So far I have this, but I am not sure if you are allowed to do this or if it is correct:
I said since $X$ and $Y$ are independent and follow the same distribution, they must have the same expectation so:
$$E(X|S=s)=E(X|X+Y=x+y)=E(Y|X+Y=x+y)=E(Y|S=s)$$ since $X$ and $Y$ are i.i.d. Then, can I say?
and since $E(X|S=s)=E(Y|S=s)$; $$E(X|S=s)+E(Y|S=s)=2E(X|S=s)=x+y,$$
This is my steps and logic, but I am not sure if this is correct, so any feedback on my work would be greatly appreciated. Thanks!