A probability-generating function (PGF) is a specific type of power series that is used to encode the probabilities of a discrete random variable. It is particularly useful in the study of probability distributions and in solving problems involving sums of independent random variables. ### Definition For a discrete random variable \( X \) that takes non-negative integer values (i.e.
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For a nonnegative integer-valued random variable , write . Its probability generating function packages the entire probability mass function into one power series:This domain is guaranteed, even for complex , because . At the constant monomial is one, including when : the convention here gives . Also . The power series may converge beyond this domain, but that needs an additional argument. Encoding a probability distribution this way turns convolution of independent random variables into multiplication.[ref]
Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist. Expected Value, §6 Generating Functions, Basic Theory, The Probability Generating Function: definition, absolute convergence, inversion, and the independent-sum identity