Consider f(x) = e, x>0, and joint probability density function fxy(x,y)=e, for 0 3 Of Y13=e³-³, y<3 Of y13=e³-y, y>3

suppose that f(y) = { 3y^2, 0<=y<=1                                      0, otherwise a) verify that f(y) is a valid probability density function.b) Determine the expected value of Yc) Determine the variance of Yd) Find the cumulative distribution function of Ye) Determine the probability that Y does not exceed 1/2

Let random variables X and Y have the joint pdf   fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1   0, otherwise   Find the joint pdf of U = X^2 and V = XY.

The joint PDF of two random variables X,Y is given byfXY(x, y) = {k e^(-y-x/4) , x > 0, y > 0                    0,                  otherwisea)  What is the value of k ?b)  Find the probability P( X<Y ).c)  Find P( 0<X<2 ).