On a generalized Eulerian distribution

✍ Scribed by Ch. A. Charalambides

• Publisher: Springer Japan
• Year: 1991
• Tongue: English
• Weight: 438 KB
• Volume: 43
• Category: Article
• ISSN: 0020-3157
• DOI: 10.1007/bf00116478

✦ Synopsis

The distribution with probability function pk(n, ~, t3) = A,~,k(a, fl)/(a + fl)[n] k = 0, 1, 2,..., n, where the parameters a and /3 are positive real numbers, An,k (~, fl) is the generalized Eulerian number and (a + fl)[n] = (a + fl)(a +/3 + 1)--. (c~ + fl + 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.