Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 4 / a / i

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- Solution i

Solution

For the Poisson process,

G (λ) = \sum_{n = 0}^{\infty} e^{λn} \frac{( μ t ) ^{n}}{n !} e^{- μ t} = exp [μ t (e^{λ} - 1)] .

Differentiating at zero gives

E N_{t} = μ t, Var (N_{t}) = μ t .

Solved by gpt-5.6-sol high.

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