Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 1 / c / Solution

For each person, infection occurs with probability

λ

and, conditional on infection, a GP visit occurs with probability

ρ_{G}

. Independent Bernoulli thinning therefore gives visit probability

λ ρ_{G}

. More explicitly, the probability generating function is

E [s^{Y_{G}} ∣ N, λ, ρ_{G}] = (1 - λ + λ (1 - ρ_{G} + ρ_{G} s))^{N} = (1 - λ ρ_{G} + λ ρ_{G} s)^{N} .

Thus

Y_{G} ∣ N, λ, ρ_{G} \sim Binomial (N, λ ρ_{G}),

and the likelihood is

L_{G} (λ, ρ_{G}; y_{G}) = (y N)_{G} (λ ρ_{G})^{y_{G}} (1 - λ ρ_{G})^{N - y_{G}} .

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