Central Limit Theorem

probability-theory

Definition

Central Limit Theorem

Let be a sequence of identically distributed random variables with finite variance

and be the sample mean of the first random variables, then:

Note that is the first central moment of the sample’s distribution.

Normal Distribution

In practice, for large , the sample mean is approximately normal-distributed:

Similarly, for the sum , we have:

Thus, the standardised and are approximately standard normal distributed: