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StochasticModelingofHeatTransferthroughHeterogeneousMaterialsStevenBrillDepartmentofAerospaceandMechanicalEngineeringUniversityofNotreDamePreparedfor:AME48491UndergraduateResearchAdvisors:Dr.
JosephM.
Powers,Dr.
SamuelPaolucci,Dr.
WaadSubberJune3,20151AbstractHeterogeneousmaterialscontainuncertaintyintheirdegreeofmixing;thisaectsthethermaldiusivity.
Inordertoquantifytheconsequencesoftheuncertaintyinmaterialpropertiesonquantitiesofinterest,suchastemperature,theonedimensionalheatequationwasmodeledusingtheMonteCarlomethod,intrusivepolynomialchaos,andnon-intrusivepolynomialchaos.
TheMonteCarlomethodcanbecomputationallyexpensive.
Inordertoreducethecomputationaltime,so-calledpolynomialchaosmethodswereused.
ItwasshownthattheintrusivepolynomialchaosmethodisabletoapproximatesolutionstothestochasticheatequationforGaussiandistributionsofthethermaldiusivity.
Itwasalsoshownthatthenon-intrusivemethodproducesasolutionthatconvergestotheMonteCarlosolution.
TheMonteCarlomethodtook4.
27hoursofcomputationtoachieveanaltemperaturewithameanof477andastandarddeviationof40;incontrastthenon-intrusivepolynomialchaosmethodachievednearlyidenticalresultsin1.
54seconds.
2Contents1Introduction52HeatEquation53Method63.
1MonteCarlo73.
2IntrusivePolynomialChaos83.
3Non-IntrusivePolynomialChaos94Results104.
1MonteCarloMethod104.
2IntrusivePolynomialChaos124.
3Non-IntrusivePolynomialChaos145DiscussionandConclusions156FutureWork187AppendixA198AppendixB229AppendixC2310AppendixD2610.
1IntroductiontoProbability2610.
1.
1RandomVariables2610.
1.
2DiscreteandContinuousRandomVariables2710.
1.
3ProbabilityDistributions2810.
2MathematicalExpectation3110.
2.
1ExpectedValue3110.
2.
2Moments3210.
2.
3Chebyshev'sInequality3510.
2.
4LawofLargeNumbers3510.
2.
5Miscellaneoustopics3610.
3SpecialProbabilityDistributions37310.
3.
1GaussianDistribution3710.
3.
2PoissonDistribution3810.
3.
3TheCentralLimitTheorem4010.
3.
4UniformDistribution4110.
4SamplingTheory4110.
4.
1Basics4141IntroductionThegoalofcomputationalscienceistodevelopmodelsthatpredictphenomenaobservedinnature.
However,thesemodelsareoftenbasedonparametersthatareuncertain.
Forexample,whenmakingaweatherprediction,itisimpossibletoknowtheentiretemperatureeldintheatmospheretopredictthecreationofclouds.
Uncertaintyintheinitialproblemformulationaectsfuturecalculationsandhasanimpactonthenalsolution.
Hence,itisnecessarytounderstandhowtheuncertaintymanifestsitselfinthesolution,thatisthefocusoftheeldofuncertaintyquantication(UQ).
InUQsomecommonmetricsaretheprobabilitydensityfunctionofthesolution,themeanandvarianceofthesolution,andthecondenceintervalofthesolution[1].
Oneclassofproblemsthatcontainsuncertaintyisthestudyofheterogeneousmaterials.
Aheterogeneousmaterialisamaterialmadeofdierentmaterials.
Heterogeneousmaterialsarecommonlyfoundindailylife.
Forexample,Swisscheeseisaheterogeneousmaterialbecauseitisamixtureofcheeseandvoids.
Whensolvingaproblemwithaheterogeneousmaterial,itisuncertainhowthematerialismixed.
Forexample,ifoneweretoselectarandompointintheSwisscheese,itisuncertainwhetheronewouldselectcheeseoravoid.
Thedegreeofmixtureofthematerialswillalsoaectthebulkmaterialpropertiesofthebody.
Thematerialpropertieswillmorecloselyresemblethematerialinhigherconcentration.
However,eveniftwobodieshadthesameconcentrationsofmaterials,itispossiblethatthematerialswouldbemorepoorlymixedinoneofthebodiesthatwouldalsoaectthematerialproperties.
Insummary,uncertaintyintheheterogeneityinamaterialaectsthematerialproperties,thatwillaectlaterpredictions.
Hencetheresultsoftheuncertaintyduetomaterialheterogeneitymustbequantied.
Inthisreport,theeectofmaterialheterogeneityonconductionheattransferwillbestudied,becauseheattransferthroughheterogeneousmaterialshasimplicationsinawiderangeofelds,likethecreationofnewmaterialsfromheterogeneousmixtures.
2HeatEquationConductionheattransferthroughamaterialisoftenwellmodeledbyTt=α2T,(1)whereTisthetemperatureofthematerial,tistime,αisthethermaldiusivityofthematerial,and2istheLaplacianoperator.
Equation(1)isaparabolicpartialdierentialequation(PDE)withrespecttospaceandtime.
Itcanbesolvedbyavarietyofanalyticandnumericalmethods.
Inthisstudy,theheat5equationwassolvedusingtheNDsolvefunctioninWolframMathematica.
Thissoftwareusesstandardnumericalmethodstodiscretizetheequationsinspaceandtimeandgenerateanapproximatesolution[2].
Thethermaldiusivityofapurematerialmaterialisdenedasα=kρcp,(2)wherekisthethermalconductivity,ρisthedensity,andcpisthespecicheatcapacityofthematerial.
Itisassumedthatthethermaldiusivityisaconstantmaterialproperty,becauseinsolidsk,ρ,andcpdonotvarymuchwithtemperature.
