9设X与Y独立,X~N(A1,o12),Y~N(2,a2),求 (1)Z1=aX+bY的数学期望与方差 (2)Z2=XY的数学期望与方差 AF(1)E(Z1=E(aX +br)=aE(X)+bE(Y)=au,+bu2 D(Zi=D(aX +bY)=a'D(X)+6 D(r)=001+602 (2)E(Z2)=E(XY)=E(X)E(Y)=12 E(X2)=D(X+E(X)=a21+121 E(Y2)=D(Y)+[E(Y)=2+2 E(XY2)=E(X2)E(Y2)=(a21+21)(a2+2) D(Z2)=D(XY)=E(XY)-E(XY) 2 O102+O 22 2217 9. 解 ( ) E Z1  E(aX  bY )  aE(X )  bE(Y )   1 2 ( ) D Z1  D(aX  bY ) ( ) ( ) 2 2  a D X  b D Y 2 2 2 1 2 2  a   b  (1) (2) ( ) E Z2  E(XY )  E(X )E(Y ) ( ) 2 E X 2  D(X)  [E(X)] 1 2 1 2     ( ) 2 E Y 2  D(Y )  [E(Y )] 2 2 2 2     ( ) 2 2 E X Y ( ) ( ) 2 2  E X E Y ( 1 ) 2 1 2     ( 2 ) 2 2 2    ( ) D Z2  D(XY ) ( ) 2 2  E X Y 2  [E(XY )] 2 2 1 2    2 2 1 2    1 2 2 2     a 1  b 2