\Header{NegBinomial}{The Negative Binomial Distribution}
\alias{dnbinom}{NegBinomial}
\alias{pnbinom}{NegBinomial}
\alias{qnbinom}{NegBinomial}
\alias{rnbinom}{NegBinomial}
\keyword{distribution}{NegBinomial}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the negative binomial distribution with parameters
\code{size} and \code{prob}.
\end{Description}
\begin{Usage}
\begin{verbatim}
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)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] vector of (non-negative integer) quantiles.
\item[\code{q}] vector of quantiles.
\item[\code{p}] vector of probabilities.
\item[\code{n}] number of observations. If \code{length(n) > 1}, the length
is taken to be the number required.
\item[\code{size}] target for number of successful trials, or dispersion
parameter (the shape parameter of the gamma mixing distribution).
\item[\code{prob}] probability of success in each trial.
\item[\code{mu}] alternative parametrization via mean: see Details
\item[\code{log, log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The negative binomial distribution with \code{size} \eqn{= n}{} and
\code{prob} \eqn{= p}{} has density
\deqn{
p(x) = \frac{\Gamma(x+n)}{\Gamma(n) x!} p^n (1-p)^x}{p(x) = Gamma(x+n)/(Gamma(n) x!) p^n (1-p)^x}
for \eqn{x = 0, 1, 2, \ldots}{}

This represents the number of failures
which occur in a sequence of Bernoulli trials before a target number
of successes is reached.

A negative binomial distribution can arise as a mixture of Poisson
distributions with mean distributed as a
gamma (\code{\Link{pgamma}}) distribution with scale parameter
\code{(1 - prob)/prob} and shape parameter \code{size}.  (This
definition allows non-integer values of \code{size}.)
In this model \code{prob} = \code{scale/(1+scale)}, and the mean is
\code{size * (1 - prob)/prob)}

The alternative parametrization (often used in ecology) is by the
\emph{mean} \code{mu}, and \code{size}, the \emph{dispersion parameter},
where \code{prob} = \code{size/(size+mu)}.  In this parametrization the
variance is \code{mu + mu\textasciicircum{}2/size}.

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

The quantile is defined as the smallest value \eqn{x}{} such that
\eqn{F(x) \ge p}{}, where \eqn{F}{} is the distribution function.
\end{Details}
\begin{Value}
\code{dnbinom} gives the density,
\code{pnbinom} gives the distribution function,
\code{qnbinom} gives the quantile function, and
\code{rnbinom} generates random deviates.
\end{Value}
\begin{SeeAlso}\relax
\code{\Link{dbinom}} for the binomial, \code{\Link{dpois}} for the
Poisson and \code{\Link{dgeom}} for the geometric distribution, which
is a special case of the negative binomial.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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)

## 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)]))
\end{ExampleCode}
\end{Examples}

