# Weibull: The Weibull Distribution

### Description

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

### Usage

 ```1 2 3 4``` ```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).

### 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

 `1` ```-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.

 ```1 2 3 4 5 6 7``` ```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)) ```