1. Problem14.15Themoment-generatingfunction of two jointly distributed ran dom variables X and Y is defined by MX,Y(v,w) = E(evX+wY), provided that this integral is finite for all (v,w) in a neighborhood of (0,0). A basic result is that MX,Y(v,w) uniquely determines the joint distribution of X and Y.†

† Using this uniqueness result, it is not difficult to verify that X and Y are independent if and only if MX,Y(v,w) = MX(v)MY(w) for all v,w.

14.2 Moment-generating functions 379

(a) What is MX,Y(v,w)if(X,Y) has a bivariate normal distribution?

(b) Suppose that the jointly distributed random variables X and Y have the property that vX + wY is normally distributed for any constants v and w.

Prove that (X,Y) has a bivariate normal density.