distributional (version 0.5.0)

dist_weibull: The Weibull distribution

Description

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

Usage

dist_weibull(shape, scale)

Arguments

shape, scale

shape and scale parameters, the latter defaulting to 1.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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):

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

See Also

stats::Weibull

Examples

dist <- dist_weibull(shape = c(0.5, 1, 1.5, 5), scale = rep(1, 4))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)