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. |

`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 `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*.
The mean and variance are
*E(X) = a*s* and
*Var(X) = a*s^2*.

`pgamma()`

uses a new algorithm (mainly by Morten Welinder) which
should be uniformly better or equal to 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, shape=1)) p <- (1:9)/10 pgamma(qgamma(p,shape=2), shape=2) 1 - 1/exp(qgamma(p, shape=1))

[Package *stats* version 2.1.0 Index]