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Given $X_1, \dots , X_n$ i.i.d. and the two sample moments $$M_1 = \frac{1}{n} \sum_{i = 1}^{n} X_i = \bar{X}$$ and $$ M_2 = \frac{1}{n} \sum_{i = 1}^{n} X_i^2$$ how can I compute: $$ S^2 = \frac{1}{n} \sum_{i = 1}^{n} (X_i - \bar{X})^2$$ such as: $$S^2 = f(M_1, M_2)$$

Thank you.

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    Hint: This has nothing to do with statistics and could have been tagged (algebra) or (polynomials).2012-06-13
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    ok, got it. Thank you did.2012-06-13
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    This has to do with statistics in that the derived formula is commonly used for computing the variance estiamte.2012-06-13

1 Answers 1

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$S^2 = \frac{1}{n}\sum_{i=1}^n (X_i - M_1)^2 = \frac{1}{n}\sum_{i=1}^n (X_i^2 - 2 X_i M_1 + M_1^2) =$

$= \frac{1}{n}\sum_{i=1}^n X_i^2 - 2 M_1 \frac{1}{n}\sum_{i=1}^n X_i + \frac{1}{n}\sum_{i=1}^n M_1^2 =$

$= M_2 - 2M_1^2 + M_1^2 = M_2 - M_1^2$