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MOMENT SOLUTION TO AN URN MODEL

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L. R. SHENTON^{ 1} AND K. O. BOWMAN^{ 2}

^{1} *Department of Statistics, University of Georgia,
Athens, GA
30602, U.S.A.*

^{2} *Computer Science and Mathematics Division, Oak Ridge
National
Laboratory,*

P.O.Box 2008, Oak Ridge, TN 37831-6367, U.S.A.
(Received August 1, 1994; revised February 17, 1995)

**Abstract.**
A subset of Bernard's RD-model
(replenishment-depletion) is considered from the viewpoint of the
calculus of finite differences. The most general case is considered
and includes an urn with balls of many colors, each color being
replenished either deterministically or stochastically. Factorial
moment generating functions (fmgfs) are employed to define probability
generating functions. A new result is given for the two color case
defining the fmgf and probability generating function (with
probabilities) when the replenishments are positive valued random
variables with given factorial moments. This result involves beta
integral transforms defining a manifold of discrete distributions.
Particular cases relate to hypergeometric discrete distributions.

*Key words and phrases*:
Bernard's urn, beta integral
transforms, finite difference calculus, generating function,
hypergeometric distributions, hypergeometric functions, moments,
replenishment-depletion urn.

**Source**
( TeX ,
DVI ,
PS )