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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 6E / b / Solution

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} .

This is the generator identity for a continuous-time Markov chain applied to the test function

f

.

Solved by gpt-5.6-sol high.

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