 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/6e/b/solution

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

Differentiate

⟨ f (n)⟩ = \sum_{n \geq 0} f_{n} p_{n}

and substitute the three master equations. Reindexing every gain term by its source state gives one contribution per possible transition:

\frac{d}{d t} ⟨ f (n)⟩ = b n = 0 \sum \infty (f_{n + 1} - f_{n}) p_{n} - d n = 2 \sum \infty (f_{n} - f_{n - 2}) n p_{n} - D (f_{1} - f_{0}) p_{1} .

f

 New to topics? Read the docs here!