Biz. $X_1,\ldots,X_n$ FAE minta:

D^{2} (\sum_{k} α_{k} X_{k}) = \sum_{k} α_{k}^{2} D^{2} (X_{k}) = σ^{2} \sum_{k} α_{k}^{2} \overset{*}{\geq} \frac{σ^{2}}{n} = D^{2} (\bar{X})

$\mathbb{D}^2\left(\sum_k \alpha_k X_k \right)= \sum_k \alpha^2_k \mathbb{D}^2\left( X_k \right)= \sigma^2 \sum_k \alpha^2_k \stackrel{*}{\ge} \frac{\sigma^2 }{n}= \mathbb{D}^2\left( \overline{X}\right)$ ahol a

*

$*$ a kvadratikus-számtani egyenlőtlenség.

Biz.:

E (\frac{\sum_{k} (X_{k} - \bar{X})^{2}}{n - 1}) = σ^{2}

$\mathbb{E}\left(\frac{\sum_k (X_k-\overline{X})^2}{n-1}\right)=\sigma^2$ Legyen

A \in R

$A\in\mathbb{R}$ , Ekkor:

f (A) = \sum_{k} (x_{k} - A)^{2} = \sum_{k} x_{k}^{2} - 2 A \sum_{k} x_{k} + n A^{2}

$f(A)=\sum_k (x_k-A)^2=\sum_k x_k^2 -2A\sum_k x_k + nA^2$ A fenti jelöléssel

E f (μ) = n σ^{2} E f (\bar{x}) = (n - 1) s_{k o r r}^{2}

$\mathbb{E}f(\mu)=n\sigma^2 \ \ \ \mathbb{E}f(\overline{x})=(n-1)s^2_{korr}$

n σ^{2} - (n - 1) s_{k o r r}^{2} =

$n\sigma^2-(n-1)s^2_{korr}=$

n E (μ^{2} - 2 μ \bar{X} + {\bar{X}}^{2}) =

$n\mathbb{E}\left( \mu^2-2\mu\overline{X}+\overline{X}^2 \right)=$

n D^{2} (\bar{X}) = σ^{2}

$nD^2\left(\overline{X}\right)=\sigma^2$