 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-3/6c/a/solution

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

n

through a birth from

n - 1

n + 2

, and leaves through either event at state

n

\frac{\partial P ( n , t )}{\partial t} = λ P (n - 1, t) + β (n + 2)^{2} P (n + 2, t) - (λ + β n^{2}) P (n, t),

with probabilities and impossible transition rates taken as zero outside the nonnegative states.

A birth changes

n

+ 1

, while a death event changes it by

- 2

. Applying these increments to the first moment gives

\frac{d}{d t} ⟨ n ⟩ = λ - 2 β ⟨ n^{2} ⟩ .

Consequently every steady state must satisfy

⟨ n^{2} ⟩ = \frac{λ}{2 β} .

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