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(30 pts) Joint pdf of random variables X and Y is given asae-(r+l)e-(v+l) (-1<1<y < &) fxx(z,y) = O.W.(10 pts) Find fx(r) and determine the value of a....

Question

(30 pts) Joint pdf of random variables X and Y is given asae-(r+l)e-(v+l) (-1<1<y < &) fxx(z,y) = O.W.(10 pts) Find fx(r) and determine the value of a. (Fully specify fx(z)) pts) Find frix(ulz) pts) Find E(e-(Y+/Y4X) X > 0 (10 pts) Find the pmfof S, in which S is the sign of XY,i.e-S = XY < 0 XY =0

(30 pts) Joint pdf of random variables X and Y is given as ae-(r+l)e-(v+l) (-1<1<y < &) fxx(z,y) = O.W. (10 pts) Find fx(r) and determine the value of a. (Fully specify fx(z)) pts) Find frix(ulz) pts) Find E(e-(Y+/Y4X) X > 0 (10 pts) Find the pmfof S, in which S is the sign of XY,i.e-S = XY < 0 XY =0



Answers

Let $X, Y,$ and $Z$ have the joint probability density function $f(x, y, z)=\left\{\begin{array}{ll}k x y^{2} z, & 0<x, y<1,0<z<2 \\ 0, & \text { elsewhere }\end{array}\right.$ (a) Find $k$. (b) Find $P\left(X<\frac{1}{4}, Y>\frac{1}{2}, 1<Z<2\right)$.

Yeah it's probably been given the following joint distribution. We would like to find the cool variance. So that's what now the co variance is equal to the expected value of X. Y minus the expected value. That's times the expected value of what? Finding this in pieces here. The expected value. That's why is the double integral. That's why times are joint distribution And notice that X&Y both go from 0 to 1. And we're going to evaluate these both from 0 to 1. Now what? You neither evaluate these my hand or type this in to a graphing utility in order to grab them Or excuse me in order to evaluate this and I'm gonna use a graphing utility here and doing so gives us the expected value. That's why is one third now finding the expected value of X. And the expected value of what? Uh huh. The expected value of X will be the double in a row of X times X plus Y. With why knowing from 0 to 1 an X. Coin From 0 to 1. Similarly the expected value of Y will be this interval. What's up with Y. Times X plus Y. All right. Mhm. Now in order to integrate this notice that these who are the exact same thing. The only difference is that actually replaced with why? But they're going to give us the exact same value which is they are symmetrical And so we just need evaluate one of them and then they would both give us the same value evaluate this top because 7 12. So that bottom is also 7 12. So come back up here to our formula. Mhm. We have the expected value. That's why minus the expected value. That's times the expected by of why. And this is 1/3 -7/12. Uh huh. Times 7/12. Mhm. This is the expected value. That's why and this is the expected value events and they expect the value of why. Pleasure. And so evaluating this gives us our co variance, Which is negative 1/1 44 In circumstances that you have won over 144.


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