Re: Moments over a Simplex

From: Robert Israel (israel_at_math.ubc.ca)
Date: 12/14/04 

Date: 14 Dec 2004 18:30:16 GMT

In article <41BE0003.9080608@netscape.net>,
Stephen J. Herschkorn <sjherschko@netscape.net> wrote:
>Han de Bruijn wrote:

>> The midpoint of a line segment (x1,x2) is (x1+x2)/2
>> The midpoint of a triangle (r1,r2,r3) is (r1+r2+r3)/3
>> The midpoint of a tetrahedron (r0,r1,r2,r3) is (r0+r1+r2+r3)/4
>> The midpoint of a simplex in N dimensions is sum(k<=N) r_k/N ?

>> The variance in x over a line segment in 1-D is: (x2-x1)^2/12
>> The variance in x over a triangle in 2-D is, if I made no errors:
>> ((x2-x1)^2 + (x3-x1)^2 + (x3-x2)^2)/36

>What do you mean by "the" variance over a multi-dimensional simplex?
>For example, suppose (X1, X2, X3) is uniformly distributed over {x
>in R^3: min x_i >= 0, x1 + x2 + x3 =1} You can come up with a
>(singular) covariance matrix, but what do you mean by one scalar variance?

What I computed in my posting was
V = E[(X - E[X]).(X - E[X])]
where X is a vector-valued random variable uniformly distributed on
the simplex, and "." is the dot product. From Han's examples, this seems
to be what he was looking for.

BTW, when I wrote R^2 for a vector R, what I meant was R.R.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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