00:52:58.920 --> 00:53:05.400
So, if we're always 0, so we know so there's no correlation then the conditional variance of actually just.

310
00:53:05.400 --> 00:53:13.679
Signal 1 script, the severity of X, knowing why didn't do anything for us but the stronger the correlation view next and why.

311
00:53:13.679 --> 00:53:18.000
Then the smaller the vary, the conditional variance is.

312
00:53:18.000 --> 00:53:27.989
That makes sense. Okay. And if the roll was was plus or minus 1, then that would mean that why it was exactly some.

313
00:53:29.130 --> 00:53:42.264
Scale scale shifted value of X so why it has to be a X plus B, if there are always 1, if that was the case, then the conditional variance of X would be 0, if we knew why it'd be no variance at all.

314
00:53:42.505 --> 00:53:47.005
So, that's what that's saying there, the larger, the correlation squared, the smaller the conditional variance.

315
00:53:47.460 --> 00:53:58.559
The mean, what's happening with the mean, is that so, and 1 was the mean for acts now, this is a conditional mean of X if we know why.

316
00:53:59.699 --> 00:54:04.019
It's what this is saying. It's a little complicated.

317
00:54:04.019 --> 00:54:09.360
But knowing why well drag us.

318
00:54:09.360 --> 00:54:17.489
It will shift if we didn't know anything about why, then the mean of act 1, if we know why, and there's a correlation.

319
00:54:17.489 --> 00:54:31.800
Knowing why we'll shift what the mean of the conditional meaning of access and this is how much it chipset the bigger role the more chipset and is a scale in here the ratio, the standard deviation. So it shifted in the direction of why.

320
00:54:31.800 --> 00:54:40.619
If it was positive and the farther wise off of why, I mean, the more the more the conditional mean of X, it shifted.

321
00:54:42.300 --> 00:54:45.539
So that's what that paragraph is saying there.

322
00:54:45.539 --> 00:54:51.150
Okay, now what's happening here is so we have our X and Y, the calcium.

323
00:54:51.150 --> 00:54:57.030
And they are, there's a correlation coefficient between them.

324
00:54:57.030 --> 00:55:05.039
So now we can say compute so coherence we defined it last chapter. It's the expectation.

325
00:55:05.039 --> 00:55:08.250
Of X minus, that's mean times Y minus it's mean.

326
00:55:09.869 --> 00:55:17.579
Okay, now and this here.

327
00:55:17.579 --> 00:55:21.239
You actually worked it out earlier I mean, working on it again.

328
00:55:23.159 --> 00:55:34.889
This is the conditional me this is also equal to the expected value this given why expected over why? Yeah, I sort of waste my hands on that detail.

329
00:55:36.989 --> 00:55:43.710
So, we look at the inner brackets as expected value as a variance given why? And then expected over.

330
00:55:43.710 --> 00:55:49.590
Over Y, so and.

331
00:55:51.900 --> 00:56:03.300
And skipping some details, this gives the CO variance as the correlation coefficient scaled up by the 2 standard deviation.

332
00:56:04.860 --> 00:56:08.849
Matches our definition of correlation coefficient.

333
00:56:08.849 --> 00:56:16.710
That's nice which we defined as a coherence scale to be dimensionalized dividing out to do sickness.

334
00:56:20.460 --> 00:56:31.590
Give an example here rainfall that 2 cities may be Albany and Boston. Let's say the random variables to the rain falls in Albany in Boston.

335
00:56:31.590 --> 00:56:42.329
And let's assume that the rainfall it's Gaussian and it's joint calcium for Albany in Boston and is a correlation.

336
00:56:45.989 --> 00:56:52.769
And what we would like here is that if we have the rainfall in Boston some year.

337
00:56:52.769 --> 00:57:00.000
So, for each value of.

338
00:57:00.000 --> 00:57:06.389
This is a little complicated here, but it's, it's worth taking time to understand this.

339
00:57:06.389 --> 00:57:11.219
And I'll pick up again in a week and a half. Now.

340
00:57:11.219 --> 00:57:17.099
It's a likelihood of things just introducing a concept of likelihood.

341
00:57:17.099 --> 00:57:24.329
For each that okay.

342
00:57:24.329 --> 00:57:28.949
For each value of Y.

343
00:57:30.150 --> 00:57:35.789
Okay, so X is Boston and why is Albany? Let's say rainfall.

344
00:57:36.960 --> 00:57:41.550
And for each value of rainfall.

345
00:57:41.550 --> 00:57:46.769
Had it backwards? Actually.

346
00:57:46.769 --> 00:57:50.969
X's Albany and why is Boston? Let's say now.

347
00:57:50.969 --> 00:57:56.280
David joint calcium probability for the rainfall in Albany in Boston.

348
00:57:56.280 --> 00:58:05.550
And breach value of rainfall in Boston, like, 10 inches this month or something. There's a density function for the rainfall.

349
00:58:05.550 --> 00:58:19.469
In Albany. Okay. And there's the most likely thing. So Boston's rainfall is 10 inches. The most likely rainfall for all ready might be 8 inches. Maybe all little drier than Boston. Perhaps.

350
00:58:19.469 --> 00:58:23.460
I don't know if it is, but let's say it is. Okay so.

