Chapter 5.2, Problem 6E

Let $X_1$ and $X_2$ have independent gamma distributions with parameters $\alpha ,\theta$ and $\beta ,\theta ,$ respectively. Let $W=\frac{X_1}{\left(X_1+X_2\right)}$.

Use a method similar to that given in the derivation of the F distribution (Example 5.2-4) to show that the pdf of W is $g\left(w\right)=\frac{\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{w}^{\alpha -1}{\left(1-w\right)}^{\beta -1},0<w<1.$ We say that W has a beta distribution with parameters $\alpha$ and $\beta$.

(See Example 5.2-3.)