概率論與數(shù)理統(tǒng)計：Chapter 5 Joint Probability Distributions and Random Samples

5JointProbabilityDistributionsandRandomSamples5.1JointlyDistributedRandomVariables5.2ExpectedValues,Covariance,andCorrelation5.3StatisticsandTheirDistributions5.4TheDistributionoftheSampleMean5.5TheDistributionofaLinearCombination

SupplementaryExercisesBibliographyInthischapter,wefirstdiscussprobabilitymodelsforthejointbehaviorofseveralrandomvariables,puttingspecialemphasisonthecaseinwhichthevariablesareindependentofoneanother.Wethenstudyexpectedvaluesoffunctionsofseveralrandomvariables,includingcovarianceandcorrelationasmeasuresofthedegreeofassociationbetweentwovariables.Introduction5.1JointlyDistributedRandomVariables

Therearemanyexperimentalsituationsinwhichmorethanonerandomvariable(rv)willbeofaninvestigator.Weshallfirstconsiderjointprobabilitydistributionsfortwodiscreterv’s,thenfortwocontinuous.TheJointProbabilityMassFunctionforTwoDiscreteRandomVariables

Theprobabilitymassfunction(pmf)ofasinglediscretervXspecifieshowmuchprobabilitymassisplacedoneachxvalue.Thejointpmfoftwodiscreterv’sXandYdescribeshowmuchprobabilitymassisplacedoneachpossiblepairofvalues(x,y).Example2Whentwofairdicearerolledinanhonestmanner,letY=(Y1,Y2),Y1----thenumbershownonfirstdice,Y2---thenumbershownonseconddice.Example3Aboxhassixtickets,labeledfrom1to6.Twoticketsareselectedfromtheboxbysamplingwithoutreplacement.LetX=(X1,X2),X1,X2,respectively,denotethelabelsofthefirstandthesecondticketsoselected.Example1wewanttostudythedistributionofpeople’sheightandweight,letX=(X1,X2),X1---theheightofpeople,X2---theweightofpeople.LetAbeanysetconsistingofpairsof(x,y)values.ThentheprobabilityisobtainedbysummingthejointpmfoverpairsinA:DEFINITION

LetXandYbetwodiscreterv’sdefinedonthesamplespaceSofanexperiment.Thejointprobabilitymassfunction

p(x,y)isdefinedforeachpairofnumbers(x,y)byp(x,y)=P(X=xandY=y)Afunctionp(x,y)canbeusedasajointpmfprovidedthatforallxandyand

Example

Whentwofairdicearerolledinanhonestmanner,letY=(Y1,Y2),Y1----thenumbershownonfirstdice,Y2---thenumbershownonseconddice.Y2Y112345611/361/361/361/361/361/3621/361/361/361/361/361/3631/361/361/361/361/361/3641/361/361/361/361/361/3651/361/361/361/361/361/3661/361/361/361/361/361/36X=thedeductibleamountonautopolicyandY=thedeductibleamountonhomeowner’spolicy.Example5.1p(x,y)X10025002000.05.15.30YDEFINITIONThemarginalprobabilitymass

functionsofXandofY,denotedbyand,respectively,aregivenbyThepmfofoneofthevariablesaloneisobtainedbysummingp(x,y)overvaluesoftheothervariable.Theresultiscalledamarginalpmfbecausewhenthep(x,y)valuesappearinarectangulartable,thesumsarejustmarginal(roworcolumn)totals.Y2Y1123456Pi.11/361/361/361/361/361/3621/361/361/361/361/361/3631/361/361/361/361/361/3641/361/361/361/361/361/3651/361/361/361/361/361/3661/361/361/361/361/361/36p.jExample5.2(Example5.1continued)

ThepossibleXvaluesarex=100andx=250,socomputingrowtotalsinthejointprobabilitiestableyieldsThemarginalpmfofXisthenand

