 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ia/paper-2/10f/v/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For

k \geq n

p_{k} (n) = (n - 1 k - 1) p^{n} (1 - p)^{k - n}

. It is the sum of

n

independent geometric variables, so the central limit theorem applies with mean

n / p

n (1 - p) / p

. Take

k_{a} (n) = ⌈ n / p + a n (1 - p) / p ⌉

and

k_{b} (n) = ⌊ n / p + b n (1 - p) / p ⌋

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