 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability/8/b/ii/solution

G_{N} (s) = j = 0 \sum \infty p (1 - p)^{j} s^{j} = \frac{p}{1 - ( 1 - p ) s} .

The series converges for

∣ (1 - p) s ∣ < 1

, in particular on

[0, 1]

. At

p = 1/2

G_{N} (s) = 1/ (2 - s)

G_{N}^{'} (s) = (2 - s)^{- 2}

, and

G_{N}^{''} (s) = 2 (2 - s)^{- 3}

. The factorial moment identities give

E (N) = G_{N}^{'} (1) = 1

and

E [N (N - 1)] = G_{N}^{''} (1) = 2

. Thus

E (N) = 1, Var (N) = G_{N}^{''} (1) + G_{N}^{'} (1) - [G_{N}^{'} (1)]^{2} = 2.

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