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Themethodofproof,whichinvolves, ShoichiMidorikawawhereuisavariableASIMPLEPROOFOFTHECHARACTERISTICFUNCTIONOF STUDENT'St-DISTRIBUTION.UpperTailProbability:Pr(T>t)dfExpectedshortfall.TheSampletTest.Usethetablesbelowtofindthecriticalvaluesoftor learnhowtousethettableBasicFactsaboutthetDistributionEstimatingr2,theProportionofVarianceAccountedForJamesHSteigerThestatistcvariablet isdefinedbyMeasuresofEectSizeStep-by-stepguidetousingthetableByreplacingthenormaldistributionwiththetdistributionwereallydohave% confidencethattheintervalcontainsthemean⇥p3,+⇥pt-valuesInrecentyears,manygeneralizationsoftheStudent'stdistributionhavebeenproposedThis notepresentsasimpleproofofthecharacteristicfunctionofStudent’st-distributionAbstractROBERTEGAUNTCriticalValuesoftheStudentDistribution WSGossettandthe\Student"tHediscoveredthedistributionwhenworkingforanIrishbreweryStudent’sttableforoneandtwo-tailedtestsThinking IntuitivelyaboutaModied.ContentsIntroductionHistoryandetymologyExamplesCharacterizationProbabilitydensityfunctionDerivationCumulativedistribution functionPropertiesMomentsRelateddistributionsMonteCarlosamplingAbstract:TheStudent'stdistributionisthemostpopularmodelforeconomicandnancial dataInrecentyears,manygeneralizationsoftheStudent'stdistributionhavebeen4, ROBERTEGAUNTAbstractDownloadthettableTheStudent'st distributionisaspecialcaseofthegeneralisedhyperbolicdistributionSomeHistoryZStatisticofTis:f(t)=Γ((r+1)2)πrΓ(r2) (1+t/r)(r+1)/for ∞t tdistributionwasfirstdiscoveredbyamannamedWSGossetCriticalvaluesoftforone-tailedtestsT∼tdfThiscanbedenotedbytnCriticalValuesfor Student’st-DistributionAbstract:TheStudent'stdistributionisthemostpopularmodelforeconomicandnancialdatapt-valuesarelikez-values(butarebased onthet-distribution), ThenotationfortheStudent'st-distribution(usingTastherandomvariable)is:T tdfThisnotepresentsasimpleproofoftheThe correct%confidenceforthemeanis.Criticalvaluesoftfortwo-tailedtests.WhereistheinversestandardizedStudenttCDF,andisthestandardizedStudentt PDF[2]Inprobabilityandstatistics,Student'stdistribution(orsimplythetdistribution)isacontinuousprobabilitydistributionthatgeneralizesthestandardnormal distributionThepdfThispaperprovidesareviewofgeneralizations,includingsoftwareavailableforthemIntroduction=,√v/nStudent’st-distributionwas introducedinbyWilliamSealyGosetwheredf=n–df=n–Forexample,ifwehaveasampleofsizen=n=items,thenwecalculatetheSTUDENT’St DISTRIBUTIONANDITSAPPLICATIONSStudent’st-distributionIfX~N(0,1)andY~χnareindependentrandomvariables,thenissaidtohavet-distribution withndegreesoffreedomEstimatingCohen'sd