随机变量的数字特征及其应用

青岛大学学士学位论文

随机变量的数字特征（期望、方差、协方差）

及其应用

学 院： 数学与统计学院 姓 名： 宋飞 专 业： 信息与计算科学 学 号： 201341702053 指导教师： 宋丽娜 职 称： 副教授

青岛大学学士学位论文

随机变量的数字特征（期望、方差、协方差）及其应用

摘要：伴随着人类思想的进步与发展，实际问题的概率化思想已经深刻的融入在

了生活的方方面面。然而，在很多事件发生的可能性的层面上来说，其结果往往会呈现出不确定性，在很多次重复试验中其结果又具有统计规律性的现象，我们将其称为随机现象。把每件事情的发生与否抽象成随机变量，于是在某些实际问题或者理论问题中人们感兴趣于某些能描述随机变量某一种特征的常数，这种由随机变量的分布所确定的，能够描述随机变量某一方面的特征的常数统称为数字特征，它在理论和实际应用中都很重要。本文对随机变量的几个重要的数字特征（包含数学期望、方差、协方差）进行了相应的研究。在探究求每个不同的数字特征所各自代表的实际意义时，通过对其定义、产生背景、实际意义等方面进行逐一分析之后，配备了相应例题进行讲解分析，达到与生活实际融会贯通的目的。最后，通过对数字特征的数学分析，可以浅谈它们各自在实际生活中的应用，已达到学以致用的目的。

关键词：随机变量；数字特征；期望；方差；协方差与相关系数

I

青岛大学学士学位论文

Digital Characteristics (Expected, Variance, Covariance) of

Random Variables and Their Applications

Abstract: With the progress and development of human thought, the probabilistic

thought of practical problems has been deeply integrated into all aspects of life. However, at the level of the likelihood of occurrence of many events, the results tend to show uncertainty, and in many times the results of repeated trials have statistical regularity, which we call random phenomena. The occurrence of each thing is abstracted as a random variable, so in some practical problems or theoretical problems in the people interested in some of the characteristics of a random variable can describe a constant, which is determined by the distribution of random variables , Constants that describe the characteristics of a particular aspect of a random variable are collectively referred to as a digital feature, which is important both in theory and in practical applications. In this paper, several important digital features (including mathematical expectation, variance, covariance) of random variables are studied. In the study of the actual meaning of each of the different digital features, through its definition, background, practical significance and other aspects of the analysis, with the corresponding examples to explain the analysis, to achieve the purpose of integration with the actual life. Finally, through the mathematical analysis of digital features, you can talk about their respective applications in real life, has reached the purpose of learning to use.

Key words: Random variables; digital characteristics; expectation; variance;

covariance and correlation coefficient

II

青岛大学学士学位论文

III

青岛大学学士学位论文

目 录

摘 要 ............................................................................................................................................. I 关键词 ............................................................................................................................................. I 英文摘要 ....................................................................................................................................... II 英文关键词 ................................................................................................................................... II 1引 言 ......................................................................................................................................... 1 2数学期望 ..................................................................................................................................... 2

2.1数学期望的引入及定义 ............................................................................................... 2 2.2研究数学期望的重要性 ............................................................................................... 3 2.3数学期望的应用问题 .................................................................................................... 4 2.3.1数学期望在经济学中的应用 …………………………………………..4 2.3.2数学期望在体育比赛中的应用………………………………………........5

3 方差???????????????????????????????7

3.1方差的引入与定义…………………………………………………………...7 3.2研究方差的重要性??????????????????????.8

3.3 方差的应用问题……………………………………………………………..9

4 协方差及相关系数………………………………………………………………..10

4.1 协方差………………………………………………………………………10 4.2 相关系数……………………………………………………………………12 4.3 协方差与相关系数的应用…………………………………………………13 总结…………………………………………………………………………………..16 参考文献………………………………………………………………………………………...17 致谢…………………………………………………………………………………………….....18

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