Covariance of $X$ and $Y^2$ where $(X,Y)$ is bivariate normal

Question

I'm trying to solve a case where there is bivariate random vector $(X,Y)$ that has the bivariate normal distribution below ($-1<\rho<1$):

$$\begin{pmatrix} X\\ Y \end{pmatrix}\sim N_{2}\left(\begin{pmatrix} 0\\ 0 \end{pmatrix},\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}\right)$$

I am trying to get covariance of $X$ and $Y^2$ but not sure how to do this. I remember that taking square of normal distribution gives chi-square distribution, but in the case of bivariate case, I am not sure how to get covariance of $X$ and $Y^2$. Could someone please help?

Thanks.

Answer 1

Hint: I will assume that $EX=EY=0$ for simplicity but the general case is not harder. If $a=\frac {EXY} {EX^{2}}$ the $X$ and $Y-aX$ are jointly normal with covariance $0$. Hence they are independent. Now $$EXY^{2}=EX((Y-aX)+aX)^{2}$$ $$=EX(Y-aX)^{2}+a ^{2} EX^{3}+2aEX^{2}(Y-aX)$$ $$=EXE(Y-aX)^{2}+a ^{2} EX^{3}+2aEX^{2}E(Y-aX)$$ which can be computed easily.