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the Weibull distribution

Peter Ralph

5 January 2021 – Advanced Biological Statistics

Stochastic minute: Weibull

The Weibull distribution

If \(T\) has a Weibull distribution, with scale \(\lambda\) and shape \(k\), then

\(T \ge 0\)
\(\P\{ T > t \} = \exp\left(- (t/\lambda)^k \right)\)
the mean is proportional to the scale: \(\E[T] = \lambda \times \Gamma(1 + 1/k)\)

It is mostly used in survival analysis, because its hazard rate is: \[\begin{aligned} h(t) = k \frac{1}{\lambda} \left(\frac{t}{\lambda}\right)^{k-1} . \end{aligned}\] which allows rates to go down (\(k<1\)), up (\(k>1\)), or stay flat (\(k=1\)) over time.

plot of chunk r plot_wei

Examples:

Time until arrival of cosmic particles: constant hazard rate, so shape = 1.
Heights of trees hit by lightning: hazard rate is higher for taller trees, so shape > 1.
Lifetime of iphones: some phones have faulty manufacturing, so will fail soon, but if not then they’ll probably last longer - decreasing hazard rate, so shape < 1.