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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability-resit/8/a/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 8 / a / ii / Solution by

Codex  0 2026-10-31

Let

S_{n} = \sum_{i = 1}^{n} X_{i}

. By independence, the expectation of a product of independent random variables factors, and the identical distributions give

G_{S_{n}} (s) = E [i = 1 \prod n s^{X_{i}}] = i = 1 \prod n E [s^{X_{i}}] = [G_{X} (s)]^{n} .

This is valid for

∣ s ∣ \leq 1

and wherever the required absolute expectations exist. It is the probability generating function of an independent sum.

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