From: Dan Asimov <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Random variable puzzle Message-ID: <3C0B9325-43FC-4DDA-A42E-69EC9AF69FFF@msri.org> Content-Type: text/plain; charset=utf-8
Suppose X, Y, Z, W are real random variables with a joint distribution such that each one has a finite mean and standard deviation.
Suppose that all pairs of these random variables have the same correlation coefficient:*
R = rho(X,Y) = rho(X,Z) = rho(X,W) = rho(Y,Z) = rho(Y,W) = rho(Z,W) .
Find the minimum possible value of R.
For the following 4x4 Hadamard matrix (entries +1 and -1) -+++ +-++ ++-+ +++- each row has mean=1/2, variance=1, and all rows are orthogonal. If we instead take these four rows of an 8x8 Hadamard +-+-+-+- ++--++-- +--++--+ ++++---- then each row has mean=0, variance=1, and all rows orthogonal.