概率论问题,随机变量X,Y独立,请问D(XY)=DX.DY吗,请给出证明.
问题描述：
概率论问题,随机变量X,Y独立,请问D(XY)=DX.DY吗,请给出证明.
答
不等于.证明如下DX=EX^2-(EX)^2DY=EY^2-(EY)^2EXY=EXEYDXY=E(XY)^2-(EXY)^2=(EX^2)(EY^2)-(EXY)(EXY)=DXDY+EX^2(EY)^2+(EX)^2EY^2-2(EX)^2(EY)^2=DXDY+(EX)^2(EY^2-(EY)^2)+(EY)^2(EX^2-(EX)^2)=D(X)D(Y)+(E(x))^2D(...