next up previous contents
Next: Normal approximation of the Up: Central limit theorem Previous: Terms and role   Contents

Distribution of a sample average

As first application of the theorem, let us remind ourselves that a sample average $ \overline{X}_n$ of $ n$ independent variables,
$\displaystyle \overline{X}_n$ $\displaystyle =$ $\displaystyle \sum_{i=1}^n\frac{1}{n}\,X_i,$ (4.74)

is normally distributed, since it is a linear combination of $ n$ variables $ X_i$, with $ c_i=1/n$. Then,
$\displaystyle \overline{X}_n$ $\displaystyle \sim$ $\displaystyle {\cal N}(\mu_{\overline{X}_n},
\sigma_{\overline{X}_n}),$ (4.75)
$\displaystyle \mu_{\overline{X}_n}$ $\displaystyle =$ $\displaystyle \sum_{i=1}^n\frac{1}{n}\,\mu = \mu,$ (4.76)
$\displaystyle \sigma^2_{\overline{X}_n}$ $\displaystyle =$ $\displaystyle \sum_{i=1}^n
\left(\frac{1}{n}\right)^2\sigma^2 = \frac{\sigma^2}{n},$ (4.77)
$\displaystyle \sigma_{\overline{X}_n}$ $\displaystyle =$ $\displaystyle \frac{\sigma}{\sqrt{n}}\,.$ (4.78)

This result, we repeat, is independent of the distribution of $ X$ and is already approximately valid for small values of $ n$.

Giulio D'Agostini 2003-05-15