 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/discrete-mathematics-probability-and-continuous-mathematics/8/a/iii/solution

G_{N}^{'} (s) = \frac{pq}{( 1 - q s ) ^{2}}, G_{N}^{''} (s) = \frac{2 p q ^{2}}{( 1 - q s ) ^{3}} .

E [N] = G_{N}^{'} (1) = q / p

and

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

. Consequently

E [N] = \frac{1 - p}{p}, Var (N) = \frac{2 q ^{2}}{p ^{2}} + \frac{q}{p} - \frac{q ^{2}}{p ^{2}} = \frac{1 - p}{p ^{2}} .

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