Suppose that X and Y are independent random variables, X has the gamma distribution with parameters 1

Suppose that X and Y are independent random variables, X has the gamma distribution with parameters α1 and β, and Y has the gamma distribution with parameters α2 and β. Let U = X/(X + Y) and V = X + Y. Show that (a) U has the beta distribution with parameters α1 and α2, and (b) U and V are independent. Look at the steps in the proof of Theorem 5.8.1.