概率统计(英文)5 Joint Probability Distributions and Random Samples

5JointProbabilityDistributionsandRandomSamples5.1JointlyDistributedRandomVariables5.2ExpectedValues,Covariance,andCorrelation5.3StatisticsandTheirDistributions5.4TheDistributionoftheSampleMean5.5TheDistributionofaLinearCombinationIntroductionInChapter3and4,westudiedprobabilitymodelsforasinglerandomvariable.Manyproblemsinprobabilityandstatisticsleadtomodelsinvolvingseveralrandomvariablessimultaneously.Inthischapter,wefirstdiscussprobabilitymodelsforthejointbehaviorofseveralrandomvariables,puttingspecialemphasisonthecaseinwhichthevariablesareindependentofoneanother.Wethenstudyexpectedvaluesoffunctionsofseveralrandomvariables,includingcovarianceandcorrelationasmeasuresofthedegreeofassociationbetweentwovariables.5.1JointlyDistributedRandomVariablesThejointprobabilitymassfunctionfortwodiscreterandomvariablesDefinition:LetXandYbetwodiscreterandomvariablesdefinedonthesamplespaceΩofanexperiment.Thejointprobabilitymassfunctionp(x,y)isdefinedforeachpairofnumbers(x,y)byLetAbeanysetconsistingofpairsof(x,y)values.ThentheprobabilityP[(X,Y)∈A]isobtainedbysummingthejointpmfoverpairsinA;Example5.1Alargeinsuranceagencyservicesanumberofcustomerswhohavepurchasedbothahomeowner’spolicyandanautomobilepolicyfromtheagency.Foreachtypeofpolicy,adeductibleamountmustbespecified.Foranautomobilepolicy,thechoicesare$100and$250,whereasforahomeowner’spolicythechoicesare0,$100,and$200.Supposeanindividualwithbothtypesofpolicyisselectedatrandomfromtheagency’sfiles.Solution:letX=thedeductibleamountontheautopolicyY=thedeductibleamountonthehomeowner’spolicy.Thejointprobabilitytableis:DetermineP(Y≥100)=Definition:ThemarginalprobabilitymassfunctionsofXandY,denotedbypX(x)andPY(y),respectively,aregivenbyExample5.2Inexample5.1,determinethemarginalprobabilitymassfunctionsofXandY.Solution:ThejointprobabilitydensityfunctionfortwocontinuousrandomvariablesDefinition:LetXandYbetwocontinuousrandomvariables.Thenf(x,y)isthejointprobabilitydensityfunctionforXandYifforanytwo-dimensionalsetAInparticular,ifAisthetwo-dimensionalrectangle{(x,y):a≤x≤b,c≤y≤d},thenForf(x,y)tobeacandidateforajointpdf,itmustsatisfyf(x,y)≥0,andWecanthinkoff(x,y)asspecifyingasurfaceatheightf(x,y)abovethepoint(x,y)inathree-dimensionalcoordinatesystem.ThenisthevolumeunderneaththissurfaceandabovetheregionA,analogoustotheareaunderacurveintheone-dimensionalcase.ThisisillustratedinFigure5.1.Example5.3Abankoperatesbothadrive-upfacilityandawalk-upwindow.Onarandomlyselectedday,letX=theproportionoftimethatthedrive-upfacilityisinuseY=theproportionoftimethatthewalk-upwindowisinuseLetthejointpdfof(X,Y)isDefinition:ThemarginalprobabilitydensityfunctionsofXandY,denotedbyfX(x)andfY(y),respectively,aregivenbyExample5.4inexample5.3,determinethemarginalprobabilitydensityfunctionsofXandYSolution:Example5.5Anutcompanymarketscansofdeluxemixednutscontainingalmonds,cashews,andpeanuts.Supposethenetweightofeachcanisexactly11b,buttheweightcontributionpfeachtypeofnutisrandom.Becausethethreeweightssumto1,ajointprobabilitymodelforanytwogivesallnecessaryinformationabouttheweightofthethirdtype.LetX=theweightofalmondsinaselectedcanandY=theweightofcashews.Thejointpdffor(X,Y)is(1)DeterminetheprobabilityP((X,Y)∈A)(2)DeterminethemarginaldensityfunctionfX(x),fY(y)(1)DeterminetheprobabilityP((X,Y)∈A)(2)DeterminethemarginaldensityfunctionfX(x),fY(y)Solution:P213Exercise9Eachfronttireonaparticulartypeofvehicleissupposedtobefilledtoapressureof26psi.Supposetheactualairpressureineachtireisarandomvariable---XfortherighttireandYforthelefttire,withjointpdf