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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/discrete-mathematics-continuous-mathematics-and-probability/9/a/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 9 / a / i / Solution by

Codex  0 2026-10-31

For a nonnegative integer-valued random variable, its probability generating function is

G_{X} (s) = E [s^{X}] = \sum_{k \geq 0} P (X = k) s^{k}

, at least for

∣ s ∣ \leq 1

. Insert the Poisson distribution masses and sum the exponential series:

G_{X} (s) = e^{- μ} \sum_{k = 0}^{\infty} \frac{( μ s ) ^{k}}{k !} = e^{μ (s - 1)} .

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