 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability-resit/8/a/ii/solution

G_{X} (s) = E (s^{X}) = \sum_{n = 0}^{\infty} s^{n} P (X = n) .

Substituting the Poisson distribution and evaluating the exponential series gives

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

The series converges for every finite complex

s

; in particular it gives the probability generating function on

0 \leq s \leq 1

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