# 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.

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.