 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2021/discrete-mathematics-continuous-mathematics-and-probability/7/b/ii/solution

G_{X} (s) = E [s^{X}] = e^{- λ} \sum_{r = 0}^{\infty} \frac{( λ s ) ^{r}}{r !} = e^{λ (s - 1)} .

The series is absolutely convergent for every finite real

s

; in particular the expression is finite for all

s > 1

, as needed for the later tail estimate.

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