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 Functio}}
By de nition: P(a 6 X < b) = Z b a f(x)dx (11:2) Any function of a random variable is itself a random variable and, if y is taken as some transformation function, y(X) will be a derived random variable. Let Y = y(X). Notice that if X = a the derived random variable Y = y(a) and if X = b, Y = y(b).
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