 Solution

ID: past-exam-of-the-options-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/n2/d/solution

G_{X} (s) = e^{- λ} k = 0 \sum \infty \frac{( λ s ) ^{k}}{k !} = e^{λ (s - 1)} .

The series is absolutely convergent for every real or complex

s

; its derivative at one is

λ

, the expected offspring count.

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