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

Approximation of binomial distribution

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

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