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

##### 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 \(H(t) = {(t/b)}^a\).

##### 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)`

```
# NOT RUN {
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 3.6.2, License: Part of R 3.6.2*