Lifetimes
Manufactured items usually fail at some time after they start to be used, and biological entities also have limited lifetimes. An item's lifetime is denoted by\(X\), with probability density function \(f(x)\) and cumulative distribution function \(F(x) = \int_0^x {f(u) \; du}\).
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 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 \(x\).
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.