求教一道概率证明题设x y是相互独立的随机变量,证明(1)若E(X)=E(Y)=0,则D(XY)=D(X)D(Y),(2)若E(X)=0或E(Y)=0,则D(XY)>=D(X)D(Y)
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![求教一道概率证明题设x y是相互独立的随机变量,证明(1)若E(X)=E(Y)=0,则D(XY)=D(X)D(Y),(2)若E(X)=0或E(Y)=0,则D(XY)>=D(X)D(Y)](/uploads/image/z/1006077-21-7.jpg?t=%E6%B1%82%E6%95%99%E4%B8%80%E9%81%93%E6%A6%82%E7%8E%87%E8%AF%81%E6%98%8E%E9%A2%98%E8%AE%BEx+y%E6%98%AF%E7%9B%B8%E4%BA%92%E7%8B%AC%E7%AB%8B%E7%9A%84%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%2C%E8%AF%81%E6%98%8E%EF%BC%881%EF%BC%89%E8%8B%A5E%28X%29%3DE%28Y%29%3D0%2C%E5%88%99D%28XY%29%3DD%28X%29D%28Y%29%2C%282%29%E8%8B%A5E%28X%29%3D0%E6%88%96E%28Y%29%3D0%2C%E5%88%99D%28XY%29%3E%3DD%28X%29D%28Y%29)
求教一道概率证明题设x y是相互独立的随机变量,证明(1)若E(X)=E(Y)=0,则D(XY)=D(X)D(Y),(2)若E(X)=0或E(Y)=0,则D(XY)>=D(X)D(Y)
求教一道概率证明题
设x y是相互独立的随机变量,证明(1)若E(X)=E(Y)=0,则D(XY)=D(X)D(Y),(2)若E(X)=0或E(Y)=0,则D(XY)>=D(X)D(Y)
求教一道概率证明题设x y是相互独立的随机变量,证明(1)若E(X)=E(Y)=0,则D(XY)=D(X)D(Y),(2)若E(X)=0或E(Y)=0,则D(XY)>=D(X)D(Y)
∵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)