Weibull: The Weibull Distribution

Description

Density, distribution function, quantile function and random generation for the Weibull distribution with parameters alpha (or shape) and beta (or scale).

This special Rlab implementation allows the parameters alpha and beta to be used, to match the function description often found in textbooks.

Usage

dweibull(x, shape, scale = 1, alpha = shape, beta = scale, log = FALSE)
pweibull(q, shape, scale = 1, alpha = shape, beta = scale,
         lower.tail = TRUE, log.p = FALSE)
qweibull(p, shape, scale = 1, alpha = shape, beta = scale,
         lower.tail = TRUE, log.p = FALSE)
rweibull(n, shape, scale = 1, alpha = shape, beta = scale)

Arguments

x, q

vector of quantiles.

vector of probabilities.

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

shape, scale

shape and scale parameters, the latter defaulting to 1.

alpha, beta

alpha and beta parameters, the latter defaulting to 1.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

Value

dweibull gives the density, pweibull gives the distribution function, qweibull gives the quantile function, and rweibull generates random deviates.

Details

The Weibull distribution with alpha (or shape) parameter $a$ and beta (or scale) parameter $\sigma$ has density given by $$f(x) = (a/\sigma) {(x/\sigma)}^{a-1} \exp (-{(x/\sigma)}^{a})$$ for $x > 0$. The cumulative is $F(x) = 1 - \exp(-{(x/\sigma)}^a)$, the mean is $E(X) = \sigma \Gamma(1 + 1/a)$, and the $Var(X) = \sigma^2(\Gamma(1 + 2/a)-(\Gamma(1 + 1/a))^2)$.

Examples

Run this code

# NOT RUN {
x <- c(0,rlnorm(50))
all.equal(dweibull(x, alpha = 1), dexp(x))
all.equal(pweibull(x, alpha = 1, beta = pi), pexp(x, rate = 1/pi))
## Cumulative hazard H():
all.equal(pweibull(x, 2.5, pi, lower=FALSE, log=TRUE), -(x/pi)^2.5, tol=1e-15)
all.equal(qweibull(x/11, alpha = 1, beta = pi), qexp(x/11, rate = 1/pi))
# }