On The Left Tail-End Probabilities and the Probability Generating Function

Abstract

In this paper, another tail-end probability function is proposed using the left tail-end probabilities, p( ≤ i ) = Πṙ The resulting function,   π_x(t), is continuous and converges uniformly within the unit circle, | t | < 1. A clear functional link is established between π_x(t) and two other well known versions of the probability generating function. When known, π_x(t) uniquely generates the components of the probability mass function of the discrete random variable, and indirectly generates moments.

Keywords: Probability Generating Function, Tail – end Probabilities, Convergence, Moments.