 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ia/paper-2/9f/iv/solution

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

Only triangle pairs sharing an edge have nonzero covariance, so

var T = (3 n) (p^{3} - p^{6}) + 2 (2 n) (2 n - 2) (p^{5} - p^{6}) .

After division by

(E T)^{2}

this is

O (n^{- 3} p^{- 3} + n^{- 2} p^{- 1}) \to 0

for

p = n^{- α}

0 < α < 1

; use part (iii).

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