dist_negative_binomial: The Negative Binomial distribution

In distributional: Vectorised Probability Distributions

The Negative Binomial distribution

Description

A generalization of the geometric distribution. It is the number of failures in a sequence of i.i.d. Bernoulli trials before a specified number of successes (size) occur. The probability of success in each trial is given by prob.

Usage

dist_negative_binomial(size, prob)

Arguments

size	The number of successful trials (target number of successes). Must be a positive number. Also called the dispersion parameter.
prob	The probability of success in each trial. Must be between 0 and 1.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_negative_binomial.html

In the following, let X be a Negative Binomial random variable with success probability prob = p and the number of successes size = r.

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

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

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

Probability mass function (p.m.f):

P(X = k) = \binom{k + r - 1}{k} (1-p)^r p^k

Cumulative distribution function (c.d.f):

F(k) = \sum_{i=0}^{\lfloor k \rfloor} \binom{i + r - 1}{i} (1-p)^r p^i

This can also be expressed in terms of the regularized incomplete beta function, and is computed numerically.

Moment generating function (m.g.f):

E(e^{tX}) = \left(\frac{1-p}{1-pe^t}\right)^r, \quad t < -\log p

Skewness:

\gamma_1 = \frac{2-p}{\sqrt{r(1-p)}}

Excess Kurtosis:

\gamma_2 = \frac{6}{r} + \frac{p^2}{r(1-p)}

See Also

stats::NegBinomial

Examples

dist <- dist_negative_binomial(size = 10, prob = 0.5)

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
support(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

distributional documentation built on June 27, 2026, 5:06 p.m.