High Quality Content by WIKIPEDIA articles! In probability theory, the probability-generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability-generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients. Probability-generating functions obey all the rules of power series with non-negative coefficients. In particular, G(1–) = 1, where G(1–) = limz->1G(z) from below, since the probabilities must sum to one. So the radius of convergence of any probability-generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients.

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