Consider a sequence of independent and identically distributed random variables with finite mean and finite variance .
Consider also the sample mean
and the sequence of variables
Then , where .
For n sufficiently large (
- 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