$X_1, X_2$ are IID $N(0,1)$. Find Cov($X_1$, $X_1 X_2$).

Question

Two variables $X_1, X_2$ are independent and identically distributed as $N(0,1)$. I need to find

Cov($X_1$, $X_1 X_2$).

Thanks!

Answer 1

\begin{align} {\rm cov}(X_1, X_1 X_2) &= E(X_1^2 X_2) - E(X_1) E(X_1 X_2) \\ &= E(X_1^2) E(X_2) - E(X_1)^2 E(X_2) \\ &= 1 \times 0 - 0^2 \times 0 \\ &= 0 \end{align}