Links

The central limit theorem

The gist of it

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

A more precise formulation

Refer to the page on independent variables
Let
x1,,xn{x_1, \ldots, x_n }
be a sample of iid random variables extracted from a distribution whose expected value is
μ\mu
and whose standard deviation is
σ\sigma
, then, as
nn \to \infty
,
n(snμ)dN(0,σ2) ,\sqrt{n} (s_n - \mu) \xrightarrow[]{\text{d}} \mathcal{N(0, \sigma^2)} \ ,
that is, the difference between the sample average
sns_n
and the population mean
μ\mu
, multiplied by
n\sqrt{n}
converges in distribution to a normal with mean 0 and variance
a=ba = b
.
Which means, the sample converges to a gaussian with mean
μ\mu
and variance
σ2\sigma^2
.

References

  1. 1.
    The Wikipedia page