2.23 This exercise provides an example of a pair of random variables X and Y for which the conditional mean of Y given X depends on X but corr(X, Y) = 0. Let X and Z be two independently distributed standard normal random variables, and let Y = X2 + Z.

a. Show that E(Y 0 X ) = X2.

b. Show that mY = 1.

c. Show that E(XY ) = 0. (Hint: Use the fact that the odd moments of a standard normal random variable are all zero.)

d. Show that cov(X, Y ) = 0 and thus corr(X, Y ) = 0.