NegBinomial {stats}  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, 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)
Arguments
x 
vector of (nonnegative integer) quantiles. 
q 
vector of quantiles. 
p 
vector of probabilities. 
n 
number of observations. If 
size 
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. 
prob 
probability of success in each trial. 
mu 
alternative parametrization via mean: see ‘Details’. 
log , log.p 
logical; if TRUE, probabilities p are given as log(p). 
lower.tail 
logical; if TRUE (default), probabilities are

Details
The negative binomial distribution with size
= n
and
prob
= p
has density
p(x) = \frac{\Gamma(x+n)}{\Gamma(n) x!} p^n (1p)^x
for x = 0, 1, 2, \ldots
, n > 0
and 0 < p \le 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 \mu = n(1p)/p
and variance n(1p)/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 noninteger
values of size
.)
An alternative parametrization (often used in ecology) is by the
mean mu
(see above), and size
, the dispersion
parameter, where prob
= 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 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.
Value
dnbinom
gives the density,
pnbinom
gives the distribution function,
qnbinom
gives the quantile function, and
rnbinom
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
rnbinom
, 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.
rnbinom
returns a vector of type integer unless generated
values exceed the maximum representable integer when double
values are returned.
Source
dnbinom
computes via binomial probabilities, using code
contributed by Catherine Loader (see dbinom
).
pnbinom
uses pbeta
.
qnbinom
uses the Cornish–Fisher Expansion to include a skewness
correction to a normal approximation, followed by a search.
rnbinom
uses the derivation as a gamma mixture of Poisson
distributions, see
Devroye, L. (1986) NonUniform Random Variate Generation. SpringerVerlag, New York. Page 480.
See Also
Distributions for standard distributions, including
dbinom
for the binomial, dpois
for the
Poisson and dgeom
for the geometric distribution, which
is a special case of the negative binomial.
Examples
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)]))