# The central limit theorem

The core of the CLT is that each sample of random variables, when their number is large enough, will converge to a gaussian distribution.

Refer to the page on independent variables

Let

${x_1, \ldots, x_n }$

be a sample of iid random variables extracted from a distribution whose expected value is $\mu$

and whose standard deviation is $\sigma$

, then, as $n \to \infty$

,$\sqrt{n} (s_n - \mu) \xrightarrow[]{\text{d}} \mathcal{N(0, \sigma^2)} \ ,$

that is, the difference between the sample average

$s_n$

and the population mean $\mu$

, multiplied by $\sqrt{n}$

converges in distribution to a normal with mean 0 and variance $a = b$

.Which means, the sample converges to a gaussian with mean

$\mu$

and variance $\sigma^2$

.- 1.