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

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