E(XY)=E(X)E(Y),是否能推出D(XY)=D(X)D(Y)希望能有详细点的推导.

Question

E(XY)=E(X)E(Y),是否能推出D(XY)=D(X)D(Y)
希望能有详细点的推导.

Answer 1

不能,
E(XY)=E(X)E(Y)
只能说明x.y独立
能够推出
D(xy)=D(X)D(Y)+(E(x))^2D(Y)+(E(Y))^2D(x)
DX=EX^2-(EX)^2
DY=EY^2-(EY)^2
EXY=EXEY
DXY=E(XY)^2-(EXY)^2=(EX^2)(EY^2)-(EXY)(EXY)
=(DX+(EX)^2)(DY+(EY)^2)-(EX)^2(EY)^2
=D(X)D(Y)+(E(x))^2D(Y)+(E(Y))^2D(x)