# 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.

• It will be helpful if you type this explicitly mathematically as a theorem. Commented Jul 3, 2021 at 10:15

$$X_1 ~|~ X_1+X_2$$ and $$X_2 ~|~ X_1+X_2$$ are not independent. They are perfectly negatively correlated distributions.
• $X_1 + X_2$ is an instance of a "collider" for $X_1$ and $X_2$. This is discussed in: Day, Felix R., et al. "A robust example of collider bias in a genetic association study." The American Journal of Human Genetics 98.2 (2016): 392-393. See also en.wikipedia.org/wiki/Collider_(statistics) . Commented Jul 6, 2021 at 11:51
It's possible if one of them is constant - for example if $$X_1$$ has a Bernoulli distribution and $$X_2$$ is always equal to zero.