Dakota Reference Manual  Version 6.15
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hypergeometric_uncertain


Aleatory uncertain discrete variable - hypergeometric

Topics

This keyword is related to the topics:

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:

\[f(x) = \frac{\left(\begin{array}{c}m\\x\end{array}\right)\left(\begin{array}{c}{N-m}\\{n-x}\end{array}\right)}{\left(\begin{array}{c}N\\n\end{array}\right)}\]

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)
  • $\left(\begin{array}{c}a\\b\end{array}\right)$ 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.