WhatisthevalueofK? Whatistheprobabilitythatbothtiresareunderfilled? Whatistheprobabilitythatthedifferenceinairpressurebetweenthetwotiresisatmost2psi.? Determinethe(marginal)distributionofairpressureintherighttirealoneP213Exercise9Eachfronttireonaparticulartypeofvehicleissupposedtobefilledtoapressureof26psi.Supposetheactualairpressureineachtireisarandomvariable---XfortherighttireandYforthelefttire,withjointpdf(b)Whatistheprobabilitythatbothtiresareunderfilled?P213Exercise9Eachfronttireonaparticulartypeofvehicleissupposedtobefilledtoapressureof26psi.Supposetheactualairpressureineachtireisarandomvariable---XfortherighttireandYforthelefttire,withjointpdf(c)Whatistheprobabilitythatthedifferenceinairpressurebetweenthetwotiresisatmost2psi.?P213Exercise9Eachfronttireonaparticulartypeofvehicleissupposedtobefilledtoapressureof26psi.Supposetheactualairpressureineachtireisarandomvariable---XfortherighttireandYforthelefttire,withjointpdf(d)Determinethe(marginal)distributionofairpressureintherighttirealone?203030IndependentRandomVariablesDefinition:TworandomvariablesXandYaresaidtobeindependentifforeverypairofxandyvalues,Otherwise,XandYaresaidtobedependentThedefinitionsaysthattwovariablesareindependentiftheirjointpmforpdfistheproductofthetwomarginalpmf’sorpdf’sExample5.6IntheinsurancesituationofExample5.1and5.2,showthatrvXandYarenotindependentSolution:Example5.7Continuous5.5.Thejointpdfof(X,Y)isDeterminewhetherXandYareindependentrv’s?Solution:Example5.8Supposethatthelifetimesoftwocomponentsareindependentofoneanotherandthatthefirstlifetime,X1,hasanexponentialdistributionwithparameterλ1whereasthesecond,X2,hasanexponentialdistributionwithparameterλ2.ThenthejointpdfisLetλ1=1/1000andλ2=1/1200.Sothattheexpectedlifetimesare1000and1200hours,respectively.Theprobabilitythatbothcomponentlifetimesareatleast1500hoursisExerciseP2111Aservicestationhasbothself-seviceandfull-serviceislands.Oneachisland,thereisasingleregularunleadedpumpwithtwohoses.LetXdenotethenumberofhosesbeingusedontheself-serviceislandataparticulartime,andletYdenotethenumberofhosesonthefull-serviceislandinuseatthattime.ThejointpmfofXandYappearsintheaccompanyingtabulation(a)WhatisP(X=1andY=1)?(b)ComputeP(X≤1andY≤1)(c)Givenaworddescriptionoftheevent{X≠0andY≠0}andcomputetheprobabilityofthisevent.(d)ComputethemarginalpmfofXandofY.WhatisP(X≤1)?(e)AreXandYindependentrv’s?Explain? P(x,y) y x 0 1 2 0 0.10 0.04 0.02 1 0.08 0.20 0.06 2 0.06 0.14 0.30Exercise2Whenanautomobileisstoppedbyarovingsafetypatrol,eachtireischeckedfortirewear,andeachheadlightischeckedtoseewhetheritisproperlyaimed.LetXdenotethenumberofheadlightsthatneedadjustment,andletYdenotethenumberofdefectivetires.(a)IfXandYareindependentwithpx(0)=0.5,px(1)=0.3,px(2)=0.2,andpy(0)=0.6,py(1)=0.1px(2)=py(3)=0.3=0.05,py(4)=0.2,displaythejointpmfof(X,Y)inajointprobabilitytable.