Eulerian Numbers in Sorting Process and Discrete Probability Distribution Function

Abstract

The purpose of this paper is to discuss the discrete probability distribution function induced from the modified bucket sorting. The distribution is related to the unique sequence called "Eulerian Numbers" proposed by Euler (1755). There are sensitive differences between a derivation process of our proposal and Eulerian numbers expressed as a number of permutation runs. Recurrence relation for the moments of the distribution is also given. The mean, variance and higher order cumulants are derived using the relation. The interesting results that the relationship among the proposed, normal and uniform distributions are shown by asymptotic expansion. The efficiency of the proposed method is illustrated through some numerical experiments.