This exercise provides an example of a pair of random variables X and Y for which the

Question:

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|X) - X2.
(b) Show that µY = 1.
(c) Show that E(XY) = 0.
(d) Show that cov(X, Y) = 0 and thus corr(X, Y) = 0.