probability-generating-functions-pgfs

A probability generating function, ,  is a polynomial in  

• The powers of are the values of 

• The coefficients are the corresponding probabilities of 

For example:

• 0 1 4 5 0.4 0.3 0.2 0.1

• The PGF is This simplifies to 

The variable  is called a dummy variable

• It is used to create a polynomial structure

• Do not confuse it with

Coefficients can never be negative

• They are probabilities!

always

This is because substituting  into a PGF:

• Turns all powers of into 1 

• Leaves the sum of all probabilities which equals 1

• For example z
  

is the formal definition of a PGF given in the Formulae Booklet

• Recall that is the expectation of 

  • Th expectation of a function of is

• Choosing the function to be gives

  • This is the sum of powers of  multiplied by their corresponding probabilities

  • That is the probability generating function of 

Fully expand the PGF

• expands to 

Read off the relevant coefficient 

• is the coefficient of 

  • 

Remember 

When PGFs can be written in the form

• Where  is a positive or negative rational number

• You may have to rearrange to get this form

Use the General Binomial Theorem to expand the PGF

• Simplify each term

• Read off probabilities

When a PGF is written as a function that is not a polynomial

For example

• 

• A Maclaurin Series is given by

• Use it to expand the PGF

  • 

• Then simplify each term