Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 5 / a / i / Solution

Writing the two functions in the question as survivor functions, $S_2(t)=S_1(t{)}^{\lambda }$. Since $H_k(t)=-\log S_k(t)$,Differentiating at times where the hazard functions exist gives $h_2(t)=\lambda h_1(t)$. Their hazard ratio is therefore the constant $\lambda$, so they form a proportional hazards family.