The Solving of the Problems with Random Division of an Interval with Use of Computer Analytic Programs

TL;DR

This paper introduces computer-assisted algorithms to solve complex probabilistic problems involving random interval division, leading to new formulas and the concept of three-dimensional generalized Catalan numbers.

Contribution

It presents novel algorithms for multidimensional integral calculations and introduces the concept of three-dimensional generalized Catalan numbers for interval division problems.

Findings

Derived new probabilistic formulas for random interval division.

Established explicit form of three-dimensional generalized Catalan numbers.

Validated algorithms through computational experiments.

Abstract

An original approach to solving rather difficult probabilistic problems arising in studying the readout of random discrete fields and having no exact analytical solutions at the moment is proposed. Several algorithms for direct, iterative, and combinatorial-recursive calculations of multidimensional integral expressions, which can describe partial solutions of these problems, are presented (these solutions are further used to search for the common closed analytical regularities). The huge volume of necessary calculations forced us to formalize completely the algorithms and to transfer all the burden of routine analytical transforms to a computer. The calculations performed helped us to establish (and to prove later) a number of new earlier unknown probabilistic formulas responsible for random division of an interval. One more important feature of this study is the fact that we introduced a new concept of 'three-dimensional generalized Catalan numbers' and found their explicit form in solving problems associated with random division of an interval.