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Under the discrete uniform distribution, the observation has probability $1/N$ when its value is an available label, and zero otherwise. Therefore the likelihood function isThis is the discrete uniform endpoint posterior model's likelihood; all endpoints smaller than the observed label are excluded.

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 ${\sum }_{N=y}^{\infty }1/N$.