Let $X_1$ and $X_2$ be jointly normal random variables with $\mathbb E[X_1] = \mathbb E[X_2] = 0$ and $\operatorname{var}(X_1) = \sigma_1^2$, $\operatorname{var}(X_2) = \sigma_2^2$. The correlation between them is $\rho$.
Find $\mathbb E\left[\left(X_1^2\right)\left(X_2^2\right)\right]$.
I know that $\rho = \frac {\mathbb E[XY]} {\sigma_1 \sigma_2}$, but I'm not sure how this equation changes when $X$ and $Y$ are squared.
Any help would be greatly appreciated!
When the $X_1$and $X_2$ are jointly normal the covariance between these two would be $2{\rho}^2$
The proog is here: covariance of two squared (not zero mean) random variables
If you don't like the proof here you an do something like this
The covariance of two squared normal variables
Then it becomes easy
$$\mathop{\mathbb{E}}\left [ {X_{1}}^{2}{X_{2}}^{2} \right ] = Cov \left( {X_1}^2 ,{X_2}^2 \right) + \mathop{\mathbb{E}}\left [ {X_{1}}^{2} \right ]\mathop{\mathbb{E}}\left [{X_{2}}^{2} \right ] = 2{\rho}^2 +{\sigma_{1}}^{2} {\sigma_{2}}^{2} $$
Hope it helps you.