Suppose we have two random variables, X and Y, which have the same distribution (Gaussian) but different parameters. We define a third RV, Z=X+Y. Of course, the mean of Z is the sum of the means of X, Y, and the S.D. of Z is the sum of the S.D.s of X, Y.
Then we observe z $\in$ Z. How can we figure out the expected component x, given z? I guess I'm wondering where in the X distribution the value that contributes to z is most likely to fall. (As a function of means and S.D.s of X, Y) I'm stumped. Thanks
HINT
It sounds like what you are looking for is $$ \mathbb{P}[X \le x|X+Y=z] = F_{X|X+Y}(x|z) $$ and then you can compute $\mathbb{E}[X|X+Y=z]$.