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. probability of success in each trial. 0 < prob <= 1.

## Details

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.

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):

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

## Examples

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

dist
#> <distribution>
#>  NB(10, 0.5)mean(dist)
#>  10variance(dist)
#>  20skewness(dist)
#>  0.6708204kurtosis(dist)
#>  0.6125
generate(dist, 10)
#> []
#>    8  9 10 11 10 14 14  5  4  7
#>
density(dist, 2)
#>  0.01342773density(dist, 2, log = TRUE)
#>  -4.310433
cdf(dist, 4)
#>  0.08978271
quantile(dist, 0.7)
#>  12