## 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 mean andvariance thenthe 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 –standard deviation   Uniform,  and are 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 limits and what is the probability that the average is less than 12?  This corresponds to a probability of 0.791. 