The expectation of the Vandermonde product squared for uniform random variables

Richard Ehrenborg

doi:10.1016/j.aam.2020.102030

The expectation of the Vandermonde product squared for uniform random variables

Richard Ehrenborg

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let X₁,X₂,…,X_n be independent random variables, each uniformly distributed on the interval (0,1). We compute the expectation of (X₁X₂⋯X_n)^k⋅∏_1≤i<j≤n(X_j−X_i)². The result relies on operators, the extended Vandermonde determinant and the hook formula for standard Young tableaux.

Original language	English
Article number	102030
Journal	Advances in Applied Mathematics
Volume	118
DOIs	https://doi.org/10.1016/j.aam.2020.102030
State	Published - Jul 2020

Bibliographical note

Funding Information:
The author thanks Theodore Ehrenborg for comments on an earlier draft of this paper. This work was supported by a grant from the Simons Foundation (# 429370 , Richard Ehrenborg).

Publisher Copyright:
© 2020 Elsevier Inc.

Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

Keywords

Hook formula
Uniform random variables
Vandermonde product

ASJC Scopus subject areas

Applied Mathematics

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10.1016/j.aam.2020.102030

Cite this

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N2 - 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(Xj−Xi)2. The result relies on operators, the extended Vandermonde determinant and the hook formula for standard Young tableaux.

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The expectation of the Vandermonde product squared for uniform random variables

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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