A generalization of the geometric distribution. It is the number of failures in a sequence of i.i.d. Bernoulli trials before a specified target number (\(r\)) of successes occurs.
Arguments
- size
The target number of successes (greater than \(0\)) until the experiment is stopped. Denoted \(r\) below.
- p
The success probability for a given trial.
pcan be any value in[0, 1], and defaults to0.5.- mu
Alternative parameterization via the non-negative mean of the distribution (instead of the probability
p), defaults tosize.
Details
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: \(\frac{(1 - p) r}{p} = \mu\)
Variance: \(\frac{(1 - p) r}{p^2}\)
Probability mass function (p.m.f.):
$$ f(k) = {k + r - 1 \choose k} \cdot p^r (1-p)^k $$
Cumulative distribution function (c.d.f.):
Omitted for now.
Moment generating function (m.g.f.):
$$ \left(\frac{p}{1 - (1 -p) e^t}\right)^r, t < -\log (1-p) $$
Alternative parameterization: Sometimes, especially when used in regression models, the negative binomial distribution is parameterized by its mean \(\mu\) (as listed above) plus the size parameter \(r\). This implies a success probability of \(p = r/(r + \mu)\). This can also be seen as a generalization of the Poisson distribution where the assumption of equidispersion (i.e., variance equal to mean) is relaxed. The negative binomial distribution is overdispersed (i.e., variance greater than mean) and its variance can also be written as \(\mu + 1/r \mu^2\). The Poisson distribution is then obtained as \(r\) goes to infinity. Note that in this view it is natural to also allow for non-integer \(r\) parameters. The factorials in the equations above are then expressed in terms of the gamma function.
See also
Other discrete distributions:
Bernoulli(),
Binomial(),
Categorical(),
Geometric(),
HurdleNegativeBinomial(),
HurdlePoisson(),
HyperGeometric(),
Multinomial(),
Poisson(),
PoissonBinomial(),
ZINegativeBinomial(),
ZIPoisson(),
ZTNegativeBinomial(),
ZTPoisson()
Examples
set.seed(27)
X <- NegativeBinomial(size = 5, p = 0.1)
X
#> [1] "NegativeBinomial(size = 5, p = 0.1)"
random(X, 10)
#> [1] 95 37 48 93 18 16 32 43 27 17
pdf(X, 50)
#> [1] 0.01629887
log_pdf(X, 50)
#> [1] -4.11666
cdf(X, 50)
#> [1] 0.6548517
quantile(X, 0.7)
#> [1] 53
## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
#> [1] "NegativeBinomial(mu = 45, size = 5)"
cdf(Y, 50)
#> [1] 0.6548517
quantile(Y, 0.7)
#> [1] 53