Machine Learning: fundamental algorithms
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
Last modified 1yr ago