Blurb::
Aleatory uncertain discrete variable - hypergeometric

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")

Topics::
discrete_variables, aleatory_uncertain_variables

Examples::

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.

Faq::

See_Also::
