This paper develops a unified framework for the study of real-order moments of arbitrary random variables. General integral representations are established in terms of cumulative distribution functions and survival functions, covering continuous, discrete, and mixed distributions supported on the whole real line. These formulas extend the classical tail-integral identities for nonnegative random variables and provide a common treatment of positive, fractional, and negative moments. For discrete distributions, explicit series representations are derived in terms of cumulative probabilities, yielding simple criteria for the existence of moments. Applications are presented for the zeta and Skellam distributions, illustrating how tail behavior determines moment finiteness and how moments can be represented geometrically through cumulative distribution functions. In addition, a representation for logarithmic moments is obtained, linking logarithmic means, Laplace transforms, and the classical Frullani identity. The results provide a unified perspective on moment representations and establish useful connections between tail probabilities, distribution functions, Laplace transforms, and moment existence.

翻译：本文建立了一个统一框架，用于研究任意随机变量的实阶矩。基于累积分布函数与生存函数，本文推导了涵盖整个实直线上的连续、离散及混合分布的一般积分表示式。这些公式推广了非负随机变量的经典尾部积分恒等式，并为正阶矩、分数阶矩与负阶矩提供了统一处理方法。针对离散分布，本文推导出基于累积概率的显式级数表示，进而得到矩存在的简洁判定准则。将上述结果应用于泽塔分布与斯凯勒姆分布，展示了尾部行为如何决定矩的有限性，以及如何通过累积分布函数对矩进行几何表示。此外，本文还获得了对数均值的一种表示，将对数均值、拉普拉斯变换与经典弗鲁拉尼恒等式联系起来。这些结果为矩表示提供了统一视角，并在尾部概率、分布函数、拉普拉斯变换与矩存在性之间建立了有用关联。