切比雪夫不等式的推廣與應(yīng)用

1、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(jìn)行推廣，討論了切比雪夫不等式在8個方面的應(yīng)用，并證明了隨機(jī)變量序列服從大數(shù)定理的1個充分條件.最后給出了切比雪夫不等式其等號成立的充要條件，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.                

2、       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

3、 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 pa

4、per 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22切比雪夫不等式的推廣 23切比雪夫不等式的應(yīng)用 53.1 利用切比雪夫不等式說明方差的意義53.2 估計事件的概率53.3  說明隨機(jī)變量取值偏離ex超過3 的概率很小 73.4 求解或證明有關(guān)概率不等式73.5 求隨機(jī)變量序列依概率的收斂值93.6 證明大數(shù)定理113.7 證明強(qiáng)大數(shù)定理123.8 證明隨機(jī)變量服從大數(shù)定理的1個充分條件204切比雪夫不等式等號成立的充要條件 225 結(jié)束語25參考文獻(xiàn)2