 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/7/a/i/solution

G_{X} (s) = E [s^{X}] = k = 0 \sum \infty P (X = k) s^{k} .

It is defined at least for

∣ s ∣ \leq 1

. The usual probability generating function is intended for nonnegative integer-valued random variables; a general discrete real-valued random variable need not have this form. The coefficient of

s^{k}

recovers

P (X = k)

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