----- Original Message ---- From: James Propp <jpropp@cs.uml.edu> To: math-fun@mailman.xmission.com Sent: Friday, August 29, 2008 5:04:52 PM Subject: Re: [math-fun] an almost-uniform random variable? Dan Asimov is right; I should have written << And here's a fun fact related to Archimedes' insight about the sphere and the cylinder: If X, Y, Z are independent Gaussians of mean 0 and variance 1, X/(X^2+Y^2+Z^2) is uniform on [-1,1].

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To elaborate on what Dan wrote: Note that if X, Y, and Z are independent Gaussians, then (X,Y,Z) is spherically symmetric in distribution, so if you map to the unit sphere by dividing by sqrt(X^2+Y^2+Z^2) you get a uniform point on the unit sphere. Now proceed as Archimedes did. Jim Propp One more typo: the denominator should be R=sqrt(XX+YY+ZZ) and not R^2. Otherwise there will be values outside [-1,1]. Also, the variances need not be 1, as long as they are equal. Gene