1. The problem statement, all variables and given/known data Let X and Y be independent exponential random variables with parameters a and b. Calculate E(X|X+Y). 2. Relevant equations 3. The attempt at a solution I'm pretty sure I have it, just want to make sure. Joint density for X and Y is abe^(-ax)e^(-by) for x,y>0. Let Z=X and W=X+Y so X=Z and Y=W-Z. The Jacobian is 1, so the joint density is abe^(-az)e^(-bw+bz)=abe^(+bz-az) e^(-bw) Marginal density for w is \int_0^ w abe^(bz-az) e^(-bw) dz=ab(e^(bw-aw)-1) /(b-a) So then conditional density is (b-a)e^(bz-az) e^(-bw)/(e^(bw-aw)-1) Then just integrate z from 0 to infinity to get e^(-bw)/(1-e^(bw-aw)) Is this okay?