Lifetime distributions

In the previous chapter, we presented the \(\ExponDistn(\lambda)\) distribution as a possible model for the lifetimes of items. However the exponential distribution's hazard function is constant, \(h(x) = \lambda\), and this is unrealistic in most applications — usually items become more likely to fail as they age and wear down.

We now describe a more general model that allows the hazard rate to increase or decrease over time.

Weibull distribution

The Weibull distribution is another standard distribution that is often used for lifetime data.

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 is particularly useful for lifetime data since its hazard function has a simple form.

Weibull hazard function

If a random variable \(X\) has a \(\WeibullDistn(\alpha, \lambda)\) distribution, its hazard function is

\[ h(x) \;\;=\;\; \alpha \lambda^{\alpha} x^{\alpha - 1} \]

The Weibull distribution's survivor function is

\[ S(x) \;\;=\;\; P(X > x) \;\;=\;\; \int_x^{\infty} {\alpha \lambda^{\alpha} u^{\alpha - 1} e^{-(\lambda u)^{\alpha}}} \; du \]

We can integrate this with a change of variable

\[ v = (\lambda u)^{\alpha} \spaced{and} dv = \alpha \lambda(\lambda u)^{\alpha - 1} \;du\]

so

\[ S(x) \;\;=\;\; \int_{ (\lambda x)^{\alpha}}^{\infty} e^{-v} \; dv \;\;=\;\; \Big[-e^{-v} \Big]_{(\lambda x)^{\alpha}}^{\infty} \;\;=\;\; e^{-(\lambda x)^{\alpha}}\]

The hazard function is

\[ h(x) \;\;=\;\; \frac{f(x)}{S(x)} \;\;=\;\; \alpha \lambda^{\alpha} x^{\alpha - 1} \]

The hazard rate changes as the age of the items, \(x\), increases,

\[ h(x) \;\;\propto\;\; x^{\alpha - 1} \]

The Weibull distribution can therefore 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.