| GammaDist {stats} | R Documentation |
The Gamma Distribution
Description
Density, distribution function, quantile function and random
generation for the Gamma distribution with parameters shape and
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
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
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
|
Details
If scale is omitted, it assumes the default value of 1.
The Gamma distribution with parameters shape =\alpha
and scale =\sigma has density
f(x)= \frac{1}{{\sigma}^{\alpha}\Gamma(\alpha)} {x}^{\alpha-1} e^{-x/\sigma}%
for x > 0, \alpha > 0 and \sigma > 0.
The mean and variance are
E(X) = \alpha\sigma and
Var(X) = \alpha\sigma^2.
pgamma() uses a new algorithm (mainly by Morten Welinder) which
should be uniformly better or equal to AS 239, see the references.
Value
dgamma gives the density,
pgamma gives the distribution function
qgamma gives the quantile function, and
rgamma generates random deviates.
Note
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) = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-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).
References
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.
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))