That integration w.r.t. angle (from the point 1) is the same* as w.r.t. arc-length. * up to a constant factor, which is irrelevant for the purpose of computing a mean (geometric or otherwise).
Sent: Tuesday, August 21, 2018 at 8:20 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "Eugene Salamin" <gene_salamin@yahoo.com>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Are these two numbers equal?
Of course, but what is the connection?
—Dan
Gene Salamin wrote: -----
Theorem: an arc of a circle that subtends angle θ from the center subtends angle θ/2 from a point on the circumference.
On Monday, August 20, 2018, 8:14:29 PM PDT, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< where f(n) is the geometric mean of chord-lengths of all chords of the unit circle containing say the point 1 >>
This is not meaningful without specifying a distribution: is the integration wrt angle, rather than (say) arc length, or some other, more obscure weight function? See https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
WFL
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