Weibull {stats}
Description
Density, distribution function, quantile function and random generation for the Weibull distribution with parameters shape and scale.
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 ≤ x], otherwise, P[X > x].
Details
The Weibull distribution with shape parameter a and scale parameter b has density given by f(x) = (a/b) (x/b)^(a-1) exp(- (x/b)^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 Γ(1 + 1/a), and the Var(X) = b^2 * (Γ(1 + 2/a) - (Γ(1 + 1/a))^2).
Values
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.
References
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1, chapter 21. Wiley, New York.
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.
See Also
Distributions for other standard distributions, including the Exponential which is a special case of the Weibull distribution.
Examples
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, tol = 1e-15) all.equal(qweibull(x/11, shape = 1, scale = pi), qexp(x/11, rate = 1/pi))
Documentation reproduced from R 2.15.0. License: GPL-2.