Suppose that two normal random variables X∼N(μx,σx2) and Y∼N(μy,σy2) are dependent. Their joint distribution can be expressed as fX,Y(x,y)=2πσxσy1−rho21e−2(1−rho2)1(zxz−2rhozxsy+zy2), where rho is the (population) correlation coefficient of X and Y,Zx and Zy are standard normal random variables computed from X and Y, respectively. (a) Derive the marginal pdf of X. (7) (b) Find the mean and variance of the conditional distribution of Y given X,[FY∣X(y∣x)]. (5) (c) Let X∼N(50,100) and Y∼N(60,400) with rho=0.75. Find the conditional distribution of Y∣X=x (4)
