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

The expected values of early and late new diagnosis counts at step $k$ areEach summand in $n_k$ corresponds to an infection that avoids early diagnosis, enters the non-early state one step later, remains there for the intervening steps and is then diagnosed. Define $h_j=0$ for $j\le 0$, and take an empty sum as zero. The early share of diagnosis intensity isIt is defined only when $e_k+n_k>0$; in particular there are no diagnoses at step 1 under the initial condition. With infection counts following a Poisson distribution and independent marking, early and late diagnosis counts in a fixed interval are independent Poisson random variables. Conditional on a positive total, the early count has a binomial distribution given the total with probability $p_{Ek}$, so this ratio also equals the expectation of the observed early fraction conditional on a nonzero total. Without that conditioning a sample fraction at zero diagnoses is undefined.