Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 36 / 4 / c

Apply independent uniforms to the $n$ proposals, and let $I_i$ indicate acceptance. Each $I_i$ is Bernoulli with success probability $p=1/M$, and the indicators are independent because the pairs $(X_i,U_i)$ are independent. HenceFor a fixed acceptance pattern, the values at accepted positions are independent with density $f$, by the single-trial conditional density calculation. The same product law holds for every pattern of a given size. Consequently, conditional on $N=m$, the ordered accepted observations have joint density ${\prod }_{j=1}^{m}f(y_j)$. This is the binomial count and iid values in fixed-budget rejection sampling; the count provides no information about those target values.

There is a finite-sample empty-output event, with probability $(1-1/M{)}^n$. Any estimator dividing by $N$ must be defined separately on $N=0$.