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1、随机变量及其分布 第三章第三章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.第一讲第一讲 随机变量的概念随机变量的概念 一、随机变量概念的产生在实际问题中，随机试验的结果可以用数 量来表
2、示，由此就产生了随机变量的概念.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.1、有些试验结果本身与数值有关（本身 就是一个数）.例如，掷一颗骰子面上出现的点数；七月份哈尔滨的最高温度；
3、每天从哈尔滨下火车的人数；昆虫的产卵数；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 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.2、在有些试验中，试验结果看来与数值无关，但我们可以引进一个变量来表示它的各 种结果.也就是说，把试
4、验结果数值化. 例如：掷一个质地均匀的硬币，用X表示试验结果任意事件A都可由随机变量X表示，如 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.正如裁判员在运动 场上不叫运动员的 名字而叫
5、号码一样 ，二者建立了一种 对应关系. 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.这种对应关系在数学上理解为定义了一种 实值函数.e.X(e)R这种实值函数与在高等数学中大家接触到
6、的函数一样吗？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.（1）它随试验结果的不同而取不同的值 ，因而在试验之前只知道它可能取值的范 围，而不能预先肯定它将取哪个值.（2）由于试验结果
7、的出现具有一定的概 率，于是这种实值函数取每个值和每个确 定范围内的值也有一定的概率.称这种定义在样本空间上的实值函数为随量机变简记为 r.v. 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 L
8、td.随机变量定义：设E是随机试验，它的样本空间是S，如果对S中的每个基本事件e，都有唯一的实数值X（e）与之对应，则称X（e）为随机变量，简记为X。特点：1随机变量X是基本事件e的函数，其定义域为S，值域为某个实数集合。2随机变量X取某个值或某些值表示事 件。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
9、 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.而表示随机变量所取的值 时,一般采用小写字母x,y,z等.随机变量通常用大写字母 X,Y,Z或希腊字母,等表示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 Aspo
10、se Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.例如，从某一学校随机选一 学生，测量他的身高.我们可以把可能的 身高看作随机变量X,然后我们可以提出关于X的各种问题.如 P(X1.7)=？ P(X1.5)=?P(1.560 即 PX20Evaluation 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.
11、Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.注意到也就是说，加油站因代营业务得到的收 入大于当天的额外支出费用的概率为0.6.PX20=PX=30+PX=40=0.6X 10 20 30 40P0.15 0.25 0.45 0.15Evaluation 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
12、Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd.二. 下面介绍几种常见的分布列。（一）. 01分布（伯努利Bernoulli分布，两点分布）若随机变量X只可能取0和1两个值，其分布列为：0p1.或X 0 1 P 1 p p 则称X服从01分布（或伯努利Bernoulli 分布或两点分布），记为XB（1，p）.Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Cl
13、ient 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7. 有100件产品,其中有95件正品,5件次品,从中任取一件产品, 定义: 求X的分布列.X 0 1 P 0.05 0.95 解:Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Clien
14、t 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.（二）、二项分布若随机变量X的分布列为： k=0,1,2,n, 0p1, q=1p, 则称X服从参数为n, p的二项分布，记为XB（n, p） 特例：n=1时 即为01分布。 Evaluation only.Evaluation only. Created with Aspose.Slides
15、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.在n重Bernoulli试验中，设成功发生的次数 为X，则X B（n, p）.性质： 1. k=0,1,2,n。2 . 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.对于固定n及p，当k增 加时 ,概率P(X=k) 先是随 之增加直至 达到最大值, 随后单调减少.二项分布的图形特点： XB(n,p)当(n+1)p不为整数时，二项概 率P(X=k)在k=(n+1)p达到最 大值；
