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Suppose that you have two random variables $X$ and $Y$, of unknown distribution and that are not independent but uncorrelated. Both $X$ and $Y$ have zero expectation.

I am looking for upper bounds or statements about $$ \mathbb{E}[X\,|\,X+Y=z]. $$

The case for Gaussianity is addressed here.

Thanks!

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    $\begingroup$ if $f$ is the joint pdf of $X$ and $Y$ (the two being not independent) $$\mathbb{E}(X | X+Y=z) = \frac{1}{\int_{-\infty}^\infty f(x,z-x) dx} \int_{-\infty}^\infty x f(x,z-x) dx$$ $\endgroup$ – reuns Jan 4 '16 at 9:31

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