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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ia/paper-2/10f/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 10F / d / Solution by

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

For the waiting time for a new record after a fixed sample, given

U = F (Y_{n})

, no overflow in the first

k

subsequent years has probability

U^{k}

. Averaging as in part (c) yields

P (τ > k) = n \int_{0}^{1} u^{n + k - 1} d u = \frac{n}{n + k}, k \geq 0.

(1)

The tail-sum formula for expectation therefore gives

E τ = \sum_{k = 0}^{\infty} P (τ > k) = n \sum_{k = 0}^{\infty} \frac{1}{n + k} = \infty.

(2)

The answer is $E τ = \infty$ . This does not contradict almost-sure finiteness: the tail probability tends to zero, but too slowly for its sum to converge.

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