GammaDist {stats}R Documentation

The Gamma Distribution


Density, distribution function, quantile function and random generation for the Gamma distribution with parameters shape and scale.


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)


x, q vector of quantiles.
p vector of probabilities.
n number of observations. If length(n) > 1, the length is taken to be the number required.
rate an alternative way to specify the scale.
shape, scale shape and scale parameters.
log, log.p logical; if TRUE, probabilities p are given as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].


If scale is omitted, it assumes the default value of 1.

The Gamma distribution with parameters shape = a and scale = s has density

f(x)= 1/(s^a Gamma(a)) x^(a-1) e^-(x/s)

for x > 0, a > 0 and s > 0. The mean and variance are E(X) = a*s and Var(X) = a*s^2.

pgamma() uses a new algorithm (mainly by Morten Welinder) which should be uniformly better or equal to AS 239, see the references.


dgamma gives the density, pgamma gives the distribution function qgamma gives the quantile function, and rgamma generates random deviates.


The S parametrization is via shape and rate: S has no scale parameter.

The cumulative hazard H(t) = - log(1 - F(t)) is -pgamma(t, ..., lower = FALSE, log = TRUE).

pgamma is closely related to the incomplete gamma function. As defined by Abramowitz and Stegun 6.5.1

P(a,x) = 1/Gamma(a) integral_0^x t^(a-1) exp(-t) dt

P(a, x) is pgamma(x, a). Other authors (for example Karl Pearson in his 1922 tables) omit the normalizing factor, defining the incomplete gamma function as pgamma(x, a) * gamma(a).


Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Shea, B. L. (1988) Algorithm AS 239, Chi-squared and Incomplete Gamma Integral, Applied Statistics (JRSS C) 37, 466–473.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

See Also

gamma for the Gamma function, dbeta for the Beta distribution and dchisq for the chi-squared distribution which is a special case of the Gamma distribution.


-log(dgamma(1:4, shape=1))
p <- (1:9)/10
pgamma(qgamma(p,shape=2), shape=2)
1 - 1/exp(qgamma(p, shape=1))

[Package stats version 2.1.0 Index]