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).

Note that for smallish values of shape (and moderate scale) a large parts of the mass of the Gamma distribution is on values of x so near zero that they will be represented as zero in computer arithmetic. So rgamma can well return values which will be represented as zero. (This will also happen for very large values of scale since the actual generation is done for scale=1.)


dgamma gives the density, pgamma gives the distribution function, qgamma gives the quantile function, and rgamma generates random deviates.
Invalid arguments will result in return value NaN, with a warning.


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).


dgamma is computed via the Poisson density, using code contributed by Catherine Loader (see dbinom).

Prior to R 2.1.0 pgamma used AS239 (Shea, 1988). It currently uses an unpublished (and not otherwise documented) algorithm `mainly by Morten Welinder'.

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. Applied Statistics, 24, 385–388.

plus a final Newton step to improve the approximation.

rgamma for shape >= 1 uses

Ahrens, J. H. and Dieter, U. (1982). Generating gamma variates by a modified rejection technique. Communications of the ACM, 25, 47–54,

and for 0 < shape < 1 uses

Ahrens, J. H. and Dieter, U. (1974). Computer methods for sampling from gamma, beta, Poisson and binomial distributions. Computing, 12, 223–246.


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))

# 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

[Package stats version 2.5.0 Index]