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

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

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

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

Digits after the decimal point: 2
Mean

Variance

Hypergeometric distribution
Cumulative distribution function
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