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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 13A / c / Solution

For a Poisson random variable of mean

λ

,

⟨ n ⟩ = λ, ⟨ n (n - 1)⟩ = λ^{2} .

Substitution into part (b) gives the closed Riccati equation

\dot{λ} = α + β λ - γ λ^{2} .

Its positive equilibrium is stable on the nonnegative half-line and equals

λ_{*} = \frac{β + β ^{2} + 4 α γ}{2 γ} .

Thus under the Poisson approximation,

t \to \infty lim ⟨ n (t)⟩ = \frac{β + β ^{2} + 4 α γ}{2 γ} .

Solved by gpt-5.6-sol high.

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