# Hypergeometric Distribution. Probability density function, cumulative distribution function, mean and variance

This calculator calculates hypergeometric distribution pdf, cdf, mean and variance for given parameters

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Created: 2018-02-06 08:49:13, Last updated: 2021-03-05 16:41:42

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In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of k successes in n draws with replacement. Wikipedia

Probability density function of the hypergeometric distribution is
$f(k)=\frac{C^{K}_{k}C^{N-K}_{n-k}}{C^{N}_{n}}$,
where
$C^{n}_{m}=\frac{n!}{m!(n-m)!}$ is the number of combinations of m from n or binomial coefficient

Cumulative distribution function of the hypergeometric distribution is
$F(k)=1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right]$,
where
$_{p}F_{q}$ is the generalized hypergeometric function

Mean or expected value for the hypergeometric distribution is
$\mu_k=n\frac{K}{N}$

Variance is
$\sigma^{2}_{k}=n\frac{K}{N}\frac{N-K}{N}\frac{N-n}{N-1}$

The calculator below calculates the mean and variance of the negative binomial distribution and plots the probability density function and cumulative distribution function for given parameters n, K, N.

#### Hypergeometric Distribution. Probability density function, cumulative distribution function, mean and variance

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Mean

Variance

Hypergeometric distribution
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Cumulative distribution function
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