Weibull distribution
/ VY-bull /
The Weibull distribution is the flexible model for time-to-failure when things age — when the chance of breaking down changes as a component, machine, or life gets older. The exponential assumes a failure rate that never changes; the Weibull relaxes that, letting the hazard rise over time (wear-out), fall over time (early-life weeding out of defects), or stay flat. This makes it the backbone of reliability engineering and a natural fit for many actuarial lifetime problems.
It has a shape parameter (often written τ or k) and a scale parameter θ. The shape is the storyteller: if it equals 1 the Weibull is just an exponential (constant hazard); if it is above 1 the hazard increases with age (things wear out, like older equipment failing faster); if it is below 1 the hazard decreases (infant mortality, where survivors of an early shaky period grow sturdier). This single dial captures the famous 'bathtub curve' of failure rates piece by piece.
Actuaries use the Weibull as a survival and severity model. As an analytic mortality law it can represent the rising force of mortality at older ages, complementing Gompertz and Makeham. As a severity distribution it offers a tail that can be tuned heavier or lighter than the exponential, useful for claim sizes. Its honest treatment of a changing hazard is exactly what the memoryless exponential cannot provide — but that flexibility means the shape parameter must be estimated with care, since it dictates whether your model believes risk grows or shrinks with age.
Model the lifetime of a machine part with a Weibull whose shape is 1.5 (>1), so the failure rate climbs as the part ages — old parts fail faster than new ones, capturing wear-out that a constant-rate exponential would miss.
The Weibull's shape parameter decides whether risk rises, falls, or holds steady with age.
The Weibull's tail is still lighter than a Pareto's; it captures aging well but should not be relied on for the most extreme catastrophe losses, where a genuine power-law tail is needed.