The Negative Binomial distribution

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.

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: \(\frac{p r}{1-p}\)

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

dist <- dist_negative_binomial(size = 10, prob = 0.5)

dist
#> <distribution[1]>
#> [1] NB(10, 0.5)
mean(dist)
#> [1] 10
variance(dist)
#> [1] 20
skewness(dist)
#> [1] 0.6708204
kurtosis(dist)
#> [1] 0.6125
support(dist)
#> <support_region[1]>
#> [1] N0

generate(dist, 10)
#> [[1]]
#>  [1]  6  9 13  7 13  4 15  5 16 22
#> 

density(dist, 2)
#> [1] 0.01342773
density(dist, 2, log = TRUE)
#> [1] -4.310433

cdf(dist, 4)
#> [1] 0.08978271

quantile(dist, 0.7)
#> [1] 12