Probability generating functions

Learning objectives

Define the probability generating function of a discrete random variable taking non-negative integer values as G(t) = E(tˣ) = Σ P(X = x)tˣ, and confirm that G(1) = 1 for any valid distribution

Derive the probability generating functions of the standard discrete distributions, namely t(1 − tⁿ)/(n(1 − t)) for the discrete uniform on 1 to n, (q + pt)ⁿ for the binomial, pt/(1 − qt) for the geometric starting at 1 and e^(λ(t − 1)) for the Poisson

Obtain the mean as G′(1) and the variance as G″(1) + G′(1) − [G′(1)]², and use these results to derive the mean and variance of a named distribution

Use the result that the generating function of a sum of independent random variables is the product of their individual generating functions, and hence show, for example, that the sum of independent Poisson variables is Poisson

Recover an individual probability P(X = r) by expanding the generating function as a power series in t and reading off the coefficient of tʳ