Lifetimes

Manufactured items usually fail at some time after they start to be used. For example, light bulbs and computer hard disks don't keep going for ever — we know that they will fail at some time after purchase but don't know when.

Biological entities (bacteria, animals, plants, ...) also have limited lifetimes — they die at some random age.

In this section, we will describe a simple model for the lifetimes of such items.

Survivor and hazard functions

Consider an item whose lifetime is denoted by \(X\), whose probability density function is \(f(x)\) and whose cumulative distribution function is \(F(x) = \int_0^x {f(u) \; du}\). We now define two further functions that are particularly meaningful for lifetime data.

The survivor function describes the probability that an item's lifetime will be greater than any constant, \(x\).

Definition

The survivor function of random variable \(X\) is

\[ S(x) \;\; = \; \; P(X \gt x) \;\;=\;\; 1 - F(x) \]

The second function that we will define describes the chance of failure, conditional on having survived until time \(x\). More precisely, the probability of failure in a very short interval of time \((x, x+\delta x]\) is

\[ P(x \lt X \le x+\delta x) \;\; \approx \; \; f(x) \times \delta x \]

The conditional probability of failing in the interval \((x, x+\delta x]\), given survival until at least time \(x\) is

\[ \begin{align} P(x \lt X \le x+\delta x \;|\; X > x) \;\;&=\;\; \frac {P(x \lt X \le x+\delta x \textbf{ and } X > x)}{P(X > x)} \\ &\approx \; \; \frac {f(x)}{S(x)} \times \delta x \end{align} \]

The quantity \(f(x) / S(x)\) therefore describes the failure rate at time \(x\), conditional on having survived until at least time \(x\). It is called the distribution's hazard function.

Definition

The hazard function of random variable \(X\) is

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

The hazard function is particularly informative — it describes how an item's age, \(x\) affects its risk of failure.