(Solved): This exercise provides an example of a pair of random variables, X and Y, for which the conditional ...
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.
c. Show that E(XY)=0. (Hint: Use the fact that the odd moments of a standard normal random variable are all 0 .) d. Show that cov(X,Y)=0 and thus corr(X,Y)=0.