distributional (version 0.2.0)

dist_negative_binomial: The Negative Binomial distribution

Description

stable

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

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.

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 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, omitted.

Moment generating function (m.g.f):

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

See Also

stats::NegBinomial

Examples

# NOT RUN {
dist <- dist_negative_binomial(size = 10, prob = 0.5)

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

generate(dist, 10)

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

cdf(dist, 4)

quantile(dist, 0.7)

# }