 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/4/a/ii/solution

G_{X} (s) = E (s^{X}) = \sum_{k \geq 1} p q^{k - 1} s^{k} = \frac{p s}{1 - q s}, ∣ q s ∣ < 1.

G_{X}^{'} (s) = \frac{p ( 1 - q s ) + pq s}{( 1 - q s ) ^{2}} = \frac{p}{( 1 - s + p s ) ^{2}} .

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