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)
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
.
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):
$$ \left(\frac{1-p}{1-pe^t}\right)^r, t < -\log p $$
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] 13 15 14 5 2 4 5 16 22 4
#>
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