MeanSquareMinusSquareMean(discrete)

Mean square minus square mean (discrete) (English)

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For a discrete random variable X,

var(X) = σ² = 〈( X − 〈X 〉) ² 〉

n
∑
i=1

p_i ( x_i − 〈X 〉) ²

n
∑
i=1

p_i x_i ² − 2 〈X 〉

n
∑
i=1

p_i x_i + 〈X 〉²

n
∑
i=1

p_i

n
∑
i=1

p_i x_i ² − 2 〈X 〉〈X 〉+ 〈X 〉〈X 〉

= 〈X ² 〉− 〈X 〉² .

i.e. the variance of X is equal to the mean of X² minus the square of the mean of X.