How to compute $\mathbb{E}[\exp(X^2Y^2)]$ when $(X,Y)\sim N(0,\Sigma)$?

Question

I'm trying to compute the following quantities

\begin{align} \mathbb{E}[\exp(X^2Y^2)],~\mathbb{E}[\exp(X^4)],~~\text{and}~\mathbb{E}[\exp(XY^2)] \end{align} where $(X,Y)\sim N(0,\Sigma)$. For simplicity, assume that the diagonal of $\Sigma$ are all ones. In particular, I'm interested in the case when $\Sigma=I_2$, the $2\times 2$ identity matrix. What is the value of the above quantities in this case? How can we compute them?

Answer 1

Assuming that $\Sigma$ is the $2\times 2$ identity matrix, here are the related simpler questions you could try:

• Do you know that if $X$ and $Y$ are jointly normal ([Edited: and when $\Sigma$ is a diagonal matrix]), then they are independent?
• Can you see how things boils down to calculating $ E(\exp(X^n))? $
• Do you know how to get the density of $X$?
• Once one has the density $f$ of $X$, one can use $$ E(g(X))=\int_\mathbb{R}g(x)f(x)\ dx. $$