I was asked the following question by my professor. Suppose $x$ and $y$ are jointly normal variables. We further suppose the marginal distribution of $x$ and $y$ are $N(0, 1)$ and $N(0, 2)$ respectively. Their correlation is $\rho$, what is $E[|x^2-y^2|]$.
I have been thinking this for a while and seem not to be able to fully calculate the expectation. What I did is to first define $a = x-y$ and $b = x+y$. As $x$ and $y$ are jointly normal, we could claim that $a$ and $b$ are jointly normal as well, and we can suppose their correlation is $\rho'$.
We can then transform the expectation to $E[|ab|] = E[|a||b|]$. And then I was thinking about two approaches. The first one is to further define $a' = |a|$ and $b'= |b|$. By some manipulation, we can probably derive the pdf of $a' b'$ and then calculate the expectation.
Another approach I was considering is to further rewrite the expectation as $$E[|ab|] = E[ab|a>0, b>0] + E[-ab|a<0, b>0] + E[-ab|a>0, b<0] + E[ab|a<0, b<0].$$
But I don't see too much prospectus in any of the approach.