I have a question concerning the bivariate normal distribution where I have been able to prove that the first covariance we are asked for is equal to $0$, but was not sure if this would be implying that the second covariance has to also equal $0$.
This is the problem:
I was able to prove that $Cov(Y-\rho X,X)=0$ and I know that $E(X^3)=0$, so with this, I can say that $E[(Y-\rho X)^{10}X^3]=Cov((Y-\rho X)^{10},X^3)$, however, I don't recall having any statement concerning the covariance with powers over the random variables. I would assume the covariance is equal to $0$, but am honestly not convinced of why would this be the case (if it is indeed equal to $0$).
Any key observation over this would be very useful.
Thank you
Best Answer
Once you proved that $\mathbb{Cov}[Y-\rho X,X]=0$ you are done because (using the fact given in the text) this is equivalent to prove that $(Y-\rho X)\perp\!\!\!\perp X $ thus
$$\mathbb{E}[(Y-\rho X)^{10}X^3]=\mathbb{E}[(Y-\rho X)^{10}]\cdot\mathbb{E}[X^3]=0$$
Being $\mathbb{E}[X^{2n+1}]=0$, $ \forall n$