# Weibull

0th

Percentile

##### The Weibull Distribution

Density, distribution function, quantile function and random generation for the Weibull distribution with parameters shape and scale.

Keywords
distribution
##### Usage
dweibull(x, shape, scale = 1, log = FALSE)
pweibull(q, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
qweibull(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
rweibull(n, shape, scale = 1)
##### Arguments
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.
shape, scale
shape and scale 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, $P[X > x]$.
##### Details

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

##### Value

dweibull gives the density, pweibull gives the distribution function, qweibull gives the quantile function, and rweibull generates random deviates.Invalid arguments will result in return value NaN, with a warning.The length of the result is determined by n for rweibull, 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. Only the first elements of the logical arguments are used.

##### Note

The cumulative hazard $H(t) = - log(1 - F(t))$ is

-pweibull(t, a, b, lower = FALSE, log = TRUE)

which is just $H(t) = (t/b)^a$.

##### Source

[dpq]weibull are calculated directly from the definitions. rweibull uses inversion.

##### References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 21. Wiley, New York.

Distributions for other standard distributions, including the Exponential which is a special case of the Weibull distribution.

• Weibull
• dweibull
• pweibull
• qweibull
• rweibull
##### Examples
library(stats) x <- c(0, rlnorm(50)) all.equal(dweibull(x, shape = 1), dexp(x)) all.equal(pweibull(x, shape = 1, scale = pi), pexp(x, rate = 1/pi)) ## Cumulative hazard H(): all.equal(pweibull(x, 2.5, pi, lower.tail = FALSE, log.p = TRUE), -(x/pi)^2.5, tolerance = 1e-15) all.equal(qweibull(x/11, shape = 1, scale = pi), qexp(x/11, rate = 1/pi)) 
Documentation reproduced from package stats, version 3.2.5, License: Part of R 3.2.5

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