dist_hypergeometric: The Hypergeometric distribution

In distributional: Vectorised Probability Distributions

The Hypergeometric distribution

Description

To understand the HyperGeometric distribution, consider a set of r objects, of which m are of the type I and n are of the type II. A sample with size k (k<r) with no replacement is randomly chosen. The number of observed type I elements observed in this sample is set to be our random variable X.

Usage

dist_hypergeometric(m, n, k)

Arguments

m	The number of type I elements available.
n	The number of type II elements available.
k	The size of the sample taken.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_hypergeometric.html

In the following, let X be a HyperGeometric random variable with success probability p = p = m/(m+n).

Support: x \in \{\max(0, k-n), \dots, \min(k,m)\}

Mean: \frac{km}{m+n} = kp

Variance: \frac{kmn(m+n-k)}{(m+n)^2 (m+n-1)} = kp(1-p)\left(1 - \frac{k-1}{m+n-1}\right)

Probability mass function (p.m.f):

P(X = x) = \frac{{m \choose x}{n \choose k-x}}{{m+n \choose k}}

Cumulative distribution function (c.d.f):

P(X \le x) = \sum_{i = \max(0, k-n)}^{\lfloor x \rfloor} \frac{{m \choose i}{n \choose k-i}}{{m+n \choose k}}

Moment generating function (m.g.f):

E(e^{tX}) = \frac{{m \choose k}}{{m+n \choose k}}{}_2F_1(-m, -k; m+n-k+1; e^t)

where _2F_1 is the hypergeometric function.

Skewness:

\frac{(m+n-2k)(m+n-1)^{1/2}(m+n-2n)}{[kmn(m+n-k)]^{1/2}(m+n-2)}

See Also

stats::Hypergeometric

Examples

dist <- dist_hypergeometric(m = rep(500, 3), n = c(50, 60, 70), k = c(100, 200, 300))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

distributional documentation built on June 27, 2026, 5:06 p.m.