发表机构
Faculty of Electrical Engineering and Computing, University of Zagreb（萨格勒布大学电气工程学院与计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究推导二项分布折叠后奇数阶绝对矩的精确约简式，得到有界格位移处的中心-尾展开式，将其应用于贝塔分布中位数的二阶展开，退化后可对应伽马分布中位数的相关展开项。
AI 中文摘要
设X服从参数为N、p的二项分布，Y=|X-Np|。我们推导了Y的每个奇数阶绝对矩的精确约简式，将其表示为有限个中心质量和中心尾概率的组合。尾系数满足求和规则且可被q-p整除，因此在对称情形下，奇数阶阶梯仅由质量项构成。随后，我们得到了有界格位移处的完整中心-尾展开式，其大参数的整数次幂具有伯努利多项式系数。作为应用，前两个尾系数给出了贝塔分布中位数的二阶展开式，其形式上的单参数退化重现了Choi关于伽马分布中位数展开的前两项。
英文摘要
Let $X\sim Bin(N,p)$ and \(Y=|X-Np|\). We derive an exact reduction for every odd absolute moment of \(Y\), expressing it through finitely many central masses and central tail probabilities. The tail coefficients satisfy a sum rule and are divisible by \(q-p\), so that in the symmetric case the odd ladder collapses to masses alone. We then obtain the complete central-tail expansion at a bounded lattice displacement, with Bernoulli-polynomial coefficients in integer powers of the large parameter. As an application, the first two tail coefficients yield a second-order expansion for the median of the beta distribution, whose formal one-parameter degeneration reproduces the first two terms of Choi's expansion for the gamma median.
Comments15 pages
