# Weibull: Create a Weibull distribution In distributions3: Probability Distributions as S3 Objects

## Description

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

## Usage

 1 Weibull(shape, scale) 

## Arguments

 shape The shape parameter k. Can be any positive real number. scale The scale parameter λ. Can be any positive real number.

## 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: λ Γ(1+1/k), where Γ is the gamma function.

Variance: λ [ Γ (1 + \frac{2}{k} ) - (Γ(1+ \frac{1}{k}))^2 ]

Probability density function (p.d.f):

f(x) = \frac{k}{λ}(\frac{x}{λ})^{k-1}e^{-(x/λ)^k}, x ≥ 0

Cumulative distribution function (c.d.f):

F(x) = 1 - e^{-(x/λ)^k}, x ≥ 0

Moment generating function (m.g.f):

∑_{n=0}^∞ \frac{t^nλ^n}{n!} Γ(1+n/k), k ≥ 1

## Value

A Weibull object.

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform()
  1 2 3 4 5 6 7 8 9 10 11 12 set.seed(27) X <- Weibull(0.3, 2) X random(X, 10) pdf(X, 2) log_pdf(X, 2) cdf(X, 4) quantile(X, 0.7)