351
00:58:23.460 --> 00:58:30.449
So, for each value of Boston rainfall, there's a density probably distribution for.

352
00:58:30.449 --> 00:58:37.260
Albany rainfall and reach value of Boston rainfall. We can find the most likely.

353
00:58:38.460 --> 00:58:41.880
Value for Albany rainfall.

354
00:58:41.880 --> 00:58:50.429
That's where the or the conditional density of already given Boston is the maximum. Okay.

355
00:58:50.429 --> 00:58:55.769
So, if boss gets 10 inches, the most likely rainfall for all, but he might be 8, perhaps.

356
00:58:55.769 --> 00:59:00.719
If Boston got 5 inches, the most likely might be.

357
00:59:02.190 --> 00:59:05.849
Was linear and before, but maybe it's that 4 and a half or something.

358
00:59:05.849 --> 00:59:09.480
Okay, so this is so want to find this.

359
00:59:09.480 --> 00:59:12.869
We want to maximize this conditional density here.

360
00:59:15.780 --> 00:59:24.929
And, okay, so you see that we have want to maximize this now, if it's and.

361
00:59:28.289 --> 00:59:31.619
Then this will work out the expected value.

362
00:59:33.869 --> 00:59:36.989
So, we have the conditional.

363
00:59:36.989 --> 00:59:42.630
Density of X, given some rainfall. Why? And it will work it out.

364
00:59:43.710 --> 00:59:50.190
And it will be, so this will be the expected value of acts of the rainfall and Albany, given Boston or.

365
00:59:50.190 --> 00:59:54.000
This is called Max, so I'm likelihood you're going to see this 1 again.

366
00:59:58.320 --> 01:00:03.960
And, oh, I got to do another communication thing here.

367
01:00:03.960 --> 01:00:07.650
Again, okay, this is.

368
01:00:07.650 --> 01:00:15.090
This is more complicated to previous 1. the transmitted signal was a binary plus or minus 1. okay.

369
01:00:15.090 --> 01:00:19.469
Here the transmitted signal is calcium. It's X, it's calcium.

370
01:00:19.469 --> 01:00:26.070
On 01 and so is the noise, but the so the both 0 1 0.

371
01:00:26.070 --> 01:00:32.340
Sorry, different variances both of means 0 and they're independent so we're at.

372
01:00:32.340 --> 01:00:37.769
Adding the 2 gals here, different variances and we get the received signal. Why.

373
01:00:39.090 --> 01:00:53.695
Okay, so I want to know what the correlation is between X and Y, if there was no noise at all, it'd be 1, because why it'd be X. the noise is much much, much bigger than X. the correlation is going to be about 0, because it's the noise told the dominate tax then.

374
01:00:55.920 --> 01:01:02.250
You can't there's no excellent. Okay. It's all noise. So correlation. So at 8 0.

375
01:01:02.250 --> 01:01:12.570
Do you want to compute and if the noise is about equal to X, then if we then we see why we can make a guess what X is but the guest could be wrong.

376
01:01:12.570 --> 01:01:26.280
Okay, in any case. So this is a continuous case. So what we would like to know is forgiven. Why sort of what's the best guess at X is.

377
01:01:26.280 --> 01:01:29.909
And this would be our maximum so we want.

378
01:01:29.909 --> 01:01:37.920
Forgiven want why we want the X, which maximizes the density function of X given Y.

379
01:01:37.920 --> 01:01:41.849
Want to compute that maximum likelihood of thing.

380
01:01:41.849 --> 01:01:46.230
And do to do, and that's what that is.

381
01:01:46.230 --> 01:01:52.619
Correlation coefficient and work it out to the answer. Actually.

382
01:01:53.940 --> 01:01:57.989
And it's fairly complicated. I'm sitting down here on the right.

383
01:01:57.989 --> 01:02:05.699
So the best guess for access is not equal to Y, in fact, that might surprise you and depends how big the noise is. So.

384
01:02:05.699 --> 01:02:10.920
And we'll work this out, and this is a good point to stop. So.

385
01:02:10.920 --> 01:02:20.519
So, what we were doing, just reminder on Monday, you can use local, any local material can't go out over the net.

386
01:02:20.519 --> 01:02:30.659
Ask for help and if you're in China and you want to write it at 3 am right. Let me know.

387
01:02:30.659 --> 01:02:35.969
Test opens up at the same time. Yeah, right. Yeah.

388
01:02:35.969 --> 01:02:43.889
And next Thursday a week from today is a holiday.

389
01:02:43.889 --> 01:02:51.150
Is it vacation and what we did for new material? Well, got it on the actually.

390
01:02:53.369 --> 01:02:59.579
Hello.

391
01:03:03.719 --> 01:03:13.170
Here we were talking about basically material starting here. Oh, okay. And I went up as far as examples.

392
01:03:16.440 --> 01:03:22.469
Up as far as that noise communications thing. Okay.

393
01:03:22.469 --> 01:03:30.420
Any final questions so, no stone now warmed up. It was snowing this morning. I'm just 10 miles off campus and.

394
01:03:30.420 --> 01:03:33.960
Colony, other than that.

395
01:03:33.960 --> 01:03:40.110
No, enjoy the weekend, get some exercise and see you.

396
01:03:40.110 --> 01:03:41.730
1 day.