2. (9 marks) Gamma random variables can be used to simulate chi-square, t, F, beta, and Dirichlet distributions, as well as being useful in their own right. Hence it is important to be able to generate gamma r.v.s as efficiently as possible. In this assignment we investigate a popular algorithm due to Marsaglia and Tsang, for o 2 1. (1) If X ~ Gamma(a, 1) then X/> ~ Gamma(a, 1). Note that Gamma(a, 3) has the pdf Box f(x) = T(a) 2-1exp (-Bx) . (2) Assume that h(x) is strictly increasing and maps the range of a onto [0, co). If X has density h(x)-le-h(@h'(x)/T(a) then Y = h(X) ~ Gamma(a, 1). (3) Given a 2 1 put d = 0 - 1/3 and c = 1/v9d and then define h : [-1/c, co) - [0, 20) by h(x) = d(1 + cr)3. Then, h(x)-le "(@)h'(x) x exp(g(x)) where g(x) = dlog((1 + cx)3) - d(1 + cx)3 + d and exp(g(x)) g g plot (qgamma (1 : 1000/1001, 1.2, 3), sort(g)) > abline (0, 1, col="red") 2