The Weibull Distribution

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

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

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


  • 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.


The cumulative hazard $H(t) = - \log(1 - F(t))$ is -pweibull(t, a, b, lower = FALSE, log = TRUE) which is just $H(t) = {(t/b)}^a$.


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


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

See Also

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

  • Weibull
  • dweibull
  • pweibull
  • qweibull
  • rweibull
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 3.3, License: Part of R 3.3

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