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

An arrival at

s \leq t

remains active with probability

1 - (t - s)

when

0 \leq t - s \leq 1

, and zero later. By Poisson thinning, the detector count is Poisson with mean

m (t) = λ \int_{m a x (0, t - 1)}^{t} [1 - (t - s)] d s = {λ (t - t^{2} /2), λ /2, 0 \leq t \leq 1, t \geq 1.

Solved by gpt-5.6-sol high.

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