distributional (version 0.5.0)

dist_hypergeometric: 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 https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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: \(\frac{km}{n+m} = kp\)

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

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):

$$ P(X \le k) \approx \Phi\Big(\frac{x - kp}{\sqrt{kp(1-p)}}\Big) $$

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)