Let $X$ and $Y$ be two indpenent r.v. How can I get an expression for:
$$E[X|X+Y=a]$$
where $a$ is a constant? In other words, is there a general rule to recover the expected value of $X$ when all I know is the value of the sum $X+Y$? Similarly, what is
$$E[X|X+Y<a] $$
Would it be easier to answer these questions if $X$ and $Y$ are not only independently but also identically distributed?
Thanks
When $X$ and $Y$ are i.i.d., $E(X\mid X+Y)=E(Y\mid X+Y)$ by symmetry hence $$E(X\mid X+Y)=\tfrac12(X+Y).$$ When $X$ and $Y$ are independent with possibly different distributions, there is no similar simple formula independent on the distributions. Even when $X$ and $Y$ are i.i.d., there exists no general formula for $E(X\mid X+Y\lt a)$, except, by the same argument, the semi-explicit identity $$E(X\mid X+Y\lt a)=\tfrac12E(X+Y\mid X+Y\lt a).$$