Suppose we have two Gaussian distributed random variable $X$~$N(0,\sigma^2)$ and $Y$~$N(0,\sigma^2)$. These variables are not independent. What will be the expected value of product of square of this random variables
$E[X^2Y^2]$ = ??
Edit 1: They are jointly Gaussian distributed with correlation coefficient $\rho$
Edit 2: $X$~$N(0,\sigma^2)$, $Y$~$N(0,\sigma^2)$
Best Answer
See Variance of product of dependent variables at CrossValidated, specifically the second answer, for the outline of a derivation. In your case, because $X$ and $Y$ have zero mean and the same variance, the formula is simpler.
As explained there, the conditional density of $Y$ given $X = x$ is normal, with mean $\rho x$ and variance $\sigma^2(1-\rho^2)$. It follows that \begin{align*} E[X^2 Y^2 \mid X]&= X^2 E[Y^2 \mid X] \\ &= X^2 \left[ \sigma^2 (1-\rho^2) + \rho^2 X^2 \right]. \end{align*} By the law of iterated expectation, $$ E[X^2Y^2] = E\left[ E[X^2Y^2 \mid X] \right] = \sigma^2(1-\rho^2) E[X^2] + \rho^2 E[X^4] $$ and you can can look up the moments $E[X^2]$ and $E[X^4]$ at the Wikipedia article on the normal distribution. You'll end up with $ \sigma^4(1+2\rho^2) $.