Central Limit Theorem: Simulations
Special Case of the Central Theorem: Normal Approximation of the Binomial
Let X be the number of successes in n independent Bernoulli trials, each with probability p of success.
,
where
is a Bernoulli random variable with probability p of success,
Then for "large n", X is approximately normal.
That is, if ~Bin(), then
~Normal()
and
~Normal()
To see this, let's look at examples.
Central Limit Theorem
Let be independent, identically distributed random variables with
Then
Normal()
and
Normal()
To see this, let's look at examples
Example
Roll a fair die once. Let be the outcome. Repeat times. Compute the sample mean, . Repeat this times and record each sample mean; plot the sample means in green along with the pdf for Normal() in blue. (See below.)