 Solution

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/i/solution

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

using

s^{0} = 1

also at

s = 0

. It is always defined for

0 \leq s \leq 1

, and indeed the power series converges absolutely for all real or complex

s

with

∣ s ∣ \leq 1

, because

\sum_{k = 0}^{\infty} p_{X} (k) ∣ s ∣^{k} \leq \sum_{k = 0}^{\infty} p_{X} (k) = 1.

It may be defined beyond the unit disk where this power series converges absolutely; such a larger domain depends on the distribution. In particular

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

and

G_{X} (1) = 1

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