AISM 54, 681-688

© 2002 ISM

## Generalized waiting time problems associated with pattern in Polya's urn scheme

### Kiyoshi Inoue and Sigeo Aki

Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

(Received July 4, 2000; revised February 21, 200)

Abstract.
Let $X_1,X_2,\ldots$ be a sequence obtained by Polya's
urn scheme. We
consider a
waiting
time problem
for the first occurrence of a pattern in the sequence
$X_1,X_2,\ldots,$ which is generalized by a notion "score".
The main part of our results is derived by the method of generalized probability
generating functions. In Polya's urn scheme, the system of equations is composed of the infinite conditional probability generating functions, which can not be solved. Then, we
present a new methodology to obtain the truncated probability generating function in a series up to an arbitrary order from the system of infinite equations. Numerical
examples are also given in order to illustrate the feasibility of our results. Our results in this paper are not only new but also a first attempt to treat the system of infinite
equations.

Key words and phrases:
Polya's urn scheme, pattern, generalized probability generating functions, conditional probability generating functions.

**Source**
(TeX , DVI )