天马哥
幼苗
共回答了15个问题采纳率:86.7% 举报
∵X,Y相互独立, ∴X^2,Y^2也相互独立
(1) D(XY)=E[XY-E(XY)]^2
=E(XY-EXEY)^2
=E(X^2Y^2)
=E(X^2)E(Y^2)
=E[(X-EX)^2]E[(Y-EY)^2]
=D(X)D(Y)
(2)不妨设E(X)=0,E(Y)可能等于0也可能不等于0
(EY)^2≥0
由(1)可知D(XY)=E[(X-EX)^2]E(Y^2)
≥E[(X-EX)^2][E(Y^2)-(EY)^2] (等号在EY=0时成立)
=E[(X-EX)^2][E(Y^2)-2(EY)^2+(EY)^2]
=E[(X-EX)^2]E[(Y-EY)^2]
=D(X)D(Y)
1年前
5