Weibull: Create a Weibull distribution

In distributions3: Probability Distributions as S3 Objects

Create a Weibull distribution

Description

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

Usage

Weibull(shape, scale)

Arguments

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

Value

A Weibull object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform()

Examples

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)

distributions3 documentation built on Sept. 30, 2024, 9:37 a.m.