(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 $\mathrm{corr}(X,Y)=0$. Let $X$ and $Z$ be two independently distributed standard normal random variables, and let $Y=X^2+Z$. 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 $\mathrm{cov}(X,Y)=0$ and thus $\mathrm{corr}(X,Y)=0$.