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PGFs of Sums & Transformations (Edexcel A Level Further Maths) : Revision Note

Mark Curtis

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PGFs of X+Y

How do I find the PGF of X + Y?

  • If Z equals X plus Y then straight G subscript Z open parentheses t close parentheses equals straight G subscript X open parentheses t close parentheses cross times straight G subscript Y open parentheses t close parentheses

    • Multiply the PGFs together

  • This only works if X and Y are independent distributions

How do I find the PGF of a repeatedly used distribution?

  • Let X be a discrete random variable with PGF straight G subscript X open parentheses t close parentheses

  • If Z equals X subscript 1 plus X subscript 2 plus... plus X subscript n where each X subscript i space end subscriptis its own independent X distribution

    • Then straight G subscript Z open parentheses t close parentheses equals straight G subscript X subscript 1 end subscript open parentheses t close parentheses straight G subscript X subscript 2 end subscript open parentheses t close parentheses... straight G subscript X subscript n end subscript open parentheses t close parentheses equals open square brackets straight G subscript X open parentheses t close parentheses close square brackets to the power of n

  • For example

    • X could be the winnings on a game played once

    • X subscript 1 comma space X subscript 2... comma X subscript 5 represent five different winnings when played five times

      • assuming each game is independent of the last

    • The winnings after five games has the PGF open square brackets straight G subscript X open parentheses t close parentheses close square brackets to the power of 5

How do I prove that the sum of two independent Poisson distributions is Poisson?

  • If X tilde Po open parentheses lambda close parentheses and Y tilde Po open parentheses mu close parentheses are independent then

    • straight G subscript X open parentheses t close parentheses equals straight e to the power of lambda open parentheses t minus 1 close parentheses end exponent and straight G subscript Y open parentheses t close parentheses equals straight e to the power of straight mu open parentheses t minus 1 close parentheses end exponent

  • So Z equals X plus Y has the PGF straight G subscript Z open parentheses t close parentheses equals straight G subscript X open parentheses t close parentheses straight G subscript Y open parentheses t close parentheses equals straight e to the power of lambda open parentheses t minus 1 close parentheses end exponent straight e to the power of mu open parentheses t minus 1 close parentheses end exponent

    • Add the powers in the exponentials

    • Factorise out open parentheses t minus 1 close parentheses

    • So straight G subscript Z open parentheses t close parentheses equals straight e to the power of open parentheses lambda plus mu close parentheses open parentheses t minus 1 close parentheses end exponent

  • This is exactly the PGF for the distribution of Po open parentheses lambda plus mu close parentheses

    • So the sum of two independent Poisson distributions is a Poisson distribution

Examiner Tips and Tricks

  • straight G subscript Z open parentheses t close parentheses equals straight G subscript X open parentheses t close parentheses cross times straight G subscript Y open parentheses t close parentheses is given in the Formulae Booklet

Worked Example

A fair three-sided spinner, labelled 2, 4 and 8, is spun. A biased coin, labelled 4 and 6, is flipped, where there is a 25% chance of landing on a 4.

Use probability generating functions to find the probability that the sum of the scores is 8.

pgfs-of-sums-1
pgfs-of-sums-2

PGFs of aX+b

How do I find the PGF of aX + b?

  • LetY equals a X plus b be a linear transformation of X

  • Then straight G subscript Y open parentheses t close parentheses equals t to the power of b straight G subscript X open parentheses t to the power of a close parentheses

    • Replace t with t to the power of a

    • Multiply the PGF by t to the power of b

  • For example

    • Take straight G subscript X open parentheses t close parentheses equals open parentheses 0.9 plus 0.1 t close parentheses to the power of 10

    • Let Y equals 8 X plus 3

    • So straight G subscript Y open parentheses t close parentheses equals t cubed open parentheses 0.9 plus 0.1 t to the power of 8 close parentheses to the power of 10

  • The formula comes from

Examiner Tips and Tricks

  • Beware, you need to learn straight G subscript Y open parentheses t close parentheses equals t to the power of b straight G subscript X open parentheses t to the power of a close parentheses as it is not given in the Formulae Booklet

Worked Example

The discrete random variable X has a probability generating function given by

straight G subscript X open parentheses t close parentheses equals fraction numerator t over denominator 5 minus 4 t end fraction

Find and simplify the probability generating function of Y, where Y equals 2 open parentheses X plus 5 close parentheses.

pgfs-of-ax-and-b

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    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.