Editing Hypergeometric distribution (section)

{{Short description|Discrete probability distribution}}
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{{Distinguish|Geometric distribution}}
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{{Infobox probability distribution
| name       = Hypergeometric
| type       = mass
| pdf_image  = [[File:HypergeometricPDF.png|300px|Hypergeometric PDF plot]]
| cdf_image  = [[File:HypergeometricCDF.png|300px|Hypergeometric CDF plot]]
| parameters = <math>\begin{align}N&\in \left\{0,1,2,\dots\right\} \\
                                 K&\in \left\{0,1,2,\dots,N\right\} \\
                                 n&\in \left\{0,1,2,\dots,N\right\}\end{align}\,</math>
| support  = <math>\scriptstyle{k\, \in\, \{\max{(0,\, n+K-N)},\, \dots,\, \min{(n,\, K)}\}}\,</math>
| pdf      = <math>\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}</math>
| cdf      = <math>1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}}\over {N \choose K}} \,_3F_2\!\!\left[\begin{array}{c}1,\ k+1-K,\ k+1-n \\ k+2,\ N+k+2-K-n\end{array};1\right],</math> where <math>\,_pF_q</math> is the [[generalized hypergeometric function]]
| mean     = <math>n {K \over N}</math>
| median   =
| mode     = <math>\left \lceil \frac{(n+1)(K+1)}{N+2} \right \rceil-1, \left \lfloor \frac{(n+1)(K+1)}{N+2} \right \rfloor</math>
| variance = <math>n{K\over N}{N-K\over N}{N-n\over N-1}</math>
| skewness = <math>\frac{(N-2K)(N-1)^\frac{1}{2}(N-2n)}{[nK(N-K)(N-n)]^\frac{1}{2}(N-2)}</math>
| kurtosis = <math> \left.\frac{1}{n K(N-K)(N-n)(N-2)(N-3)}\cdot\right.</math><br />
<math>\big[(N-1)N^{2}\big(N(N+1)-6K(N-K)-6n(N-n)\big)</math><br />
<math>{}+6 n K (N-K)(N-n)(5N-6)\big]</math>
| entropy =
| mgf     = <math>\frac{\binom{N-K}{n} \,_2F_1(-n, -K\,;\, N - K - n + 1\,;\, e^{t})}{\binom{N}{n}}</math>
| char = <math>\frac{\binom{N-K}{n} \,_2F_1(-n, -K\,;\, N - K - n + 1\,;\, e^{it})}
{\binom{N}{n}}</math>
}}

In [[probability theory]] and [[statistics]], the '''hypergeometric distribution''' is a [[Probability distribution#Discrete probability distribution|discrete probability distribution]] that describes the probability of <math>k</math> successes (random draws for which the object drawn has a specified feature) in <math>n</math> draws, ''without'' replacement, from a finite [[Statistical population|population]] of size <math>N</math> that contains exactly <math>K</math> objects with that feature, wherein each draw is either a success or a failure. In contrast, the [[binomial distribution]] describes the probability of <math>k</math> successes in <math>n</math> draws ''with'' replacement.