How to express expectation of product in terms of other moments?

Question

I'm having trouble with this problem. $X$ and $Y$ are random variables and $a$ and $b$ are constants. Assume that $E(Y|X) = aX + b$, where $E(.)$ is the expected value operator. How do I express $E(YX)$ as a function only of the first two moments of $X$? What should be my approach?

Answer 1

Use the fact that $XE(Y|X)=E(XY|X)$. this gives $E(XY)=E(E(XY|X))=E(aX^{2}+bX)=aEX^{2}+bEX$.