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1、Chapter1ProbabilityandDistributions1.1IntroductionManykindsofinvestigationsmaybecharacterizedinpartbythefactthatrepeatedexperimentation,underessentiallythesameconditions,ismoreorlessstandardprocedure.Forinstance,inmedicalresearch,interestmaycenterontheeffectofadrugthatistobeadministered;oraneconomis
2、tmaybeconcernedwiththepricesofthreespecifiedcommoditiesatvarioustimeintervals;ortheagronomistmaywishtostudytheeffectthatachemicalfertilizerhasontheyieldofacerealgrain.Theonlywayinwhichaninvestigatorcanelicitinformationaboutanysuchphenomenonistoperformtheexperiment.Eachexperimentterminateswithanoutcom
3、e.Butitischaracteristicoftheseexperimentsthattheoutcomecannotbepredictedwithcertaintypriortotheperformanceoftheexperiment.Supposethatwehavesuchanexperiment,buttheexperimentisofsuchanaturethatacollectionofeverypossibleoutcomecanbedescribedpriortoitsperformance.Ifthiskindofexperimentcanberepeatedunde
4、rthesameconditions,itiscalledarandomexperiment,andthecollectionofeverypossibleoutcomeiscalledtheexperimentalspaceorthesamplespace.Example1.1.1.Inthetossofacoin,lettheoutcometailsbedenotedbyTandlettheoutcomeheadsbedenotedbyH.Ifweassumethatthecoinmayberepeatedlytossedunderthesameconditions,thenthetos
5、softhiscoinisanexampleofarandomexperimentinwhichtheoutcomeisoneofthetwosymbolsTandH;thatis,thesamplespaceisthecollectionofthesetwosymbols.Example1.1.2.Inthecastofonereddieandonewhitedie,lettheoutcomebetheorderedpair(numberofspotsuponthereddie,numberofspotsuponthewhitedie).Ifweassumethatthesetwodice
6、mayberepeatedlycastunderthesameconditions,thenthecastofthispairofdiceisarandomexperiment.Thesamplespaceconsistsofthe36orderedpairs:(1,1),...,(1,6),(2,1),...,(2,6),...,(6,6).LetCdenoteasamplespace,letcdenoteanelementofC,andletCrepresentacollectionofelementsofC.If,upontheperformanceoftheexperiment,th
7、eoutcome12ProbabilityandDistributionsisinC,weshallsaythattheeventChasoccurred.NowconceiveofourhavingmadeNrepeatedperformancesoftherandomexperiment.Thenwecancountthenumberfoftimes(thefrequency)thattheeventCa
