Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration

TL;DR

This paper derives closed-form expressions for integrals involving incomplete beta functions and applies these results to problems in probability, random walks, and combinatorics, providing new analytical tools for these areas.

Contribution

It introduces a general closed-form solution for integrals of products of incomplete beta functions using Kampé de Fériet functions, with applications to probability and combinatorics.

Findings

Closed-form expressions for integrals of incomplete beta functions.

Applications to moments of beta-distributed maxima and string enumeration.

Results on expected exit times for conditioned random walks.

Abstract

We study the probability that one beta-distributed random variable exceeds the maximum of two others, allowing all three to have general parameters. This amounts to studying Euler transforms of products of two incomplete beta functions. We provide a closed form for the general problem in terms of Kamp\'e de F\'eriet functions and a variety of simpler closed forms in special cases. The results are applied to derive the moments of the maximum of two independent beta-distributed random variables and to find inner products of incomplete beta functions. Restricted to positive integer parameters, our results are applied to determine an expected exit time for a conditioned random walk and also to a combinatorial problem of enumerating strings comprised of three different letters, subject to constraints.