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1、 在前面的课程中，我们讨论了随机变量及其分 布，如果知道了随机变量X的概率分布，那么X的 全部概率特征也就知道了.然而，在实际问题中，概率分布一般是较难 确定的. 而在一些实际应用中，人们并不需要知道随机变量的一切概率性质，只要知道它的某些 数字特征就够了.Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2
2、004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.因此，在对随机变量的研究中，确定某些数 字特征是重要的 .在这些数字特征中，最常用的是数学期望、方差、协方差和相关系数Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 As
3、pose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.第一节 数学期望离散型随机变量的数学期望连续型随机变量的数学期望随机变量函数的数学期望Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004
4、-2011 Aspose Pty Ltd.一、离散型随机变量的数学期望 1、概念的引入：X的数学期望E(X)Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.若统计100天, 32天没有出废
5、品; 30天每天出一件废品; 17天每天出两件废品; 21天每天出三件废品;可以得到这100天中每天的平均废品数为这个数能否作为 X的平均值呢？（假定小张每天至多出 现三件废品 ）例如： 某车间对工人的生产情况进行考察. 车 工小张每天生产的废品数X是一个随机变量. 如何 定义X的平均值呢？我们先观察小张100天的生产情况Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Cl
6、ient Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.可以想象，若另外统计100天，车工小张不出废品， 出一件、二件、三件废品的天数与前面的100天一般 不会完全相同，这另外100天每天的平均废品数也不 一定是1.27.n0天没有出废品; n1天每天出一件废品; n2天每天出两件废品; n3天每天出三件废品.可以得到n天中每天的平均废品数为(假定小张每天至多出 三件废品) 一般来说, 若统计n天 ,Evaluation only.Evaluation only. C
7、reated with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.这是 以频率为权的加权平均当N很大时，频率接近于概率 ，所以我们在求废品数X 的平均值时，用概率代替 频率，得平均值为这是 以概率为权的加权平均这样得到一个确定的数. 我们就用这个数作为随机变 量X 的平均值
8、 .Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.定义1 设X是离散型随机变量，它的分布率是: PX=xk=pk , k=1,2,请注意 :离散型随机变量的数学期望是一个绝对收 敛的级数
9、的和.数学期望简称期望，又称为均值。若级数绝对收敛，则称级数即的和为随机变量X的数学期望，记为 ,Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.例10 1 200.2 0.80 1 20
10、.60.3 0.1Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.数学期望的性质(4) 设X、Y 相互独立，则 E(XY)=E(X)E(Y);请注意: 由E(XY)=E(X)E(Y) 不一
11、定能推出X,Y 独立Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.当X为离散型时,它的分布率为P(X= xk)=pk ;定理 (1)设Y是随机变量X的函数:Y=g (X) (g是连续函数
12、)Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.可见，服从参数为n和p的二项分布的随机变量X 的数学期望是 n p.XB(n,p), 若设则 X= X1+X2+Xn= npi=1,2，n
13、因为 P(Xi =1)= p,P(Xi =0)= 1-p所以 E(X)=则X表示n重贝努里试验中的“成功” 次数.E(Xi)= = p例2 求二项分布 XB(n,p) 的数学期望Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2
14、011 Aspose Pty Ltd.Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.例 设(X,Y)的分布律Evaluation only.Evaluation only. Create
15、d with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with
16、Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.二、连续型随机变量的数学期望设X是连续型随机变量，其密度函数为f (x),在 数轴上取很密的分点x0 0, D(Y)0,称在不致引起混淆时，记 为 .Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2
