Probability generating functions for discrete real-valued random variables

The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real-valued discrete random variables, the well-known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5, allow us to extend this technique to general discrete random variables. Being so, the purpose of this work is twofold. First, we show that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions. Second, we intend to draw attention to some practical ways of performing the necessary calculations.