Question

Let X and Y be

independent random variables and both exponentially distributed with mean 1/μ. Prove that the distribution of a new random variable Z= X/(X+Y)  is uniform over(0,1), i.e., prove P{Z ≤ z} = z for 0 < z ≤ 1. 

Note: There are at least two ways to show this. One is to directly integrate over some domain based on the joint pdf of X, Y , and the other is to utilize the conditional expectation theorem, i.e., first to make one of X, Y constant and deal with it, and