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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-12/6/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 12 / 6 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The first right endpoint is the smallest of

n

independent pair maxima. A maximum exceeds

r

with probability

1 - r^{2}

, so for

0 \leq x \leq n

,

Pr (R_{1} \geq x / n) = (1 - \frac{x ^{2}}{n})^{n} .

Taking the limit gives the requested, correctly normalized result

Pr (n R_{1} \geq x) ⟶ e^{- x^{2}} (x > 0) .

Thus

n R_{1}

converges to a continuous nonnegative limit

W

with survival function

Pr (W \geq x) = e^{- x^{2}}

. In particular the scaled first right endpoint is tight. This calculation uses maxima of pairs, not the first order statistic of all

2 n

individual endpoints, which would have a different scale. The converted TeX loses the formula in this part; the displayed limit has been recovered directly from the PDF.

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