NegBinomial {base}R Documentation

The Negative Binomial Distribution

Description

Density, distribution function, quantile function and random generation for the negative binomial distribution with parameters size and prob.

Usage

dnbinom(x, size, prob, log = FALSE)
pnbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
qnbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)
rnbinom(n, size, prob)

Arguments

x, q vector of quantiles representing the number of failures which occur in a sequence of Bernoulli trials before a target number of successes is reached, or alternately the probability distribution of a compound Poisson process whose intensity is distributed as a gamma (pgamma) distribution with scale parameter (1-prob)/prob and shape parameter size (this definition allows non-integer values of size).
x vector of (non-negative integer) quantiles.
q vector of quantiles.
p vector of probabilities.
n number of observations to generate.
size target for number of successful trials, or shape parameter of gamma distribution.
prob probability of success in each trial, or scale of gamma distribution (prob = scale/(1+scale)).
log, log.p logical; if TRUE, probabilities p are given as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

Details

The negative binomial distribution with size = n and prob = p has density

p(x) = Gamma(x+n)/(Gamma(n) x!) p^n (1-p)^x

for x = 0, 1, 2, ...

If an element of x is not integer, the result of dnbinom is zero, with a warning.

The quantile is defined as the smallest value x such that F(x) >= p, where F is the distribution function.

Value

dnbinom gives the density, pnbinom gives the distribution function, qnbinom gives the quantile function, and rnbinom generates random deviates.

See Also

dbinom for the binomial, dpois for the Poisson and dgeom for the geometric distribution, which is a special case of the negative binomial.

Examples

x <- 0:11
dnbinom(x, size = 1, prob = 1/2) * 2^(1 + x) # == 1
126 /  dnbinom(0:8, size  = 2, prob  = 1/2) #- theoretically integer

## Cumulative ('p') = Sum of discrete prob.s ('d');  Relative error :
summary(1 - cumsum(dnbinom(x, size = 2, prob = 1/2)) /
                  pnbinom(x, size  = 2, prob = 1/2))

x <- 0:15
size <- (1:20)/4
persp(x,size, dnb <- outer(x,size,function(x,s)dnbinom(x,s, pr= 0.4)),
      xlab = "x", ylab = "s", zlab="density", theta = 150)
title(tit <- "negative binomial density(x,s, pr = 0.4)  vs.  x & s")

image  (x,size, log10(dnb), main= paste("log [",tit,"]"))
contour(x,size, log10(dnb),add=TRUE)
[Package Contents]