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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 $Y=X_{2}+Z$.
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 $cov(X,Y)=0$ and thus $corr(X,Y)=0$.

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