# HyperGeometric: Create a HyperGeometric distribution In distributions3: Probability Distributions as S3 Objects

## 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. 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).

## Usage

 1 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://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)}, …, \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)

Moment generating function (m.g.f):

Not useful.

## Value

A HyperGeometric object.

Other discrete distributions: Bernoulli, Binomial, Categorical, Geometric, Multinomial, NegativeBinomial, Poisson
  1 2 3 4 5 6 7 8 9 10 11 12 set.seed(27) X <- HyperGeometric(4, 5, 8) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 4) quantile(X, 0.7)