Generalization of the gamma distribution. Often used in survival and time-to-event analyses.
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):
$$\sum_{n=0}^\infty \frac{t^n\lambda^n}{n!} \Gamma(1+n/k), k \ge 1$$
Examples
set.seed(27)
X <- Weibull(0.3, 2)
X
#> [1] "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] 0.05518192
log_pdf(X, 2)
#> [1] -2.89712
cdf(X, 4)
#> [1] 0.7080417
quantile(X, 0.7)
#> [1] 3.713233