切比雪夫不等式的推廣與應用
摘要：在估計某些事件的概率的上下界時，常用到著名的切比雪夫不等式.本文從4個方面對切比雪夫不等式進行推廣，討論了切比雪夫不等式在8個方面的應用，並證明了隨機變量序列服從大數定理的1個充分條件.最後給出了切比雪夫不等式其等號成立的.充要條件，並用現代概率方法重新證明了切比雪夫不等式.
關鍵詞：切比雪夫不等式；隨機變量序列；強大數定理；幾乎處處收斂；大數定理.
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 錄
中文標題……………………………………………………………………………………………1
中文摘要、關鍵詞…………………………………………………………………………………1
英文標題……………………………………………………………………………………………1
英文摘要、關鍵詞…………………………………………………………………………………1
正文
§1 引言……………………………………………………………………………………………2
§2切比雪夫不等式的推廣 ………………………………………………………………………2
§3切比雪夫不等式的應用 ………………………………………………………………………5
3.1 利用切比雪夫不等式說明方差的意義………………………………………………………5
3.2 估計事件的概率………………………………………………………………………………5
3.3 說明隨機變量取值偏離EX超過3 的概率很小 ……………………………………………7
3.4 求解或證明有關概率不等式…………………………………………………………………7
3.5 求隨機變量序列依概率的收斂值……………………………………………………………9
3.6 證明大數定理…………………………………………………………………………………11
3.7 證明強大數定理………………………………………………………………………………12
3.8 證明隨機變量服從大數定理的1個充分條件………………………………………………20
§4切比雪夫不等式等號成立的充要條件 ………………………………………………………22
§5 結束語…………………………………………………………………………………………25
參考文獻……………………………………………………………………………………………26
致謝…………………………………………………………………………………………………27
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