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{{MSC|62E}}
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{{TEX|done}}
  
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A
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[[Probability distribution|probability distribution]] of a random variable $X$ which takes non-negative integer values, defined by the formula
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\begin{equation}\label{*}
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P(X=k)=\frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*}
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\end{equation}
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where the parameters <math>N,M,m</math> are non-negative integers which satisfy the condition <math>m\leq M\leq N</math>. A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size <math>N</math>, there are <math>M</math>  "marked"  and <math>N-M</math>  "unmarked"  elements, and if the sampling (without replacement) is performed until the number of  "marked"  elements reaches a fixed number <math>m</math>, then the random variable <math>X</math> — the number of  "unmarked"  elements in the sample — has a negative hypergeometric distribution \eqref{*}. The random variable <math>X+m</math> — the size of the sample — also has a negative hypergeometric distribution. The distribution \eqref{*} is called a negative hypergeometric distribution by analogy with the
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[[Negative binomial distribution|negative binomial distribution]], which arises in the same way for sampling with replacement.
  
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The mathematical expectation and variance of a negative hypergeometric distribution are, respectively, equal to
  
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\begin{equation}
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m\frac{N-M} {M+1}
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\end{equation}
  
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and
  
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\begin{equation}
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m\frac{(N+1)(N-M)} {(M+1)(M+2)}\Big(1-\frac{m}{M+1}\Big) \, .
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\end{equation}
  
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When <math>N, M, N-M \to \infty</math> such that <math>M/N\to p</math>, the negative hypergeometric distribution tends to the
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[[negative binomial distribution]] with parameters <math>m</math> and <math>p</math>.
  
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The distribution function <math>F(n)</math> of the negative hypergeometric function with parameters <math>N,M,m</math> is related to the
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[[Hypergeometric distribution|hypergeometric distribution]] <math>G(m)</math> with parameters <math>N,M,n</math> by the relation
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\begin{equation}
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F(n) = 1-G(m-1) \, .
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\end{equation}
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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
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[[Statistical quality control|statistical quality control]].
  
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====References====
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{|
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|-
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==Major contributions==
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|valign="top"|{{Ref|Be}}||valign="top"|  Y.K. Belyaev,  "Probability methods of sampling control", Moscow  (1975)  (In Russian) {{MR|0428663}} 
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|-
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By [[User:Boris Tsirelson|Boris Tsirelson]], December 2011-April 2012:
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|valign="top"|{{Ref|BoSm}}||valign="top"|  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) {{MR|0243650}} {{ZBL|0529.62099}}
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|-
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[[Measurable space]]
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|valign="top"|{{Ref|JoKo}}||valign="top"|  N.L. Johnson,  S. Kotz,  "Distributions in statistics, discrete distributions", Wiley  (1969) {{MR|0268996}} {{ZBL|0292.62009}}
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+ [[Standard Borel space]]
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|-
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+ [[Analytic Borel space]]
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|valign="top"|{{Ref|PaJo}}||valign="top"|  G.P. Patil,  S.W. Joshi,   "A dictionary and bibliography of discrete distributions", Hafner  (1968) {{MR|0282770}}
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+ [[Universally measurable]]
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|-
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+ [[Measure space]]
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|}
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+ [[Standard probability space]]
 
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+ [[Measure algebra (measure theory)]]
 
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[[User:Jjg]] (no)
 
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[[User:Jjtorrens]] (no)
 
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By [[User:Yakovenko|Yakovenko]], April-May 2012:
 
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[[Regular singular point]]
 
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+ [[Fuchsian singular point]]
 
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+ [[Normal form]]
 
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+ [[Local normal forms for dynamical systems]]
 
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+ [[Algebraic decidability of local classification problems]]
 
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+ [[Darboux theorem]]
 
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+ [[Multi-index notation]]
 
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+ [[Whitney extension theorem]]
 
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+ [[Borel theorem]]
 
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+ [[Darboux integral]]
 
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+ [[Distribution of tangent subspaces]]
 
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+ [[Separatrix]]
 
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+ [[Limit cycle]]
 
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+ [[Topological equivalence]]
 
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+ [[Parallel transport]]
 
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+ [[Bundle]]
 
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+ [[Tangent space]];
 
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see also [[User:Yakovenko/sandbox1|Connection]]
 
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[[User:Camillo.delellis]], July-? 2012:
 
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[[Convergence of measures]]
 
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+ [[Radon measure]]
 
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+ [[Riesz representation theorem]]
 
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+ [[Signed measure]]
 
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+ [[Absolute continuity]]
 
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+ [[Baire space]]
 
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+ [[Density of a set]]
 
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+ [[Rectifiable set]]
 
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+ [[Differentiation of measures]]
 
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+ [[Rademacher theorem]]
 
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+ [[Jordan decomposition (of a function)]]
 
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+ [[Jordan decomposition (of a signed measure)]]
 
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+ [[Function of bounded variation]]
 
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+ [[Cantor ternary function]]
 
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+ [[Variation of a function]]
 
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+ [[Atom]]
 
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+ [[Outer measure]]
 

Latest revision as of 14:46, 5 June 2017



2020 Mathematics Subject Classification: Primary: 62E [MSN][ZBL]


A probability distribution of a random variable $X$ which takes non-negative integer values, defined by the formula \begin{equation}\label{*} P(X=k)=\frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*} \end{equation} where the parameters \(N,M,m\) are non-negative integers which satisfy the condition \(m\leq M\leq N\). A negative hypergeometric distribution often arises in a scheme of sampling without replacement. If in the total population of size \(N\), there are \(M\) "marked" and \(N-M\) "unmarked" elements, and if the sampling (without replacement) is performed until the number of "marked" elements reaches a fixed number \(m\), then the random variable \(X\) — the number of "unmarked" elements in the sample — has a negative hypergeometric distribution \eqref{*}. The random variable \(X+m\) — the size of the sample — also has a negative hypergeometric distribution. The distribution \eqref{*} 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

\begin{equation} m\frac{N-M} {M+1} \end{equation}

and

\begin{equation} m\frac{(N+1)(N-M)} {(M+1)(M+2)}\Big(1-\frac{m}{M+1}\Big) \, . \end{equation}

When \(N, M, N-M \to \infty\) such that \(M/N\to p\), the negative hypergeometric distribution tends to the negative binomial distribution with parameters \(m\) and \(p\).

The distribution function \(F(n)\) of the negative hypergeometric function with parameters \(N,M,m\) is related to the hypergeometric distribution \(G(m)\) with parameters \(N,M,n\) by the relation \begin{equation} F(n) = 1-G(m-1) \, . \end{equation} 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

[Be] Y.K. Belyaev, "Probability methods of sampling control", Moscow (1975) (In Russian) MR0428663
[BoSm] 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) MR0243650 Zbl 0529.62099
[JoKo] N.L. Johnson, S. Kotz, "Distributions in statistics, discrete distributions", Wiley (1969) MR0268996 Zbl 0292.62009
[PaJo] G.P. Patil, S.W. Joshi, "A dictionary and bibliography of discrete distributions", Hafner (1968) MR0282770
How to Cite This Entry:
Boris Tsirelson/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox&oldid=28400
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