Given $2$ random variables $X, Y$ S.T: $X \sim \operatorname{Exp}(2)$ and $E(Y|X=x)=3x+1$
I want to calculate $E(XY)$
My Approach:
I know that $E(X)=0.5$, $\operatorname {var}(X)=0.25$, Plus: $E(Y|X)=3X+1$ which means: $E(Y)=E(E(Y|X))=E(3X+1)=2.5$
So $E(X)E(Y)=0.5\times 2.5=1.25$
I know that $E(XY)\geq E(X)E(Y)$ but I'm stuck here. How can I continue?