Editing Hypergeometric distribution (section)

=== Probability mass function ===
The following conditions characterize the hypergeometric distribution:
* The result of each draw (the elements of the population being sampled) can be classified into one of [[Binary variable|two mutually exclusive categories]] (e.g. Pass/Fail or  Employed/Unemployed).
* The probability of a success changes on each draw, as each draw decreases the population (''[[sampling without replacement]]'' from a finite population).

A [[random variable]] <math>X</math> follows the hypergeometric distribution if its [[probability mass function]] (pmf) is given by<ref>{{Cite book
| edition   = Third
| publisher = Duxbury Press
| last      = Rice
| first     = John A.
| title     = Mathematical Statistics and Data Analysis
| year      = 2007
| page      = 42
}}</ref>

:<math> p_X(k) = \Pr(X = k) 
= \frac{\binom{K}{k} \binom{N - K}{n-k}}{\binom{N}{n}},</math>

where

*<math>N</math> is the population size,
*<math>K</math> is the number of success states in the population,
*<math>n</math> is the number of draws (i.e. quantity drawn in each trial),
*<math>k</math> is the number of observed successes,
*<math display="inline">\textstyle {a \choose b}</math> is a [[binomial coefficient]].

The {{Abbr|pmf|probability mass function}} is positive when <math>\max(0, n+K-N) \leq k \leq \min(K,n)</math>.

A random variable distributed hypergeometrically with parameters <math>N</math>, <math>K</math> and <math>n</math> is written <math display="inline">X \sim \operatorname{Hypergeometric}(N,K,n)</math> and has [[probability mass function]] <math display="inline"> p_X(k)</math> above.