Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 8 / b / ii

Table of contents
- Solution ii

Solution

For

0 \leq s \leq 1

, condition on

N = k

. Independence of the descendant total progeny variables gives

E [s^{\sum_{i = 1}^{N} Z_{i}} ∣ N = k] = \prod_{i = 1}^{k} E [s^{Z_{i}}] = G_{Z} (s)^{k},

including the empty product

1

when

k = 0

. Taking expectation over

N

proves the probability generating function of an independent random sum formula

E [s^{\sum_{i = 1}^{N} Z_{i}}] = k = 0 \sum \infty P (N = k) G_{Z} (s)^{k} = G_{N} (G_{Z} (s)) .

The same identity holds for complex

∣ s ∣ < 1

by absolute convergence.

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