Let $E(X)=\mu$ and $\operatorname{Var}(X)=\sigma^2$. If $E(Y|X)=a+bX$, find $E(XY)$ as a function of $\mu$ and $\sigma$.

Question

I can't figure out the answer for a question on my econometrics course. Somehow it seems simple, but still I can't seem to figure it out. Maybe I am thinking the wrong way about it. Could someone perhaps help me to figure this out? The question is a follows:

Let $E(X)=\mu$ and $\operatorname{Var}(X)=\sigma^2$. If $E(Y|X)=a+bX$, find $E(XY)$ as a function of $\mu$ and $\sigma$.

Now I see that $y$ is linearly dependent on $x$, but that is the conditional expectation of $y$. Of course, I would be happy with the answer. I was wondering, however, if someone could also perhaps attempt to explain how I should approach such a problem? This would help me a lot.

Thank you in advance!

Answer 1

We can use the law of total expectation. We have that $$ \operatorname E[XY]=\operatorname E[\operatorname E[XY\mid X]]=\operatorname E[X\operatorname E[Y\mid X]]. $$ Hence, $$ \operatorname E[XY]=\operatorname E[X(a+bX)]=a\operatorname E[X]+b\operatorname E[X^2]=a\mu+b(\sigma^2+\mu^2) $$ using the fact that $\operatorname EX^2=\operatorname{Var}X+(\operatorname EX)^2$.