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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 28L / a / Solution

Let

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

. The infinitesimal definition gives the forward equations

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 !},

so

X_{t} \sim Poisson (λ t)

.

Solved by gpt-5.6-sol high.

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