Transcription of The Central Limit Theorem
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Example: bankruptcy
The Central Limit Theorem
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).
n are i.i.d. (independent and identically distributed) random variables having the same distribution with mean , variance ˙2, and moment generating function M X(t), then if n!1 the limiting distribution of the random variable Z= T n ˙ p n (where T= X 1 +X 2 + +X n) is the standard normal distribution N(0;1). Proof: M Z(t) = M T n ˙ p n (t ...
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