[M@gpie]

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4. Let $X_1, X_2, X_3, X_4$ be real random variables with Gaussian joint probability. Show that \[ E[X_1X_2X_3X_4] = E[X_1X_2]E[X_3X_4]+E[X_1X_3]E[X_2X_4]+E[X_1X_4]E[X_2X_3]\]
5. Let $X$ be a Gaussian random variable with zero mean and unit variance. Let a new random variable $Y$ be defined as follows: If $X=\zeta$, then
\[ Y = \left\{ \begin{array}{cc} \zeta & \text{with probability} \frac{1}{2} \\ -\zeta & \text{with probability} \frac{1}{2}\end{array}\right.\]
Determine the joint pdf of $X$ and $Y$ and the pdf of $Y$ alone.