 Solution

ID: past-exam-of-the-options-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/n2/a/solution

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

where the series is defined, with the usual power-series convention

s^{0} = 1

even at

s = 0

. In particular

G_{X} (0) = p_{0}

and

G_{X} (1) = 1

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