## The Central Limit Theorem

The Central Limit Theorem by itself makes the NormalDistribution the most important probability distribution. It statesthat any distribution can be approximated to some extent by thenormal distribution. Specifically, if a sample of size n is takenfrom a population with meanandvariancethenthe sample mean has the approximate distribution The mean and variance of some common distributions of given in thetable below.

Distribution | Parameters and Meaning of Parameters | Mean | Variance | Approximate Distribution of Sample Mean for Sample of Size n |

-mean â€“standard deviation | ||||

Uniform, | andare the minimum and maximum possible values of the random variable | |||

Binomial, | is the number of trials,is the probability of success | |||

Geometric, | is the probability of success | |||

Poisson, | is the average number of events per time period |

Example: For a sample of size 10 taken from a uniform distribution between the limitsandwhat is the probability that the average is less than 12?

This corresponds to a probability of 0.791.