Given Independent $X, Y$, Prove $X+Y=W$ and $\frac{X}{X+Y}=Z$ are independent if $X$ and $Y$ are identical exponential distributions [duplicate]
Solution 1:
You ca use jacobian method. You already know that $W\sim\text{Gamma}[2;\lambda]$ thus when you get $f(w;z)$ you realize that $Z\sim U(0;1)$ independent from W
Another way to prove independence is to use Basu's theorem.
First observe that exponential distribution is a scale family. $W$ is Complete and Sufficient while $Z$, scale invariant, is Ancillary. Thus invoking Basu's theorem, $Z;W$ are independent