Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 1 / b / Solution

Multiplying the likelihood function by the Pareto distribution prior givesThe integral of this kernel is $m^{-(\alpha +n)}/(\alpha +n)$, so the normalized posterior distribution isThus it is $\mathrm{Pareto}(\alpha +n,m)$. This proves uniform-Pareto conjugacy: applying Bayes theorem preserves the family of prior distributions.