论文标题
方便的二项式时刻公式
Handy Formulas for Binomial Moments
论文作者
论文摘要
尽管二项式分布与概率理论和应用统计推断的相关性，但其高阶时刻的理解很少。现有的公式要么不够笼统，要么不够结构化和简化，以实现预期的应用程序。 本文以\ emph {polyenmials的形式引入了新的二项式矩，而不是在成功概率中。与先前的作品相比，获得的公式可以说是更好的结构化，更简单和优越的。此外，本文提出了算法来得出这些公式以及在Python符号代数软件包中的工作实现。 新颖的方法是一种组合论证，再加上依赖对称理论的巧妙代数简化。作为一个有趣的副产品，我们建立了\ emph {中央二项式矩的渐近估计值}，从而改善了先前工作的部分结果。
Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications. This paper introduces novel formulas for binomial moments, in form of \emph{polynomials in the variance} rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in the Python symbolic algebra package. The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct we establish \emph{asymptotically sharp estimates for central binomial moments}, improving upon partial results from prior works.
