Compute $\mathbb{E}(X|X+Y)$ for $X, Y$ IID standard exponential

Question

How to find the random variable $\mathbb{E}(X|X+Y)$ where $X$ and $Y$ are IID exponential random variables with parameter $1$?

Answer 1

Hint: Compare $E[X\mid X+Y]$ and $E[Y\mid X+Y]$. Compute $E[X+Y\mid X+Y]$. Conclude.

Thus, the result uses only that $X$ and $Y$ are i.i.d. (and integrable), not their exact distribution. To be more precise, the result uses only that $(X,Y)$ is exchangeable, that is, that the random vectors $(X,Y)$ and $(Y,X)$ have the same distribution.