Also,αmustbepositive,becausephysically,anegativethermaldiusivitywouldviolatethesecondlawofthermodynamicsbyallowingtemperaturetoowfromacoldbodytoahotbody.
Also,mathematically,solutionstoEq.
(1)areunstablewhenαisnegative.
3MethodEquation(1)isdeterministic.
Itdoesnotcontainanyuncertainty.
Becauseαisamaterialproperty,theuncertaintyduetothematerialheterogeneityisaddedtotheheatequationbydeningthethermaldiusivityα(ξ)=α+σαξ,(3)whereαisthemeanofthermaldiusivity,σαisthestandarddeviationofthethermaldiusivity,andξisarandomvariablewithaprobabilitydistributionfunction(pdf)ofaGaussiandistributionwithameanof0andastandarddeviationof1.
TheGaussiandistributionindescribedinAppendixD.
Hence,α(ξ)hasapdfofaGaussianwithmeanofαandastandarddeviationσα.
EachξsampledfromtheGaussianrepresentsadierentthermaldiusivity.
However,becausethereisnospatialcomponentintheformulationofα(ξ),eachsampleofξisequivalenttosamplinganewmaterial.
Forexample,iftheheterogeneityofamaterialisdescribedbyaGaussiandistribution,likeinEq.
(3),eachξvaluewouldrepresentadierentmixtureofthematerials.
However,itassumesthesamethermaldiusivitythroughoutthewholebody.
Inordertosimplifytheproblem,itisassumedthattheproblemisonedimensional.
Hence,thestochasticheatequationismodeledasT(x,t,ξ)t=α(ξ)2T(x,t,ξ)x2.
(4)6ThesetofinitialandboundaryconditionsischosentobeT(0,t)=T0,(5)T(1,t)=T0,(6)T(x,0)=T0+T0sin(πx),(7)foradomainofx∈(0,1)andT0=300.
BecauseTisnowafunctionofξthatisdescribedbyapdf,thereisalsoapdfthatrepresentsTateachpointinspaceandtime.
Equation(4)isnotadeterministicproblem,soitcannotbesolvedusingtraditionalnumericalmethods.
However,ifaspecicξ0wassampledfromthepdf,Eq.
(4)couldbesolveddeterministicallylikeEq.
(1)[3].
Inordertodevelopageneralsolutiontothestochasticheatequation,threetechniquesareused:theMonteCarlomethod,theintrusivepolynomialchaosmethod,andthenon-intrusivepolynomialchaosmethod.
3.
1MonteCarloTherstmethodusedtosolvethestochasticheatequationistheMonteCarlomethod.
ThismethodinvolvessamplingalargenumberofvaluesofξandsubstitutingthemintoEq.
(4).
Onceaξvalueissubstituted,thestochasticheatequationbecomesdeterministic.
Henceforeachsampledξ,theheatequationissolved,andthetemperatureisevaluatedataspecicpoint(x0,t0).
OncealargenumberofξsaresampledandtheresultingvaluesforT(x0,t0)arecalculated,ahistogramoftheT(x0,t0)iscreated.
Asthenumberofsamplesincreases,thehistogramofT(x0,t0)approachestheprobabilitydensityfunctionofT(x0,t0).
ThenthemeanandstandarddeviationofT(x0,t0)arecalculatedtocharacterizethedistribution[4].
TheMonteCarlomethodisusedthroughoutUQasastandardtocompareothermethodsagainst.
Be-causetheMonteCarlomethodsimplyinvolvessamplingrandomvariablesandsolvingthePDEnumerically,itprovidesareliablesolution,becausethenumericalmethodsusedtosolvethePDEarechosensothattheerrorsduetospatialandtemporaldiscretizationarenegligible.
AlthoughthesolutionstothePDEsarereliable,aninnitenumberofξsmustbesampledinordertoobtainthetruepdfofthetemperatureatapoint.
However,alargenumberofsampleswillleadtoameanandstandarddeviationagainstwhichothermethodscanbecompared.
AlthoughtheMonteCarlomethodprovidesreliablesolutions,itiscomputa-tionallyexpensivebecauseitreliesonsolvingalargenumberofpartialdierentialequations.
Hence,lesscomputationallyexpensivemethodsaredesirable.
73.
2IntrusivePolynomialChaosTheuncertaintyinthethermaldiusivityisapproximatedusingaP+1termFourierexpansionα(ξ)=Pp=0αpφp(ξ),(8)whereαpisasetofcoecientsandφp(ξ)isasetofknownbasisfunctions.
Asξvaries,thevalueofα(ξ)varies,thatcausestheoutputTtovary.
Hence,theoutputtemperatureisalsoapproximatedwithaFourierexpansionT(x,t,ξ)=Pp=0Tp(x,t)φp(ξ),(9)whereTp(x,t)isasetofamplitudesthatareafunctionofspaceandtime.
Itisworthnotingthatthetermamplitudeisnotatraditionalamplitudethatusuallyconnotesthedierencebetweenthemaximumandminimumofafunction.
Withthisapproximation,aseparationofvariablestechniquecanbeperformedsothattheuncertaintyinξiscontainedinthebasisfunctionsandthespatialandtemporaldependenceiscontainedintheamplitudes.
ThedetailedderivationofthismethodisincludedinAppendixA.
Thebasisfunctionsarechosentobetheprobabilists'Hermitepolynomials.
TheGalerkinmethodwasusedtotaketheinnerproductofbothsidesofEq.
(8)tocalculatethecoecients.
ThenEqs.
(8-9)aresubstitutedintoEq.
(4).
SubsequentlytheamplitudesoftheboundaryconditionsarefoundandasystemofcoupledPDEsiscreatedtosolvefortheamplitudes.
Thetwo-termformulation(P=1)wasperformed.
