Find the Expected value and variance of X

Question

I need to find the find expected value of $x$ using iterated expectation and then find the variance of $x$.

So far, I have found the mean and variance of $X$ conditional on $Y=y$ as: mean$=y/2$ and Variance$=y^2/12$

Answer 1

You have here all the pieces you need. You have found $E(X|Y)$, $V(X|Y)$, and $f_Y$.

"Iterated expectation", which you say you are to use, tells us that $E(X)=E(E(X|Y))$.

By what you have told us, this in turn equals $E(y/2)$. Since you have $f_Y$, you can use it to calculate the expectation of $y/2$.

Another useful fact here is that $V(X)=E(V(X|Y)) + V(E(X|Y))$. This is especially useful since, again, you have already found all the pieces needed to fill in this formula. By what you have given us, this quantity equals $E(y^2/12) + V(y/2)$. And once again, since you have already found $f_Y$, you can use it to calculate both $E(y^2/12)$ and $V(y/2)$.