Gaussian integrals involving absolute value functions

Abstract

We provide general formulas to compute the expectations of absolute value and sign of Gaussian quadratic forms, i.e. $\mathbb{E}$ |〈X, AX〉+〈b, X〉+c| and $\mathbb{E}$ sgn(〈X, AX〉+〈b, X〉+c) for centered Gaussian random vector X, fixed matrix A, vector b and constant c. Products of Gaussian quadratics are also discussed and followed with several interesting applications.