(b)ComputeP(X≤1andY≤1)fromthejointprobabilitytableandverifythatitequalstheproduceP(X≤1).P(Y≤1)(c)WhatisP(X+Y=0)(d)ComputeP(X+Y≤1)Exercise9Eachfronttireonaparticulartypeofvehicleissupposedtobefilledtoapressureof26psi.Supposetheactualairpressureineachtireisarandomvariable---XfortherighttireandYforthelefttire,withjointpdf WhatisthevalueofK? Whatistheprobabilitythatbothtiresareunderfilled? Whatistheprobabilitythatthedifferenceinairpressurebetweenthetwotiresisatmost2psi.? Determinethe(marginal)distributionofairpressureintherighttirealone AreXandYindependentrv’s?DEFINITION:LetXandYbetwocontinuousrv’swithjointpdff(x,y)andmarginalXpdffX(x).ThenforanyXvaluexwhichfX(x)>0,theconditionalprobabilitydensityfunctionofYgiventhatX=xisIFXandYarediscrete,replacingpdf’sinthisdefinitiongivenstheconditionalprobabilitymassfunctionofYwhenX=x.Example5.12Reconsiderthesituationofexample5.3and5.4letX=theproportionoftimethatthedrive-upfacilityisbusyY=theproportionoftimethatthewalk-upwindowisinuseLetthejointpdfof(X,Y)is DeterminetheconditionalpdfofYgiventhatX=0.8Solution:Example5.12Reconsiderthesituationofexample5.3and5.4letX=theproportionoftimethatthedrive-upfacilityisbusyY=theproportionoftimethatthewalk-upwindowisinuseLetthejointpdfof(X,Y)is(b)Theprobabilitythatthewalk-infacilityisbusyatmosthalfthetimegiventhatX=0.8Solution:Example5.12Reconsiderthesituationofexample5.3and5.4letX=theproportionoftimethatthedrive-upfacilityisbusyY=theproportionoftimethatthewalk-upwindowisinuseLetthejointpdfof(X,Y)is(c)Theexpectedproportionoftimethatthewalk-infacilityisbusygiventhatX=0.8Solution:Exercise6LetXdenotethenumberofbrandXVCRssoldduringaparticularweekbyacertainstore.ThepmfofXisSixtypercentofallcustomerswhopurchasebrandXVCRsalsobuyanextendedwarranty.LetYdenotethenumberofpurchasersduringthisweekwhobuyanextendedwarranty. WhatisP(X=4,Y=2)? CalculateP(X=Y) DeterminethejointpmfofXandYandthenthemarginalpmfofY. X 0 1 2 3 4 4PX(x) 0.1 0.2 0.3 0.25 0.15Exercise17Anecologistwishestoselectapointinsideacircularsamplingregionaccordingtoauniformdistribution(inpracticethiscouldbedonebyfirstselectingadirectionandthenadistancefromthecenterinthatdirection).LetX=thexcoordinateofthepointselectedandY=theycoordinateofthepointselected.Ifthecircleiscenteredat(0,0)andhasradiusR,thenthejointpdfofXandYis WhatistheprobabilitythattheselectedpointiswithinR/2ofthecenterofthecircularregion? WhatistheprobabilitythatbothXandYdifferfrom0byatmostR/2? WhatisthemarginalpdfofX?ofY?areXandYindependent?5.2ExpectedValues,Covariance,andCorrelationPROPOSITION:LetXandYbejointlydistributedrv’swithpmfp(x,y)accordingtowhetherthevariablesarediscreteorcontinuous.Thentheexpectedvalueofafunctionh(X,Y),denotedbyE[h(X,Y)]orμh(X,Y),isgivenbyE[h(X,Y)ifXandYarecontinuousifXandYarediscreteP220Exercise22Aninstructorhasgivenashortquizconsistingoftwoparts.Forarandomlyselectedstudent,letX=thenumberofpointsearnedonthefirstpartandY=thenumberofpointsearnedonthesecondpart.SupposethatthejointpmfofXandYisgivenintheaccompanyingtable.y(A)ComputeEXandEY x 0 5 10 15 0 0.02 0.06 0.02 0.10 5 0.04 0.15 0.20 0.10 10 0.01 0.15 0.14 