EvaluatingtheinnerproductsleadstothefollowingcoupledPDEs:T0(0,t)=300,T1(0,t)=300,T0(1,t)=300,T1(1,t)=300,(10)T0(x,0)=300+300sin(πx),T1(x,0)=0,(11)T0(x,t)t=2T0(x,t)x2+σ2T1(x,t)x2,(12)T1(x,t)t=σ2T0(x,t)x2+2T1(x,t)x2.
(13)ThesystemofPDEsissolvednumericallyinMathematicafortheamplitudesT0(x,t)andT1(x,t),buttheseequationsmayhaveananalyticsolution.
ThenT0(x,t)andT1(x,t)aresubstitutedintoEq.
(9)togivethesolutionT(x,t,ξ)=T0(x,t)+T1(x,t)ξ.
(14)8InasimilarmannertotheMonteCarlomethod,oncetheexpressionforT(x,t,ξ)isfound,alargenumberofξvaluesaresampledandsubstitutedintoit.
Thentheresultsataspecicx0andt0couldbecalculatedandstatisticallyanalyzed.
ThetermT(x0,t0,ξ)isapolynomialofξbecausethebasisfunctionsareHermitepolynomials.
AlthoughboththeintrusivepolynomialchaosandtheMonteCarlomethodssamplealargenumberofξs,thepolynomialchaosmethodinvolvescalculatingthevaluesofpolynomialsforeachξ,andtheMonteCarlomethodinvolvessolvingaPDEforeachξvalue.
Also,inthepolynomialchaosmethodthePDEsaresolvedonce,whichisadvantageousbecausesolvingthePDEsisthemostcomputationallyexpensivepartofthemethod.
Hence,onewouldexpectthattheintrusivepolynomialchaosmethodisamoreecientmethodofquantifyingtheuncertainty[4].
Thismethodiscalledanintrusivemethod,becauseitonlyrequirestheformulationofastochasticversionoftheoriginalmodelandnotmultiplesolutionsoftheoriginalmodel[1].
However,thismethodonlyrequiresonetosolveP+1PDEs,soonewouldexpectthatitislesscomputationallyexpensivethantheMonteCarlomethod.
3.
3Non-IntrusivePolynomialChaosNon-intrusivepolynomialchaosisanothermethodthatmanipulatesstochasticproblemstosolvethemmoreecientlythantheMonteCarlomethod.
Likeintheintrusivepolynomialchaosmethod,thetemperatureisapproximatedbyasetoflinearlyindependentbasisfunctionsandamplitudestoseparatetherandomvariablefromthedeterministicvariables:T(x,t,ξ)=Pp=0Tp(x,t)φp(ξ).
(15)ThedetailsofthismethodareincludedinAppendixB.
Again,thebasisfunctionsarechosentobetheprobabilists'Hermitepolynomials.
TheGalerkinprocedureisusedtotaketheinnerproductofbothsidesofEq.
(15).
Intheend,anexpressionfortheamplitudesisdetermined:Tp(x,t)=∞∞T(x,t,ξ)φp(ξ)w(ξ)dξp!
≈Ii=0WiT(x,t,ξi)φp(ξi)w(ξi)p!
.
(16)Gaussianquadratureisusedtoapproximatetheintegral.
ThevaluesofT(x,t,ξi)areevaluatedforallGausspointsξivaluesbysubstitutingthevaluesintoEq.
(4),thestochasticheatequation,anditcanbesolveddeterministically.
Hence,foraspecicnumberofterms,P,theheatequationissolveddeterministicallyforeachofIξivaluesandTp(x,t)isformedbyperformingaweightedsummation.
Hence,throughGaussianquadrature,theintegralisapproximated,andtheresultingamplitudesaremultipliedbythebasestoapproximatethetemperaturedistributionlikeinEq.
(15).
Onceaspecicpointinspace9andtime,(x0,t0)ischosen,theresultingapproximationissimplyapolynomialfunctionofξ.
Likeintheintrusivepolynomialchaosmethod,oncethepolynomialapproximationiscreated,alargenumberofξvaluesaresampledandtheresultingtemperaturevaluescanbestatisticallyanalyzed.
Thismethodisnon-intrusivepolynomialchaos,becausethesolutionisdependentonsolvingtheoriginalPDEmultipletimesforeachξivalueforGaussianquadrature[1].
4ResultsInthisstudy,theinitialconditions,boundaryconditions,andthermaldiusivityapproximationwerearbi-trarilydened.
TheinitialconditionsinEqs.
(5-7)werearbitrarilychosen.
Also,thethermaldiusivityasdenedinEq.
(3)wasarbitrarilyselectedwith=1,σ=0.
75,andξasarandomvariablefromaGaussiandistributionwithameanof0andastandarddeviationof1.
Theratioofstandarddeviationtomeanforαwasselectedtobe0.
75sotherewouldbealargerangeofαvalues.
Inthisformulation,itwasassumedthatαisdescribedbyaGaussiandistribution,butwhenmodelingarealmaterial,onewouldcharacterizeitsheterogeneitytodeterminethecorrectdistributiontomodelthematerial.
However,becausetheGaussiandistributionisfullydescribedbythemeanandstandarddeviation,theheterogeneityinamaterialcouldbeapproximatedbyaGaussianbycalculatingthemeanandstandarddeviationofthethermaldiusivityofthematerial.
Thesolutionwasstudiedbycalculatingthetemperatureataspecicpointinspaceandtime.
Ataspecicspatialandtemporalpoint,thetemperatureissimplyafunctionofξ.
Ineachmethod,alargenumberofξvaluesweresampledandtheresultingsolutionswereplottedinahistogram.
Themeanandstandarddeviationofthesolutionsatthespecicpointwerecalculatedinordertocharacterizethesolution.
Fromtheseproperties,anapproximatingGaussiancouldbecreated.
Ifthisapproximationhasalargeamountoferror,otherdistributionscouldalsobeusedtoapproximatethesolutionbycalculatingthehighermomentsofthesolution.
