distributions3 (version 0.1.2)

NegativeBinomial: Create a Negative Binomial distribution

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

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.

Value

A NegativeBinomial object.

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):

Omitted for now.

Moment generating function (m.g.f):

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

See Also

Other discrete distributions: Bernoulli(), Binomial(), Categorical(), Geometric(), HyperGeometric(), Multinomial(), Poisson()

Examples

# NOT RUN {
set.seed(27)

X <- NegativeBinomial(10, 0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
# }