out of curiosity, I wonder if there is a solution for the expectation of the product of two chi-square variables (or the variance of the product of the normals).
Say: Let $(X,Y)$ be jointly normal, with mean 0 and covariance matrix $\Sigma = \begin{pmatrix} \sigma_{1}^{2} & \sigma_{12}\\ \sigma_{12} & \sigma_{2}^{2} \end{pmatrix} $. I know the expecation of $XY$, $\mathbb{E}[XY]=\sigma_{12}$, but how about the variance of $XY$, i.e. $\mathbb{E}[(XY-\sigma_{12})^2]$?
Any thoughts on this?