Afteranapproximatepdfiscreated,acondenceintervaliscalculated.
4.
1MonteCarloMethodItwasarbitrarilychosenthatthetemperaturewouldbeanalyzedatthepointx0=0.
3,t0=0.
3.
However,becausetheMonteCarlomethodinvolvessamplingalargenumberofξvalues,thereisachancethatanegativethermaldiusivityvalueisselected.
Specically,whenξ1,becausetheHermitebasisfunctionsareorthogonal.
ItisalsoclearfromEq.
(37)thatα0=andα1=σ.
Hence,theintegraldidnothavetobecalculatedforthiscaseandactuallycouldhavebeendeterminedbyinspection[4].
Next,Eqs.
(9,8)aresubstitutedintotheEq.
(4):tPp=0Tp(x,t)φp(ξ)=Pp=0αpφp(ξ)2x2Pm=0Tm(x,t)φm(ξ).
(38)Theboundaryconditionsmustalsobeapproximatedusingaseries,soT(0,t,ξ)=300=Pp=0Tp(0,t)φp(ξ),(39)T(1,t,ξ)=300=Pp=0Tp(1,t)φp(ξ),(40)T(x,0,ξ)=300+300sin(πx)=Pp=0Tp(x,0)φp(ξ).
(41)Theinitialamplitudes,Tp(x,t)couldbecalculatedusingthesameprocessasEq.
(36),butbyinspectionbecauseφ0(ξ)=1,itisclearbyinspectionthatT0(0,t)=300,T1(0,t)=0,T2(0,t)=0,.
.
.
,TP(0,t)=0,(42)T0(1,t)=300,T1(1,t)=0,T2(1,t)=0,.
.
.
,TP(1,t)=0,(43)T0(x,0)=300+300sin(πx),T1(x,0)=0,T2(x,0)=0,.
.
.
,TP(x,0)=0.
(44)Now,manipulatingEq.
(38)givesPp=0Tp(x,t)tφp(ξ)=Pp=0Pm=0αp2Tm(x,t)x2φp(ξ)φm(ξ).
(45)Onceagain,thedependenceontherandomvariable,ξ,mustberemovedinordertocalculatetheamplitudesTp(x,t)toapproximatethetemperature.
Onceagain,theGalerkinmethodisusedbytakingtheinner21productofbothsidesofEq.
(45)withrespecttothesetofthebasisfunctions,φl(ξ):φl(ξ),Pp=0Tp(x,t)tφp(ξ)=φl(ξ),Pp=0Pm=0αp2Tm(x,t)x2φp(ξ)φm(ξ)l=0,1,.
.
,P,(46)Pp=0Tp(x,t)tφl(ξ),φp(ξ)=Pp=0Pm=0αp2Tm(x,t)x2φl(ξ),φp(ξ)φm(ξ)l=0,1,.
.
.
,P,(47)Pp=0Tp(x,t)t(l!
δlp)=Pp=0Pm=0αp2Tm(x,t)x2φl(ξ),φp(ξ)φm(ξ)l=0,1,.
.
.
,P,(48)l!
Tl(x,t)t=Pp=0Pm=0αp2Tm(x,t)x2φl(ξ),φp(ξ)φm(ξ)l=0,1,.
.
.
,P,(49)Tl(x,t)t=1l!
Pp=0Pm=0αp2Tm(x,t)x2φl(ξ),φp(ξ)φm(ξ)l=0,1,.
.
.
,P.
(50)Thedependenceonξhasbeensuccessfullyremovedbecauseinnerproductexpressionissimplytheintegralφl(ξ),φp(ξ)φm(ξ)=∞∞φl(ξ)φp(ξ)φm(ξ)1√2πeξ2/2dξ,(51)thatcanbecalculatedandtabulated.
Becausethereisnolongerdependenceonξ,Eq.
(50)isasystemofP+1coupledpartialdierentialequations[4].
Thesystemcanbesolvedusingnumericalschemes,likeNDSolveinMathematica.
8AppendixBThederivationofthenon-intrusivepolynomialchaosmethodiscontinuedfromEq.
(15).
Onceagain,thebasisfunctionswerechosentobetheprobabilists'HermitepolynomialsbecausetheyareorthogonalwithrespecttotheGaussianweightingfunction.
ThenaGalerkinformulationisused,andtheinnerproductofbothsidesofEq.
(15)istakenwithrespecttothebasisfunctions.
Thentheorthogonalityofthebasisfunctionsisusedtosolvefortheamplitudes[6]:T(x,t,ξ),φm(ξ)=Pp=0Tp(x,t)φp(ξ),φm(ξ)m=1,2,.
.
,P,(52)T(x,t,ξ),φm(ξ)=Pp=0Tp(x,t)φp(ξ),φm(ξ)m=1,2,.
.
,P,(53)T(x,t,ξ),φm(ξ)=Pp=0Tp(x,t)m!
δpmm=1,2,.
.
,P,(54)T(x,t,ξ),φm(ξ)m!
=Tm(x,t)m=1,2,.
.
,P.
(55)22BecausetheamplitudesTm(x,t)aresimplyequaltoaninnerproduct,theinnerproductcanbeexpandedtoshowthattheamplitudesareequaltointegrals:Tp(x,t)=∞∞T(x,t,ξ)φp(ξ)w(ξ)dξp!
p=1,2,.
.
,P,(56)w(ξ)=1√2πeξ2/2.
(57)IntheintegralinEq.
(56),φp(ξ)andw(ξ)areknown,buttheintegralcannotbeevaluatedoutrightbecauseT(x,t,ξ)isunknown.
However,theintegralcanbeapproximatedusingGaussianquadrature.
Gaussianquadratureisamethodofapproximatinganintegralasaseriesofweights,Wiandfunctionvaluesatspecicpoints,ξi:∞∞f(ξ)dξ≈Ii=0Wif(ξi).
