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.

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):

Too nasty, ommited.

Moment generating function (m.g.f):

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

See also

Examples

set.seed(27) X <- NegativeBinomial(10, 0.3) X
#> Negative Binomial distribution (size = 10, p = 0.3)
random(X, 10)
#> [1] 45 25 27 41 13 11 16 22 12 23
pdf(X, 2)
#> [1] 0.0001591371
log_pdf(X, 2)
#> [1] -8.745745
cdf(X, 4)
#> [1] 0.001665663
quantile(X, 0.7)
#> [1] 27