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.
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
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)]))
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