I'm facing this question while looking for the variance of the product of two normal distributions. Let say we have $x$ and $y$ which are normally distributed with mean = 0 and sd = 1 and correlated by $\rho$. Is their a way to compute the expected covariance of $x$ and $y^2$? My simulations show that it should always be 0 in these conditions, but i'm looking for some examples, proof or references.
Thank you,
Define $Z:=X-\rho Y$ so $Z\sim N(0,\,1-\rho^2)$ is uncorrelated with $Y$. If $X,\,Y$ are joint Normally distributed (which is a stronger condition than their each being Normally distributed), so are $Y,\,Z$, and $Z$ is also independent of $Y$ so$$\begin{align}\operatorname{Cov}(X,\,Y^2)&=\Bbb E(XY^2)-\Bbb EX\Bbb EY^2\\&=\Bbb E(ZY^2+\rho Y^3)\\&=\Bbb EZ\Bbb EY^2\\&=0.\end{align}$$In particular, the penultimate $=$ works because $Z$ will, on the above assumptions, be uncorrelated with $Y^2$.
