Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration
Abstract
We study the probability that one betadistributed 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 Kampé de Fériet 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 betadistributed 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.12216
 Bibcode:
 2021arXiv210412216C
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Probability;
 33C20 (Primary) 05A15;
 05A10 (Secondary)
 EPrint:
 Accepted for publication in Brazilian Journal of Probability and Statistics