(i) 如果(X1,X2,......,Xn)是离散型随机变量，它的分布律为P?x1,x2,......,xn?，若
(x1,x2,......,xn)?g(x1,x2,......,xn)P?X1?x1,X2?x2,......,Xn?xn?绝对收敛，则
E?g(X1,X2,......,Xn)??(x1,x2,......,xn)?g(x1,x2,......,xn)P?X1?x1,X2?x2,......,Xn?xn?
(ii) 如果(X1,X2,......,Xn)是连续型随机变量，它的联合概率密度函数为f(x1,x2,......,xn)， 若
??????????......?????g(x1,x2,......,xn)f(x1,x2,......,xn)dx1dx2......dxn绝对收敛，则称
????E(Y)???????????......?g(x1,x2,......,xn)f(x1,x2,......,xn)dx1dx2......dxn为随机变量
Y?g(X1,X2,......,Xn)的数学期望. 5.随机变量的数学期望的性质
假设下面出现的期望都存在. (1) 设C为常数，则E(C)?C.
(2) 设X是一个随机变量，则E(CX)?CE(X). (3) 设X，Y是随机变量，则E(X?Y)?E(X)?E(Y). (4) 设X，Y独立，则E(XY)?E(X)E(Y).
二.随机变量的方差
1.方差的定义
设X是随机变量，若E为DX或Var(X).
??X?E(X)??存在，则称E??X?E(X)??为X的方差.，记
222.标准差(均方差)的定义
称?(X)?D(X)为X的标准差或者均方差.
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3.方差的计算
①DX?E②
??X?E(X)??
22DX?E?X?E(X)????2?EX2?2E(X)X??E(X)?2??E(X2)?2E(X)E(X)??E(X)?2 ?E(X2)??E(X)?4.随机变量的标准化
*设E(X)??，D(X)??2?0，??0.记X?X???，则E(X)?*E(X)????0，
2??X????E(X)????2????X???X??X??????????*D(X)?D??E?????E??????E??????????????????????
??22??X?E(X)???1 ?E????2E?X?E(X)??2?1??????????X??*称X?为X的标准化变量.
?2??5.方差的性质
假设如下出现的方差都存在. (1) 设C是常数，则D(C)?0. 事实上，D(C)?E?C?E(C)??2C?C???0. ???E???2(2) 设X是随机变量，C是常数，则有D(CX)?CDX.
事实上，
2D(CX)?E?CX?E(CX)??E?CX?CE(X)??EC2?X?E(X)??C2E?X?E(X)? ?C2D(X)(3) 设X，Y是随机变量，则D(X?Y)?DX?DY?2E?(X?EX)(Y?EY)?.特别地，若X，Y相互独立，则D(X?Y)?DX?DY.
事实上，
?2??2??2??2? 32
D(X?Y)?E??X?Y?E(X?Y)?2??E?X?E(X)?Y?E(Y)?2?2?2??
?E??X?E(X)???E??Y?E(Y)???2E??X?E(X)??Y?E(Y)?? ?E?X?E(X)??2?X?E(X)??Y?E(Y)???Y?E(Y)?22 ?DX?DY?2E??X?E(X)??Y?E(Y)??若X，Y相互独立，则X?E(X)与Y?E(Y)相互独立.则
E??X?E(X)??Y?E(Y)???E?X?E(X)?E?Y?E(Y)??(?E(X)?E(X?)(?E(Y)?E(Y?)?0于是，D(X?Y)?DX?DY.
(4)D(X)?0的充要条件是X以概率1取常数E(X)，即DX()?0?PXEX()??1??.
事实上，P?X?E(X)??1?P?X?E(X)?0??1.令Y?X?E(X)，则P?Y?0??1，
2即PY?0?1.这样，D(X)?E????X?E(X)???E(Y2)?0?1?0.
2必要性由切比雪夫不等式证得.若D(X)?0，则PX???????D(X)?2?0.于是，
P?X??????0，P?X??????1.这样，P?????X??????1，即
P?X??????P?X??????1，于是，P?X??????1，P?X??????0.即
F(???)?0.于是，1?P?X??????P?X??????1.这样，
P?X????P?X????P?X????F(?)?F(??)?F(?)?F(??)?1. 6.切比雪夫不等式
设随机变量X具有数学期望E(X)??，方差D(X)??2.则对任意??0，不等式
?2P?X??????2成立.
