Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 44 / 1 / a / Solution

The paper uses $F$ for the survivor function, rather than for the cumulative distribution function. For a nonnegative survival time $T$ with an absolutely continuous probability distribution, defineWhere $F(t)>0$, the hazard function is the instantaneous event rate conditional on survival:The integrated hazard is $H(t)={\int }_0^{t}h(u)du$. Since $h=-F^{\prime }/F$, integration gives $H(t)=\log F(0)-\log F(t)$. With the usual time origin and no event at zero, $F(0)=1$, soConversely, differentiating $F=e^{-H}$ gives $F^{\prime }=-hF$, so these relationships determine each of the survivor function, hazard function and integrated hazard from either of the others. A positive limiting survivor function is possible if the integrated hazard remains finite; a proper finite event time instead has $F(t)\to 0$.