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ON A GENERALIZED EULERIAN DISTRIBUTION

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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*, *alpha*, *beta*) =
*A*_{n,k}(*alpha*, *beta*)/(*alpha* + *beta*)^{[n]}, *k* = 0, 1, 2,...., *n*,
where the parameters *alpha* and *beta* are positive real numbers,
*A*_{n,k}(*alpha*, *beta*) is the generalized Eulerian number and
(*alpha* + *beta*)^{[n]} = (*alpha* + *beta*)(*alpha* + *beta* + 1)···(*alpha* + *beta* + *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.

**Source**
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