Let X be a Gamma(n, β) rv; n is a positive integer and β is a known number, e.g., β = 17. Show that V = 2X/β has χ2 with 2n degrees of freedom distribution.

1. use the brute force approach

2. use the mgf approach

3. Let Y be a Gamma(n, β) rv independent of X. What is the distribution of X/Y ? Justify your answer by showing that the distribution of X/Y is the same as that of ratio of independent χ2 rvs. Hint: use your result in subproblems 1 or 2. Do not try to do the transformation using Jacobians or conditioning.