dirttee (version 1.0.2)

gumbel: The Gumbel Distribution.

Description

Density, distribution function, quantile function and random generation for the gumbel distribution with the two parameters location and scale.

Usage

dgumbel(x, location = 0, scale = 1)
pgumbel(q, location = 0, scale = 1)
qgumbel(p, location = 0, scale = 1)
rgumbel(n, location = 0, scale = 1)

Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, and rgumbel generates random deviates.

The length of the result is determined by n for rgumbel, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result.

Arguments

q

vector of quantiles.

location

location parameter and mode of the distribution.

scale

scaling parameter, has to be positive.

x

vector of locations.

p

vector of probabilities.

n

number of observations. If \(length(n) > 1\), the length is taken to be the number required.

Details

The gumbel distribution has the following density and cdf
\(f(x) = (1/scale)*exp((x - location)/scale - exp((x - location)/scale))\), \(F(x) = 1 - exp(- exp((x - location)/scale))\). The mode of the distribution is location, the variance is \(\pi^{2/6} * \code{scale}\).

References

Collett, D. (2015). Modelling survival data in medical research, chapter 6. CRC press.

Examples

hist(rgumbel(1000))

qg <- qgumbel(0.1, 1, 2)

pgumbel(qg, 1, 2)

ax <- c(1:1000)/100-5
plot(ax,dgumbel(ax), type = 'l')