Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 6 / a

For a nonnegative continuous event time $T$, the survivor function is $F(t)=\mathbb{P}(T>t)$, and its probability density function isThe hazard function is the instantaneous event rate conditional on being event-free at that time:If the event is certain to occur in finite time, the density is proper: ${\int }_0^{\infty }f(t)dt=1$, so $F(t)\to 0$. Since $F(0)=1$ and $F^{\prime }=-hF$, the cumulative hazard function is $H(t)=-\log F(t)$. Therefore a certain eventual event requires unbounded integrated hazard:If the lifetime has a finite upper endpoint, the same divergence occurs as that endpoint is approached, and $H$ is interpreted as infinite beyond it.