(58)FortheI+1termapproximationoftheintegral,thevaluesforWiandξiaretakenfromAppendixC.
Forthisspecicproblem,theGaussianquadratureapproximationoftheintegralisTp(x,t)=∞∞T(x,t,ξ)φp(ξ)w(ξ)dξp!
≈Ii=0WiT(x,t,ξi)φp(ξi)w(ξi)p!
.
(59)9AppendixCGaussianquadratureisamethodtoapproximateanintegralasasum,similartothetrapezoidalrule.
IngeneralthequadratureformulaisI=baf(ξ)dξ≈Ii=1Wif(ξi),(60)wheref(ξ)isageneralfunction,Wiareweights,andξiarespecicabscissas.
Theweightsandabscissasarechosenbasedonthemethod.
Forthetrapezoidalrulethesevaluesarechosensothatalineisintegratedbetweenthetwoadjacentpointsintheapproximation.
However,inGaussianquadrature,theWiandξivaluesarechosenforthemostaccurateapproximationforaspecicnumberofterms.
Onaninniteinterval,(a,b)theabscissavaluesarechosentobethezerosofthenthphysicists'Hermitepolynomial.
Thephysicists'Hermitepolynomialsaredierentthantheprobabilists'Hermitepolynomialsdiscussedin23thisreport:H0(ξ)=1,(61)H1(ξ)=2ξ,(62)H2(ξ)=4ξ22,(63)H3(ξ)=8ξ312ξ,(64).
.
.
(65)Hn(ξ)=(1)neξ2dneξ2dξn.
(66)However,withasmalleort,onecouldalsoformulatetheGaussianquadraturefortheprobabilists'Hermitepolynomials.
TheweightsforaninnitedomainareWi=2n1n!
√πn2[Hn1(ξi)]2.
(67)TheabscissasandweightsforGaussianquadratureforuptoI=10aretabulatedinTables1-9[7].
Table1:GaussianquadratureξiandWivaluesforI=2±ξiWi0.
7071067811865481.
4611411826611Table2:GaussianquadratureξiandWivaluesforI=3±ξiWi0.
0000000000000001.
18163590060371.
2247448713315891.
3239311752136Table3:GaussianquadratureξiandWivaluesforI=4±ξiWi0.
5246476232752901.
05996448289501.
6506801238857851.
240225817695824Table4:GaussianquadratureξiandWivaluesforI=5±ξiWi0.
0000000000000000.
94530872048290.
9585724646138190.
98658099675142.
02018428704560861.
1814886255360Table5:GaussianquadratureξiandWivaluesforI=6±ξiWi0.
4360774119276170.
87640133443621.
3358490740136970.
93558055763122.
3506049736744921.
1369083326745Table6:GaussianquadratureξiandWivaluesforI=7±ξiWi0.
0000000000000000.
81026461755680.
8162878828589650.
82868730328361.
6735516287674710.
89718460022522.
6519613568352331.
1013307296103Table7:GaussianquadratureξiandWivaluesforI=8±ξiWi0.
3811869902073220.
76454412865171.
1571937124467800.
79289004838641.
9816567566958430.
86675260656342.
9306374202572441.
071930144248025Table8:GaussianquadratureξiandWivaluesforI=9±ξiWi0.
0000000000000000.
72023521560610.
7235510187528380.
73030245274511.
4685532892166680.
76460812509462.
2665805845318430.
84175270147873.
1909932017815281.
0470035809767Table9:GaussianquadratureξiandWivaluesforI=10±ξiWi0.
3429013272237050.
68708185395131.
0366108297895140.
70329632310491.
7566836492998820.
74144193194362.
5327316742327900.
82066612640483.
4361591188377381.
025451691365710AppendixDHereisanoverviewofsomebasicconceptsofprobabilitytheory.
10.
1IntroductiontoProbability10.
1.
1RandomVariablesArandomvariableisafunctionofallofthepointsinasamplespace.
Randomvariablesareusuallydenotedwithcapitalletters,likeX.
Arandomvariablecanalsobethoughtofasamethodofassigningvaluestooutcomesofarandomprocess.
Example1Denearandomvariable,X,thatisthevalueoftherollofadie.
26Solution:X=1,ifthevalueoftherollis12,ifthevalueoftherollis23,ifthevalueoftherollis34,ifthevalueoftherollis45,ifthevalueoftherollis56,ifthevalueoftherollis6.
Anotherwaytowritethesolutionis:X=x,wherexisthevalueoftheroll.
Randomvariablesarenotusedliketraditionalvariablesbecausetheydonothaveaxedvalue.
Hencetheexpression7=X+2,(68)whereXisarandomvariableisnotmeaningful,becauseitlimitsthepossibilitiesoftherandomvariable.
Instead,randomvariablesaredescribedwithprobabilities.
Forexample,theprobabilitythattherandomvariableXfromExample1isgreaterthan3couldbewrittenP(X>3).
ThisexampleshowcasesthenotationalconvenienceofrandomvariablesbecausewithouttherandomvariableonewouldhavetowritethatprobabilityasP(Therollofadie>3).
Hence,randomvariablesarenotusedinequationslikealgebraicvariables,butareusedasspecicfunctions.
10.
1.
2DiscreteandContinuousRandomVariablesArandomvariableiscalledadiscreterandomvariableifittakesonaniteorcountablenumberofvalues.
Arandomvariableiscalledacontinuousrandomvariableifittakesonaninnitenumberofvalues.
Example2Arethefollowingrandomvariablesdiscreteorinnite1.
X=Thevalueoftherollofadie2.
W=Theweightofanelephant273.
Y=TheexacttimeofthewinninghorseintheKentuckyDerby4.
O=Thenumberoforangesonanorangetree5.
