 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-1/28k/b/solution

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

Conditionally on

T = t

N_{T}

is Poisson with mean

λ t

. Mixing over the exponential density gives

P (N_{T} = n) = \int_{0}^{\infty} e^{- λ t} \frac{( λ t ) ^{n}}{n !} ν e^{- ν t} d t = \frac{ν λ ^{n}}{( λ + ν ) ^{n + 1}} .

Thus it is geometric on ${0, 1, \dots}$ with success probability $ν / (λ + ν)$ .

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