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$ \left( \begin{array}{c} X_1 \\ X_2 \end{array} \right) \sim N\left( \left( \begin{array}{c} 0 \\ 0 \end{array} \right) , \left( \begin{array}{cc} 1 & r \\ r & 1 \end{array} \right) \right) $

How do you to calculate Cov$(X_1^2,X_2^2)$?

I know Cov$(X_1^2,X_2^2)=E(X_1^2X_2^2)-E(X_1^2)E(X_2^2)$ and I could calculate $E(X_1^2)$ and $E(X_2^2)$. However, I got stuck at the $E(X_1^2X_2^2)$.

Any thought on how to do that part? Thanks!

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    Given the value of $X_2$, $X_1$ is normal with known mean and variance, and so you can get $E[X_1^2X_2^2 \mid X_2] = X_2^2 E[X_1^2 \mid X_2]$ which should be a quartic $X_2$, and then get the expected value of the result?2011-10-18

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