## 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 Sampleof Size n |

-mean â€“standard deviation | ||||

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

Binomial, | isthe number of trials,isthe probability of success | |||

Geometric, | isthe probability of success | |||

Poisson, | isthe average number of events per time period |

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

This corresponds to a probability of 0.791.