# Probability Generating Function

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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