Any discrete distribution with support on $\{0,\ldots, d\}$ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of $d$-dimensional Bernoulli variables $\boldsymbol{X}=(X_1,\ldots, X_d)$ whose sums $\sum_{i=1}^dX_i$ have the same distribution $p$ is a convex polytope $\mathcal{P}(p)$ and we analytically find its extremal points. Our main result is to prove that the Hausdorff measure of the polytopes $\mathcal{P}(p), p\in \mathcal{D}_d,$ is a continuous function $l(p)$ over $\mathcal{D}_d$ and it is the density of a finite measure $\mu_s$ on $\mathcal{D}_d$ that is Hausdorff absolutely continuous. We also prove that the measure $\mu_s$ normalized over the simplex $\mathcal{D}$ belongs to the class of Dirichlet distributions. We observe that the symmetric binomial distribution is the mean of the Dirichlet distribution on $\mathcal{D}$ and that when $d$ increases it converges to the mode.

翻译：任何定义在 $\{0,\ldots, d\}$ 上的离散分布都可以构造为伯努利变量之和的分布。我们证明了，其和 $\sum_{i=1}^dX_i$ 具有相同分布 $p$ 的 $d$ 维伯努利变量 $\boldsymbol{X}=(X_1,\ldots, X_d)$ 的类是一个凸多面体 $\mathcal{P}(p)$，并且我们解析地找到了它的极值点。我们的主要结果是证明，对于 $p\in \mathcal{D}_d$，多面体 $\mathcal{P}(p)$ 的豪斯多夫测度是定义在 $\mathcal{D}_d$ 上的连续函数 $l(p)$，并且它是豪斯多夫绝对连续的有限测度 $\mu_s$ 的密度。我们还证明了在单纯形 $\mathcal{D}$ 上归一化的测度 $\mu_s$ 属于狄利克雷分布类。我们观察到对称二项分布是 $\mathcal{D}$ 上狄利克雷分布的均值，并且当 $d$ 增加时，它收敛于众数。