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:

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

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.