Let f(x) and g(x) be probability density functions, and suppose that for some constant

c, f(x) = cg(x) for all x. Suppose we can generate random variables having density function g, and consider the following algorithm. Step 1: Generate Y, a random variable having density function g. Step 2: Generate U, a uniform (0, 1) random variable. Step 3: If U (Y) cg(Y) set X = Y. Otherwise, go back to Step 1. Assuming that successively generated random variables are indepen- dent, show that:

(a) X has density function f

(b) the number of iterations of the algorithm needed to generate X is a geometric random variable with mean c.