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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-1/26j/e/solution

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Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 26J / e / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Assign each arriving car independently to a uniformly chosen lot. The Poisson thinning theorem gives three independent Poisson processes, each of rate $λ /3$ . For an interval of length

t

, their joint probability generating function is

exp [λ t (\frac{z _{a} + z _{b} + z _{c}}{3} - 1)] = \prod_{i = a, b, c} exp [\frac{λ t}{3} (z_{i} - 1)] .

(1)

The factorization proves independence of the counts in that interval; independence over disjoint intervals proves independence of the processes.

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