# dist_hypergeometric: The Hypergeometric distribution In distributional: Vectorised Probability Distributions

\lifecycle

stable

## Usage

 1 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

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.

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)}, …, \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 ≤ k) \approx Φ\Big(\frac{x - kp}{√{kp(1-p)}}\Big)

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 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)