Q:

Joint Probability distribution for $Z=X/(X+Y)$

Suppose X and Y are two independent random variables with exponential distributions Exp(1)
$Z=X/(X+Y)$
Find $P(Z<z)$ and show the random variable Z has uniform distribution. 
Thanks

A:

For $z\in\left(0,1\right)$ we find:
$$P\left(Z\geq z\right)=\int_{0}^{\infty}P\left(Z\geq z\mid Y=y\right)f_{Y}\left(y\right)dy=\int_{0}^{\infty}e^{\frac{-zy}{1-z}}e^{-y}dy=\int_{0}^{\infty}e^{\frac{-y}{1-z}}dy=1-z$$

