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I am asked to prove that for two integrable independent and identical distributed random variables E[X|X +Y] = E[Y|X +Y] and then compute it .

What I have done and the way I am thinking about it is to let X+Y=Z so we can have something of the form E(Z-Y/Z) so we can use the 'distributive' property but nothing really happens from there except from finding that for the second part the outcome is E(X+Y)/2

Can anyone give me a hint for the first part? Thank you

Greg
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1 Answers1

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Because $X$ and $Y$ are i.i.d., $E[X|X+Y]=E[Y|X+Y]$ follows from symmetry. Your second part isn't quite correct either: from the first part: \begin{align*} E[X|X+Y]=E[Y|X+Y]&=\frac{1}{2}\left(E[X|X+Y]+E[Y|X+Y]\right)\\ &=\frac{1}{2}E[X+Y|X+Y]\\ &=\frac{1}{2}(X+Y). \end{align*}

yurnero
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