 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/8/a/ii/solution

By hypothesis the power series converges at every positive real argument, so its radius of convergence is infinite. Termwise differentiation of a power series is therefore valid in a neighborhood of

s = 1

. For

k \geq 1

G^{(k)} (s) = \sum_{j = k}^{\infty} j (j - 1) \dots (j - k + 1) p (j) s^{j - k} .

s = 1

, the right side is the expectation of the falling factorial of

X

G^{(k)} (1) = E [X (X - 1) \dots (X - k + 1)] .

k = 1

gives

G^{'} (1) = E [X] .

For

k = 0

1

; the equality becomes

G (1) = 1

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