en>LindsayH: WPCleaner v1.27 - Headlines start with three "=" and later with level two (Fixed using WP:WCW)

2013-06-21T19:23:42Z

WPCleaner v1.27 - Headlines start with three "=" and later with level two (Fixed using WP:WCW)

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{{Disputed|date=November 2013}}
{{orphan|date=June 2013}}


{{Probability distribution |
name =Negative hypergeometric|
type =mass|
pdf_image =|
cdf_image =|
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\, \left\{\max{(0,\, n+K-N)},\, \dots,\, \min{(K,\, n )}\right\}}\,</math>|
pdf =<math>{{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose 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>|
mean =<math>n {K\over N}</math>|
median =|
mode =<math>\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>
<math>\Big[(N-1)N^{2}\Big(N(N+1)-6K(N-K)-6n(N-n)\Big)+</math>
<math>6 n K (N-K)(N-n)(5N-6)\Big]</math>|
entropy =|
mgf =<math>\frac{{N-K \choose n} \scriptstyle{\,_2F_1(-n, -K; N - K - n + 1; e^{t}) } }
{{N \choose n}} \,\!</math>|
char =<math>\frac{{N-K \choose n} \scriptstyle{\,_2F_1(-n, -K; N - K - n + 1; e^{it}) }}
{{N \choose n}} </math>
}}
In [[probability theory]] and [[statistics]], the '''hypergeometric distribution''' is a discrete [[probability distribution]] that describes the probability of <math>k</math> successes in <math>n</math> draws ''without'' replacement from a finite [[population]] of size <math>N</math> containing a maximum of <math>K</math> successes. This is in contrast to the [[binomial distribution]], which describes the probability of <math>k</math> successes in <math>n</math> draws ''with'' replacement.

==Definition==

The '''negative hypergeometric distribution''' describes the probability of the number of elements taken without replacement from a finite population whose elements can be classified into two mutually exclusive categories like Pass/Fail, Male/Female or Employed/Unemployed that stops when a fixed number of elements of certain class have been taken. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw.

== References ==
*{{cite doi|10.1016/j.jda.2006.01.001}}

* Skala, M. (2011) [http://ansuz.sooke.bc.ca/professional/hypergeometric.pdf "Hypergeometric tail inequalities: ending the insanity"], unpublished note

== External links ==
* [http://demonstrations.wolfram.com/TheHypergeometricDistribution/ The Hypergeometric Distribution] and [http://demonstrations.wolfram.com/BinomialApproximationToAHypergeometricRandomVariable/ Binomial Approximation to a Hypergeometric Random Variable] by Chris Boucher, [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Hypergeometric Distribution |urlname=HypergeometricDistribution}}

{{ProbDistributions|discrete-finite}}
{{Common univariate probability distributions}}

{{DEFAULTSORT:Hypergeometric Distribution}}
[[Category:Discrete distributions]]
[[Category:Factorial and binomial topics]]
[[Category:Probability distributions]]

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en>LindsayH: WPCleaner v1.27 - Headlines start with three "=" and later with level two (Fixed using WP:WCW)