The gamma-Transform Approach: a New Method for the Study of a Discrete and Finite Random Variable

In this paper, we present a new method for the study of a discrete random variable whose probability distribution has a finite support. The approach is based on the introduction of a transform of the probability density function, named gamma-transform, which better suits the finite nature of the random variable than the traditional probability generating function. In particular, in addition to the transformation/anti-transformation pair, a simple formula is presented for computing the factorial moments of a random variable directly from the gamma-transform of its probability density function. Moreover, it is shown how the gamma-transform can be determined from the nature of the combinatorial problem under study thanks to its physical meaning. Examples and applications to estimation problems relevant for computer science are provided, in which the simple construction of a gamma-transform gives immediate access to the complete characterization of the underlying probability distribution (density function and moments).