From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, January 1, 2009 12:35:00 PM Subject: [math-fun] Correlation puzzle Phil wrote in the "recent illogical conflation" thread: << It's rather "the enemy of my enemy is my friend"-like, which is also commonly used.
This reminds me of an interesting statistics thang: (I don't recall posting this here, but if I did please forgive me.) PUZZLE ------ Given real-valued random variables X,Y,Z in a joint distribution, suppose that the correlation coefficient* between any two of them is the same number C. QUESTION: What is the minimum value, over all such joint distributions, that C can take? (We assume all means and variances are finite.) --Dan _____________________________________________________________ * There are several types of correlation coefficient, but this refers to what is by far the commonest one: CC(U,V) = E(U_s * V_s) where W_s denotes the standardized version of the random variable W: W_s := (W - E(W)) / sd(W) where E is expectation and sd is standard deviation. _____________________________________________________________________ The minimum C is -1/2. Without changing C, we may shift x, y, and z so that each has zero mean, and then scale them so that each has unit variance. Then 0 <= E((x + y + z)^2) = 3 + 6 C, so C >= -1/2. To actually achieve this value, let the distribution consist of the three points (2, -1, -1), (-1, 2, -1), and (-1, -1, 2). We have E(x) = 0, etc., E(x^2) = 2, etc., E(xy) = -1, etc. Then C = E(xy)/(sqrt(E(x^2)) sqrt(E(y^2))) = -1/2. Gene