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The Central Limit Theorem - Main Concepts

The Central Limit TheoremSuppose that a sample of sizenis selected from a population that has mean and standarddeviation . LetX1,X2, ,Xnbe thenobservations that are independent and identicallydistributed ( ). Define now the sample mean and the total of thesenobservations asfollows: X= ni=1 XinT=n i=1 XiThecentral Limit theoremstates that the sample mean Xfollows approximately the normaldistribution with mean and standard deviation n, where and are the mean and stan-dard deviation of the population from where the sample was selected. The sample sizenhasto be large (usuallyn 30) if the population from where the sample is taken is the population follows the normal distribution then the sample sizencan be either smallor summarize: X N( , n).To transform Xintozwe use:z= x nExample: LetXbe a random variable with = 10 and = 4. A sample of size 100 is takenfrom this population. Find the probability that the sample mean of these 100 observations isless than 9.

Central limit theorem - proof For the proof below we will use the following theorem. Theorem: Let X nbe a random variable with moment generating function M Xn (t) and Xbe a random variable with moment generating function M X(t). If lim n!1 M Xn (t) = M X(t) then the distribution function (cdf) of X nconverges to the distribution function of Xas ...

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