The Bernoulli structure of discrete distributions

Any discrete distribution with support on {0, …, d} can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of d-dimensional Bernoulli variables X = (X1, …, Xd) whose sums ∑di=1 Xi have the same distribution p is a convex polytope P(p) and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes P(p), p ∈ Dd, is a continuous function l(p) over Dd and it is the density of a finite measure µs on Dd that is Hausdorff absolutely continuous. We also prove that the measure µs normalized over the simplex Dd belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on Dd and that when d increases it converges to the mode.