 Solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

A particle arriving at time

s \leq t

survives until

t

e^{- μ (t - s)}

. Independently retaining each Poisson arrival with this time-dependent probability is Poisson thinning. Hence the number alive at time

t

is Poisson with mean

λ \int_{0}^{t} e^{- μ (t - s)} d s = \frac{λ}{μ} (1 - e^{- μ t}) .

Thus

N_{t}^{alive} \sim Poisson (\frac{λ}{μ} (1 - e^{- μ t})) .

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