解:D∑aX)=E∑ax1-E(∑aX)=ED∑a(X1-EX EC∑q(X-EXx-E,-∑∑ aa: cov(x,X) ∑aD(X)+2∑∑aa1cov(x,X) 若X1,H2,…,Xn独立,则 D∑ax)=∑aD(X) 例321:设(XY)的密度函数为 (x+y),0≤x≤1,0≤y≤2 p(x, y) 其它 求D(2X-3+8) 解:D(2X-3y+8)=D(2X-3)=4DX+9D-12cov(X,y P2(x)=」2(x+y)d=(x+1)0≤x≤1 E(X)=x3(x+1)d E(x2)=x23(x+1xs 9 18 D(X)=E(x2)-(E)213同理:D()81 E(r) 1 (x+y)dydx cov(x, Y)=E(XY)-EXEY=-L D(2X-3Y+8)245 81-3.025 例322:设(X,Y)~N(1,42,01,02,P),求Pxy Mr: cOV(X, Y)=E(X-EX)(Y-E解： ( ) = = 1 ∑= n i D aiXi 2 1 1 [ ( )] i n i i i n i E ∑aiX E ∑a X = = − 2 1 [ ( )] i i n i E ∑ai X − EX = = [ ( )( )]= 1 1 i i j j n i n j E ∑∑ai aj X − EX X − EX = = cov( , ) 1 1 i j n i n j ∑∑ai aj X X = = = ( ) 2 cov( , ). 1 2 j i j i j i i n i ∑ai D X ∑∑a a X X = < + 若 X1, X2 ,", Xn 独立，则 ( ) = 1 ∑= n i D aiXi ( ) 1 2 i n i ∑ai D X = 例 3.21：设(X,Y)的密度函数为 ⎪ ⎩ ⎪ ⎨ ⎧ + ≤ ≤ ≤ ≤ = 0 , 其它 ( ) , 0 1, 0 2 3 1 ( , ) x y x y p x y 求 D(2X − 3Y + 8) 。 解： D(2X − 3Y + 8) = D(2X − 3Y ) = 4DX + 9DY −12cov(X ,Y ) ( 1) ,0 1 3 2 ( ) 3 1 ( ) 2 0 = + = + ≤ ≤ ∫ p x x y dy x x X 9 5 ( 1) 3 2 ( ) 1 0 = + = ∫ E X x x dx 18 7 ( 1) 3 2 ( ) 1 0 2 2 = + = ∫ E X x x dx 162 13 ( ) ( ) ( ) 2 2 D X = E X − EX = 同理： 81 23 D(Y ) = 3 2 ( ) 3 1 ( ) 1 0 2 0 = + = ∫∫ E XY xy x y dydx 81 1 cov(X ,Y ) = E(XY ) − EXEY = − 3.025 81 245 D(2X − 3Y + 8) = = 例 3.22：设（X,Y）～ ( , , , , ) ，求 2 2 2 N µ1 µ2 σ1 σ ρ ρ X Y 。 解：cov(X ,Y ) = E(X − EX )(Y − EY )