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Abstract:
In a previous paper [Far East J. Theo. Stat. 5(1) (2001), 113-142], we have studied cumulants of the geometric distribution, showing that they lead to Eulerian numbers such as the sets

These Eulerian numbers (of the first kind) relate to partitions of s from the factorial s!, and as discrete distributions which appear to be nearly the normal distribution when s is large.

Eulerian numbers of the 2nd kind are from the central moments of the geometric distribution and contain sets such as These are symmetric partitions of where refers to the truncated series for the negative exponential The basic results spring from a finite difference equation of the first order, this being solved by usage of the finite difference operator

As with Eulerian numbers of the 1st kind, there is the property of normality; actually the Eulerian numbers of the 2nd kind, viewed as discrete distribution, are asymptotically normally distributed.

Keywords and phrases: asymptotic moments, asymptotic series, difference-differential equation, maximum likelihood estimator, power series distributions, recurrence relations.

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