Journal of Iranian Statistical Society
Volume:2 Issue: 1, 2003

Bioinformatics is an emerging field of science emphasizing the application of mathematics, statistics, and informatics to study and analysis of very large molecular biological (mostly, genetic and genomic) systems (data sets). In a comparatively broader setup oflarge biological systems without necessarily having a predominant genetic undercurrent, and having genesis in biometry to biostatistics, biostochastics has evolved as the primary vehicle for the much needed statistical reasoning. It is intended to point out the genuine need for statistical reasoning in this evolving interdisciplinary field, and in that way, to appraise the limitations of current (mostly, algorithm based) statistical resolutions.

Given an i.i.d. sequence of n letters from a finite alphabet, we consider the length of the longest run of any letter. In the equiprobable case, results for this run turn out to be closely related to the well-known results for the longest run of a given letter. Forcoin-tossing, tail probabilities are compared for both kinds of runs via Poisson approximation.

This paper reviews P´olya urn models and their connection to random trees. Basic results are presented, together with proofs that underly the historical evolution of the accompanying thought process. Extensions and generalizations are given accordingto chronology:• P´olya-Eggenberger’s urn• Bernard Friedman’s urn• Generalized P´olya urns• Extended urn schemes• Invertible urn schemesConnections to random trees are surveyed. Numerous applications to trees common in computer science are discussed, including:

This paper establishes the first four moment expansions to the order o(a−1) of S0ta/pta, where S0n= Pni=1 Yi is a simplerandom walk with E(Yi) = 0, and ta is a stopping time given byta = inf {n≥ 1: n + Sn + n > a}where Sn = Pni=1 Xi is another simple random walk with E(Xi) = 0,and {n, n≥ 1} is a sequence of random variables satifying certain assumptions. These moment expansions complement the classical central limit theorem for a random number of i.i.d. random variables when the random number has the form ta, which arises from