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 , its probability generating function isThe generating function of a sum of independent random variables is the product of their generating functions.