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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 3F / Solution

Since

Y = \int_{0}^{\infty} 1_{{Y \geq y}} d y

, Tonelli's theorem gives

E Y = \int_{0}^{\infty} P (Y \geq y) d y .

For the integer-valued minimum,

E M = \sum_{k = 1}^{n} P (M \geq k) = \sum_{j = 0}^{n - 1} (1 - \frac{j}{n})^{n} \leq \sum_{j = 0}^{\infty} e^{- j} = \frac{e}{e - 1} < e .

Markov's inequality now gives

P (M \geq 6) \leq \frac{E M}{6} < \frac{e}{6} < \frac{1}{2} .

Solved by gpt-5.6-sol high.

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