Re: Distance Between 2 Randomly-chosen Points on a Sphere

From: Stephen J. Herschkorn (herschko_at_rutcor.rutgers.edu)
Date: 10/10/04 

Date: Sun, 10 Oct 2004 12:48:46 GMT

Again, the original question:

> What is the most likely straight-line distance between two randomly
> selected points on a sphere?

The explicit formal model that others and I have been using: Let S be
the unit sphere in R^3. Define a probability distribution P on S
by PA = (surface area of A) / (4 pi); P is called the uniform
distribution on S. Let X and Y be two independent S-valued random
variables each with this uniform distribution. Equvalently, (X, Y)
has a uniform distribution on S^2. Let Z = || X - Y||_2, a random
variable which takes values in the real inteval [0, 2]. What is the
mode of the distribution of Z?

While one might claim that the phrase "randomly selected" is ambiguous,
in practice it is not. Unless otherwise qualified, such a phrase is to
be interpreted as indicating a uniform distribution on the sample space.

Nor is the phrase "most likely" ambigous; it refers ot the mode of a
distribution (not the expectation). 

-- 
Stephen J. Herschkorn                        herschko@rutcor.rutgers.edu

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