The expectation of the Vandermonde product squared for uniform random variables

Abstract

Let X1,X2,…,Xn be independent random variables, each uniformly distributed on the interval (0,1). We compute the expectation of (X1X2⋯Xn)k⋅∏1≤i<j≤n(Xj−Xi)2. The result relies on operators, the extended Vandermonde determinant and the hook formula for standard Young tableaux.