First, give the probability that a draw is accepted, and the distribution of the number of draws...

The aim of this exercise is to prove that Algorithm 8.1 samples from Q. First, give the probability that a draw is accepted, and the distribution of the number of draws required have one draw accepted. Then, let q˜ denote the density of the distribution generated by this Algorithm. Use a recursive argument to show that q(x) ˜ = m(x){q(x)/Cm(x)}+ ˜q(x)(1 - 1/C), and deduce the result. See also Exercise 8.3 for an alternative proof