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If $(X_1,X_2)$ form a bivariate normal distribution such that $X_1$ and $X_2$ are marginally Standard normal but $\text{Cov}(X_1,X_2)$ is $c$. Then find the correlation between ${X_1}^2$ and ${X_2}^2$

We can find $E(X_1 X_2)$ as covariance is given. But to find Covariance of ${x_1}^2$ and ${x_2}^2$ we need to find $E\left({(X_1 X_2)}^2\right)$ which I found difficult to find. I tried to manually find the distribution of $X_1X_2$ and do it but it is computationally very difficult. Please help

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    $\begingroup$ One way to do it is Cholesky decomposition - note that $(X_1, X_2)$ has the same distribution as $(Z_1, cZ_1 + \sqrt{1 - c^2}Z_2)$, where $Z_1, Z_2$ are independent standard normal random variables. $\endgroup$ – BGM Apr 18 '17 at 14:54
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    $\begingroup$ @user385655 Did I edit your question correctly? $\endgroup$ – Arbuja Apr 18 '17 at 15:28
  • $\begingroup$ One way to calculate it is using the following formula for centered multivariate normal $(X_1,X_2,X_3,X_4)$: $\mathsf{E}[X_1X_2X_3X_4]=\mathsf{E}[X_1X_2]\mathsf{E}[X_3X_4]+\mathsf{E}[X_1X_3]\mathsf{E}[X_2X_4]+\mathsf{E}[X_1X_4]\mathsf{E}[X_2X_3]$ $\endgroup$ – JGWang Apr 19 '17 at 3:47

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