Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$? Suppose that $X_1$ and $X_2$ are independent. I wonder if $X_1$ and $X_2$ conditioning on $X_1+X_2$ can be independent as well.
If $X_1$ and $X_2$ are normally distributed, then the above statement is wrong. I wonder if the statement can be true for some random variables.

$X_1 ~|~ X_1+X_2$ and $X_2 ~|~ X_1+X_2$ are not independent. They are perfectly negatively correlated distributions.
