hypergeometric_uncertain

Aleatory uncertain discrete variable - hypergeometric

Topics

discrete_variables, aleatory_uncertain_variables

Specification

  • Alias: None

  • Arguments: 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:

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

where the three distribution parameters are:

  • \(N\): the total population

  • \(m\): the number of items in the selected population (e.g. the number of white balls in the full urn of \(N\) items)

  • \(n\) 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.