Chapter 5.2, Problem 7E

Let $X_1$ and $X_2$ be independent chi-square random variables with $r_1$ and $r_2$ degrees of freedom, respectively. Show that

(a) $U=\frac{X_1}{\left(X_1+X_2\right)}$ has a beta distribution with $\alpha =\frac{r_1}{2}$ and $\beta =\frac{r_2}{2}$.

(b) $V=\frac{X_2}{\left(X_1+X_2\right)}$ has a beta distribution with $\alpha =\frac{r_2}{2}$ and $\beta =\frac{r_1}{2}$.