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.)