Statistics Solutions

[See] Let X 1 , X 2 , X 3 ,…, X n be identically independently distributed random variables with mean μ and variance σ ^2. Let S^2=(1)/(n-1)∑limits_i=1^n(X_i-X̄)^2

Question: Let X 1 , X 2 , X 3 ,…, X n be identically independently distributed random variables with mean \(\mu \) and variance \({{\sigma }^{2}}\). Let

\[{{S}^{2}}=\frac{1}{n-1}\sum\limits_{i=1}^{n}{{{\left( {{X}_{i}}-\bar{X} \right)}^{2}}}\]

where \(\bar{X}=\frac{1}{n}\sum\limits_{i=1}^{n}{{{X}_{i}}}\). Show that \({{S}^{2}}\) is an unbiased estimator for \({{\sigma }^{2}}\)

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