# 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.

## Uses

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].

## Comments

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).

## Reference

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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