cov(X,XY)? if X,Y is not independent

Question

For two normal random variables , $X$ and $Y$ whose mean are not zero,

If $ cov(X,Y) $ is given as $\sigma_{XY}^2 $ , are there any simple way to calculate $ cov(X,XY) ?$

Answer 1

Let us see... $$\begin{align} \mathsf{Cov}(X,XY) ~=~& \mathsf E(X^2Y)-\mathsf E(X)~\mathsf E(XY) \\ ~=~& \mathsf E(X^2Y)-\mathsf E(X)~\big(\mathsf{Cov}(X,Y)+\mathsf E(X)~\mathsf E(Y)\big) \\ ~=~& \mathsf E(X^2Y)-\mathsf E(X)~\mathsf{Cov}(X,Y)-\mathsf E(X)^2~\mathsf E(Y) \\ ~=~& \mathsf E(X^2Y)-\mu_{\small X}~\sigma_{\small XY}^{~2}-\mu_{\small X}^{~2}~\mu_{\small Y} \end{align}$$ Uhm, ... no. That's about as simple as it gets. If you have the means and covariance of $X$ and $Y$, then you will still need to evaluate the expectation of $X^2Y$.