# NegativeBinomial: Create a Negative Binomial distribution In distributions3: Probability Distributions as S3 Objects

## Description

A generalization of the geometric distribution. It is the number of successes in a sequence of i.i.d. Bernoulli trials before a specified number (r) of failures occurs.

## Usage

 1 NegativeBinomial(size, p = 0.5)

## Arguments

 size The number of failures (an integer greater than 0) until the experiment is stopped. Denoted r below. p The success probability for a given trial. p can be any value in [0, 1], and defaults to 0.5.

## 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 Negative Binomial random variable with success probability p = p.

Support: \{0, 1, 2, 3, ...\}

Mean: \frac{p r}{1-p}

Variance: \frac{pr}{(1-p)^2}

Probability mass function (p.m.f):

f(k) = {k + r - 1 \choose k} \cdot (1-p)^r p^k

Cumulative distribution function (c.d.f):

Too nasty, ommited.

Moment generating function (m.g.f):

\frac{(1-p)^r}{(1-pe^t)^r}, t < -\log p

## Value

A NegativeBinomial object.