Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 7 / b / ii

Table of contents
- Solution ii

Solution

For

0 \leq s < 1

, differentiate the probability generating function term by term and multiply by

s / μ

:

G_{Y} (s) = k \geq 1 \sum \frac{k p _{k}}{μ} s^{k} = \frac{s}{μ} k \geq 1 \sum k p_{k} s^{k - 1} = \frac{s}{μ} G_{Y}^{'} (s) .

This is the size-biased probability generating function identity. At

s = 1

, the finite-mean assumption gives the limiting value one on both sides; at zero, positive support gives value zero.

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