切比雪夫不等式的推广与应用.doc

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 chebysters inequalityabstract:the famous chebyshevs 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 chebyshevs ineuquality. furthermore, the paper makes a demonstration again for chebyshevs 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22切比雪夫不等式的推广 23切比雪夫不等式的应用 53.1 利用切比雪夫不等式说明方差的意义53.2 估计事件的概率53.3  说明随机变量取值偏离ex超过3 的概率很小 73.4 求解或证明有关概率不等式73.5 求随机变量序列依概率的收敛值93.6 证明大数定理113.7 证明强大数定理123.8 证明随机变量服从大数定理的1个充分条件204切比雪夫不等式等号成立的充要条件 225 结束语25参考文献2