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# Properties of The Hazard Function $h(t)$

- The hazard function relates the instantaneous, conditional probability of failure in $t+\Delta t$ given $T>t$ 

$$h(t)=\lim_{\Delta t \to 0} \frac{Pr(t < T \le t+\Delta t|T \ge t)}{\Delta t}$$

- Alternate names for the hazard function

    + The hazard rate function

    + The failure rate function

    + Force of mortality (common in actuarial science)

- The bathtub-shaped hazard curve (Figure 2.2) is often used to described the lifetimes of systems or people

    + Infant Mortality: Systems (people) can have a relatively high probability of failure in early life

    + Useful Life: Systems (people) that survive the early stages of life typically experience a long period in which the probability of failure is relatively low

    + Wearout: As systems (people) age the probability of failure begins to increase

