Expected value of product of square of random variable

Question

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: $X$~$N(0,\sigma^2)$, $Y$~$N(0,\sigma^2)$

Answer 1

Try using the Law of total expectation - set $Z = X^2Y^2$ and use:

$$\mathbb{E}[X^2Y^2]=\mathbb{E}[Z]=\int_{y}\mathbb{E}[Z\mid Y=y]\cdot\Pr[Y=y]$$