 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/28l/a/solution

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

Let

p_{n} (t) = P (X_{t} = n)

p_{0}^{'} = - λ p_{0}, p_{n}^{'} = λ p_{n - 1} - λ p_{n} (n \geq 1),

with

p_{0} (0) = 1

. Solving recursively gives

P (X_{t} = n) = e^{- λ t} \frac{( λ t ) ^{n}}{n !},

X_{t} \sim Poisson (λ t)

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