P=ThestockmarketvalueofashareoftheDowJonesSolution:1.
Xisadiscreterandomvariablebecausethereareonly6possiblevalues.
2.
Wisacontinuousrandomvariablebecausetheexactweightofanelephantcouldbeanynumberbetween12,000lbfand12,001lbf.
3.
Yisacontinuousrandomvariablebecausetheexactwinningtimecouldbemeasuredin2:03.
66oritcouldbe2:03.
65764oritcouldbe2:03.
6576498012.
4.
Oisadiscreterandomvariablebecausethenumberoforangesonatreeiscountable.
5.
PisadiscreterandomvariablebecausethestockmarketvalueoftheDowJonesisroundedtothenearestcent.
Therefore,thereareacountablenumberofvaluesitcantake.
10.
1.
3ProbabilityDistributionsForadiscreterandomvariableX,letx1,x2,.
.
.
xnbethenpossiblevaluesofX.
EachofthesevalueshaveprobabilitiesP(X=xi)=f(xi).
i=1,2,.
.
.
,n(69)TheprobabilityfunctionorprobabilitydistributionisdenedasP(X=x)=f(x).
(70)Itisclearthatforx=xiEq.
(70)becomesEq.
(69)andforotherxvalues,f(x)=0.
Foradiscreterandomvariable,theprobabilitydistributionwillbeapiecewisefunction.
Twopropertiesofdiscreteprobabilitydistributionsare:f(x)≥0,(71)ni=1f(xi)=1.
(72)Thesepropertiesarederivedfromthefactthataprobabilityisalwaysgreaterthanorequaltozeroandthefactthatallofthediscretevaluesareincludedinthedistribution.
28Foracontinuousrandomvariable,theresultingprobabilitydistributionwillbecontinuousbecausethereareaninnitenumberofpossiblevalues.
Hence,thepropertiesofacontinuousprobabilitydistributionare:f(x)≥0,(73)∞∞f(x)=1.
(74)Example3WhatistheprobabilitydistributionofthevalueofthesumoftwodicerollsSolution:Whenrollingtwodice,thereare36possibleoutcomes:1,11,21,31,41,51,62,12,22,32,42,52,63,13,23,33,43,53,64,14,24,34,44,54,65,15,25,35,45,55,66,16,26,36,46,56,6Whentheseresultsaresummedtheybecome23456734567845678956789106789101178910111229Hence,thereare11possibleresults.
LetYbethediscreterandomvariableofthesumofthetwodicerolls.
f(2)=P(Y=2)=111f(3)=P(Y=3)=211f(4)=P(Y=4)=311f(5)=P(Y=5)=411f(6)=P(Y=6)=511f(7)=P(Y=7)=611f(8)=P(Y=8)=511f(9)=P(Y=9)=411f(10)=P(Y=10)=311f(11)=P(Y=11)=211f(12)=P(Y=12)=111Hence,f(y)canbeplottedasshowninFigure10.
1.
3.
Figure9:Exampleofaprobabilitydistribution.
3010.
2MathematicalExpectation10.
2.
1ExpectedValueTheexpectedvalueormeanofadiscreterandomvariableisE(X)=ni=1xif(xi),(75)wherenisthenumberofpossibilitiesandtheexpectedvalueforacontinuousrandomvariableisE(X)=∞∞xf(x)dx.
(76)TheexpectedvalueforafunctionofarandomvariableisE[g(x)]=ni=1g(xi)f(xi),(77)E[g(x)]=∞∞g(x)f(x)dx.
(78)SometheoremsaboutexpectedvaluesareE(cX)=cE(X),(79)E(X+Y)=E(X)+E(Y),(80)E(XY)=E(X)E(Y).
(81)Themeanisgenerallydenotedby[8].
Example4WhatistheexpectedvalueofthesumofvefairdiceSolution:Becauseadiehas6discretevalues,wewilluseEquation75.
IfU,V,W,X,andYarethevaluesofthedice,foronedie,E(U)=E(V)=E(W)=E(X)=E(Y)=116+216+316+416+516+616=72.
Thentheexpectedvalueofthesumcanbewritten:E(U+V+W+X+Y)=E(U)+E(V)+E(W)+E(X)+E(Y)=572=352.
31Example5WhatistheexpectedvalueinminutesofclockthatstopsarandomtimeduringanhourSolution:Theprobabilitydistributionfunctionforthisproblemis:f(x)=160,0≤x≤600,otherwise.
Becauseitcouldstopatanyfractionbetweentheminutevalues,thispdfiscontinuous.
ThereforewewilluseEquation(76).
E(X)=∞∞xf(x)dx=600x160dx=11206020=30.
(82)10.
2.
2MomentsThevarianceisdenedasVar(X)=E[(X)2]=σ2X,(83)whereσXisthestandarddeviation.
σ2=ni=1(xi)2f(xi),(84)σ2=∞∞(x)2f(x).
(85)Thevarianceisameasureofthedispersion.
Ifthevaluesareconcentratednearthemean,thevarianceissmall.
32Figure10:Exampleofanormaldistributionwithdierentvariances.
ArandomvariableXwithmeanandstandarddeviationσhasanassociatedstandardizedrandomvariableX=Xσ.
(86)Standardizedrandomvariablesareusedforcomparingdierentdistributions.
Themomentsofadistributionareusedtodeterminesimilaritybetweentwodistributions.
Therthcentralmomentisdenedr=E[(X)r],(87)r=ni=1(xi)rf(xi),(88)r=∞∞(x)rf(x)dx,(89)thatconnotes0=1,1=,and2=σ2,because0isthetotalareaundertheprobabilitydistributionfunction,1isthemeanand,2isthevariance.
Therthrawmomentisr=E(Xr).
(90)ThemomentgeneratingfunctionofXisdenedbyMX(t)=EetX.