0.01ExampleReconsiderthesituationofexample5.3and5.4letX=theproportionoftimethatthedrive-upfacilityisbusyY=theproportionoftimethatthewalk-upwindowisinuseLetthejointpdfof(X,Y)isDeterminetheexpectedvalueofXandY,thatistocalculateEXandEYExample5.13Fivefriendshavepurchasedticketstoacertainconcert.Iftheticketsareforseats1-5inaparticularrowandtheticketsarerandomlydistributedamongthefive,whatistheexpectednumberofseatsseparatinganyparticulartwoofthefive?LetXandYdenotetheseatnumberofthefirstandsecondindividuals,respectively.Possible(X,Y)pairsare{(1,2),(1,3),…,(5,4)},andthejointpmfof(X,Y)isThenumberofseatsseparatingthetwoindividualsish(X,Y)=|X-Y|-1.Theaccompanyingtablegivesh(x,y)foreachpossible(x,y)pair.ThusExample5.14InExample5.5,thejointpdfoftheamountXofalmondsandamountYofcashewsina1-1bcanofnutswasIf1lbofalmondscoststhecompany$1.00,1lbofcashewscosts$1.50,and1lbofpeanutscosts$.50,thenthetotalcostofthecontentsofacanish(X,Y)=(1)X+(1.5)Y+(.5)(1-X-Y)=.5+.5X+YTheexpectedtotalcostisCovarianceWhentworandomvariablesXandYarenotindependent,itisfrequentlyofinteresttoassesshowstronglytheyarerelatedtooneanother;DEFINITON:Thecovariancebetweentworv’sXandYisTherationaleforthedefinitionisasfollows:SupposeXandYhaveastrongpositiverelationshiptooneanother,bywhichwemeanthatlargevalueofXtendtooccurwithlargevalueofYandsmallvalueofXwithsmallvaluesofY.Thenmostoftheprobabilitymassordensitywillbeassociatedwith(x-μX)and(y-μY)eitherbothpositive(bothXandYabovetheirrespectivemeans)orbothnegative,sotheproduct(x-μX)(y-μY)willtendtobepositive.Thus,forastrongpositiverelationship,Cov(X,Y)shouldbequitepositive.Forastrongnegativerelationship,thesignsof(X-μX)and(Y-μY)willtendtobeopposite,yieldinganegativeproduct.Thus,forastrongnegativerelationship,Cov(X,Y)shouldbequitenegative.IfXandYarenotstronglyrelated,positiveandnegativeproductswilltendtocanceloneanother,yieldingacovariancenear0.Example5.15Thejointandmarginalpmf’sforX=automobilepolicydeductibleamountandY=homeownerpolicydeductibleamountinExample5.1wereFromwhichμX=∑xpX(x)=175andμY=125.Therefore.ThefollowingshortcutformulaforCov(X,Y)simplifiesthecomputations.PROPOSITION:Example5.16Thejointandmarginalpdf’sX=amountofandY=amountofcashewswereWithfY(y)obtainedbyreplacingxbyyinfX(x).ItiseasilyverifiedthatμX=μY=2/5,andThusCov(X,Y)=2/15-(2/5)2=2/15-4/25=-2/75.Anegativecovarianceisreasonableherebecausemorealmondsinthecanimpliesfewercashews.CorrelationDEFINITONThecorrelationcoefficientofXandY,denotedbyCorr(X,Y),ρXYorjustρ,isdefinedbyExample5.17ItiseasilyverifiedthatintheinsuranceproblemE(X2)=36,250,σ2x=36,250-(175)2=5625,σX=75,E(Y2)=22,500,σ2y=6875,andσY=82.92.Thisgivesρ=1875/(75)(82.92)=.301FromwhichμX=∑xpX(x)=175andμY=125.Therefore.PROPOSITION1.IfaandcareeitherbothpositiveorbothnegativeCorr(aX+b,cY+d)=Corr(X,Y)2.Foranytworv’sXandY,-1≤Corr(X,Y)≤1.Proof:wesetThatisPROPOSITION1.IfXandYareindependent,thenρ=0,butρ=0doesnotimplyindependence.2.ρ=1or-1ifY=aX+bforsomenumbersaandbwitha≠0,ThispropositionsaysthatρisameasureofthedegreeoflinearrelationshipbetweenXandY,andonlywhenthetwovariablesareperfectlyrelatedinalinearmannerwillρbeaspositiveornegativeasitcanbe.Aρlessthan1inabsolutevalueindicatesonlythattherelationshipisnotcompletelylinear,buttheremaystillbeaverystrongnonlinearrelation.Also,ρ=0doesnotimplythatXandYareindependent,butonlythatthereiscompleteabsenceofalinearrelationship.Whenρ=0,XandYaresaidtobeuncorrelated.Twovariablescouldbeuncorrelatedyethighlydependentbecausethereisastrongnonlinearrelationship,sobecarefulnottoconcludetoomuchfromknowingthatρ=0Example5.18LetXandYbediscreterv’swithjointpmfThispointsthatreceivepositiveprobabilitymassareidentifiedonthe(x,y)coordinatesysteminFigure5.5.ItisevidentfromthefigurethatthevalueofXiscompletelydeterminedbythevalueofYandviceversa,sothetwovariablesarecompletelydependent.However,bysymmetryμX=μY=0andE(XY)=(-4)1/4+(-4)1/4+(4)1/4=0,soCov(X,Y)=E(XY)-μXμY=0andthusρXY=0.Althoughthereisperfectdependence,thereisalsocompleteabsenceofanylinearrelationship!P220Exercise24Sixindividuals,includingAandB,takeseatsaroundacirculartableinacompletelyrandomfashion.Supposetheseatsarenumbered1,….,6.LetX=A’sseatnumberandY=B’sseatnumber.IfAsendsawrittenmessagearoundthetabletoBinthedirectioninwhichtheyareclosest,howmanyindividuals(includingAandB)wouldyouexpecttohandlethemessage?Exercise26Considerasmallferrythatcanaccommodatecarsandbuses.Thetollforcarsis$3,andthetollforbusesis$10.LetXandYdenotethenumberofcarsandbuses,respectively,carriedonasingletrip.SupposethejointdistributionofXandYisasgiveninthetable7.Computetheexpectedrevenuefromasingletrip. yx 0 1 0 0.025 0.015 0.010 1 0.050 0.030 0.020 2 0.125 0.075 0.050 3 0.150 0.090 0.060 4 0.1 0.06 0.04 5 0.050 0.030 0.020Exercise27AnnieandAlviehaveagreedtomeetforlunchbetweennoon(0:00P.M.)and1:00P.M.DenoteAnnie’sarrivaltimebyX,Alvie’sbyY,andsupposeXandYareindependentwithpdf’sWhatistheexpectedamountoftimethattheonewhoarrivesfirstmustwaitfortheotherperson?**TheDistributionoffunctionsofRandomVariables**AfunctionofrandomvariableisdefinedbyY=f(X),whereXisarandom,fisafunction. FordiscretevariableX,iftheprobabilityfunctionisThentheprobabilityfunctionofY=f(X)isgivenbySupplementcontents X x1 x2 …… xn P p1 p2 …… pn Y=f(X) f(x1) f(x2) …… f(xn) P p1 p2 …… pnExample1:TheprobabilityfunctionofdiscreterandomvariableXisgivenbyDeterminetheprobabilityfunctionof(1)X+2,(2)X2,(3)-X+1Solution: X -2 -1 0 2 4 P 1/8 1/4 1/8 1/6 1/3 X+2 0 1 2 4 6 P 1/8 1/4 1/8 1/6 1/3 X2 0 1 4 16 P 1/8 1/4 7/24 1/3 -X+1 3 2 1 -1 -3 P 1/8 1/4 1/8 1/6 1/3 ForcontinuousvariableX,ifthedensityfunctionisf(x),andY=g(X),thedistributionfunctionofYisgivenbyWhereg-1(y)istheinversefunctionofg(y)ThenthedensityfunctionofYiscapturedasfollow:Example2:LetXbeuniformlydistributedon(a,b).IdentifythedistributionofSolution:thedensityfunctionofXisThedistributionfunctionofYisThedensityfunctionofYisSolution:thedensityfunctionofXisThedistributionfunctionofYisSo,ify<0,thenF(y)=0ThedensityfunctionofYisExample4:SupposeXhavestandardnormaldistribution,determinethedensityfunctionofY,whenYis Y=eX,Y=2X2+1,Y=|X|Solution:thedensityfunctionofXis(1)TheprobabilityfunctionisSo,ify<0,thenF(y)=0Therefore,thedensityfunctionofYisSimilarly(2)Thedensityfunctionis(3)ThedensityfunctionisExercise1:LetXbeuniformlydistributedon(0,1).IdentifythedistributionofExercise2:LetXhavetheexponentialdistributionwithmean1.DeterminethedensityfunctionofY,whenYis(1)Y=eX(2)Y=X2**TheCumulativeDistributionFunctionof(X,Y)**ForcontinuousrandomvectorX=(X1,X2),weconcernitsjointdistributionfunctionanditsjointdensityfunction. BivariatedistributionfunctionAbivariatedistributionforarandomvector(X,Y)withtwocomponentsisanassignmentofrelativefrequenciestothevaluesandcollectionsofvaluesofX,Y.ThisassignmentiscompletelydeterminedbytheunderlyingexperimentandthedefinitionofX,Y,anditcanbesummarizedbyabivariatedistributionfunction,whichisgivenby:(x,y) BivariatedensityfunctionThecontinuousrandomvectorX=(X,Y),itsdensityfunctionisgivenbyf(x,y),satisfying: