例4149机变量X,Y-(x)=8x+y)0<x<20<y<2 其它 求EX,EY,E(X2),E(XY) ++ 解:应用定理4.2:Eg(X,Y) g(x, y)f(x, y)dxdy ++o E(X) xf(x,y)dcd小y x(x+y)dxdy=776 00 -oO 8 E(X2)= 厂=x(x,y)(x+y)b=58 8 同理由对称性:E()=(x,y)dtd=76E2583 E(XY)=(xy)f(x, y)dxdy xy(x+y)dxdy=4/3 8例4.1.4.随机变量(X,Y)~f(x,y)=      +     0 其 它 ( ) 0 2 0 2 8 1 x y x y 求EX,EY,E(X2 ),E(XY) 解:应用定理4.1.2:   +  − +  − E(X) = xf (x, y)dxdy  +  − +  − E[g(X,Y )] = g(x, y) f (x, y)dxdy   = + 2 0 2 0 ( ) 8 1 x x y dxdy =7/6   = + 2 0 2 0 2 ( ) 8 1 x x y dxdy =5/3 同理由对称性: =7/6 EY2=5/3   +  − +  − E(X ) = x f (x, y)dxdy 2 2   +  − +  − E(Y) = yf (x, y)dxdy   +  − +  − E(XY) = (xy) f (x, y)dxdy   = + 2 0 2 0 ( ) 8 1 xy x y dxdy =4/3