Similarly,themarginalpmfofYisobtainedfromcolumntotalsassoasbefore.ExampleTherearethreewhiteballsandthreeredballs,putthemintothreeboxes,theboxesarelabeled1,2,3.LetXdenotethenumberofwhiteballsinfirstbox,Ydenotethenumberofredballsinsecondbox.Determinejointprobabilitydistributionof(X,Y)XY0123012364/72996/72948/7298/72996/72916/818/8112/72948/7298/814/816/7298/72912/7296/7291/729Example5Putthreeballs(theseballsareundistinguished)intothreeboxes,whichlabeled1,2,3.LetXdenotethenumberofballsinthefirstbox,Ydenotethenumberofballsinthesecondbox.Determinethejointprobabilitydistributionof(X,Y).XY012301/271/91/91/2711/92/91/9021/91/90031/27000Exercise1:Twocardsaredrawnfromaspecialdeckconsistingoftwoheartsandtwodiamonds,andtheyareplacedfacedowninfrontofus.Thetwocardsarethenturnedoverandtheirsuitsareobserved.LetX1andX2bedefinedasfollows:X1X20101/61/311/31/6DeterminethejointprobabilitydistributionofX=(X1,X2)TheJointProbabilityDensityFunctionforTwoContinuousRandomVariablesDEFINITIONLetXandYbecontinuousrv’s.Thenf(x,y)isthejointprobabilitydensityfunctionforXandYifforanytwo-dimensionalsetAInparticular,ifAistwo-dimensionalrectanglethenForf(x,y)tobeacandidateforajointpdf,itmustsatisfy.ThenisthevolumeunderneaththissurfaceandabovetheregionA,analogoustotheareaunderacurveintheone-dimensionalcase.ThisisillustratedinFigure5.1.f(x,y)xySurfacef(x,y)A=Shadedrectangle

Wecanthinkoff(x,y)asspecifyingasurfaceatheightf(x,y)abovethepoint(x,y)inathree-dimensionalcoordinatesystem.Toverifythatthisisalegitimatepdf,notethatandExample5.3Supposethejointpdfof(X,Y)isgivenbyExampleThepdfofvector(X,Y)isgivenbyDetermine:(1)theconstantc

(2)theprobability

P{(X,Y)∈G}G11x+y=1Solution:(1)(2)G11x+y=1Example:Thepdfforthevector(X,Y)whosecomponentsarepositiverandomvariablesisgivenbyDeterminetheprobabilitythatY>X.Solution:

DefinitionThemarginalprobabilitydensityfunctionsofXandY,denotedbyand,respectively,aregivenbyAswithjointpmf’s,fromthejointpdfofXandY,eachofthetwoMarginaldensityfunctionscanbecomputed.Example:ThepdfofXandYisgivenbyDeterminethemarginaldensityfunctions.Solution:Example:Supposethepdfofrandomvector(X,Y)isgivenby:Determine(1)themarginaldensityfunctionsfX(x),fY(y)

(2)theprobabilityofP{X+Y>1},P{Y>X}(1)Themarginaldensityfunctionare:SoSimilarlySolution:(2)Computertheprobability121x+y=1D00121y=xG0Exercise1:Theprobabilitydensityfunctionforthecontinuousrandomvector(X,Y)isgivenbyDeterminethemarginaldensityfunctionsfX(x),fY(y)Exercise2:Theprobabilitydensityfunctionforthecontinuousrandomvector(X,Y)isgivenbyDeterminethemarginaldensityfunctionsfX(x),fY(y)CalculatetheprobabilitythatX>2YIndependentrandomvariables

Inchapter2wepointedoutthatonewayofdefiningindependenceoftwoeventsistosaythatAandBareindependentif.Wenowuseananalogousdefinitionfortheindependenceoftworv’s.orDEFINITIONTworandomvariablesXandYaresaidtobeindependentifforeverypairofxandyvaluesotherwiseisnotsatisfiedforall(x,y),thenXandYaresaidtobedependent.whenXandYarecontinuouswhenXandYarediscreteExample5.6IntheinsurancesituationofExample5.1and5.2,soXandYarenotindependent.Example5.8Supposethatthelifetimesoftwocomponentsareindependentofoneanotherandthatthefirstlifetime,,hasanexponentialdistributionwithparameterwhereasthesecond,,hasanexponentialdistributionwithparameter.ThenthejointpdfisLet=1/1000and=1/1200,sothattheexpectedlifetimesare1000hoursand1200hours,respectively.Theprobabilitythatbothcomponentlifetimesareatleast1500hoursisExpectedValues,Covariance,andCorrelation5.2PROPOSITION

LetXandYbejointlydistributedrv’swithpmfp(x,y)accordingtowhetherthevariablesarediscreteorcontinuous.Thentheexpectedvalueofafunctionh(X,Y),denotedbyE[h(X,Y)]orμh(X,Y),isgivenbyE[h(X,Y)]Example5.13

Fivefriendshavepurchasedticketstoacertainconcert.Iftheticketsareforseats1-5inaparticularrowandtheticketsarerandomlydistributedamongthefive,whatistheexpectednumberofseatsseparatinganyparticulartwoofthefive?LetXandYdenotetheseatnumberofthefirstandsecondindividuals,respectively.Possible(X,Y)pairsare{(1,2),(1,3),…,(5,4)},andthejointpmfof(X,Y)is

