 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ia/paper-2/12f/a/solution

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

G_{X} (t) = E [t^{X}] = \sum_{k = 0}^{\infty} p_{k} t^{k}, 0 \leq t \leq 1.

G_{X}^{'} (t) = \sum_{k \geq 1} k p_{k} t^{k - 1}, G_{X}^{''} (t) = \sum_{k \geq 2} k (k - 1) p_{k} t^{k - 2} .

The coefficients are nonnegative, and

P (X \geq 2) > 0

, so both derivatives are positive and nondecreasing on

(0, 1]

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