Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 3 / c

For the stated scale-parameterisation of the Weibull distribution, its incubation survivor function is $S_T(s)=\exp [-(s/\lambda {)}^{\kappa }]$ for $s\ge 0$, with $\lambda ,\kappa >0$. The continuous back-calculation of infection incidence equation becomesSet $h(t-s)=0$ before the infection process's start if one is specified. For the equal-width endpoint approximation, integrate the incubation density over each delay bin:Thus the discrete Weibull equation isFor unequal intervals replace $q_{k-i}$ by $S_T(\max \{0,t_{k-1}-t_i\})-S_T(\max \{0,t_k-t_i\})$. These are probabilities, not point evaluations of a density; no additional factor of $\Delta$ is needed after bin integration.