Hypergeometric

Suppose that, from a population of M elements, of which x possess a certain attribute, you draw a sample of n items without replacement. The number of items that possess the certain attribute in such a sample is a hypergeometric variate. The hypergeometric cumulative distribution function is a discrete function defined for nonnegative integers x. The hypergeometric distribution with M elements in the population, K successes in the population, and sample size n is defined by the following summation of quotients of binomial coefficients for 0≤xn.

$\displaystyle \func$HypergeomDist$\displaystyle \left(\vphantom{ x;M,K,n}\right.$x;M, K, n$\displaystyle \left.\vphantom{ x;M,K,n}\right)$ = $\displaystyle \sum_{{k=0}}^{{x}}$$\displaystyle {\frac{{\binom{K}{k}%
\binom{M-K}{n-k}}}{{\binom{M}{n}}}}$

For x < 0, the distribution function is 0, and for xn, the function is 1. The hypergeometric probability density function is

$\displaystyle \limfunc$HypergeomDen$\displaystyle \left(\vphantom{ k;M,K,n}\right.$k;M, K, n$\displaystyle \left.\vphantom{ k;M,K,n}\right)$ = $\displaystyle {\frac{{\binom{K}{k}\binom{M-K}{%
n-k}}}{{\binom{M}{n}}}}$

for integers k , K , n, and M satisfying 0≤kn, 0≤KM, and 0 < nM.


\begin{example}
What is the probability of at most five successes when you draw...
...sses,
or $\func{HypergeomDist}\left( 5;100,30,10\right) =.96123$
\end{example}

The hypergeometric distribution is the model for sampling without replacement. The hypergeometric distribution can be approximated by the binomial distribution when the sample size is relatively small.

These plots (created as polygonal plots) depict $\func$HypergeomDen$\left(\vphantom{
x;100,30,10}\right.$x;100, 30, 10$\left.\vphantom{
x;100,30,10}\right)$  and $\func$HypergeomDist$\left(\vphantom{
x;100,30,10}\right.$x;100, 30, 10$\left.\vphantom{
x;100,30,10}\right)$ for 0≤x≤10.

itbpFU2.1491in1.7495in0inHypergeometric density functionPlot itbpFU2.1491in1.7495in0inHypergeometric distribution functionPlot