Fred>Bill --- why multiply by pi/4 to define your "mean"? WFL So that mean(x,x) = x instead of 4 x/π. It's legally required by the Federal Bureau of Weighs and Means. --rwg On 3/1/12, Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote:> Jim Propp>>> One tends to think that the random variable associated with the larger> circle would have more variance.> But as many of you have probably figured out by now, the two random> variables have the exact same distribution.>> A more symmetrical way to describe this random variable is as the length of> the vector u-v (or if you prefer u+v), where u and v are plane vectors of> length 1 and 2, respectively, chosen uniformly and independently.>> If you exploit symmetry by choosing u to be a fixed vector of length 1 and> letting v vary over all vectors of length 2, you get one of the two> asymmetric descriptions; if you instead choose v to be a fixed vector of> length 2 and let u vary over all vectors of length 1, you get the other one.>> Jim Propp>>> It's easier than that. Uniform on the circles means uniform angles> wrt x-axix, e.g.> The segment joining (1,0) to 2(cos t, sin t) is obviously congruent to the> one> joining (2,0) to (cos t, sin t), length = sqrt(5 - 4 cos t).>> Interesting: The expected length of the third side of SAS triangle x,t,y> for t uniform in (0,π) is>> (2*(x + y)*EllipticE[(4*x*y)/(x + y)^2])/Pi .>> So (1/2)*(x + y)*EllipticE[(4*x*y)/(x + y)^2] is an interesting sort of> mean.>> --rwg