?这里分别对连续型、离散型随机变量予以说明.
(1) 若X为离散型随机变量，其分布律为P?X?xk??pk，则
P?X??????xk??p???k?xk??p???(xk??)2k?2?2?2?pk(xk??)?2 ?k?12 33
(2) 若X为连续型随机变量，设其概率密度函数为f(x)，则
P?X??????
x?????f(x)dx?x?????f(x)(x??)2?2dx?1?2x?????f(x)(x??)2dx?1?2?2三.随机变量的协方差
1.协方差、相关系数的定义
量E??X?E(X)??Y?E(Y)??称为随机变量X与Y的斜方差.记为cov(X,Y).而
D(X)D(Y)称为X，Y的相关系数.
??E??X?E(X)??Y?E(Y)??2.协方差的计算
covX(Y,?)EY??)???X?E(X ?E(XY)?E(X)E(Y)E??(Y)(E?X)Y(E)X(?E)Y(E)Y?(E)X ?E?XY?E(X)Y?(E)Y?X(E)X(Y?E)?(E)covX(X,?)E???)X??X?E(X3.协方差的性质 (1)对称性 covX(Y,?)事实上，
coYvX( ,E?()?X???E?X?E(?X)?2(X) Dcov(X,Y)?E??X?E(X)??Y?E(Y)???E??Y?E(Y)??X?E(X)???cov(Y,X).
,Y?)abcovX( Y,(2)齐次性 cova(Xb事实上，
cov(aX,bY)?E??aX?E(aX)??bY?E(bY)???Eab??X?E(X)?Y?E(Y)????abcov(X,Y).
??(?YZ,?)(3)分配律 covXcoXv(Z?,)Ycov Z 34
事实上，
cov(X?Y,Z)?E??X?Y?E(X?Y)??Z?E(Z)???E??X?E(X)??Z?E(Z)???Y?E(Y)??Z?E(Z)?? ?E??X?E(X)??Z?E(Z)???E??Y?E(Y)??Z?E(Z)???cov(X,Z)?cov(Y,Z)(4)不等式性质 cov(X,Y)?D(X)D(Y).并且当且仅当存在a，b，使得
P?Y?aX?b??1时，等式成立. 事实上，任取a?R，
E???Y?E(Y)??a?X?E(X)??2??E??Y?E(Y)??2a?Y?E(Y)??X?E(X)??a?X?E(X)??222 ?E?Y?E(Y)??2aE??Y?E(Y)??X?E(X)???a2E?X?E(X)?2???2? ?D(X)a2?2cov(X,Y)a?D(Y)?0这样，
2 ????2cov(X,Y)??4D(X)D(Y)?4cov2(X,Y)?4D(X)D(Y)?4?cov(X,Y)?D(X)D(Y)????02于是，cov(X,Y)?D(X)D(Y). 若cov(X,Y)?D(X)D(Y)，若cov(X,Y)?D(X)D(Y).若cov(X,Y)?D(X)D(Y)，则
D(X)D(Y)cov(X,Y)D(Y)，于是， a???D(X)D(X)D(X)2????D(Y)D(Y)??.于是， E??Y?E(Y)?X?E(X)?D(Y?X)?0????????D(X)D(X)????????D(Y)D(Y)??P?Y?X?E(Y)?E(X)??1.
D(X)D(X)????若cov(X,Y)??D(X)D(Y)，则a?cov(X,Y)?D(X)D(Y)D(Y)，于是， ???D(X)D(X)D(X)2???D(Y)D(Y)???E??Y?E(Y)?X?E(X)?D(Y?X)?0.于是， ????????D(X)D(X)????????D(Y)D(Y)??P?Y?X?E(Y)?E(X)??1.于是，存在a，bD(X)D(X)????，使得P?Y?aX?b??1.
另一方面，若P?Y?aX?b??1，则P?Y?aX?b??1.于是，D(Y?aX)?0.即
a2D(X)?2acov(X,Y)?D(Y)?0.又由于cov(X,Y)?D(X)D(Y).但方程 a2D(X)?2acov(X,Y)?D(Y)?0要有实根必须cov(X,Y)?D(X)D(Y).因此，
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