(3) Cov(X, + X2,Y) = Cov(Xi,Y) + Cov(X2,Y)证明Cov(X, + X2,Y)= E[(X + X,)Y]- E(X, + X,)E(Y)= E(X,Y)+ E(X,Y)-[E(X,)+ E(X2)]E(Y)=[E(XY)- E(X)E(Y)]+[E(X,Y)- E(X2)E(y)= Cov(X,, Y) +Cov(X2, Y)
(3) Cov( , ) Cov( , ) Cov( , ). X1 + X2 Y = X1 Y + X2 Y 证明 E(X X )Y E(X X )E(Y ) = 1 + 2 − 1 + 2 E(X Y ) E(X Y ) E(X ) E(X )E(Y ) = 1 + 2 − 1 + 2 E(X Y ) E(X )E(Y ) E(X Y ) E(X )E(Y ) = 1 − 1 + 2 − 2 ( , ) ( , ) = Cov X1 Y +Cov X2 Y ( , ) Cov X1 + X2 Y
期望、方差、协方差的性质对比期望方差协方差E(c)=CD(c)=0Cov(c,X)=0D(aX)=a-D(X),Cov(aX,bY)E(aX)=aE(X),=abCov(X,Y)E(X+Y)D(X+Y)=D(X)+Cov(X+Y,Z)=Cov(X,Z)=E(X)+E(Y)D(Y)+2Cov(X,Y)+Cov(Y,Z)当X与Y独立时当X与Y独立时E(XY)-E(X)E(Y)D(X+Y)=D(X)+DY)沈阳师范大学ShentangNomal Univesth
期望、方差、协方差的性质对比 期望 方差 协方差 E(c)=C D(c)=0 Cov(c,X)=0 E(aX)=aE(X), D(aX)=a2D(X), Cov(aX,bY) =abCov(X,Y) E(X+Y) =E(X)+E(Y) D(X+Y)=D(X)+ D(Y)+2Cov(X,Y) 当X与Y独立时 D(X+Y)=D(X)+D(Y) Cov(X+Y,Z) =Cov(X,Z) 当X与Y独立时 +Cov(Y,Z) E(XY)=E(X)E(Y)