Abstract
The distribution with probability function p k(n, α, β) = A n, k(α, β)/(α+ β)[p], 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.
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References
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Charalambides, C.A. On a generalized Eulerian distribution. Ann Inst Stat Math 43, 197–206 (1991). https://doi.org/10.1007/BF00116478
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DOI: https://doi.org/10.1007/BF00116478