.. _variables-hypergeometric_uncertain:

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hypergeometric_uncertain
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Aleatory uncertain discrete variable - hypergeometric




**Topics**


discrete_variables, aleatory_uncertain_variables



.. toctree::
   :hidden:
   :maxdepth: 1

   variables-hypergeometric_uncertain-total_population
   variables-hypergeometric_uncertain-selected_population
   variables-hypergeometric_uncertain-num_drawn
   variables-hypergeometric_uncertain-initial_point
   variables-hypergeometric_uncertain-descriptors


**Specification**

- *Alias:* None

- *Arguments:* INTEGER

- *Default:* no hypergeometric uncertain variables


**Child Keywords:**

+-------------------------+--------------------+-------------------------+-----------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword          | Dakota Keyword Description                    |
|                         | Group              |                         |                                               |
+=========================+====================+=========================+===============================================+
| 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                      |
+----------------------------------------------+-------------------------+-----------------------------------------------+

.. __: variables-hypergeometric_uncertain-total_population.html
__ variables-hypergeometric_uncertain-selected_population.html
__ variables-hypergeometric_uncertain-num_drawn.html
__ variables-hypergeometric_uncertain-initial_point.html
__ variables-hypergeometric_uncertain-descriptors.html



**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:

.. math:: 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:

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

In addition,

- :math:`x`, the abscissa of the density function, indicates the number of successes (e.g. drawing a white ball)
- :math:`\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 :math:`N` balls, and we know that :math:`m` of the balls are white and the remaining balls are green. If we draw :math:`n` balls from the urn without replacement, the hypergeometric distribution describes the probability of drawing :math:`x` white balls.