Past exam of the simulation and statistical programming course of the University of Oxford / 2023 / 1 / b / ii / Solution

Stop rejection sampling when exactly $N=66,564$ target draws have been accepted. The total proposal count $T_N$ is a sum of $N$ independent waiting times with geometric distribution, equivalently a negative binomial distribution counting trials. ConsequentlyThe accepted-sample Monte Carlo estimator then has the approximate $99\%$ precision from (a)(iii). The displayed proposal cost is an expected value, which need not be an integer. Spending this many proposals in advance does not ensure that $N$ target draws will have been accepted.