Thenumberofseatsseparatingthetwoindividualsish(X,Y)=|X-Y|-1.Theaccompanyingtablegivesh(X,Y)foreachpossible(x,y)pair.--h(x,y)Y15432X12345--30120--0121001210--03210--Thus,CovarianceWhentworandomvariablesXandYarenotindependent,itisfrequentlyofinteresttoassesshowstronglytheyarerelatedtooneanother;DEFINITONThe

covariancebetweentworv’sXandYisExample5.15Thejointandmarginalpmf’sforX=automobilepolicydeductibleamountandY=homeownerpolicydeductibleamountinExample5.1wereyp(x,y)x10025002000.05.15.30250.5xpX(x)100250.5.5ypY(y)100.250.25FromwhichμX=∑xpX(x)=175andμY=125.Therefore.PROPOSITION

ThefollowingshortcutformulaforCov(X,Y)simplifiesthecomputations.IftheXiareindependent,then,andwehaveanothercorollary.

IftheXiareindependent,

thenExample5.16

Thejointandmarginalpdf’sX=amountofandY=amountofcashewswereWithfY(y)obtainedbyreplacingxbyyinfX(x).ItiseasilyverifiedthatμX=

μY=2/5,and

ThusCov(X,Y)=2/15-(2/5)2=2/15-4/25=-2/75.

ExampleSupposearandomvariableXBin(12,0.5),andanotherrandomvariableYN(0,1),COV(X,Y)=-1,findvarianceandcovarianceofV=4X+3Y+1andW=-2X+4YExample:LetXhavethebinomialdistributionwithparametersn,p,determineV(X).Solution:becauseabinomialdistributionisnBernoullidistribution,andaBernoullidistributionisXi01PqpBecauseX1,X2,…,Xnareindependent,thenExample5.17ItiseasilyverifiedthatintheinsuranceproblemofExample5.15,E(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)=.301CorrelationDEFINITION

ThecorrelationcoefficientofXandY,denotedbyCorr(X,Y),orjust,isdefinedbyPROPOSITION1.Ifaandcareeitherbothpositiveorbothnegative

Corr(aX+b,cY+d)=Corr(X,Y)2.Foranytworv’sXandY,-1≤Corr(X,Y)≤1.PROPOSITION1.IfXandYareindependent,thenρ=0,butρ=0doesnotimplyindependence.2.ρ=1or–1iffY=aX+bforsomenumbersaandbwitha≠0Example5.18

LetXandYbediscreterv’swithjointpmfHowever,thetwovariablesarecompletelydependent.Althoughthereisperfectdependence,thereisalsocompleteabsenceofanylinearrelationship!5.3

StatisticsandTheirDistributions

TheobservationsinasinglesampleweredenotedinChapter1byx1,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.DEFINITIONAstatisticisanyquantitywhosevaluecanbecalculatedfromsampledata.Priortoobtainingdata,thereisuncertaintyastowhatvalueofanyparticularstatisticwillresult.astatisticisarandomvariableandwillbedenotedbyanuppercaseletter;

alowercaseletterisusedtorepresentthecalculatedorobservedvalueofthestatistic.RandomSamplesTheprobabilitydistributionofanyparticularstatisticdependsnotonlyonthepopulationdistribution(normal,uniform,etc.)andthesamplesizenbutalsoonthemethodofsampling.DEFINITION

Therv’sX1,X2,…,Xn

aresaidtoforma(simple)randomsampleofsizenif1.TheXi’sareindependentrv’s.2.EveryXihasthesameprobabilitydistribution.DerivingtheSamplingDistributionofaStatistic

