TITLE: Covariance of two chi-square random variables
QUESTION [5 upvotes]: Let $(X,Y)$ follow bivariate normal distribution where $X,Y$ both follow $N(0,1)$ distribution and $\operatorname{cov}(X,Y)=c$. Determine the $\operatorname{cov}(X^2,Y^2)$.
I think that the answer will be $c^2$. But I can't make it out. Any help is appreciated.

REPLY [13 votes]: As usual with normal random variables, a direct approach is to realize $(X,Y)$ using some independent standard normal random variables, in this case $$(X,Y)=(X,cX+aZ),$$ where $a^2=1-c^2$ and $(X,Z)$ are i.i.d. standard normal. Thus, $$X^2Y^2=c^2X^4+2acZX^3+a^2X^2Z^2.$$
Using $E(X^2)=E(Z^2)=1$, $E(Z)=0$ and $E(X^4)=3$, one gets
$$
E(X^2Y^2)=3c^2+a^2=2c^2+1,
$$
hence
$$
\mathrm{Cov}(X^2,Y^2)=E(X^2Y^2)-1=2c^2.
$$
Note that this does not use the joint density of $(X,Y)$, not even the density of a single standard normal random variable $X$, only the first moments of $X$.