The Gumbel Distribution

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

dGumbel(x, loc=0, scale=1, log = FALSE)
pGumbel(q, loc=0, scale=1, lower.tail = TRUE)
qGumbel(p, loc=0, scale=1, lower.tail = TRUE)
rGumbel(n, loc=0, scale=1)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

loc, scale

Location and scale parameters (can be given as vectors).

log

Logical; if TRUE, the log density is returned.

lower.tail

Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

Details

The Gumbel distribution function with parameters \(loc = a\) and \(scale = b\) is $$G(z) = \exp\left\{-\exp\left[-\left(\frac{z-a}{b}\right) \right]\right\}$$ for all real \(z\), where \(b > 0\).

Value

dGumbel gives the density function, pGumbel gives the distribution function, qGumbel gives the quantile function, and rGumbel generates random deviates.

See also

rFrechet, rGenExtrVal, rRevWeibull

Author

Alec Stephenson <alec_stephenson@hotmail.com>

Examples

dGumbel(-1:2, -1, 0.5)
#> [1] 0.735758882 0.236409903 0.035966459 0.004945231
pGumbel(-1:2, -1, 0.5)
#> [1] 0.3678794 0.8734230 0.9818511 0.9975243
qGumbel(seq(0.9, 0.6, -0.1), 2, 0.5)
#> [1] 3.125184 2.749970 2.515465 2.335863
rGumbel(6, -1, 0.5)
#> [1] -0.89493155 -0.03361349 -0.40927660 -0.78263401 -1.17527107 -0.44974342
p <- (1:9)/10
pGumbel(qGumbel(p, -1, 2), -1, 2)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9