I saw this question in UIUC's probability class homework and it got me curious on how to solve it.
Suppose $X$ and $Y$ are jointly Gaussian such that $X \sim N(0, 9)$, $Y \sim N(0, 4)$; and the correlation coefficient is denoted by $\rho$. The solutions to the questions below may depend on $\rho$ and may fail to exist for some values of $\rho$.
For what values of $d$ is $X + dY$ independent of $(X - dY)^3$?
However I wasn't able to find a solution. How do you obtain the covariance of $(X - dY)^3$? I am trying to do $Cov(Z-X,X^3)$ turns into $Cov(Z,X^3) - Cov(X,X^3)$ but I don't know what $Cov(X,X^3)$ solves to?