Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 2 / f / Solution

Multiplying the reciprocal improper prior by the likelihood function givesThus this discrete uniform endpoint posterior is proper. Its exact cumulative probability is $S_y^{-1}{\sum }_{k=y}^n{k}^{-2}$. Integral approximation gives $S_y\approx {\int }_y^{\infty }{t}^{-2}dt=1/y$ and ${\sum }_{k=y}^n{k}^{-2}\approx {\int }_y^n{t}^{-2}dt=1/y-1/n$, soThese approximations are useful when $y$ is moderately large and are not exact discrete identities. For example the exact untruncated median at $y=100$ is $199$. Although its quantiles are finite, its posterior mean is infinite, since its expectation numerator is the divergent sum ${\sum }_{N=y}^{\infty }1/N$.