# Weibull

##### The Weibull Distribution

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

and `scale`

.

- Keywords
- distribution

##### 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 $P[X \le x]$, otherwise, $P[X > x]$.

##### Details

The Weibull distribution with `shape`

parameter $a$ and
`scale`

parameter $b$ has density given by
$$f(x) = (a/\sigma) {(x/\sigma)}^{a-1} \exp (-{(x/\sigma)}^{a})$$ for $x > 0$.
The cumulative distribution function is
$F(x) = 1 - exp(- (x/b)^a)$
on $x > 0$, the
mean is $E(X) = b \Gamma(1 + 1/a)$, and
the $Var(X) = b^2 * (\Gamma(1 + 2/a) - (\Gamma(1 + 1/a))^2)$.

##### 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

-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.

##### See Also

Distributions for other standard distributions, including the Exponential which is a special case of the Weibull distribution.

##### Examples

`library(stats)`

```
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))
```

*Documentation reproduced from package stats, version 3.2.2, License: Part of R 3.2.2*