Difference between revisions of "Negative hypergeometric distribution"
(formula formatting - still nicer) |
Erel Segal (talk | contribs) (replace some images with latex equations) |
||
| Line 3: | Line 3: | ||
P(X=k)=\frac{ {k+m-N \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*} | P(X=k)=\frac{ {k+m-N \choose k}{N-m-k \choose M-m} } { {N \choose M} } \tag{*} | ||
\end{equation} | \end{equation} | ||
| − | where the parameters < | + | 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 [[Negative binomial distribution|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 | 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 | and | ||
| − | + | \begin{equation} | |
| + | m\frac{(N+1)(N-M)} {(M+1)(M+2)}[1-\frac{m}{M+1}] | ||
| + | \end{equation} | ||
| − | When < | + | When <math>N, M, N-M \to \infty</math> such that <math>M/M\to p</math>, the negative hypergeometric distribution tends to the [[negative binomial distribution]] with parameters <math>m</math> and <math>p</math>. |
| − | The distribution function < | + | The distribution function <math>F(n)</math> of the negative hypergeometric function with parameters <math>N,M,m</math> is related to the [[Hypergeometric distribution|hypergeometric distribution]] <math>G(m)</math> with parameters <math>N,M,n</math> 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|statistical quality control]]. | 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|statistical quality control]]. | ||
Revision as of 07:10, 26 March 2015
A probability distribution of a random variable 
which takes non-negative integer values, defined by the formula
\begin{equation}
P(X=k)=\frac{ {k+m-N \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)}[1-\frac{m}{M+1}] \end{equation}
When \(N, M, N-M \to \infty\) such that \(M/M\to p\), 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
| [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) |
Negative hypergeometric distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_hypergeometric_distribution&oldid=36355