Probabilityrulescanbeusedtoobtainthedistributionofastatisticprovidedthatitisa“fairlysimple”functionoftheXi’sandeithertherearerelativelyfewdifferentXvaluesinthepopulationorelsethepopulationdistributionhasa“nice”form.Thesecondmethodofobtaininginformationaboutastatistic’ssamplingdistributionistoperformasimulationexperiment.Thismethodisusuallyusedwhenaderivationviaprobabilityrulesistoodifficultorcomplicatedtobecarriedout.Suchanexperimentisvirtuallyalwaysdonewiththeaidofacomputer.Thefollowingcharacteristicsofanexperimentmustbespecified:Thestatisticofinterest(,S,aparticulartrimmedmean,etc.)2.Thepopulationdistribution(normalwithμ=100andσ=15,uniformwithlowerlimitA=5andupperlimitB=10,etc)SimulationExperiments(omit)3.Thesamplesizen(e.g.,n=10orn=50)4.Thenumberofreplicationk(e.g.,k=500)Thenuseacomputertoobtainkdifferentrandomsamples,eachofsizen,fromthedesignatedpopulationdistribution.Foreachsuchsample,calculatethevalueofthestatisticandconstructahistogramofthekcalculatedvalues.Thishistogramgivestheapproximatesamplingdistributionofthestatistic.Thelargerthevalueofk,thebettertheapproximationwilltendtobe(theactualsamplingdistributionemergesask→∞).Inpractice,k=500or1000isusuallyenoughifthestatisticis“fairlysimple”.Example5.23Considerasimulationexperimentinwhichthepopulationdistributionisquiteskewed.Figure5.12showsthedensitycurveofacertaintypeofelectroniccontrol(thatisactuallyalognormaldistributionwithE(ln(X))=3andV(ln(X)))=.4).Againthestatisticofinterestisthesamplemean.Theexperimentutilized500replicationsandconsideredthesamefoursamplesizesasinExample5.22.TheresultinghistogramsAlongwithanormalprobabilityplotfromMINITABforthe500valuesbasedonn=30areshowninFigure5.13).f(x)x0255075.01.03.02Figure5.12DensitycurveforthesimulationexperimentofExample5.23[E(X)=μ=21.7584,V(X)=σ2=82.1449]PROPOSITION5.4TheDistributionoftheSampleMeanIfthesamplesizeincreaseston=100,E(X)isunchanged,butσX=500,halfofitspreviousvalue(thesamplesizemustbequadrupledtohalvethestandarddeviationofX).Example5.24Inanotchedtensilefatiguetestonatitaniumspecimen,theexpectednumberofcyclestofirstacousticemission(usedtoindicatecrackinitiation)isμ=28,000,andthestandarddeviationofthenumberofcyclesisσ=5000.LetX1,X2,…,X25bearandomsampleofsize25,whereeachXiisthenumberofcyclesonadifferentrandomlyselectedspecimen.ThentheE(X)=μ=28,000,andtheexpectedtotalnumberofcyclesforthe25specimensisE(T0)=nμ=25(28,000)=700,000.ThestandarddeviationsofXandT0areTheCaseofaNormalPopulationDistributionLookingbacktothesimulationexperimentofExample5.22,weseethatwhenthepopulationdistributionisnormal,eachhistogramofXvaluesiswellapproximatedbyanormalcurve.Thepreciseresultfollows.PROPOSITIONForNormalpopulation,Example5.25Thetimethatittakesarandomlyselectedratofacertainsubspeciestofinditswaythroughamazeisanormallydistributedrvwithμ=1.5minandσ=.35min.Supposefiveratsareselected.THEOREMTheCentralLimitTheorem(CLT)TheCentralLimitTheoremForIfn>30,theCLTcanbeusedExample5.26Whenabatchofacertainchemicalproductisprepared,theamountofaparticularimpurityinthebatchisarandomvariablewithmeanvalue4.0gandstandarddeviation1.5g.If50batchesareindependentlyprepared,whatisthe(approximate)probabilitythatthesampleaverageamountofimpurityisbetween3.5and3.8g?Accordingtotheruleofthumbtobestatedshortly,n=50islargeenoughfortheCLTtobeapplicable.thenhasapproximatelyanormaldistributionwithmeanvalueandsoOtherApplicationsoftheCentralLimitTheorem(omit)TheCLTcanbeusedtojustifythenormalapproximationtothebinomialdistributiondiscussedinChapter4.PROPOSITIONLetX1,X2,…,Xn

bearandomsamplefromadistributionforwhichonlypositivevaluesarepossible[P(Xi>0)=1].Thenifnissufficientlylarge,theproductY=X1X2…..Xnhasapproximatelyalognormaldistribution.Toverifythis,notethat

ln(Y)=ln(X1)+ln(X2)+….+ln(Xn)TheDistributionofaLinearCombination5.5DefinitionPROPOSITIONWhetherornottheXi’sareindependent,IfX1,X2,…,Xnareindependent,3.ForanyX1,X2,…,Xn,andExample5.28Agasstationsellsthreegradesofgasoline:regularunleaded,extraunleaded,andsuperunleaded.Thesearepricedat$1.20,$1.35,and$1.50pergallon,respectively.LetX1,X2andX3denotetheamountsofthesegradespurchased(gallon)onaparticularday.SupposetheXi’sareindependentwithμ1=1000,μ