切比雪夫不等式的推广与应用

切比雪夫不等式的推广与应用

摘要：在估计某些事件的概率的上下界时，常用到著名的切比雪夫不等式.本文从4个方面对切比雪夫不等式进行推广，讨论了切比雪夫不等式在8个方面的应用，并证明了随机变量序列服从大数定理的1个充分条件.最后给出了切比雪夫不等式其等号成立的充要条件，并用现代概率方法重新证明了切比雪夫不等式.

关键词：切比雪夫不等式；随机变量序列；强大数定理；几乎处处收敛；大数定理.

The Popularization and Application of Chebyster’s Inequality

Abstract:The famous Chebyshev’s Inequality is usually used when estimating the boundary from above or below of probability . The paper presents popularization from four respects. First, the paper discusses its application in eight aspects and demonstrates a complete condition that the foundation of random number sequence coconforms to he Law of Large Numbers theorem. And then , the author analyzes its complete and necessary condition for foundation of Chebyshev’s Ineuquality. Furthermore, the paper makes a demonstration again for Chebyshev’s Inequality with the method of modern probability.

Key words: Cherbyshev’ Inequality; Random number sequence; Law of Large Numbers; Almost Everywhere Convergence；Law of Strong Large Numbers.

目 录

中文标题……………………………………………………………………………………………1

中文摘要、关键词…………………………………………………………………………………1

英文标题……………………………………………………………………………………………1

英文摘要、关键词…………………………………………………………………………………1

正文

§1 引言……………………………………………………………………………………………2

§2切比雪夫不等式的推广 ………………………………………………………………………2

§3切比雪夫不等式的应用 ………………………………………………………………………5

3.1 利用切比雪夫不等式说明方差的意义………………………………………………………5

3.2 估计事件的概率………………………………………………………………………………5

3.3 说明随机变量取值偏离EX超过3 的概率很小 ……………………………………………7

3.4 求解或证明有关概率不等式…………………………………………………………………7

3.5 求随机变量序列依概率的收敛值……………………………………………………………9

3.6 证明大数定理…………………………………………………………………………………11

3.7 证明强大数定理………………………………………………………………………………12§4切比雪夫不等式等号成立的充要条件 ………………………………………………………22

§5 结束语…………………………………………………………………………………………25

参考文献……………………………………………………………………………………………26

致谢…………………………………………………………………………………………………27

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