Transcription of 11 | TRANSFORMING DENSITY FUNCTIONS
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11 | TRANSFORMING DENSITY FUNCTIONS
11 | TRANSFORMINGDENSITYFUNCTIONSIt canbe expedient to useatransformationfunctionto transformoneprobability densityfunctioninto thistopic,it is helpfulto recapitulatethemethod of integrationby substitutionof a Substitutionofa newVariableImaginethata newcomerto integrationcomesacrossthefollowing:Zp 202xcosx2dxAssumingthatthenewcomerdoesn' tnoticethattheintegrandis thederivative of sinx2,oneway to proceedwouldbe to substitutea newvariableyforx2:Lety=x2 Replacethelimitsx= 0 andx=p 2byy= 0 andy= 2 Replace2xcosx2by 2pycosyNotethatx=pyandhencedxdy=12pyands o replacedxbydy2pyTheoriginalproblemis thereby transformedinto thefollowingintegration:Z 20cosy dy=hsinyi 20= 1 TheGeneralCaseIt is instructive to developthegeneralcasealongsidetheabove example:General CaseAboveExampleZbaf(x)dxZp 202xcosx2dxChoosea transformationfunctiony(x)y(x) =x2 Noteitsinversex(y)x(y) =pyReplacethelimitsbyy(a) andy(b)0 and 2 Replacef(x) byf x(y) 2pycosyReplacedxbydxdydy12pydyResultisZy (b)y(a)f x(y) dxdydyZ 20cosy dy{ {ApplicationtoProbability DENSITY FunctionsTheprevioussectioninformallylea dsto thegeneralformulaforintegrationby substitutionof a newvariable:Zbaf(x)dx=Zy(b)y(a)f x(y) dxdydy(11:1)Thisformulahasdirectapplicat iontotheprocessof transformingprobability densityfunctions: : :SupposeXis a randomvariablewhoseprobability DENSITY functionisf(x).}}
Observations and Constraints The crucial step is (11.3). One imagines noting a sequence of values of a random variable X and for each value in the range a to b using a transformation function y(x) to compute
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