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8 Let X and Y be continuous random variables with joint probability density function given bye-r(y+1) '0 < % J! 0 < y < e - 1 =(6'c) f {e elsewhe...

Question

8 Let X and Y be continuous random variables with joint probability density function given bye-r(y+1) '0 < % J! 0 < y < e - 1 =(6'c) f {e elsewhere.Calculate E(X | Y = y).

8 Let X and Y be continuous random variables with joint probability density function given by e-r(y+1) '0 < % J! 0 < y < e - 1 =(6'c) f {e elsewhere. Calculate E(X | Y = y).



Answers

Find the covariance of random variables $X$ and $Y$ having the joint probability density function $$ f(x, y)=\left\{\begin{array}{ll} x+y, & 0<x<1,0<y<1 \\ 0, & \text { elsewhere } \end{array}\right. $$

Well that's probably been doing the following joint distribution. Now, the first thing we invite to find is the value of K and find the value of day. We know that the triple integral of K X y squared Z Must be equal to one. In order for us to a probability distribution. No extras from 0 to 1. Why goes from 0 to 1 and z It goes from 0 to 2. This means we have K times the integral from 0-2, brazil is easy. The integral from 0 to 1 of y squared. Dy the admiral From 0 to 1 of X dx Is equal to one. Now the integral From 0 to 2 of Z. Z is equal to The integral from 0 to 1 of why squared is one third And the integral from 0 to 1 Of that is 1/2. Yeah. Mhm. Mhm Two and one half canceled. So this tells us that K over three is one. Yeah, That's OK. Is equal to three. Yeah. Some things are joint distribution is really F of X. Y. Z. Mhm. This three X Y squared Z. With X&Y. Between zero and 1 Lindsey. Between zero and 2. That it helps us on B. Because one of the probability That acts as less than 1/4, Why is greater than 1/2 And Z is between one and 2. So finding this probability we want to integrate the triple integral of three at Weiss Birds. E Z goes from 1 to 2, Hind rose from 1/2 to 1 because it has an upper bound at one. Then Ash goes from 0 to 1/4 because it has a lower bound at zero. So this is equal to three times the integral From 0 to 1 4th versus the arts Times. The integral from 1/2 to 1 of Y squared dy Yeah, I'm gonna go from 1 to 2 of Z dizzy. So this is three times now. If we end great From 01 4th of X. Dx We get 1/32. If we integrate, why squared From 1/2 to 1, What do you 7/24? And then if we integrate Z DZ from one to We get three hands and then we multiply all these together. Mhm. This is 21/5, 12. Yeah.

All right. So we've got this joint density function here and were we need to find the value sea. So we know that Ah, this entire density function over the region, eh? Of, ah f of x y z dee's equal to one Because it's a joint density function for end of variables. So we know that we know that one is equal Interval from zero to two to go on reserve two into going on a journey to of C x y z Deac steal I d z. All right, so that sea of the integral and ah, we can just do this as all these separate into girls because they're multiplying and one does not depend on the other. Okay, easy. Okay. And if we just look at one of these into girls X squared over to charity too, we see that that's able to to so that one is equal to C times two to the three. Two c is equal to one of the great. All right, so now we want to look at part be here, so part B, we're looking at everything less than or equal to one settle. We're into girl. Our shuttle into girls are all going to go from one toe zero one that way wth x y z d x y z And very similarly like our previous problem. For the previous part of this problem, at least we can waken you can do the same thing basically right One a into girls You're the one of X d. Jax into girls here to one of why do I do girl zero one of Z D And just looking at one separate one of these, the at one over to so that one over eight times one over two. Cute. It's one over twenty stuck two to the six she and over eight time whenever sixty four. Fantastic. Percy, that explains why Lizzie Lesson or equal to one. You won't find this probability. So we know Z is equal to one minus x minus y. And, uh, what else do we know here? Way. Know that Z is equal to zero. Then Y is equal to one minus X and ah, these are all bounded below zero. Next is around it or above, uh, extended by one. So we're doing this triple in general now, Okay? And one eighth x y Z First we do easy do I and DX eso This is a pretty standard straightforward into girl. So let's go ahead and do this out and ah x y times one minus x minus y squared. Do you know why Dia and ah, I'm actually in line to just leave this the rest of this to you? There's a lot of algebra here in integration, but it's pretty straightforward. It it's not that difficult. Okay, I will write a few of these. I guess you got zero one of one over twelve x to the fifth, minus one over three x to the fourth this one over to certain whenever two x to the third guy won over sixteen. Out here on there's one third X squared minus one twelve packs DX they And once you do this integration, you'LL get one over five seven six zero Ah, as are finally ends

Yeah, that's probably been doing this distribution function and we all like to find the expected value of route X squared false watch where the expected value of root X squared plus Y squared is going to be the double integral of route X squared plus Weisberg times or probability distributions of times for X. Y. The arts do I. Next is greater than zero. And so extras from zero to infinity. Yeah. Why is less than one and two? Why does from negative infinity? Mhm. Up to one. That's what we're wanting to evaluate this interval. Because our bounds are not dependent upon any of the variables. We can rewrite it. So this will be the integral from the of insanity to one four. While we can bring those out times in a row from zero to infinity of eggs. Route next word. Both wash word. Yeah. Taking our anti derivative here will still have this negative infinity to one on the outside this before. Why times? Okay, X squared plus Y squared to the three house over three evaluated for magical zero, two and three. A little misunderstanding my apartment said zero is less than X. Comma, Y is less than one. That means they're both between your own one. So I need to change these bounds here and you go from 0 to 1. Sorry about that 0 to 1. And both of these. I mean they're both between your own one. That'll make this work out better for us here. So extras from 0 to 1. So now we're going to evaluate that we put in one. We put in zero and then we subtract them. And in doing this, yeah. We're left with the integral from 0 to 1 of four Y. Time one of the third over three. So that's what we get when we plug in zero in the fucking one. Yeah. We have Y squared plus one 23 house over three minus. Why do the third over three dy distributed in the four Y. Means we have the integral from 0 to 1 of four thirds. Y. Times widespread plus one to the three house minus four thirds. Why do the fourth dy take your anti derivative now? Mhm. And this will give us 4 15 widespread plus one. The five halves minus 4/15. Why do the fifth evaluated for more equal 0 to 1? And then now we plug in one. We plug in zero maybe some tracks. And this gives us the lovely number of 16 route to over 15 minus 8/15. And so that's our expected value.

In this problem, it's required to find the value off New X and you Why? Let's start by New X. Your fax is integration off our boundaries from 0 to 1 for X and Y six exports three to play. Why the X do you want will be equal integration from 0 to 1 three over toe? Why do you, Roy, which is equal to 3/4 and for m y or mule boy? Thesis is mule Milton knew why so Integration from 0 to 1 seeks experts toe Why poor toe g x d y all these years? Thanks. So it will be integration from 0 to 1 two boy pose to do y which is equal to over three. So me, why is two or 3 a.m. u X is 3/4. Thank you.


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