Distribution to estimate a population size from hypergeometric sampling | Vose Software

Distribution to estimate a population size from hypergeometric sampling


Format: HypergeoM(s, n, M, max)


The HypergeoM(s, n, M, max) is a discrete bounded distribution used to estimate the size of a population  in a Hypergeometric process.


The HypergeoM distribution models the size of a population M when one knows the sub-population size D, the size of a random sample taken n, and the number in that sample that were from the sub-population s.

Since we have already observed n from this population, and (D - s)  are known to exist though they weren’t in the sample, the HypergeoM is bounded on [n + D - s, max].


There are four parameters for a Hypergeometric process: s, n, D, M.  Knowing any three allows us to construct a distribution to estimate the fourth – see Hypergeometric process.

With knowledge of s,n and D we still have no information that allows us to place a maximum bound on the population size M. The max parameter gets round this, but places the emphasis on you – the user – to select a suitable maximum value.

The distribution was first described in Vose (2000).



Vose, David (2000). Risk analysis - a quantitative guide

ModelRisk functions added to Microsoft Excel for the Hypergeometric-M distribution

VoseHypergeoM generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseHypergeoMObject constructs a distribution object for this distribution.

VoseHypergeoMProb returns the probability mass or cumulative distribution function for this distribution.

VoseHypergeoMProb10 returns the log10 of the probability mass or cumulative distribution function.


HypergeoM distribution equations



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