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)**:

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

## 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
```