Probability Generating Function

Naming: Let's call it pgf in short.

Introduction: In probability theory, the pgf of a discrete random variable (r.v) is a power series representation (the generating function) of the probability mass function (pmf) of the random variable.

Univariate Case: If X is a discrete r.v taking values in the non-negative integers {0,1,2,...} then the probability function of X is defined as-

where, p is the pmf of X.

Multivariate Case:

If X = (X1,...,Xd ) is a discrete random variable taking values in the d-dimensional non-negative integer lattice {0,1, ...}d, then the probability generating function of X is defined as

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where p is the probability mass function of X. The power series converges absolutely at least for all complex vectors z = (z1,...,zd ) ∈ ℂd with max{|z1|,...,|zd |} ≤ 1.

Definition of Probability Generating Function:

If X is a discrete r.v which can take values 0, 1, 2... with the corresponding probabilities Po, P₁, P₂....then pgf denoted by p(s) is defined as follows-

Theorem:

For a r.v X, which assumes only integral values the E( X) and V(X) can be calculated as follows in the terms of the probability generating function-

E(X) = P' (1)

and V (X) = P''(1) + P('1) - {P'(1)}²