covariance of two squared (not zero mean) random variables

Question

Let $X\sim N(\mu_1,\sigma_1^2)$ and $Y\sim N(\mu_2,\sigma_2^2)$ and $\operatorname{Cov}(X,Y)=c$. How can we compute $\operatorname{Cov}(X^2,Y^2)?$

Thanks.

Answer 1

I'll assume $X$ and $Y$ are jointly normal. Their covariance matrix is $$ C = \pmatrix{\sigma_1^2 & c\cr c & \sigma_2^2}$$ so their joint density is $$ \eqalign{f(x,y) &= \dfrac{1}{2\pi \sqrt{\det(C)}} \exp\left( - \pmatrix{x-\mu_1 & y-\mu_2\cr} C^{-1} \pmatrix{x-\mu_1 \cr y-\mu_2}/2\right)\cr &= \dfrac{1}{2\pi \sqrt{\sigma_1^2 \sigma_2^2 - c^2}} \exp\left(\frac{(x-\mu_1)^2 \sigma_2^2 - 2 c (x-\mu_1) (y-\mu_2) + (y-\mu_2)^2 \sigma_1^2}{2(c^2-\sigma_1^2\sigma_2^2)}\right) }$$ Doing the integrations (with Maple's help), I get $$ \text{Cov}(X^2, Y^2) = {\mathbb E}[X^2 Y^2] - {\mathbb E}[X^2] {\mathbb E}[Y^2] = 2 c^2 + 4 c \mu_1 \mu_2 $$