 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/5/a/iii/solution

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

Since these are

E [N]

and

E [N (N - 1)]

, respectively,

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

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