Create a Negative Binomial distribution
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)
size |
The number of failures (an integer greater than 0) until the experiment is stopped. Denoted r below. |
p |
The success probability for a given trial. |
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
A NegativeBinomial object.
Other discrete distributions: Bernoulli,
Binomial, Categorical,
Geometric, HyperGeometric,
Multinomial, Poisson
set.seed(27) X <- NegativeBinomial(10, 0.3) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 4) quantile(X, 0.7)
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