This paper develops a unified probabilistic framework based on distributional derivatives and Dirac delta representations for the analysis of conditional and transportation-related quantities. General identities are established for arbitrary random variables, encompassing absolutely continuous, discrete, and mixed distributions. The proposed approach yields unified formulas for conditional expectations, conditional distributions, hazard functions, and improper distributions, revealing a common localization mechanism underlying these classical concepts. The framework is further combined with copula methods to investigate transportation and dispersion functionals through dependence structures. Exploiting the extremal properties of the Fréchet--Hoeffding bounds together with expectation inequalities induced by $Δ$-antitonic functions, sharp bounds are derived for absolute difference moments under fixed marginals. These results lead to concise derivations of quantile representations for the Wasserstein distance and a corresponding upper transportation functional, as well as survival-function representations and bounds for generalized absolute difference moments. As a particular case, new representations are obtained for the bivariate Gini mean difference and the associated bivariate Gini index. Applications are given to Wasserstein-type functionals arising in the normal approximation of standardized counting distributions, including Poisson, Binomial, and Negative Binomial models, for which explicit quantile representations are derived. Overall, the results establish explicit links among conditional structures, dependence modeling, dispersion measures, normal approximation, and optimal transport, providing a unified perspective on several fundamental constructions in probability and mathematical statistics.

翻译：本文基于分布导数与狄拉克Delta表示，建立了分析条件量和运输相关量的统一概率框架。针对任意随机变量——包括绝对连续型、离散型和混合型分布——建立了普适恒等式。所提方法对条件期望、条件分布、风险函数及非正常分布给出了统一公式，揭示了这些经典概念背后的共同局部化机制。该框架进一步与Copula方法相结合，通过依赖结构研究了运输与离散泛函。利用Fréchet-Hoeffding边界的极值性质及Δ-反序函数诱导的期望不等式，在固定边际分布下导出了绝对差矩的紧界。这些结果给出了Wasserstein距离的分位数表示及其对应上运输泛函的简洁推导，以及广义绝对差矩的生存函数表示与边界。作为特例，获得了二元基尼均值差及相关二元基尼指数的全新表示。在标准化计数分布的正态逼近中——包括泊松、二项和负二项模型——给出了显式分位数表示的Wasserstein型泛函应用。总体上，这些结果建立了条件结构、依赖建模、离散测度、正态逼近与最优输运之间的显式联系，为概率论与数理统计学中的若干基本构造提供了统一视角。