Gamma | R Documentation |
Density, distribution function, quantile function and random
generation for the Gamma distribution with parameters alpha
(or shape
) and beta
(or scale
or 1/rate
).
This special Rlab implementation allows the parameters alpha
and beta
to be used, to match the function description
often found in textbooks.
dgamma(x, shape, rate = 1, scale = 1/rate, alpha = shape, beta = scale, log = FALSE) pgamma(q, shape, rate = 1, scale = 1/rate, alpha = shape, beta = scale, lower.tail = TRUE, log.p = FALSE) qgamma(p, shape, rate = 1, scale = 1/rate, alpha = shape, beta = scale, lower.tail = TRUE, log.p = FALSE) rgamma(n, shape, rate = 1, scale = 1/rate, alpha = shape, beta = scale)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
rate |
an alternative way to specify the scale. |
alpha, beta |
an alternative way to specify the shape and 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 P[X <= x], otherwise, P[X > x]. |
If beta
(or scale
or rate
) is omitted, it assumes
the default value of 1
.
The Gamma distribution with parameters alpha
(or shape
)
= a and beta
(or 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. The mean and variance are E(X) = a*s and Var(X) = a*s^2.
pgamma()
uses algorithm AS 239, see the references.
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.
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) = 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)
.
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, alpha=1)) p <- (1:9)/10 pgamma(qgamma(p,alpha=2), alpha=2) 1 - 1/exp(qgamma(p, alpha=1))
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