Chapter 6.6, Problem 63E

In Example 6.14, Y₁ and Y₂ were independent exponentially distributed random variables, both with mean β. We defined U₁ = Y₁/(Y₁ + Y₂) and U₂ = Y₁ + Y₂ and determined the joint density of (U₁, U₂) to be

$f_{U_1,U_2}(u_1,u_2)=\{\begin{array}{ll}\frac{1}{{\beta }^2}{u}_2{e}^{-u_2/\beta },\hfill & 0<u_1<1,0<u_2,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$

1. a Show that U₁ is uniformly distributed over the interval (0, 1).
2. b Show that U₂ has a gamma density with parameters α = 2 and β.
3. c Establish that U₁ and U₂ are independent.