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

- The cumulative hazard function relates the "accumulated" hazard rate over time

$$H(t)=\displaystyle \int_{0}^{t} h(t)dx$$

- The cumulative hazard function may also be used to compute the average hazard rate $T \in (t_1,t_2]$ 

$$
\begin{aligned}
AHR(t_1,t_2)&=\frac{\int_{t_{1}}^{t_{2}}h(u)du}{t_2-t_1}=\frac{H(t_2)-H(t_1)}{t_2-t_1}\\\\\\\\ AHR(0,t)&=\frac{\int_{0}^{t}h(u)du}{t}=\frac{H(t)}{t}\approx\frac{F(t)}{t}\;\;\text{ for }\; F(t) < 0.10 
\end{aligned}
$$
