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$X$ and $Y$ are two random independent variables, they have the same distribution and have finite expected value. Find $ \mathbb{E} [X|X+Y] $

My attempt:

$$ \mathbb{E}[X|X+Y] + \mathbb{E}[Y|X+Y] = X + Y $$Whats more $$ \mathbb{E}[X|X+Y] = \mathbb{E}[Y|X+Y] $$ because $X$ and $Y$ have the same distribution.

Hence:$$ \mathbb{E}[X|X+Y] = \frac{1}{2}(X+Y) $$

Edit:

From comments below I conclude that this is a sufficient solution.

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