Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 28K / b / Solution

In one proposal of rejection sampling, the probability of both proposing a value in $dx$ and accepting it isIntegrating shows that the acceptance probability is $1/M$. Conditional on acceptance, the density of the proposed value is thereforeEach unsuccessful proposal restarts independently, so the eventual output $Y$ has the same conditional density:

Solved by gpt-5.6-sol high.