distributional (version 0.5.0)

dist_negative_binomial: 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

target for number of successful trials, or dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer.

prob

probability of success in each trial. 0 < prob <= 1.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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: $\frac{pr}{1-p}$

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

Probability mass function (p.m.f):

$$f(k)=(\genfrac{}{}{0ex}{}{k+r-1}{k})\cdot (1-p{)}^r{p}^k$$

Cumulative distribution function (c.d.f):

Too nasty, omitted.

Moment generating function (m.g.f):

$${\left(\frac{1-p}{1-p{e}^t}\right)}^r,t<-\log p$$

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)