Dakota Reference Manual: hypergeometric_uncertain

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hypergeometric_uncertain

Aleatory uncertain discrete variable - hypergeometric

## Topics

This keyword is related to the topics:

- discrete_variables
- aleatory_uncertain_variables

## Specification

Alias: none

Argument(s): INTEGER

Default: no hypergeometric uncertain variables

Child Keywords:

| Required/Optional | Description of Group | Dakota Keyword | Dakota Keyword Description
--- | --- | --- | --- | ---
| Required | total_population | Parameter for the hypergeometric probability distribution describing the size of the total population
| Required | selected_population | Distribution parameter for the hypergeometric distribution describing the size of the population subset of interest
| Required | num_drawn | Distribution parameter for the hypergeometric distribution describing the number of draws from a combined population
| Optional | initial_point | Initial values for variables
| Optional | descriptors | Labels for the variables

## Description

The hypergeometric probability density is used when sampling without replacement from a total population of elements where

- The resulting element of each sample can be separated into one of two non-overlapping sets
- The probability of success changes with each sample.

The density function for the hypergeometric distribution is given by:

where the three distribution parameters are:

- N is the total population
- m is the number of items in the selected population (e.g. the number of white balls in the full urn of N items)
- n is the size of the sample drawn (e.g. number of balls drawn)

In addition,

- x, the abscissa of the density function, indicates the number of successes (e.g. drawing a white ball)
- indicates a binomial coefficient ("a choose b")

## Theory

The hypergeometric is often described using an urn model. For example, say we have a total population containing N balls, and we know that m of the balls are white and the remaining balls are green. If we draw n balls from the urn without replacement, the hypergeometric distribution describes the probability of drawing x white balls.