 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/5/b/ii/solution

For an independent nonnegative integer-valued count

N

and identically distributed independent summands, the random sum probability generating function is

G_{Z} (s) = G_{N} (G_{X_{1}} (s))

. Conditioning on

N = n

proves this, since the conditional probability generating function is

G_{X_{1}} (s)^{n}

Here

G_{X_{1}} (s) = 1 - r + rs

, so

G_{Z} (s) = \frac{p}{1 - ( 1 - p ) ( 1 - r + rs )} = \frac{p}{p + ( 1 - p ) r - ( 1 - p ) rs} .

This identity holds throughout

∣ s ∣ \leq 1

, and its power-series extension has radius

[p + (1 - p) r] / [(1 - p) r]

 New to topics? Read the docs here!