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6.5 Central Limit Theorem

The central limit theorem (CLT) is the theoretical reason for the practical importance of the Normal distribution in probability and statistics. The central limit theorem proves mathematically that, for distributions with means and standard deviations, linear combinations of random variables, such as sums and weighted averages, tend to be normally distributed.

A simple version of the CLT is as follows: Take a sample Y1? Y2,..., Yn of n independent observations from any distribution with Mean[Yi] = μ and Standard Deviation[Yi] = σ Then as the sample size n approaches infinity, the distribution of the standardized sample mean, i.e.,


  

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