ON A GENERALIZED EULERIAN DISTRIBUTION

CH. A. CHARALAMBIDES

Statistical Unit, University of Athens, Panepistemiopolis, Athens 157 84, Greece

(Received April 24, 1989; revised February 13, 1990)

Abstract. The distribution with probability function p_{k}(n, α, β) = A_{n,k}(α, β)/(α + β)^{[n]}, k = 0, 1, 2, …, n, where the parameters α and β are positive real numbers, A_{n,k}(α, β) is the generalized Eulerian number and (α + β)^{[n]} = (α + β)(α + β + 1) ⋯ (α + β + n − 1), introduced and discussed by Janardan (1988, Ann. Inst. Statist. Math., 40, 439–450), is further studied. The probability generating function of the generalized Eulerian distribution is expressed by a generalized Eulerian polynomial which, when expanded suitably, provides the factorial moments in closed form in terms of non-central Stirling numbers. Further, it is shown that the generalized Eulerian distribution is unimodal and asymptotically normal.

Key words and phrases: Eulerian numbers, Eulerian polynomials, Stirling numbers, random permutations, unimodality, asymptotic normality.