样本X1,X2,……Xn平均数为5,方差为3,则3(X1-1),3(X2-1),……,3(Xn-1)标准差为?

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  • 样本X1,X2,……Xn平均数为5,方差为3,

    则,

    5 = [X1 + X2 + ...+ Xn]/n

    3 = [(X1 - 5)^2 + (X2 - 5)^2 + ...+ (Xn - 5)^2]/n

    因此,

    [3(X1 - 1) + 3(X2 - 1) + ...+ 3(Xn - 1)]/n

    = 3[X1 + X2 + ...+ Xn]/n - 3

    = 15 - 3 = 12

    {[3(X1 - 1) - 12]^2 + [3(X2 - 1) - 12]^2 + ...+ [3(Xn - 1) - 12]^2]/n

    = {[3X1 - 15]^2 + [3X2 - 15]^2 + ...+ [3Xn - 15]^2]/n

    = {9[X1 - 5]^2 + 9[X2 - 5]^2 + ...+ 9[Xn - 5]^2]/n

    = 9[(X1 - 5)^2 + (X2 - 5)^2 + ...+ (Xn - 5)^2]/n

    = 9*3 = 27.

    则3(X1-1),3(X2-1),……,3(Xn-1)标准差为

    (27)^(1/2) = 3*(3)^(1/2)