Lifetime distributions
The \(\ExponDistn(\lambda)\) distribution is an appropriate model for the lifetime of an item if its hazard function is constant, \(h(x) = \lambda\). This is unrealistic in most applications — usually items become more likely to fail as they age and wear down.
The Weibull distribution is a more general model that allows the hazard rate to increase or decrease over time.
Definition
A random variable \(X\) is said to have a Weibull distribution with parameters \(\alpha \gt 0\) and \(\lambda \gt 0\),
\[ X \;\;\sim\;\; \WeibullDistn(\alpha,\; \lambda) \]if its probability density function is
\[ f(x) \;\;=\;\; \begin{cases} \alpha \lambda^{\alpha} x^{\alpha - 1} e^{-(\lambda x)^{\alpha}} & x \gt 0 \\[0.4em] 0 & \text{otherwise} \end{cases} \]The Weibull distribution's hazard function has a particularly simple form.
Weibull hazard function
If a random variable \(X\) has a \(\WeibullDistn(\alpha, \lambda)\) distribution, its hazard function is
\[ h(x) \;\;=\;\; \alpha \lambda^{\alpha} x^{\alpha - 1} \](Proved in full version)
Since \(h(x) \;\;\propto\;\; x^{\alpha - 1}\), the Weibull distribution can be used as a model for items that either deteriorate or improve over time.