PropertiesofDistributionFunctionExample5Thedistributionfunctionof(X,Y)isgivenbyDeterminethedensityfunction.Solution:Example6Thedensityfunctionofvector(X,Y)isgivenbyDetermine:(1)theconstantc(2)thedistributionfunctionF(x,y)(3)theprobabilityP{(X,Y)∈G}x+y=1G11Solution:(1)(2)(3)Example7:Thedensityfunctionforthevector(X,Y)whosecomponentsarepositiverandomvariablesisgivenbyDeterminetheprobabilitythatY>X.Solution: MarginalDistributionsFromabivariatedensityfunction,itispossibletorecapturetheunivariatedistributionsforarandomvector’sscalarcomponents.Namely,forvector(X,Y),ifwewanttoknowtheprobabilitydistributionofrandomvariableXortheprobabilitydistributionofrandomvariableY,wecancaptureasfollow:FX(x),FY(y)aredefinedasthemarginaldistributionfunctionofXandY,respectively.ThesepairoffunctionsarethemarginaldensityfunctionofXandY,respectively.Example8:ThejointdensityofXandYisgivenbyDeterminethemarginaldensityfunctions.Solution:Example9:Supposethedensityfunctionofrandomvector(X,Y)isgivenby:Determine(1)thejointdistributionfunctionF(x,y)(2)themarginaldensityfunctionsfX(x),fY(y)(3)theprobabilityofP{X+Y>1},P{Y>X}Solution:(1)a.whenx<0,y<0,F(x,y)=0b.whenx>1andy>2,F(x,y)=1c.when0≤x≤1,0≤y≤2d.when0≤x≤1,y>2e.whenx>10≤y≤2Sothejointdistributionfunctionis(2)Themarginaldensityfunctionare:(3)Computertheprobability121x+y=1D121y=xG00Exercise1:Theprobabilitydensityfunctionforthecontinuousrandomvector(X,Y)isgivenby DeterminethemarginaldensityfunctionsfX(x),fY(y) DeterminethejointdistributionfunctionF(x,y)Exercise2:Theprobabilitydensityfunctionforthecontinuousrandomvector(X,Y)isgivenby DeterminethemarginaldensityfunctionsfX(x),fY(y) DeterminethejointdistributionfunctionF(x,y) CalculatetheprobabilitythatX>2YExercise3:Thejointdensityforamixedrandomvector(N,X)isgivenbyDeterminetheprobabilitythatN<2andtheprobabilitythatX>4Solution:ThemarginalprobabilitydistributionfunctionisThus,theprobabilitythatN<2isThemarginaldensityfunctionforXisgivenbyThus,theprobabilitythatX>4is5.3StatisticsandTheirDistributionsTheobservationsinasinglesampleweredenotedinChapter1byx1,x2,…xn.Considerselectingtwodifferentsamplesofsizenfromthesamepopulationdistribution.Thexi’sinthesecondsamplewillvirtuallyalwaysdifferatleastabitfromthoseinthefirstsample.Forexample,afirstsampleofn=3carsofaparticulartypemightresultinfuelefficienciesx1=30.7,x2=29.4,x3=31.1,whereasasecondsamplemaygivex1=28.8,x2=30.0,andx3=31.1.Beforeweobtaindata,thereisuncertaintyaboutthevalueofeachxi.Becauseofthisuncertainty,beforethedatabecomesavailablewevieweachobservationasarandomvariableanddenotethesamplebyX1,X2,…,Xn(uppercaselettersforrandomvariables).Thisvariationinobservedvaluesinturnimpliesthatthevalueofanyfunctionofthesampleobservations,suchasthesamplemean,samplestandarddeviation,orsamplefourthspread,alsovariesfromsampletosample.Thatis,priortoobtainingx1,…,xn,thereisuncertaintyastothevalueofx,thevalueofs,andsoon.DEFINITONAstatisticisanyquantitywhosevaluecanbecalculatedfromsampledata.Priortoobtainingdata,thereisuncertaintyastowhatvalueofanyparticularstatisticwillresult.Therefore,astatisticisarandomvariableandwillbedenotedbyanuppercaseletter;alowercaseletterisusedtorepresentthecalculatedorobservedvalueofthestatisti