E(r2)=[y /(x, y)dxdy=Cdxy-dy3 D(1)=41 6 0 example.2-台设备由三大件组成,载设备的运转过程中需要调整的概率分别为oo 0.30,假设各部分相互独立,X表示需要调整的部件数,试求X的分布,E(X),D(x) Solution设A={部件需要调整ki=1.23),P(A)=0.P(A2)=02,P(41)=03 由于各部件相互独立,则有 P(X=0)=P(A1A2A3)=0.9×0.8×0.7=0.504 P(X=D)=P(A, A2 A3+ A1A, A3+ A1 A2 A3) =0.1×0.8×0.7+09×0.2×0.7+09×08×0.3=0.398 P(X=2)=P(A1A243+A1A2A3+A1A2A3) =0.9×02×0.3+0.1×0.8×0.3+0.1×0.2×07=0.092 P(X=3)=P(AA2A3)=0.1×0.2×0.3=0006 E(X)=0×0.504+1×0.398+2×0.092+3×0.006=0.6 E(X2)=02×0.504+12×0.398+22×0092+32×0006=0.82 D(X)=0.82-062=046 §3.3协方差及相关系数、矩 (Covariance, Correlation coefficient and Moment 我们除了讨论X与Y的数学期望和方差外,还需讨论描述X与Y之间相互关系的数字特 征.本节讨论这方面的数字特征 协方差及相关系数的定义 Covariance and correlation coefficient) efinition34设有二维随机变量(X,Y),如果EX-E(X川[Y-E(Y)存在,则称 E[X-E(X川[Y-E()为随机变量X与Y的协方差.记为Co(X,Y),即 Cov(X, =ELX-ECXJLY-E(I 称p Cov(X,n) 为随机变量X与Y的相关系数.若Cov(X,Y)=0,称X与Y不 D(x)√DY) exist, then it is called covariance of random variable X and Y, and written Cov(X, Y), namer WB .( Suppose there are two dimension random variable(x, y), if ELX-E(XIY-ECY) Cov(X, =ELX -E(XJLY-E(n] Cov(X,Y) P is called correlation coefficient of random variable X and D(X)√D(Y) Cov(X, y)=0, then X and Y is not correlational. 、协方差与相关系数的性质( Property of covariance and correlation coefficient) 协方差的性质 (1)Cov(X, =Cov(r, X) (2)Cov(X, Y=E(XY-E(XE(): D(X±Y)=D(X)+D(Y)±2Cov(X,)44 6 1 3 2 ( ) ( , ) 1 0 2 3 1 0 2 2 = = = =   −  x x D E Y y f x y dxdy dx y dy x dx 6 1 0 6 1 , ( ) 18 1 9 4 2 1 D(X ) = − = D Y = − = ． Example 3.22 一台设备由三大件组成，载设备的运转过程中需要调整的概率分别为 0.10， 0.20，0.30，假设各部分相互独立， X 表示需要调整的部件数，试求 X 的分布， E(X ), D(X ) ． Solution 设 Ai = 部件i需要调整(i =1,2,3), P(A1 ) = 0.1,P(A2 ) = 0.2,P(A3 ) = 0.3 ， 由于各部件相互独立，则有 P(X = 0) = P(A1 A2 A3 ) = 0.9 0.8 0.7 = 0.504 ( 1) ( ) 3 3 1 2 2 2 3 1 P X = = P A1 A A + A A A + A A A = 0.10.80.7 +0.90.20.7 +0.90.80.3 = 0.398 ( 2) ( 3 ) 3 1 2 2 2 3 1 P X = = P A1A A + A A A + A A A = 0.90.20.3+ 0.10.80.3+ 0.10.20.7 = 0.092 P(X = 3) = P(A1A2A3 ) = 0.10.20.3 = 0.006 E(X ) = 0 0.504 +1 0.398 + 2 0.092 + 3 0.006 = 0.6 ( ) 0 0.504 1 0.398 2 0.092 3 0.006 0.82 2 2 2 2 2 E X =  +  +  +  = ( ) 0.82 0.6 0.46 2 D X = − = §3.3 协方差及相关系数、矩 (Covariance, Correlation coefficient and Moment) 我们除了讨论 X 与 Y 的数学期望和方差外，还需讨论描述 X 与 Y 之间相互关系的数字特 征．本节讨论这方面的数字特征． 一、 协方差及相关系数的定义(Covariance and correlation coefficient) Definition 3.4 设有二维随机变量 (X,Y) ，如果 E[X − E(X )][Y − E(Y)] 存在，则称 E[X − E(X )][Y − E(Y)] 为随机变量 X 与 Y 的协方差．记为 Cov(X ,Y) ，即 Cov X Y ( , ) = E[X − E(X )][Y − E(Y)] 称 ( ) ( ) ( , ) D X D Y Cov X Y  XY = 为随机变量 X 与 Y 的相关系数．若 Cov X Y ( , ) 0 = ，称 X 与 Y 不 相关．(Suppose there are two dimension random variable (X,Y) , if E[X − E(X )][Y − E(Y)] is exist, then it is called covariance of random variable X and Y , and written Cov(X ,Y) , namely Cov X Y ( , ) = E[X − E(X )][Y − E(Y)], ( ) ( ) ( , ) D X D Y Cov X Y  XY = is called correlation coefficient of random variable X and Y . If Cov(X ,Y) = 0 , then X and Y is not correlational..) 二、 协方差与相关系数的性质(Property of covariance and correlation coefficient) 1． 协方差的性质 (1) Cov X Y ( , ) = Cov(Y, X ) ； (2) Cov X Y ( , ) = E(XY) − E(X )E(Y) ； (3) D(X  Y) = D(X ) + D(Y)  2Cov(X,Y) ；