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

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