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.

\(\alpha \gt 1\)
The hazard function \(h(x)\) is an increasing function of \(x\) so the item becomes less reliable as it gets older.
\(\alpha \lt 1\)
The hazard function \(h(x)\) is a decreasing function of \(x\) so the item becomes more reliable as it gets older.
\(\alpha = 1\)
The hazard function \(h(x)\) is constant and the lifetime distribution is an exponential distribution.