Central Limit Theorem

Law of Large Numbers

Law

Consider a sequence of IID random variables where and for all . Let , then:

Note

Theorem

Let where are independently drawn observations from a population (for ):

then for sufficiently large , we have that the distribution of is approximately , regardless of the distribution of .

That is,
As :

It does not matter what the starting distribution is, we will get a distribution that is approximately normal.

Remember each is the average of a bunch of experiments. Each of these averages forms a normal distribution.

Notes:

The CLT allows us to use methods based on the normal distribution in a wide variety of situations
It is a major reason why the normal distribution is used so often in statistical inference
In most practice situations, can be viewed as a reasonable rough guideline but it is not a strict rule

Example with 120 students, since we are asking how the average () will behave, we can use CLT.

Result

If , then

Example

Suppose , then calculate .

solution
We have and .
Then by CLT:

Result

If , then .