Assume that two independent random variables $X$ and $Y$ are Gamma-distributed such that $X \sim \Gamma(a,c)$ and $Y \sim \Gamma(b,c)$ with $a, b, c > 0$.
How can we see that two random variables $Z = \frac{X}{X+Y}$ and $W = X+Y$ are also independent?
I am trying to see that $P(Z \leq p, W \leq q) = P(Z \leq p)P(W\leq q)$, but I can not see how to compare the both sides. Is there any other way?
(This problem is one of exercises of my textbook without any answer.)
Note that $(Z,W)=\varphi(X,Y)$ , where $\varphi(x,y)=\left(\frac{x}{x+y},x+y\right) $ and $\varphi$ is a diffeomorphism, so we can use a formula for density of $\varphi(X,Y) $ : $$g(z,w)=\left| J\varphi^{-1}(z,w) \right|f(\varphi^{-1}(z,w))$$ , where $f$ is density of $(X,Y)$ and $J$ is Jacobian determinant and after some calculations we get: $$g(z,w)=\frac{1}{B(a,b)} z^{a-1}(1-z)^{b-1}I_{(0,1)}(z) \frac{c^{a+b}}{\Gamma(a+b)}w^{a+b-1}e^{-cw}I_{(0,\infty)}(w) $$ and we see that $Z$ and $W$ are independent and $Z \sim Beta(a,b)$ and $W \sim \Gamma(a+b,c)$
