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1、ComparisonofDi erentTechniquestoGenerateNormalRandomVariablesRitabrataRoyNovember4,2002AbstractThisexerciseaimsatexploringdi erenttechniquesforcreatingarandomvariableXaccordingtoanormaldistributionwithzeromeanandunitvariance.Themethodsincludetheuseof
2、aninversecumulativedistributionfunction,theBox{Mullermethod,thepolartechniqueandtheapplicationoftheCentralLimitTheoremstouni-formrandomvariables.Thenormalrandomvariablesgeneratedbythesemethodsarethencomparedaccordingtodi erentperformancemetrics,includ
3、ingtheirmean,varianceandkurtosis,andconclusionsaredrawnabouttheperformanceofeachofthetechniques.1IntroductionTheGaussiandistribution,ornormaldistribution,isacontinuousdistribu-tionfunctionthatisde nedcompletelybythemeanandvarianceofthedistribution.The
4、probabilitydensityfunction(pdf)ofaGaussianrandomvariableXisgivenby:21 (x )fX(x)=pe22(1)2Submittedaspartialful llmentofcourserequirementinStochasticSignalsandSys-tems(ECE330:541)toDr.WadeTrappe,AssistantProfessor,Rutgers,TheStateUni-versityofNewJe
5、rsey.1where,:meanofthedistribution2:varianceofthedistributionTheGaussiandistributionisreal-valuedandsymmetricaboutthemean.Astandardnormaldistributionisobtainedbyputting=0and2=1in(1)andistypicallyrepresentedasN(0;1).TheGaussiandistribution,inthefor
6、mofabell-shapedcurve,appearsinseveralman-madeandnaturalphenomenaandhasbeenchristenednormalasatributetoitsubiquity.Thefrequentocurrenceofthedistributionfol-lowsfromtheCentralLimitTheorems,whichstatethatthemeanofasetofvariateswithanydistributionhavinga
7、nitemeanandvariancetendstotheGaussiandistribution.Aspointedoutin[1],innature,manymacroscopicphenomenaresultfromtheadditionofnumerousindependent,microscopicprocesses;thisgivesrisetotheGaussianrandomvariable.Historically,thenormaldistributionwas rstintr
8、oducedbydeMoivreinthesecondeditionofhisDoctrineofChances(1718),inthecontextofapproximationsoflargebinomialcoecients.HisresultwasextendedbyLaplaceinhisbookAnalyticalTheoryofProbabilities(1812),andisnowcalledTheoremofdeMoivre-Laplace.Aroundthat
