Density, distribution function, quantile function and random generation for the negative binomial distribution with parameters
dnbinom(x, size, prob, mu, log = FALSE) pnbinom(q, size, prob, mu, lower.tail = TRUE, log.p = FALSE) qnbinom(p, size, prob, mu, lower.tail = TRUE, log.p = FALSE) rnbinom(n, size, prob, mu)
- vector of (non-negative integer) quantiles.
- vector of quantiles.
- vector of probabilities.
- number of observations. If
length(n) > 1, the length is taken to be the number required.
- 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.
- alternative parametrization via mean: see ‘Details’.
- log, log.p
- logical; if TRUE, probabilities p are given as log(p).
- logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].
The negative binomial distribution with
size = n and
prob = p has density for x = 0, 1, 2, ..., n > 0 and 0 < p ≤ 1.
This represents the number of failures which occur in a sequence of Bernoulli trials before a target number of successes is reached. The mean is n(1-p)/p and variance n(1-p)/p^2.
A negative binomial distribution can also arise as a mixture of Poisson distributions with mean distributed as a gamma distribution (see
pgamma) with scale parameter
(1 - prob)/prob and shape parameter
size. (This definition allows non-integer values of
An alternative parametrization (often used in ecology) is by the mean
size, the dispersion parameter, where
size/(size+mu). The variance is
mu + mu^2/size in this parametrization.
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.
prob will result in return value
NaN, with a warning. The length of the result is determined by
rnbinom, and is the maximum of the lengths of the numerical parameters for the other functions. The numerical parameters other than
n are recycled to the length of the result. Only the first elements of the logical parameters are used.
require(graphics) 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, prob = 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) ## Alternative parametrization x1 <- rnbinom(500, mu = 4, size = 1) x2 <- rnbinom(500, mu = 4, size = 10) x3 <- rnbinom(500, mu = 4, size = 100) h1 <- hist(x1, breaks = 20, plot = FALSE) h2 <- hist(x2, breaks = h1$breaks, plot = FALSE) h3 <- hist(x3, breaks = h1$breaks, plot = FALSE) barplot(rbind(h1$counts, h2$counts, h3$counts), beside = TRUE, col = c("red","blue","cyan"), names.arg = round(h1$breaks[-length(h1$breaks)]))
Documentation reproduced from R 3.0.1. License: GPL-2.