𝜇-estimator

Central limits theorem

The central limits theorem states that as the sample size increases, the sample mean converges in distribution to a normal distribution, regardless of the underlying distribution of . #m/thm/stat That is,

or equivalently

as . In the case where itself is normally distributed, is already normal for all . Otherwise, is generally taken as a good guide.

Proof1

Consider a set of independent, similarly distributed random variables with expected value and probability density function . It is useful to introduce the Random function

which by Distribution has distribution

as required.

#state/tidy | #SemBr | #lang/en

Footnotes

  1. 2006, Statistische Mechanik, p. 8 ↩