 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-2/3f/solution

Codex  0 Created 2026-09-23 Updated 2026-10-03

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 .

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

 New to topics? Read the docs here!