Transcription of 1 Introduction to reducing variance in Monte Carlo …
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1 Introduction to reducing variance in Monte Carlo …
Copyrightc 2007 by Karl Sigman1 Introduction to reducing variance in Monte Carlo Review of confidence intervals for estimating a meanIn statistics, we estimate an unknown mean =E(X) of a distribution by collectingniidsamples from the distribution,X1, .. , Xnand using the sample meanX(n) =1nn j=1Xj.(1)Letting 2=V ar(X) denote the variance of the distribution, we conclude thatV ar(X(n)) = 2n.(2)Thecentral limit theoremasserts that asn , the distribution ofZndef= n (X(n) ) tends toN(0,1), the unit normal distribution. LettingZdenote aN(0,1) rv, we conclude that fornsufficiently large,Zn Zin distribution.
So, in practice we would use s(n) is place of σ when constructing our confidence intervals. For example, a 95% confidence interval is given by X(n)±(1.96)s√(n) n
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