 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-4/27k/a/solution

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

A \subseteq R^{d}

, let

N (A) = # (Π \cap A)

λ

is characterized by:

N (A) \sim Poisson (Λ (A)), Λ (A) = \int_{A} λ (x) d x;

the counts $N (A_{1}), \dots, N (A_{m})$ are independent random variables whenever $A_{1}, \dots, A_{m}$ are pairwise disjoint bounded Borel sets.

Because

Λ

has a density with respect to Lebesgue measure, it is atomless, so the process is simple: almost surely no location occurs more than once.

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