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% File src/library/stats/man/GammaDist.Rd
% Part of the R package, http://www.R-project.org
% Copyright 1995-2007 R Core Development Team
% Distributed under GPL 2 or later

\name{GammaDist}
\alias{GammaDist}
\alias{dgamma}
\alias{pgamma}
\alias{qgamma}
\alias{rgamma}
\concept{incomplete gamma function}
\title{The Gamma Distribution}
\description{
  Density, distribution function, quantile function and random
  generation for the Gamma distribution with parameters \code{shape} and
  \code{scale}.
}
\usage{
dgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE)
pgamma(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE,
       log.p = FALSE)
qgamma(p, shape, rate = 1, scale = 1/rate, lower.tail = TRUE,
       log.p = FALSE)
rgamma(n, shape, rate = 1, scale = 1/rate)
}
\arguments{
  \item{x, q}{vector of quantiles.}
  \item{p}{vector of probabilities.}
  \item{n}{number of observations. If \code{length(n) > 1}, the length
    is taken to be the number required.}
  \item{rate}{an alternative way to specify the scale.}
  \item{shape, scale}{shape and scale parameters.  Must be positive,
    \code{scale} strictly.}
  \item{log, log.p}{logical; if \code{TRUE}, probabilities/densities \eqn{p}
    are returned as \eqn{log(p)}.}
  \item{lower.tail}{logical; if TRUE (default), probabilities are
    \eqn{P[X \le x]}, otherwise, \eqn{P[X > x]}.}
}
\value{
  \code{dgamma} gives the density,
  \code{pgamma} gives the distribution function,
  \code{qgamma} gives the quantile function, and
  \code{rgamma} generates random deviates.

  Invalid arguments will result in return value \code{NaN}, with a warning.
}
\details{
  If \code{scale} is omitted, it assumes the default value of \code{1}.

  The Gamma distribution with parameters \code{shape} \eqn{=\alpha}{= a}
  and \code{scale} \eqn{=\sigma}{= s} has density
  \deqn{
    f(x)= \frac{1}{{\sigma}^{\alpha}\Gamma(\alpha)} {x}^{\alpha-1} e^{-x/\sigma}%
  }{f(x)= 1/(s^a Gamma(a)) x^(a-1) e^-(x/s)}
  for \eqn{x \ge 0}, \eqn{\alpha > 0}{a > 0} and \eqn{\sigma > 0}{s > 0}.
  (Here \eqn{\Gamma(\alpha)}{Gamma(a)} is the function implemented by \R's
  \code{\link{gamma}()} and defined in its help.  Note that \eqn{a=0}
  corresponds to the trivial distribution with all mass at point 0.)

  The mean and variance are
  \eqn{E(X) = \alpha\sigma}{E(X) = a*s} and
  \eqn{Var(X) = \alpha\sigma^2}{Var(X) = a*s^2}.

  The cumulative hazard \eqn{H(t) = - \log(1 - F(t))}{H(t) = - log(1 - F(t))}
  is \code{-pgamma(t, ..., lower = FALSE, log = TRUE)}.

  Note that for smallish values of \code{shape} (and moderate
  \code{scale}) a large parts of the mass of the Gamma distribution is
  on values of \eqn{x} so near zero that they will be represented as
  zero in computer arithmetic.  So \code{rgamma} can well return values
  which will be represented as zero.  (This will also happen for very
  large values of \code{scale} since the actual generation is done for
  \code{scale=1}.)
}
% Have caught all currently known problems; hence no longer say:
%   Similarly, \code{qgamma} has a very hard job for
%   small \code{scale}, and warns of potential unreliability for
%   \code{scale < 1e-10}.
\note{
  The S parametrization is via \code{shape} and \code{rate}: S has no
  \code{scale} parameter.

  \code{pgamma} is closely related to the incomplete gamma function.  As
  defined by Abramowitz and Stegun 6.5.1 (and by \sQuote{Numerical
    Recipes}) this is
  \deqn{P(a,x) = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} dt}{P(a,x) =
      1/Gamma(a) integral_0^x t^(a-1) exp(-t) dt}
  \eqn{P(a, x)} is \code{pgamma(x, a)}.  Other authors (for example
  Karl Pearson in his 1922 tables) omit the normalizing factor,
  defining the incomplete gamma function as \code{pgamma(x, a) * gamma(a)}.
  A few use the \sQuote{upper} incomplete gamma function, the integral
  from \eqn{x} to \eqn{\infty}{infinity} which can be computed by
  \code{pgamma(x, a, lower=FALSE) * gamma(a)}, or its normalized
  version. See also
  \url{http://en.wikipedia.org/wiki/Incomplete_gamma_function}.
}
\source{
  \code{dgamma} is computed via the Poisson density, using code contributed
  by Catherine Loader (see \code{\link{dbinom}}).

  \code{pgamma} uses an unpublished (and not otherwise documented)
  algorithm \sQuote{mainly by Morten Welinder}.

  \code{qgamma} is based on a C translation of

  Best, D. J. and D. E. Roberts (1975).
  Algorithm AS91. Percentage points of the chi-squared distribution.
  \emph{Applied Statistics},  \bold{24}, 385--388.

  plus a final Newton step to improve the approximation.

  \code{rgamma} for \code{shape >= 1} uses

  Ahrens, J. H. and Dieter, U. (1982).
  Generating gamma variates by a modified rejection technique.
  \emph{Communications of the ACM}, \bold{25}, 47--54,

  and for \code{0 < shape < 1} uses

  Ahrens, J. H. and Dieter, U. (1974).
  Computer methods for sampling from gamma, beta, Poisson and binomial
  distributions. \emph{Computing}, \bold{12}, 223--246.
}
\references{
  Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
  \emph{The New S Language}.
  Wadsworth & Brooks/Cole.

  Shea, B. L. (1988)
  Algorithm AS 239,  Chi-squared and incomplete Gamma integral,
  \emph{Applied Statistics (JRSS C)} \bold{37}, 466--473.

  Abramowitz, M. and Stegun, I. A. (1972)
  \emph{Handbook of Mathematical Functions.} New York: Dover.
  Chapter 6: Gamma and Related Functions.
}
\seealso{
  \code{\link{gamma}} for the gamma function, \code{\link{dbeta}} for
  the Beta distribution and \code{\link{dchisq}} for the chi-squared
  distribution which is a special case of the Gamma distribution.
}
\examples{
-log(dgamma(1:4, shape=1))
p <- (1:9)/10
pgamma(qgamma(p,shape=2), shape=2)
1 - 1/exp(qgamma(p, shape=1))

# even for shape = 0.001 about half the mass is on numbers
# that cannot be represented accurately (and most of those as zero)
pgamma(.Machine$double.xmin, 0.001)
pgamma(5e-324, 0.001)  # on most machines this is the smallest
                       # representable non-zero number
table(rgamma(1e4, 0.001) == 0)/1e4
}
\keyword{distribution}