The Probabilistic Evaluation Of Nested Multiple Integrals With Applications

The evaluation of multiple integrals is an important issue of classic analysis, commonly used methods of the variable substitution and change of the order of integration to deal with. This article discusses a special class of the nested multiple integral, which is related to the theories of probability distribution of order statistics and the combinatorial mathematics of Young tableaux counting formula. Using order statistics and the mathematics recursive method, we get the determinant expression of the nested integral.This paper mainly studies the following nested multiple integral: here I0,k=1, i=1,2,…,k.We calculate and get the results of the In,1, In,2, In,3, In,, then guess the determinant form of In,k, finally prove it with the method of mathematical induction.This article also discusses some applications of the nested integral multiple in combinatorial mathematics.