Probability Generating Functions

Definition and Usage

A Probability Generating Function (PGF) is a way to encode the probabilities of discrete random variables.
It is simply a polynomial where the coefficients indicate the probabilities in the distribution.
Typically used for its beneficial properties in handling various calculations related to random variables.

For a discrete random variable X, the PGF, denoted G_X(s), is defined as Σs^x P(X = x), where the summation extends over all possible values x that the random variable X can take.
Here, “s” is usually a real or complex number between -1 and 1, and P(X = x) is the probability that the random variable X equals x.
The PGF is essentially a series expansion of probabilities, where each term corresponds to some outcome of the random variable.

The n-th derivative of the PGF evaluated at s=1 gives the expected value of the n-th power of the random variable X, i.e., E(X^n).
A PGF uniquely identifies the discrete probability distribution from which it is derived, i.e., if two random variables have the same PGF, then they have the same distribution.
If we multiply the generating functions of two independent random variables together, we get the generating function of their sum.

PGFs are widely used to solve probabilistic problems in fields like telecommunication, computer science, and physics.
They offer a systematic way to deal with operatively complicated sums.
With PGFs, we get a compact and elegant representation of a random variable’s entire probability distribution.

PGFs can be derived only for discrete random variables. They are not applicable for continuous random variables.
Calculating the PGF and its derivatives can be computationally intensive, especially for large values of n.
Decoding a distribution from its PGF may sometimes require knowledge of specific techniques, such as power series or partial fraction decomposition.