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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 28K / b / Solution

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)

.

Solved by gpt-5.6-sol high.

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