For any positive integer $n$ and positive real number $\theta$, the Gamma distribution $\Gamma(n, \theta)$ has density $f_{\Gamma}$ defined on $(0, \infty)$ by

$$f_{\Gamma}(x)=\frac{\theta^{n}}{(n-1) !} x^{n-1} e^{-\theta x} .$$

For any positive integers $a$ and $b$, the Beta distribution $B(a, b)$ has density $f_{B}$ defined on $(0,1)$ by

$$f_{B}(x)=\frac{(a+b-1) !}{(a-1) !(b-1) !} x^{a-1}(1-x)^{b-1}$$

Let $X$ and $Y$ be independent random variables with respective distributions $\Gamma(n, \theta)$ and $\Gamma(m, \theta)$. Show that the random variables $X /(X+Y)$ and $X+Y$ are independent and give their distributions.