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1、Chapter2FundamentalsofStatisticsThischapterdiscussessomefundamentalconceptsofmathematicalstatis-tics.Theseconceptsareessentialforthematerialinlaterchapters.2.1Populations,Samples,andModelsAtypicalstatisticalproblemcanbedescribedasfollows.Oneoraseriesofrandomexperimentsisperformed;
2、somedatafromtheexperiment(s)arecollected;andourtaskistoextractinformationfromthedata,interprettheresults,anddrawsomeconclusions.Inthisbookwedonotconsidertheproblemofplanningexperimentsandcollectingdata,butconcentrateonstatisticalanalysisofthedata,assumingthatthedataaregiven.Adescr
3、iptivedataanalysiscanbeperformedtoobtainsomesummarymeasuresofthedata,suchasthemean,median,range,standarddevia-tion,etc.,andsomegraphicaldisplays,suchasthehistogramandbox-and-whiskerdiagram,etc.(see,e.g.,HoggandTanis(1993)).Althoughthiskindofanalysisissimpleandrequiresalmostnoassum
4、ptions,itmaynotallowustogainenoughinsightintotheproblem.Wefocusonmoresophisticatedmethodsofanalyzingdata:statisticalinferenceanddecisiontheory.2.1.1PopulationsandsamplesInstatisticalinferenceanddecisiontheory,thedatasetisviewedasareal-izationorobservationofarandomelementdefinedonap
5、robabilityspace(Ω,F,P)relatedtotherandomexperiment.TheprobabilitymeasurePiscalledthepopulation.Thedatasetortherandomelementthatproduces91922.FundamentalsofStatisticsthedataiscalledasamplefromP.Thesizeofthedatasetiscalledthesamplesize.ApopulationPisknownifandonlyifP(A)isaknownvalue
6、foreveryeventA∈F.Inastatisticalproblem,thepopulationPisatleastpartiallyunknownandwewouldliketodeducesomepropertiesofPbasedontheavailablesample.Example2.1(Measurementproblems).Tomeasureanunknownquan-tityθ(forexample,adistance,weight,ortemperature),nmeasurements,x1,...,xn,aretakenin
7、anexperimentofmeasuringθ.Ifθcanbemeasuredwithouterrors,thenxi=θforalli;otherwise,eachxihasapossiblemea-surementerror.Indescriptivedataanalysis,afewsummarymeasuresmaybecalculated,forexample,thesamplemeanXn1x¯=xini=1andthesamplevarianceXn212s=(xi−x¯).n−1i=1However,whatistherelations
8、hipbetween¯xandθ?Aretheyclose(ifn
