. . xx大学毕 业 论 文题目： 大数定律及其应用 院 部 信息科学与工程学院 专业班级 信息与计算科学1班 届 次 x 届 学生姓名 xx 学 号 xx 指导教师 xxx 二O一一 年 六 月 十 日装订线. . . 大数定律及其应用 Law of large numbers and its application专业Speciality信息与计算科学Information and Computing Science学生Undergraduatexxxx指导教师Supervisorxxxxxxx大学xx年六月xxx UniversityJune, xxxx大学学士学位论文摘要对于随机现象而言,其统计规律性只有在基本相同的条件下进行大量的重复试验才能显现出来.本文主要是通过大数定律来讨论随机现象最根本的性质平均结果稳定性的相关内容.大数定律,描述当试验次数很大时所呈现的概率性质的定律,是随机现象统计规律性的具体表现.本文首先介绍了大数定律涉及的一些基础知识,以便于对文中相关知识的理解.通过比较,就不同条件下存在的大数定律做了具体的分析,介绍了几种较为常见的大数定律和强大数定律,总结了大数定律的应用,主要有大数定律在数学分析中的应用,大数定律在生产生活中的应用,大数定律在经济如：保险、银行经营管理中的应用等等,将理论具体化,将可行的结论用于具体的数学模型中,使大家对大数定律在实际生活中的应用价值有了更深的认识.关键词：大数定律；强大数定律；数学分析；经济.AbstractTo random phenomemon, its statisticses law just can present when a great deal of repeated test are carried on under the basic same condition. This text mainly is pass law of large numbers to talk about random phenomenons most basic quality-related contents of average result stability .Law of large numbers presents the law of probability quality when test the number of times is very big.This text firstly introduces some basics which are involved in law of large numbers in order to make it easier to understand the corresponding knowledge in this paper.Through comparison, this article analyzes some conditions of the law of large numbers, introduces several kinds of more familiar law of large numbers and strong law of large numbers,tallying up application of law of large numbers,mainly including application of law of large numbers in mathematical analysis, application of law of large numbers in production and living,application of law of large numbers in economy,such as insurance, bank management and so on.It makes mathematical theory concretely,considers some viable conclusions in concrete mathematical model.Thus we can have deeper understanding on the law of large numbers in the real life.Key words:Law of large numbers,strong law of large numbers,mathematical analysis,economy.山东农业大学学士学位论文目 录引言11大数定律21.1 大数定律的定义21.2 常用的大数定律21.2.1 伯努利大数定律21.2.2 泊松大数定律31.2.3 切比雪夫大数定律31.2.4 马尔可夫大数定律31.2.5 辛钦大数定律41.3 强大数定律41.3.1 博雷尔强大数定律51.3.2 科尔莫戈罗夫强大数定律51.4 几个大数定律的关系及适用场合61.4.1 各个大数定律之间的关系61.4.2 大数定律适用条件的分析71.4.3 几个大数定律的应用场合分析72大数定律的应用112.1大数定律在数学分析中的应用112.1.1 在积分方面的应用112.1.2 证明一致收敛122.1.3 在极限中的应用132.2大数定律在生产生活中的应用152.2.1 误差方面的应用152.2.2 估计数学期望和方差162.3大数定律在经济中的应用162.3.1 大数定律在保险中的应用162.3.2 大数定律在银行经营管理中的应用18参考文献21致谢22Contents Introduction11 Law of large numbers21.1 Definition of law of large numbers.21.2 Common law of large numbers21.2.1 Bernoullis Law of Large Numbers.21.2.2 Poisson Law of Large Numbers.31.2.3 Chebyshev Law of Large Numbers31.2.4 Markov Law of Large Numbers.31.2.5 Khintchine Law of Large Numbers.41.3 Strong Law of Large Numbers.41.3.1 Borel Strong Law of Large Numbers.51.3.2 Kolmogorov Strong Law of Large Numbers.51.4 Relationship and occasions of several law of large numbers.61.4.1 Relationship between the various law of large numbers.61.4.2 Analysis of the conditions of the law of large number.71.4.3 Several application occations of the law of large number.72 Application of law of large numbers.112.1 Application of law of large numbers in mathematical analysis112.1.1 Application of the integral.112.2.2 Proof of uniform convergence.122.2.3 Application of limiton132.2 Law of large numbers of application in the production and living.152.2.1 Application of error152.2.2 Mathematical expectationand varianceestimation.162.3 Law of large numbers of applications in the economy.162.3.1 Application of law of large numbers in insurance.162.3.2 Application of law of large numbers in bank management.18Reference.21Acknowledgement22山东农业大学学士学位论文引言概率论与数理统计是研究随机现象的统计规律的科学,而随机现象的统计规律性只有在相同条件下进行大量重复试验或观察才呈现出来.在随机事件的大量重复出现中,往往呈现几乎必然的规律,这个规律就是大数定律.大数定律是概率论中一个非常重要的课题,而且是概率论与数理统计之间一个承前启后的重要纽带.大数定律阐明了大量随机现象平均结果具有稳定性,证明了在大样本条件下,样本平均值可以看作总体平均值,它是“算数平均值法则”的基本理论,通俗地说,这个定理就是在试验不变的条件下,重复试验多次,随机事件的频率以概率为稳定值.在现实生活中,经常可以见到这一类型的数学模型,比如,我们向上抛一枚硬币,硬币落下后哪一面朝上本来是偶然的,但当我们向上抛硬币的次数足够多时,达到上万次甚至几十万几百万次之后,我们会发现,硬币向上的次数约占总次数的二分之一,偶然中包含着必然.又如：在分析天平上称重量为a的物品,若以 表示n次重复称量的结果,经验告诉我们,当n充分大时,它们的算术平