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 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

For a nonnegative integer-valued random variable, its probability generating function is

G_{X} (s) = E [s^{X}] = k = 0 \sum \infty P (X = k) s^{k} .

The series converges absolutely at least for

∣ s ∣ \leq 1

, with

0^{0} = 1

in the

k = 0

term; it may converge on a larger domain.

ii

Solution

This is the geometric distribution on

{0, 1, 2, \dots}

with success parameter

1/2

. Its probability generating function is a geometric series:

G_{N} (s) = \frac{1}{2} k = 0 \sum \infty (\frac{s}{2})^{k} = \frac{1}{2 - s}, ∣ s ∣ < 2.

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