(91)33TheconvergentdiscreteformisMX(t)=ni=1etXf(xi),(92)andthecontinuousformisMX(t)=∞∞etXf(x).
(93)MX(t)generatesmomentsbecausetherthderivativeevaluatedatt=0istherthmoment,r.
WhenthemomentgeneratingfunctioniswrittenasaTaylorseries,thecoecientoftherthtermistherthmoment[8].
ThecharacteristicfunctionisφX(ω)=MX(iω)=EeiωX.
(94)forcharacteristicfunctionstheseriesandintegralconvergeabsolutely.
ThecharacteristicfunctioncanberelatedtothefunctionusingFouriertransforms.
f(x)=12π∞∞eiωxφX(ω)dω.
(95)Example6Forafairdie,nd(a)themomentgeneratingfunction,(b)therstthreemoments.
Solution:Becauseitisadie,thereare6discretepossibilities,sowewilluseEquation(92)MX(t)=etXf(x)=et(1)16+et(2)16+et(3)16+et(4)16+et(5)16+et(6)16MX(t)=et6+e2t6+e3t6+e4t6+e5t6+e6t6.
ThentoevaluatethemomentswemustndMX(0),MX(0),MX(0),andMX(0).
MX(t)=et6+e2t3+e3t2+2e4t3+5e5t6+e6tMX(t)=et6+2e2t3+3e3t2+8e4t3+25e5t6+6e6tMX(t)=et6+4e2t3+9e3t2+32e4t3+125e5t6+36e6t0=MX(0)=16+16+16+16+16+16=11=MX(0)=16+13+12+23+56+1=722=MX(0)=16+23+32+83+256+6=9163=MX(0)=16+43+92+323+1256+36=473634Notethatthe0=1asexpectedandtherstmoment,1isthemean,asshowninExample4.
Nowlet'sconrmthat2isthevariance.
UsingEquation(84)wendσ2=E[X2]=1216+2216+3216+4216+5216+6216=91610.
2.
3Chebyshev'sInequalityChebyshev'sInequalitystatesthatifXisarandomvariablehavingmeanandvarianceσ2thatarenitethenifisanypositivenumberP(|Xσ22,(96)or,with=kσ,P(|X|≥kσ)≤1k2.
(97)Chebyshev'sInequalitytellsustheprobabilitythatXwilldierbymorethankstandarddeviationsfromthemean.
Thisisusefulbecausewehavenotspeciedtheprobabilitydistribution[8].
Example7Forexample,whatistheprobabilitythatrandomvariableX(withanydistribution)diersfromitsmeanbymorethan4standarddeviations.
UsingEquation(97)wherek=4,P(|X|)≥4σ)≤142=0.
062510.
2.
4LawofLargeNumbersAsaresult,theLawofLargeNumbersstatesthatifPn=X1,X2,.
.
.
,Xnaremutuallyindependentrandomvariableswithanitemeanofandvarianceσ2thenlimn→∞PPnn≥=0,(98)thatmeansthattheprobabilitythatthearithmeticmeanofX1,X2,.
.
.
,Xndiersfromitsexpectedvaluebymorethananypositivenumberapproacheszeroasn→∞.
Thestronglawoflargenumbersstatesthatlimn→∞(X1+.
.
.
+Xn)/n=withprobabilityone.
Example8Wewilltestthelawoflargenumbersforafairdie.
AsshowninExample4,themeanofonerollofadieis=72=3.
5.
Table10showsthatasn→∞,Snn→3.
5asexpected.
Hencebecause,|Snn|approaches0thentheprobabilitythatthatquantityisgreaterthananypositivenumberapproacheszeroasEquation(98)states.
35Table10:Thelawoflargenumberswastestedtosimulateafairdieforarangeofn.
n51020501001000100001000001000000Snn2.
84.
33.
153.
343.
533.
5693.
46773.
50343.
4996|Snn|0.
70.
80.
350.
160.
030.
0690.
03230.
00340.
000410.
2.
5MiscellaneoustopicsThemeandeviation(M.
D.
)isdenedasM.
D.
(X)=E[|X|]=ni=1|xi|f(xi),(99)fordiscretevariablesorM.
D.
(X)=E[|X|]=∞∞|x|f(x)dx,(100)forcontinuousvariables.
Themeandeviationisametricusedtodescribethedispersionofthepdf.
Intuitively,itisthemeandistancefromthemeanofpdf.
Skewnesscanbemeasuredwithcoecientsofskewness,likeα3=E[(X)3]σ3=3σ3,(101)whereσ3willbepositiveornegativedependingonifthedistributionisskewedrightorleft.
σ3=0forasymmetricdistribution.
Kurtosisdescribeshowlargeorsmallthepeakis.
Anormalcurvehasacoecientofkurtosisequalto3[8].
α4=E[(X)4]σ4=4σ4.
(102)Example9WhenrandomvariableXisoneoftheset12,13,13,7,6,20,11,10calculatethemeandeviation.
Solution:Firstwemustcalculatethemean=12+13+13+7+6+20+11+108=11.
5.
36ThenusingEquation(99)wecalculatethemeandeviationtobeM.
D.
(X)=|1211.
5|+|1311.
5|+|1311.
5|+|711.
5|+|611.
5|+|2011.
5|+|1111.
5|8+|1011.
5|8M.
D.
(X)=3.
10.
3SpecialProbabilityDistributions10.
3.
1GaussianDistributionAbinomialdistributionisgivenbytheprobabilitydensityfunctionf(x)=P(X=x)=nxpxqnx=n!
x!
(nx)!
pxqnx,(103)wherepistheprobabilityofsuccess,q=1pistheprobabilityoffailure,Xisarandomvariablethatdenotesthenumberofsuccessesofntrials,andxisthenumberofsuccesses.
Ifn=1,thisiscalledaBernoullidistribution.
