Computing the covariance of two Brownian area integrals J. A. Wellner* University of Washington, Statistics, Box 354322, Seattle, Washington 98195-4322, U.S.A. R. T. Smythe Department of Statistics, Oregon State University, Corvallis, Oregon 97331-4606, U.S.A. We compute the expected product of two correlated Brownian area integrals, a problem that arises in the analysis of a popular sorting algorithm. Along the way we ®nd three different formulas for the expectation of the product of the absolute values of two standard normal random variables with correlation h. These two formulas are found: (a) via conditioning and the non-central chi-square distribution; (b) via Mehler's formula; (c) by representing the correlated normal random variables in terms of independent normal's and integration using polar coordinates. Key Words and Phrases: bivariate normal distribution, Brownian bridge, correlation, expectation, Mehler's formula, non-central chi-square, pro- duct of absolute values. 1 The problem Suppose that B j; j  1; 2; 3 are independent Brownian bridge processes on [0, 1]; recall that a Brownian bridge process B is a mean zero Gaussian process on [0, 1] with covariance CovBs; Bt  s ^ t  st; s; t 20; 1: De®ne two random variables A 1 and A 2 by A 1  Z 1 0 jB 1 t B 2 tj dt; A 2  Z 1 0 jB 1 t B 3 tj dt: The following question arose in the course of trying to analyze the behavior of a certain sorting algorithm; see SMYTHE and WELLNER (1999): *jaw@stat.washington.edu smythe@stat.orst.edu 101 Statistica Neerlandica (2002) Vol. 56, nr. 1, pp. 101±109 Ó VVS, 2002. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.