Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 43 / 1 / a / iii

The acceptance probability is $1/C$, soFor the given proposal probability density function $g$, every valid envelope must have $C\ge \sup f/g=M$. Rejection increases with $C$, so the optimal envelope is $C=M$, giving rejection probability $1-1/M$. This optimizes the constant for the fixed proposal; it does not assert that $g$ is the best possible proposal. For densities specified only almost everywhere, an essential supremum is sufficient.