generating moment and expecting value

Question

Let X and Y be independent geometric random variables with parameter p and let $Z=E(X|X+Y)$.

1. Compute the moment generating function $M_X$ of X.
2. Compute the moment generating function $M_{X+Y}$ of $X+Y$.
3. Compute $E(Z)$.
4. Compute $Var(Z)$.

For 1. I got $M_x=pe^t/((1-e^t(1-p))$ 2. $M_{X+Y}$ of $X+Y$ is $M_x(t)M_y(t)$ as they are independent so I just need $M_y$ that is the same as in case 1, right?

3. I don't know what means $Z=E(X|X+Y)$, is just $X+Y$? Thanks

Answer 1

$E(X|X+Y)=E(Y|X+Y)$ since $X,Y$ are i.i.d. Adding these two we get $1$. Hence, $Z=E(X|X+Y)=\frac 1 2$ and $EZ =\frac 1 2, var (Z)=0$.