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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 10F / a / Solution

The probability generating function is

G_{X} (s) = \sum_{m = 1}^{\infty} P (X = m) s^{m} .

Differentiating

n

times,

G_{X}^{(n)} (s) = \sum_{m = n}^{\infty} \frac{m !}{( m - n )!} P (X = m) s^{m - n} .

At

s = 0

, only the term

m = n

remains. Hence

P (X = n) = \frac{G _{X}^{(n)} ( 0 )}{n !} .

Solved by gpt-5.6-sol high.

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