Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 descri ...Full description
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.