The R Project SVN R

\name{NegBinomial}
\title{The Negative Binomial Distribution}
\usage{
dnbinom(x, size, prob)
pnbinom(q, size, prob)
qnbinom(p, size, prob)
rnbinom(n, size, prob)
}
\alias{dnbinom}
\alias{pnbinom}
\alias{qnbinom}
\alias{rnbinom}
\arguments{
\item{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 (\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}).}
\item{x}{vector of (non-negative integer) quantiles.}
\item{q}{vector of quantiles.}
\item{p}{vector of probabilities.}
\item{n}{number of observations to generate.}
\item{size}{target for number of successful trials /\cr
    shape parameter of gamma distribution.}
\item{prob}{probability of success in each trial /\cr
    determines scale of gamma distribution
    (\code{prob} = \code{scale/(1+scale)}).}
}
\description{
  These functions provide information about the negative binomial
  distribution with parameters \code{size} and \code{prob}.  \code{dnbinom}
  gives the density, \code{pnbinom} gives the distribution function,
  \code{qnbinom} gives the quantile function and \code{rnbinom}
  generates random deviates.
}
\details{
  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}

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

  The quantile is left continuous: \code{qnbinom(q, \dots)} is the 
  largest integer x such that P(X <= x) < q.
}
\seealso{
  \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.
}
\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)))
title(tit <- "negative binomial density(x,s, pr = 0.4)  vs.  x & s")
## if persp() only could label axes ....

image  (x,size, log10(dnb), main= paste("log [",tit,"]"))
contour(x,size, log10(dnb),add=TRUE)
}
\keyword{distribution}