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

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- Solution ii

Solution

Summing a geometric series gives the probability generating function of the zero-based geometric distribution:

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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