Generalization of the gamma distribution. Often used in survival and time-to-event analyses.

Weibull(shape, scale)

## Arguments

shape The shape parameter $$k$$. Can be any positive real number. The scale parameter $$\lambda$$. Can be any positive real number.

## Value

A Weibull object.

## Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3, where the math will render with additional detail and much greater clarity.

In the following, let $$X$$ be a Weibull random variable with success probability p = $$p$$.

Support: $$R^+$$ and zero.

Mean: $$\lambda \Gamma(1+1/k)$$, where $$\Gamma$$ is the gamma function.

Variance: $$\lambda [ \Gamma (1 + \frac{2}{k} ) - (\Gamma(1+ \frac{1}{k}))^2 ]$$

Probability density function (p.d.f):

$$f(x) = \frac{k}{\lambda}(\frac{x}{\lambda})^{k-1}e^{-(x/\lambda)^k}, x \ge 0$$

Cumulative distribution function (c.d.f):

$$F(x) = 1 - e^{-(x/\lambda)^k}, x \ge 0$$

Moment generating function (m.g.f):

$$\sum_{n=0}^\infty \frac{t^n\lambda^n}{n!} \Gamma(1+n/k), k \ge 1$$

Other continuous distributions: Beta, Cauchy, ChiSquare, Exponential, FisherF, Gamma, LogNormal, Logistic, Normal, StudentsT, Tukey, Uniform

## Examples


set.seed(27)

X <- Weibull(0.3, 2)
X#> Weibull distribution (shape = 0.3, scale = 2)
random(X, 10)#>  [1] 1.440254e-05 4.128282e+01 2.513340e-03 2.840554e+00 7.792913e+00
#>  [6] 1.472187e+00 4.985175e+01 7.900541e+02 1.972819e+01 1.063212e+01
pdf(X, 2)#> [1] 0.05518192log_pdf(X, 2)#> [1] -2.89712
cdf(X, 4)#> [1] 0.7080417quantile(X, 0.7)#> [1] 3.713233