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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 (科教范文网http://fw.nseac.com)
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 (转载自http://zw.NSEaC.com科教作文网)
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
3.8 证明随机变量服从大数定理的1个充分条件………………………………………………20
§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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