二维随机变量的边缘分布与联合分布关系探讨

摘要

本文首先理解二维随机变量的联合分布的概念、性质及其两种基本表达形式：离散型二维随机变量联合概率分布和连续型二维随机变量联合概率密度。掌握已知两个随机变量的联合分布时分别求它们的边缘分布的方法。在文献研究的基础上，运用随机事元和随机事元集合，建立了二维随机变量分布和边缘分布的形式化可拓模型。利用可拓变换和传导变换，结合形式化的可拓推理知识，对二维随机变量在可拓变换下的传导分布模型进行了研究。将随机事元、随机事元集合、可拓变换、可拓推理知识等引入到二维随机变量分布的研究中，使分析更加形式化，逻辑性更强。运用随机事元和随机事元集合建立了二维随机变量分布的可拓模型。本文对这种特例作了深入研究，分析了具有这种性质的二维密度f(x，y)的结构特点与本质，有助于我们更好地了解正态分布的特殊性质。

关键词：二维随机变量；边缘分布；联合分布

Abstract

In this paper，we first understand the concept and properties of the joint distribution of two-dimensional random variables and their two basic expressions: joint probability distribution of discrete two-dimensional random variables and joint probability density of continuous two-dimensional random variables. The method of finding the edge distribution of the joint distribution of two known random variables is mastered. On the basis of literature research， a formal extension model of two-dimensional random variable distribution and edge distribution is established by using random event element and random element set. By using extension transformation and conduction transformation combined with formalized knowledge of extension reasoning，the conduction and distribution models of two-dimensional random variables under extension transformation are studied. The random event element，random event set，extension transformation and extension reasoning knowledge are introduced into the study of two-dimensional random variable distribution，making the analysis more formalized and logical. The extension model of the distribution of two dimensional random variables is established by using the random event element and the set of random element. This special case is studied in depth. The structure and nature of the two-dimensional density f (x，y) with this property is analyzed，which helps us to better understand the special properties of normal distribution.

Key words:two-dimensional random variables; edge distribution; joint distribution

目录

摘要...........................................................................................................................I Abstract......................................................................................................................... II 1 随机变量独立性及其判定. (1)

1.1 随机变量独立性定义 (1)

1.1.1随机变量及随机变量独立性的定义 (1)

1.1.2随机变量独立性的两个简单定理 (2)

1.2 离散型随机变量独立性的判定 (4)

1.2.1离散型随机变量判别法一 (4)

1.2.2离散型随机变量判别法二 (8)

1.3 连续型随机变量独立性的判定 (12)

1.3.1连续型随机变量判别法一 (12)

1.3.2连续型随机变量判别法二 (13)

2 边缘分布与联合分布关系探讨 (16)

2.1 二维随机变量的分布函数 (16)

2.2 二维离散型随机变量 (17)

2.3 二维连续型随机变量 (18)

2.4 随机变量的独立性 (18)

2.5条件分布 (19)

2.6 二维随机变量函数的分布 (20)

结论 (21)

致谢 (21)

参考文献 (22)