1.Probability generating function 

The probability generating function (PGF) is a useful tool for dealing with discrete random variables taking

values 0, 1, 2, . . .. Its particular strength is that it gives us an easy way of characterizing 

the distribution of X +Y when X and Y are independent. In general it is difficult to find the distribution 

of a sum using the traditional probability function. The PGF transforms a sum into a product and enables

it to be handled much more easily. The name probability generating function also

gives us another clue to the  role  of  the  PGF. 

The  PGF  can  be  used  to  generate  all  the probabilities of the distribution.

This is generally tedious and is not often an efficient way of calculating probabilities.

However, the fact that it can be done demonstrates that the PGF tells us everything there is to know about the distribution.

Let X be a discrete random variable taking values in the non-negative integers {0, 1, 2, . . .}.

The probability generating function (PGF)  of  X  is   for  all 𝑠∈𝐼𝑅  for  which  the  sum converges. 

Calculating the probability generating function 

Properties of the PGF:

Binomial Distribution

Poisson Distribution

*Using  the  probability  generating  function  to  calculate probabilities

The probability generating function gets its name because the power series can be

expanded and differentiated to reveal the individual probabilities. Thus, given only the

PGF  we can recover all probabilities P(X = x).

For shorthand, write = P(X = x). Then 

Let X be a discrete random variable with PGF 

Find the distribution of X. 

Thus 

*Uniqueness of the PGF

The formula  shows that the whole sequence

of probabilities 

is determined by the values of the PGF and its derivatives at s = 0.

It follows that the PGF specifies a unique set of probabilities. 

Practical use: If we can show that two random variables have the same PGF in some

interval containing 0, then we have shown that the two random variables have the same distribution.

Another way of expressing this is to say that the PGF of X tells us everything there is to know about the distribution of X. 

*Probability  generating  function  for  a  sum  of  independent random variables

One of the PGF’s greatest strengths is that it turns a sum into a product:

This makes the PGF useful for finding the probabilities and moments of a sum of independent random variables. 

Theorem:Suppose that  are independent random

variables, and let  Then 

Suppose  that  X  and  Y  are  independent  with X ~ Poisson(𝜆).

and Y~Poisson(𝜇). Find the distribution of X + Y.

But this is the PGF of the Poisson(𝜆+𝜇)distribution.

So, by the uniqueness of PGFs, X+Y ~ Poisson(𝜆+𝜇).

(We can use the PGF to calculate the moments of the distribution of X)

As  well  as  calculating  probabilities,  we  can  also  use  the  PGF  to calculate  the 

moments  of  the  distribution  of  X.  The  moments  of  a distribution are the mean, variance, etc. 

Theorem: Let X be a discrete random variable with PGF𝐺௑(s). Then: 

(This is the kth factorial moment of X.)

1-

2-

Let X~Poisson(𝜆). The PGF of Find 𝐸(𝑋)𝑎𝑛𝑑 𝑉𝑎𝑟(𝑋).

For the variance, consider