The Negative Binomial Distribution (Version 2)
Usage
dnbinom2(x, size, prob, mu, log = FALSE)
pnbinom2(q, size, prob, mu, lower.tail = TRUE, log.p = FALSE)
qnbinom2(p, size, prob, mu, lower.tail = TRUE, log.p = FALSE)
rnbinom2(n, size, prob, mu)Arguments
- x
vector of (non-negative integer) quantiles.
- size
total number of successful trials (failures + successes), 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.
- mu
alternative parametrization via mean: see ‘Details’.
- log, log.p
logical; if TRUE, probabilities p are given as log(p).
- q
vector of quantiles.
- lower.tail
logical; if TRUE (default), probabilities are \(P[X \le x]\); otherwise, \(P[X > x]\).
- p
vector of probabilities.
- n
number of observations. If length(n) > 1, the length is taken to be the number required.
Value
dnbinom2 gives the density, pnbinom2 gives the distribution function, qnbinom2 gives the quantile function, and rnbinom2 generates random deviates.
Invalid size or prob will result in return value NaN, with a warning.
The length of the result is determined by n for rnbinom2, and is the maximum of the lengths of the numerical arguments for the other functions.
The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.
rnbinom2 returns a vector of type integer unless generated values exceed the maximum representable integer when double values are returned since R version 4.0.0.
Details
The negative binomial distribution with size = r and prob = p has density $$ f(x) = \Gamma(x)/(\Gamma(r)(x-r)!)p^{r}(1-p)^{x-r}, $$ for \(r >0\) and x = r, r+1, r+2, ...
This represents the total number of trials (failures and sucesses) which occur in a sequence of Bernoulli trials. The mean is \(\mu = n/p\) and variance \((1-p)/p^2\).
If an element of x is not integer, the result of dnbinom is zero, with a warning.
The case size == 0 is the distribution concentrated at zero. This is the limiting distribution for size approaching zero, even if mu rather than prob is held constant. Notice though, that the mean of the limit distribution is 0, whatever the value of \(mu\).
The quantile is defined as the smallest value x such that \(F(x) \ge p\), where F is the distribution function.