PGFs of X+Y
How do I find the PGF of X + Y?
- If then
- Multiply the PGFs together
- This only works if and are independent distributions
How do I find the PGF of a repeatedly used distribution?
- Let be a discrete random variable with PGF
- If where each is its own independent distribution
- Then
- For example
- could be the winnings on a game played once
- represent five different winnings when played five times
- assuming each game is independent of the last
- The winnings after five games has the PGF
How do I prove that the sum of two independent Poisson distributions is Poisson?
- If and are independent then
- and
- So has the PGF
- Add the powers in the exponentials
- Factorise out
- So
- This is exactly the PGF for the distribution of
- So the sum of two independent Poisson distributions is a Poisson distribution
Exam Tip
- 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.