| GammaDist {stats} | R Documentation |
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.
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))