Central Limit Theorem

Consider a sequence of independent and identically distributed random variables with finite mean and finite variance .

Consider also the sample mean

the sum
and the sequence of variables

Then , where .


For n sufficiently large (

  • Standardization:
  • Mean:
  • Sum sequence:


Consider the sequence

where is a sample function such that as . Then where .

Applications of the Central Limit Theorem[edit | edit source]

Approximation of binomial distribution[edit | edit source]

For the approximation of a binomial distribution we use the correction of continuity which is 0.5.

Assessment of the error of the sample mean[edit | edit source]

For assessing the error of the sample mean we consider the strong law of large numbers and the CLT, as well as :

  • Margin of error:
  • Sample variance:

  • independent and identically distributed :


Conclusive, to assess the error one can use the following interval