# Upper bounds for $\mathbb E[X | X+Y = z]$

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!

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