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 \(X\). For example, consider that in a set of 20 car parts, there are 4 that are defective (type I). If we take a sample of size 5 from those car parts, the probability of finding 2 that are defective will be given by the HyperGeometric distribution (needs double checking).

HyperGeometric(m, n, k)

m | The number of type I elements available. |
---|---|

n | The number of type II elements available. |

k | The size of the sample taken. |

A `HyperGeometric`

object.

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.

In the following, let \(X\) be a HyperGeometric random variable with
success probability `p`

= \(p = m/(m+n)\).

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

**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)**:

$$
P(X \le k) \approx \Phi\Big(\frac{x - kp}{\sqrt{kp(1-p)}}\Big)
$$
**Moment generating function (m.g.f)**:

Not useful.

Other discrete distributions: `Bernoulli`

,
`Binomial`

, `Categorical`

,
`Geometric`

, `Multinomial`

,
`NegativeBinomial`

, `Poisson`

#> HyperGeometric distribution (m = 4, n = 5, k = 8)#> [1] 3 4 3 4 4 4 4 4 4 4#> [1] 0#> [1] -Inf#> [1] 1#> [1] 4