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Difference between revisions of "Negative hypergeometric distribution"

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A [[Probability distribution|probability distribution]] of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662201.png" /> which takes non-negative integer values, defined by the formula
 
A [[Probability distribution|probability distribution]] of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662201.png" /> which takes non-negative integer values, defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">
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<math>P(X=k)=\frac{ {k+m-N \choose k}{N-m-k \choose M-m} } { {N \choose M} }</math>  
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</td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
  
 
where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662203.png" /> are non-negative integers which satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662204.png" />. A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662205.png" />, there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662206.png" />  "marked"  and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662207.png" />  "unmarked"  elements, and if the sampling (without replacement) is performed until the number of  "marked"  elements reaches a fixed number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662208.png" />, then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662209.png" /> — the number of  "unmarked"  elements in the sample — has a negative hypergeometric distribution (*). The random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n06622010.png" /> — the size of the sample — also has a negative hypergeometric distribution. The distribution (*) is called a negative hypergeometric distribution by analogy with the [[Negative binomial distribution|negative binomial distribution]], which arises in the same way for sampling with replacement.
 
where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662203.png" /> are non-negative integers which satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662204.png" />. A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662205.png" />, there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662206.png" />  "marked"  and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662207.png" />  "unmarked"  elements, and if the sampling (without replacement) is performed until the number of  "marked"  elements reaches a fixed number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662208.png" />, then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n0662209.png" /> — the number of  "unmarked"  elements in the sample — has a negative hypergeometric distribution (*). The random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066220/n06622010.png" /> — the size of the sample — also has a negative hypergeometric distribution. The distribution (*) is called a negative hypergeometric distribution by analogy with the [[Negative binomial distribution|negative binomial distribution]], which arises in the same way for sampling with replacement.

Revision as of 19:30, 24 March 2015

A probability distribution of a random variable which takes non-negative integer values, defined by the formula

\(P(X=k)=\frac{ {k+m-N \choose k}{N-m-k \choose M-m} } { {N \choose M} }\) 
(*)

where the parameters are non-negative integers which satisfy the condition . A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size , there are "marked" and "unmarked" elements, and if the sampling (without replacement) is performed until the number of "marked" elements reaches a fixed number , then the random variable — the number of "unmarked" elements in the sample — has a negative hypergeometric distribution (*). The random variable — the size of the sample — also has a negative hypergeometric distribution. The distribution (*) is called a negative hypergeometric distribution by analogy with the negative binomial distribution, which arises in the same way for sampling with replacement.

The mathematical expectation and variance of a negative hypergeometric distribution are, respectively, equal to

and

When such that , , , the negative hypergeometric distribution tends to the negative binomial distribution with parameters and .

The distribution function of the negative hypergeometric function with parameters is related to the hypergeometric distribution with parameters by the relation

This means that in solving problems in mathematical statistics related to negative hypergeometric distributions, tables of hypergeometric distributions can be used. The negative hypergeometric distribution is used, for example, in statistical quality control.

References

[1] Y.K. Belyaev, "Probability methods of sampling control" , Moscow (1975) (In Russian)
[2] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)


Comments

References

[a1] N.L. Johnson, S. Kotz, "Distributions in statistics, discrete distributions" , Wiley (1969)
[a2] G.P. Patil, S.W. Joshi, "A dictionary and bibliography of discrete distributions" , Hafner (1968)
How to Cite This Entry:
Negative hypergeometric distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_hypergeometric_distribution&oldid=19018
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article