| 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 algorithm 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))