I have what I'm afraid is a simple stats problem that is stumping me. I have two random variables, X and Y, independently normally distributed:
X ~ N(0, sigmaX) Y ~ N(0, sigmaY).
I observe the sum of these two variables, Z = X+Y, and want to develop a conditional expectation on X given the sum. A colleague said, "ah, yes, classic signal-extraction problem. Solution is:"
E[X|X+Y] = (X + Y) * sigmaX / (sigmaX + sigmaY)
This looked about right so I thanked him and figured I work it out at home. It appears I'm a little rusty here, though. I can give verbal reasoning why this would be true but can't write down the math. What is the mathematical reason this is true?