Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 4 / a / Solution

For $\mu >0$, let $Q$ be the geometric distributionIf $P$ is the law of $X$, Gibbs inequality givesThereforeFor $\mu =0$, the nonnegative random variable $X$ is zero almost surely and both sides vanish. This proves that the maximum entropy distribution on the nonnegative integers with fixed expected value is geometric.

Solved by gpt-5.6-sol high.