I am facing the following problem(s). Suppose we have two independent random variables $X$ and $Y$, such that $\mathbb{E}[X] = 0$. I am looking for sufficient conditions on the joint distribution of $X$ and $Y$, say $P_{(X,Y)}$, which lead to $\mathbb{E}[X \mid X+Y <0] = 0$. My suspicion is for example that if the marginal distributions are symmetric around 0, this should be true. Is this so? What if they are both standard normal?
I am not mathematician and I am struggling to write the last expectation in an integral form. This would be already of a great help for me.
Thanks in advance,
Peter