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% File src/library/stats/man/GammaDist.Rd

% Part of the R package, https://www.R-project.org

% Copyright 1995-2025 R Core 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

    are given as logarithms.}

  \item{lower.tail}{logical; if \code{TRUE} (default), probabilities are

    \eqn{P[X < x]}, otherwise, \eqn{P[X \ge x]} (exceptionally not

    \emph{cadlag} here, to ensure, e.g., \code{pgamma(0, shape=0)} remains zero).}

}

\value{

  \code{dgamma} gives the density,

  \code{pgamma} is the cumulative distribution function, and

  \code{qgamma} is the quantile function of the Gamma distribution.

  \code{rgamma} generates random deviates.

  Invalid arguments will result in return value \code{NaN}, with a warning.

  The length of the result is determined by \code{n} for

  \code{rgamma}, and is the maximum of the lengths of the

  numerical arguments for the other functions.

  The numerical arguments other than \code{n} are recycled to the

  length of the result.  Only the first elements of the logical

  arguments are used.

}

\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

\preformatted{

    -pgamma(t, ..., lower = FALSE, log = TRUE)

}

  Note that for smallish values of \code{shape} (and moderate

  \code{scale}) a large part 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} may 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 \bibcitep{R:Becker+Chambers+Wilks:1988} parametrization was via \code{shape}

  and \code{rate}: S had no \code{scale} parameter. It is an error

  to supply both \code{scale} and \code{rate}.

  \code{pgamma} is closely related to the incomplete gamma function.

  As defined by \bibcitet{|R:Abramowitz+Stegun:1972|section 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 \eqn{\gamma(a,x)} as

  \eqn{\gamma(a,x) = \int_0^x t^{a-1} e^{-t} dt,}{gamma(a,x) =

    integral_0^x t^(a-1) exp(-t) dt,} i.e., \code{pgamma(x, a) * gamma(a)}.

  Yet other use the \sQuote{upper} incomplete gamma function,

  \deqn{\Gamma(a,x) = \int_x^\infty t^{a-1} e^{-t} dt,}{Gamma(a,x) = integral_x^Inf t^(a-1) exp(-t) dt,}

  which can be computed by

  \code{pgamma(x, a, lower = FALSE) * gamma(a)}.

  Note however that \code{pgamma(x, a, ..)} currently requires \eqn{a > 0},

  whereas the incomplete gamma function is also defined for negative

  \eqn{a}.  In that case, you can use \code{gamma_inc(a,x)} (for

  \eqn{\Gamma(a,x)}) from package \CRANpkg{gsl}.

  See also

  \url{https://en.wikipedia.org/wiki/Incomplete_gamma_function}, or

  \url{https://dlmf.nist.gov/8.2#i}.

}

\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

  \bibcitet{R:Best+Roberts:1975} plus a final Newton step to improve the

  approximation.

  \code{rgamma} for \eqn{\code{shape} \ge 1} uses

  \bibcitet{R:Ahrens+Dieter:1982a}

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

  \bibcitet{R:Ahrens+Dieter:1974}.

}

\references{

  \bibinfo{R:Abramowitz+Stegun:1972}{footer}{Chapter 6: Gamma and Related Functions.}

  \bibshow{*,

    R:Shea:1988}

  NIST Digital Library of Mathematical Functions.

  \url{https://dlmf.nist.gov/}, section 8.2.

}

\seealso{

  \code{\link{gamma}} for the gamma function.

  \link{Distributions} for other standard distributions, including

  \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))

\donttest{# 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 5e-324 is the smallest

                       # representable non-zero number

table(rgamma(1e4, 0.001) == 0)/1e4

}}

\keyword{distribution}