 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/13a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

n

and sum. Equivalently, a birth changes

n

+ 1

and a death by

- 1

, so

\frac{d}{d t} ⟨ n ⟩ = ⟨ b_{n} - d_{n} ⟩ = α + β ⟨ n ⟩ - γ ⟨ n (n - 1)⟩ .

The equation is not closed in the mean because the quadratic death rate introduces the second factorial moment.

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