TheGaussianornormaldistributionthatisgiventheprobabilitydensityfunctionf(x)=1σ√2πe(x)2/2σ2,(104)whereisthemeanandσisthestandarddeviation.
ThedistributionfunctionisF(x)=P(X≤x)=1σ√2πx∞e(v)2/2σ2dv.
(105)WedeneZ=Xσ,(106)whereZisthestandardizedvariableorstandardscore.
IfthemeanofZis0andthevarianceis1,thenthedensityfunctionisf(z)=1√2πez2/2,(107)thatisknownasthestandardnormaldensityfunction.
Figure10.
2.
2showsanexampleofanormaldistri-butionwithvaryingstandarddeviations.
IfZ=Xnp√npq,(108)37thentheGaussiandistributioncloselyapproximatesthebinomialdistributionwherenisthenumberoftrials,pistheprobabilityofsuccess,andqistheprobabilityoffailure.
Thegeneralruleisthatifnp>5andqp>5thenitisagoodapproximationofthebinomialdistributionandasnapproachedinnity,thebinomialdistributionapproachestheGaussiandistribution.
Writtenanotherway,limn>∞Pa≤Xnp√npq≤b=1√2πbaeu2/2du.
(109)Thenormaldistributionisoftencalleda"bellcurve"andisseenavarietyofelds.
Forexamplethedistributionofhumanheightsiswellapproximatedbyanormaldistribution.
Table11showsthemeanandvarianceofthedistribution[8].
Table11:ThepropertiesofthenormaldistributionasdescribedinEquation(104).
MeanVarianceσ2MomentGeneratingFunctionM(t)=eut+(σ2t2/2)CharacteristicFunctionφ(ω)=eiω(σ2ω2/2Example10Showthatthebinomialdistributioncanbecloselyapproximatedbythenormaldistributionforincreasingnforippingafaircoin.
Solution:Ifafaircoinisipped,p=0.
5.
Figure11showsthatasnincreases,theGuassiandistributionsbecomesmorelikethebinomialdistribution.
10.
3.
2PoissonDistributionThePoissondistributionhasapdff(x)=P(X=x)=λxeλx!
,x=0,1,2,.
.
.
(110)whereλisapositiveconstantandXisarandomvariablethatcanbeanon-negativeinteger.
ThePoissondistributionapproachesthenormaldistributionasλ→∞where(Xλ)/√λisthestandardizedrandomvariable.
ThePoissondistributiondescribesmanysituationslikethenumberofmutationsinastrandsofDNAorthenumberofcarsgoingthroughatraclight.
Figure10.
3.
2showsanexampleofaPoissondistributionforavarietyofλvalues.
Table12showsthemeanandvarianceofthedistribution[8].
38Figure11:TheguresshowthattheGaussiandistributionmorecloselyapproximatesthebinomialdistri-butionasn→∞withp=0.
5.
Table12:ThepropertiesofthePoissondistributionasdescribedinEquation(110).
Mean=λVarianceσ2=λMomentGeneratingFunctionM(t)=eλ(et1)CharacteristicFunctionφ(ω)=eλ(eiω1)39Figure12:ExampleofPoissondistributionswithdierentmeans.
Example11DemonstratethatthePoissondistributionapproachestheGaussiandistributionasλ→∞.
Solution:AsshowninFigure10.
3.
2,thenormaldistributionmorecloselyapproximatesthePoissondistri-butionasλ→∞.
Figure13:Asλincreases,theGaussiandistributionwith=λandσ=√λmorecloselyapproximatesthePoissondistribution.
10.
3.
3TheCentralLimitTheoremTheCentralLimitTheoremstatesthatifX1,X2,.
.
.
,Xnareindependentrandomvariableswiththesameprobabilitydistributionfunctionthathasanitemeanandanitevarianceσ2,andSn=ni=1Xithen,limn→∞Pa≤Snnσ√n≤b=1√2πbaeu2/2du.
(111)40ThistheoremshowsthatanydistributionwithanitemeanandvariancecanbeapproximatedwiththeGaussiandistributionasn→∞.
10.
3.
4UniformDistributionTheuniformdistributionforarandomvariableXisgivenbythedensityfunctionf(x)=1ba,a≤x≤b0,otherwise.
(112)Althoughtheuniformdistributionseemstrivial,itisimportanttonoticethatasshowinFigure10.
3.
4becausetheareaunderthefunctionmustalwaysbe1,thedistributionislargerwhenthedierencebetweentheboundariesissmaller.
Table13showsthemeanandvarianceofthedistribution.
Table13:ThepropertiesoftheuniformdistributionasdescribedinEquation(112).
Mean=12(a+b)Varianceσ2=112(ba)2Figure14:Theprobabilitydistributionfunctionisshownforauniformdistributionwitha=0.
10.
4SamplingTheory10.
4.
1BasicsSamplingischoosingonesamplefromapopulationthatisdenedbyaprobabilitydistribution.
Samplingwithreplacementiswhenasamplecanbechosenmultipletimes.
Samplingwithoutreplacementdoesnot41allowamembertobechosenmultipletimes.
Asamplestatisticisanumberobtainedfromasampleinordertoestimateaparameterofthepopulation.
SamplestatisticsforannsizedsamplecanbewrittenasafunctionoftherandomvariablesX1,X2,.
.
.
,Xn.
Usually,GreeklettersareusedforpopulationparametersandRomanlettersareusedfortherelatedsamplestatistics.
Thesamplingdistributionistheprobabilitydistributionofthesamplestatistic[8].
Example12Ifthereisabagcontainingonlyoneredmarble,onegreenmarble,andonebluemarbleandyoutakeasamplemarbleandthenputitbackinthebag,whattypeofsamplingdidyoudoSolution:Thissituationisanexampleofsamplingwithreplacementbecausethemarblewasreturnedtothebag,soitcouldbechosenagain.
42