关于条件分布与条件数字特征的研究

Research on the Conditional Distribution and the Conditional Digital Characteristics

摘要: 近年来，随着人们对随机现象的不断观察和研究，条件分布及条件期望在各个领域中得到了广泛的应用，但是现有理论仍存在很多不足的地方。本文对条件分布及其数字特征进行了研究，不仅介绍了经典条件分布和扩展条件分布的定义，而且给出了全概率公式的密度函数形式和分布函数形式。此外，通过对条件分布的数字特征进行探讨，我们证明并推广得到了有关条件期望的新的性质，例如，重期望公式的全概率形式，同时还给出了条件方差的定义和其性质的证明。

Abstract: In recent years, with the continuous observation and study of random phenomena, conditional distribution and conditional expectation have been widely used in various fields, but there are still a lot of shortcomings in the existing theories. In this paper, the conditional distribution and its digital characteristics are studied. Not only the definition of classical conditional distribution and extended conditional distribution is introduced, but also the density function form and the distri-bution function form of the full probability formula are given. In addition, by discussing the digital features of the conditional distribution, we prove and generalize the new properties of the condi-tional expectation, for example, the full probability form of the heavy expectation formula, and also give the definition of the conditional variance and the proof of its properties.