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. 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):

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

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

cdf(X, 4)#>  0.001665663quantile(X, 0.7)#>  27