两种方法求证E(x-Ex)^2=E(x^2)+[E(x)]^2
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E(x-Ex)^2 = E[x^2 - 2xEx + (Ex)^2] = E(x^2) - 2(Ex)(Ex) + (Ex)^2 = E(x^2) - (Ex)^2
E(x^2) = E(x-Ex+Ex)^2 = E[(x-EX)^2 + 2(x-Ex)(Ex) + (Ex)^2]
= E(x-Ex)^2 + 2E[(x-Ex)](Ex) + (Ex)^2
= E(x-Ex)^2 + 2(Ex)[Ex - Ex] + (Ex)^2
= E(x-Ex)^2 + (Ex)^2,
E(x-Ex)^2 = E(x^2) - (Ex)^2
E(x^2) = E(x-Ex+Ex)^2 = E[(x-EX)^2 + 2(x-Ex)(Ex) + (Ex)^2]
= E(x-Ex)^2 + 2E[(x-Ex)](Ex) + (Ex)^2
= E(x-Ex)^2 + 2(Ex)[Ex - Ex] + (Ex)^2
= E(x-Ex)^2 + (Ex)^2,
E(x-Ex)^2 = E(x^2) - (Ex)^2
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