Probability generating function

$$\Pi_X(s) = E(s^X)$$

probability generating functions are a characteristic function of distributions, which take advantage of various properties of power series to derive useful properties.

The p.g.f if expressed in terms of an arbitrary variable s, which is arbitrary in the sense that the p.g.f. should hold for any particular value of s in the range |s| <= 1 (also called a dummy variable)

Useful properties can be written, by fixing s to some particular value such as;

$$s = 0$$ $$\Pi(0) = p(0)$$

Hence the P(X=0) is the value that the p.g.f. takes with parameter 0

when

$$s = 1$$ $$\Pi(s) = \Sum_{x=0}{\infty} p(x) = 1$$    since p is a probability function

= Mean =

$$\mu = \Pi'(1)$$

= Variance =

$$\sigma^2 = \Pi''(1) + \mu - \mu^2$$

= Compound distribution =

when some random variable can be described as a random sum of random variables, that variable is said to have a compound distribution

the p.g.f. of a compound distribution is given by

$$\Pi_{Z}(s) = \Pi_{Z}(\Pi_{X}(s))$$