Find $E(X_1^2X_2^2)$

Question

Let $X_1$and $X_2$ be jointly normal, with $EX_1=EX_2=0$, $var(X_1)=\sigma_1^2, var(X_2)=\sigma_2^2$.The correlation between them is $\rho$.

(i) Find $E(2X_1+X_2)^2$.

(ii) Find $E(X_1^2X_2^2)$

(iii) Find $E(X_1^4(2X_1+X_2)^2)$

I know how to solve for (i)

$cov(X_1,X_2)= E(X_1X_2)−E(X_1)E(X_2) = E(X_1X_2)$

$E(2X_1+X_2)^2 =4E(X_1^2) + 4E(X_1X_2) + E(X_2^2)$

$E(X^2) = var(X)+ E(X)^2$

My question is for (ii) &(iii) $E(X_1^2X_2^2) = E[(X_1X_2)^2]$ How can I find $var(X_1X_2)$?

Answer 1

The best way to do this is to observe that $U=X_1$ and $V=X_2+aX_1$ are independent for some $a$. Indeed, we only need covariance of these two variables to be $0$ (because of joint normality). Once you find this $a$ just write $EX_1^{2}X_2^{2}=EU^{2}(V-aU)^{2}=EU^{2}EV^{2}+a^{2}EU^{4}-2aEU^{3}EV$. Use the same idea for iii).