$X,Y,Z$ are standard normal R.V.
What is the value of $\operatorname{E}[X|X+Y+Z=1]$ and $\operatorname{Var}(X|X+Y+Z=1)$?
I think the first one should be $1/3$ by symmetry but don't know how to tackle the second part.
Any thought on this? Thanks a lot!