Dan has it exactly right. To see why the solution is unique, write all the equations as linear equations over the ring of polynomials of degree n. With respect to the monomial basis, the system becomes upper triangular. I apologize for the extra head-scratching caused by the over-terseness of my postings. Sometime over the next year I'll write all this up in a careful (and hopefully clear) way. Jim On 6/25/12, Dan Asimov <dasimov@earthlink.net> wrote:
Although I don't understand Jim's comment about dithering random variables, I've been able to find a linear map from the simplex S_n (defined as the points in the non-negative orthant of R^(n+1) whose sum of coordinates = 1) of all probability distributions on the set {0,1, . . .,n} of n+1 points (i.e., all possible sets of striking probabilities on the plane diagonal of integer points with i+j=n in the first quadrant), to S_(n-1), that takes the binomial distribution Binomial(p,n) to Binomial(p,n-1) (for each p in [0,1]).
I guess these are what Jim is referring to.
Since the binomial distributions for all p just form a curve in the simplex S_n, I may have to think a little harder to see that these linear maps
L_n : S_n -> S_(n-1)
are uniquely determined by the property of taking Binomial(p,n) in S_n to Binomial(p,n-1) in S_(n-1) for each p.
--Dan
Jim wrote:
I'll pose the key idea in the form of a puzzle: Find the linear map that sends probability distributions on {0,1,...,n} to probability distributions on {0,1,...,n-1} with the property that for every p, the map sends the distribution Binomial(p,n) to the distribution Binomial(p,n-1). (Here we treat probability distributions as vectors in the obvious way.)
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