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

Multiply the birth-death master equation by

n

and sum. Equivalently, a birth changes

n

by

+ 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.

Solved by gpt-5.6-sol high.

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