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1、7MathematicsYoucandoalotofmathsinR.Hereweconcentrateonthekindsofmathematicsthatfindmostfrequentapplicationinscientificworkandstatisticalmodelling:functions;continuousdistributions;discretedistributions;matrixalgebra;calculus;differentialequations.7.1MathematicalfunctionsForthekindsoffuncti
2、onsyouwillmeetinstatisticalcomputingthereareonlythreemathematicalrulesthatyouneedtolearn:theseareconcernedwithpowers,exponentsandlogarithms.Intheexpressionxbtheexplanatoryvariableisraisedtothepowerb.Inextheexplanatoryvariableappearsasapower–inthisspecialcase,ofe=2.71828,ofwhichxistheexponent.T
3、heinverseofexisthelogarithmofx,denotedbylog(x)–notethatallourlogsaretothebaseeandthat,forus,writinglog(x)isthesameasln(x).Itisalsousefultorememberahandfulofmathematicalfactsthatareusefulforworkingoutbehaviouratthelimits.Wewouldliketoknowwhathappenstoywhenxgetsverylarge(e.g.x→∞)andwhathappensto
4、ywhenxgoesto0(i.e.whattheinterceptis,ifthereisone).Thesearethemostimportantrules:Anythingtothepowerzerois1:x0=1.Oneraisedtoanypowerisstill1:1x=1.Infinityplus1isinfinity:∞+1=∞.Oneoverinfinity(thereciprocalofinfinity,∞–1)iszero:1=0.∞Anumber>1raisedtothepowerinfinityisinfinity:1.2∞=∞.Afraction(e.
5、g.0.99)raisedtothepowerinfinityiszero:0.99∞=0.TheRBook,SecondEdition.MichaelJ.Crawley.©2013JohnWiley&Sons,Ltd.Published2013byJohnWiley&Sons,Ltd.MATHEMATICS259Negativepowersarereciprocals:x−b=1.xb√Fractionalpowersareroots:x1/3=3x.Thebaseofnaturallogarithms,e,is2.71828,soe∞=∞.Last,butperhapsm
6、ostusefully:e−∞=1=1=0.e∞∞Therearebuilt-infunctionsinRforlogarithmic,probabilityandtrigonometricfunctions(p.17).7.1.1LogarithmicfunctionsThelogarithmicfunctionisgivenbyy=aln(bx).Herethelogarithmistobasee.Theexponentialfunction,inwhichtheresponseyistheantilogarithmofthecontinuousexplanatoryvaria
7、blex,isgivenbyy=aebx.Boththesefunctionsaresmoothfunctions,andtodrawsmoothfunctionsinRyouneedtogenerateaseriesof100ormoreregularlyspacedxvaluesbetweenmin(x)andmax(x):x<-seq(0,10,0.1)InRtheexponentialfunctionisexpandthenaturallogfunction(