# Inverse Hypergeometric distribution

Format: InvHypergeo(s, D, M,)

The Inverse Hypergeometric distribution InvHypergeo(s,D,M) models the number of failures one would have before achieving the sth success in a hypergeometric sampling where there are D individuals of interest (their selection is a 'success') in a population of size M. Four examples of the Inverse Hypergeometric distribution are shown below:

## Uses

It should be used in any situation where there is Hypergeometric sampling and one is asking the question: "How many failures will I observe before I get s successes?", or alternatively "How many samples do I need to have s successes?".

## Example

Imagine that a credit card company estimates that 20 of its 450 platinum card holders have gained their wealth illegally. A banking regulator is to perform an audit. How many randomly selected platinum card holders will they have to investigate before finding a criminal? Answer: InvHypergeo(1,20,450) + 1.

## Comments

The Inverse Hypergeometric goes by a variety of other names: Negative Hypergeometric distribution, Hypergeometric Waiting Time distribution, and the Markov-Pólya distribution.

## Approximations

The computational burden of constructing the Inverse Hypergeometric distribution makes it an appealing candidate to be approximated. Where the Hypergeometric process is closely approximated by a Binomial process (roughly, where the sample size is less than 10% of the population size), the Inverse Hypergeometric distribution is approximated by a Negative Binomial. Similarly, where D/M is very small, the Inverse Hypergeometric distribution is approximated by the Gamma.

## Zero-modified version

When modeling or analyzing counting data, it is often desirable to modify the probability of zero of the discrete distribution we use, to more accurately model the probability of "no event occurring". We can make two types of modifications to our distribution for this:

• Zero-inflated model - we increase the probability of zero.

• Zero-truncated model - we entirely remove the probability of zero events occurring.

## ModelRisk functions added to Microsoft Excel for the Inverse Hypergeometric distribution

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

VoseInvHypergeoObject constructs a distribution object for this distribution.

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

VoseInvHypergeoProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseInvHyperGeoFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseInvHyperGeoFitObject constructs a distribution object of this distribution fitted to data.

VoseInvHyperGeoFitFitP returns the parameters of this distribution fitted to data.

## ModelRisk functions added to Microsoft Excel for the Zero-Inflated Inverse Hypergeometric distribution

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

VoseZIInvHypergeoObject constructs a distribution object for this distribution.

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

VoseZIInvHypergeoProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZIInvHyperGeoFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZIInvHyperGeoFitObject constructs a distribution object of this distribution fitted to data.

VoseZIInvHyperGeoFitFitP returns the parameters of this distribution fitted to data.

## ModelRisk functions added to Microsoft Excel for the Zero-Truncated Inverse Hypergeometric distribution

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

VoseZTInvHypergeoObject constructs a distribution object for this distribution.

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

VoseZTInvHypergeoProb10 returns the log10 of the probability mass or cumulative distribution function.

VoseZTInvHyperGeoFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseZTInvHyperGeoFitObject constructs a distribution object of this distribution fitted to data.

VoseZTInvHyperGeoFitFitP returns the parameters of this distribution fitted to data.

## Zero-Truncated Inverse Hypergeometric distribution equations

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