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切比雪夫不等式的推廣與應(yīng)用
切比雪夫不等式的推廣與應(yīng)用
摘要：在估計(jì)某些事件的概率的上下界時(shí)，常用到著名的切比雪夫不等式.本文從4個(gè)方面對(duì)切比雪夫不等式進(jìn)行推廣，討論了切比雪夫不等式在8個(gè)方面的應(yīng)用，并證明了隨機(jī)變量序列服從大數(shù)定理的1個(gè)充分條件.最后給出了切比雪夫不等式其等號(hào)成立的充要條件，并用現(xiàn)代概率方法重新證明了切比雪夫不等式.
關(guān)鍵詞：切比雪夫不等式；隨機(jī)變量序列；強(qiáng)大數(shù)定理；幾乎處處收斂；大數(shù)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(biāo)題……………………………………………………………………………………………1
中文摘要、關(guān)鍵詞…………………………………………………………………………………1
英文標(biāo)題……………………………………………………………………………………………1
英文摘要、關(guān)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(yīng)用 ………………………………………………………………………5
3.1 利用切比雪夫不等式說(shuō)明方差的意義………………………………………………………5
3.2 估計(jì)事件的概率………………………………………………………………………………5
3.3 說(shuō)明隨機(jī)變量取值偏離EX超過(guò)3 的概率很小 ……………………………………………7
3.4 求解或證明有關(guān)概率不等式…………………………………………………………………7
3.5 求隨機(jī)變量序列依概率的收斂值……………………………………………………………9
3.6 證明大數(shù)定理…………………………………………………………………………………11
3.7 證明強(qiáng)大數(shù)定理………………………………………………………………………………12
3.8 證明隨機(jī)變量服從大數(shù)定理的1個(gè)充分條件………………………………………………20
§4切比雪夫不等式等號(hào)成立的充要條件 ………………………………………………………22
§5 結(jié)束語(yǔ)…………………………………………………………………………………………25
參考文獻(xiàn)……………………………………………………………………………………………26
致謝…………………………………………………………………………………………………27
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