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. Must be strictly positive.
log, log.p logical; if TRUE, probabilities/densities p are returned 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. (Here Gamma(a) is the function implemented by R's gamma() and defined in its help.)

The mean and variance are E(X) = a*s and Var(X) = a*s^2.

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


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.

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).

As from R 2.1.0 pgamma() uses a new algorithm (mainly by Morten Welinder) which should be uniformly as accurate as AS 239.


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 2.2.1 Index]