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Weibull {stats}

The Weibull Distribution
Package: 
stats
Version: 
R 2.15.3

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.

The length of the result is determined by n for rweibull, and is the maximum of the lengths of the numerical parameters for the other functions. The numerical parameters other than n are recycled to the length of the result. Only the first elements of the logical parameters are used.

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 2.15.3. License: GPL-2.