Sea \(X \sim N(\mu,\sigma^2)\), verificar que \(S^2 = \frac{\sum(X_i-\overline{X})^2}{n-1}\) es un estimador insesgado de \(\sigma^2\)
Primero vamos a calcular, usando la fórmula alternativa de la varianza:
$$E(X_i^2) = Var(X_i) + E(X_i)^2 = \sigma^2 + \mu^2$$
$$E(\overline{X}^2) = Var(\overline{X}) + E(\overline{X})^2 = \frac{\sigma^2}{n} + \mu^2$$
Luego, para ver de S^2 es insesgado tenemos que calcular su esperanza:
$$E(S^{ 2 })=E\left( \frac { \sum (X_{ i }-\overline { X } )^{ 2 } }{ n-1 } \right) = E\left(\frac { \sum X_{ i }^{ 2 }-2X_{ i }\overline { X } +\overline { X } ^{ 2 }) }{ n-1 } \right)$$
$$= E\left( \frac { \sum X_{ i }^{ 2 }-2\sum X_{ i }\overline { X } +\sum \overline { X } ^{ 2 }) }{ n-1 } \right)$$
$$= E\left( \frac { \sum X_i^2-2 \overline { X } \sum X_{ i }+\sum \overline { X } ^{ 2 }) }{ n-1 } \right)$$
$$= E\left( \frac { \sum X_i^2-2 \overline { X } n \overline { X }+n \overline { X } ^{ 2 }) }{ n-1 } \right)$$
$$= E\left( \frac { \sum X_i^2-2 n \overline { X }^2 +n \overline { X } ^{ 2 }) }{ n-1 } \right)$$
$$= E\left( \frac { \sum X_i^2- n \overline { X }^2}{n-1} \right)$$
$$= \frac{\sum E(X_i^2)-nE(\overline X^2)}{n-1}$$
$$= \frac{n ( \sigma^2 + \mu^2) - n (\frac{\sigma^2}{n}+\mu^2)}{n-1}$$
$$= \frac{n \sigma^2 + n \mu^2 - \sigma^2 - n \mu^2}{n-1}$$
$$= \frac{n\sigma^2 - \sigma^2}{n-1}$$
\(= \frac{\sigma^2 (n-1)}{n-1} = \sigma^2\)