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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 12F / a / Solution

The probability generating function is

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

Termwise differentiation gives

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]

.

Solved by gpt-5.6-sol high.

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