| Weibull {stats} | R Documentation |
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. |
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]. |
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), 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).
dweibull gives the density,
pweibull gives the distribution function,
qweibull gives the quantile function, and
rweibull generates random deviates.
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.
dexp for the Exponential which is a special case of a
Weibull distribution.
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=FALSE, log=TRUE), -(x/pi)^2.5, tol=1e-15) all.equal(qweibull(x/11, shape = 1, scale = pi), qexp(x/11, rate = 1/pi))