Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 7 / a / Solution

Choose an individual uniformly from the population. The law of total probability gives the population density and survivor function as mixtures:A hazard function is a ratio of density to survival, so averaging the densities does not average those ratios. Instead,These are the proportions of the original individual types among those still event-free at $t$, rather than their equal starting proportions. For example, a mixture of different exponential distributions progressively favours the lower-rate individuals. This population hazard of a survival mixture generally differs from $n^{-1}{\sum }_i{h}_i(t)$.