AWinningStrategyforRoulette

Size: px
Start display at page:

Download "AWinningStrategyforRoulette"

Transcription

1 AWinningStrategyforRoulette logreturn. Keywordsandphrases:Roulette,Bayesstrategy,Dirichletprior,convexloss,expected UniversityofWisconsinatMadison JeromeH.Klotz logcapitalafternplaysforlossfunction,weshowthatthebayesstrategyfora Dirichletpriorisasymptoticallyoptimal.Wesimulatetoillustratethebehavior ofthestrategyforabiasedfavorablewheelandanunbiasedunfavorablewheel. thegameofroulettewithanunbalancedwheel.usingnegativeaverageexpected Weexaminethestatisticalproblemofcomputingafavorablebettingstrategyfor Abstract 1IntroductionandNotation. favorablebetduetowheelimbalance.wilson(1965),presentsanicediscussionofthe Considertheproblemofbettingonaroulettegamewiththepossibilityofanunknown problemwithanecdotesaboutgroupsexploitingsuchimbalancetomakemoneyandgiving dataonhisownexperiences.morerecently,barnhart(1992)givesinterestingstoriesabout largecasinowins.ethier(1982),considerstherelatedproblemofhypothesistestingfor favorablenumbers.ofmoredirectinteresttothegambler,istheproblemofdeterminingan Xik=1iftheballfallsintocellnumberkontheithplay,andXik=0withprobability optimalbettingstrategytomaximizecapitalgain. 1?pkifitdoesnot. Xik=0,or1,PKk=1Xik=1,p=(p1;p2;:::;pK)T,PKk=1pk=1.Thuswithprobabilitypk, X1;X2;:::;XnhaveamultinomialdistributionM(1;p)whereXi=(Xi1;Xi2;:::;XiK)T,?$1:00foraloss(Xik=0).ForrouletteintheUS,thereusuallyareatotalofK=38cells Asastatisticalmodelforrouletteoutcomes,letindependentrandomcolumnvectors forawheellabeled00;0;1;2;:::;36withpayomk=$35forallsinglenumberbets. Theithplaypayofor$1:00betoncellnumberkisMkdollarsforawin(Xik=1)and 1

2 letthestrategybethecolumnvectorn=(n0;n1;:::;nk)twheren0istheproportion ofcn?1thatisnotbetandnkfork=1;2;:::;kistheproportionbetoncellnumberk forthenthgamblewithpkk=0nk=1. Startingwithinitialdollarcapital,C0,letCnbethecapitalattheendofthenthbet,and WriteX[n]=(X1;X2;:::;Xn)andassumei=(i0;i1;:::;iK)Tonlydependson (X1;X2;:::;Xi?1)(F(X[i?1])measurable). Thecapitalattheendofngamblesforthisstrategywillbe n,forknownp,tomaximizetheexpectedlogcapitalasgivenbykelly(1956).kelly's useofthelogpenalizesextremewagersthatbeteverythingonpropersubsetsandprevents Werstdiscussthesolutiontotheprobabilityproblemofndingtheoptimalstrategy Cn=Cn?1(n0+KXk=1nk(Mk+1)Xnk)=C0nYi=1(i0+KXk=1ik(Mk+1)Xik) (1) andexaminethepropertiesofthebayesstrategyforbalanceddirichletpriors. bankruptcy.wethenconsiderthestatisticalproblemofndingastrategyforunknownp 2AnOptimalStrategyforKnown(p1;p2;:::;pK). Whenthetruecellfrequenciespareknown,Kelley(1956),givestheoptimalstrategy. theorems.becauseofthelackofdetailinkelley'sderivationwepresentthesolutioninour Breiman(1960),andFinkelsteinandWhitley(1981)alsodiscusstheproblemandgivelimit notation. subjecttotheinequalityconstraintsk0fork=0;1;:::;kandequalityconstraint PKk=0k=1. initialcapital,weequivalentlymaximizetheexpectedlogreturnforasinglegamble Forknownpanoptimalstrategyn=(p)isindependentofn.Forourlossandunit Rockafellar(1970)corollary28.3.1)wecanminimizetheLagrangian UsingtheKuhnTuckertheorem(seeforexampleMangasarian(1969)section7.2.2or ()=KXk=1pkln(0+(Mk+1)k) where=(0;1;:::;k;+)tisthevectoroflagrangemultipliers. 2 (2)

3 fork=0;1;:::;k, KXk=0kk=0 fortheequalityconstraint.solvingequation(2)k=1;2;:::;kforkintermsof0and withk0fortheinequalityconstraintsk=0;1;:::;k,and KXk=0k=1 (4) (3) gives IfwedeneG+=fk:1kK;k>0gthensubstitutingthesekvaluesinequation Ifk>0,thenk=0becauseofequation(3)andthen k=pk +?k?0 +?0 (Mk+1): (Mk+1): (5) abovevalueof0weobtain+=1. For0>0equation(3)gives0=0.Usingequation(2)fork=0,aftersubstitutingthe (4)gives Whenk=0,equation(5)with+=1gives k=1?pk(mk+1)=00 0=1?Pk2G+pk=+ 1?Pk2G+(Mk+1)?1: (7) (6) wherep0=1?pk2g+pk. themsothatp1(m1+1)p2(m2+1):::pk(mk+1).wethenhavetwocases. sok=0implies0pk(mk+1).thenforthissolution TodeterminethesetG+=fk:1kK;k>0grstsortthevaluesandrelabel ()=X (6)with+=1that01pk(Mk+1)forallk=1;2;:::;Kand()willbemaximized If1p1(M1+1)thenpk1=(Mk+1)forallk=1;2;:::;K.Itfollowsfromequation k2g+pkln(pk(mk+1))+p0ln(0) forg+empty(k=0fork=1;2;:::;k,0=1).thusforthiscasewedon'tbet. contrary,thatj62g+forj<k2g+.thenj=0and0pj(mj+1)byequation(7). contradiction. Butpj(Mj+1)pk(Mk+1)so0pk(Mk+1)and0<k=pk?0=(Mk+1)0,a If1p1(M1+1)weprovethatifk2G+thenj2G+for1jk.Assume,tothe ThusG+=f1;2;:::;rgwherer1isthelargestintegerwith1?Prk=1(Mk+1)?1>0 pr(mr+1)>0[r] 3 (8)

4 fork=1;2;:::;r,andk=0fork=r+1;:::;k.wealsohavethemaximum wherep0=pkk=r+1pk.callthismaximum(p). where0[k]=(1?pkj=1pj)=(1?pkt=1(mt+1)?1).thenwith0=0[r] ()=rxk=1pkln(pk(mk+1))+p0ln(0) k=pk?0=(mk+1) (10) (9) Forknownp,theoptimalstrategyisconstantsothatforasingleplay(n=1),wehave 3AverageReturnECnandVariance. Similarly Usingindependence,forgeneraln, E(C1=C0)=E(0+KXk=1k(Mk+1)X1k)=0+KXk=1pkk(Mk+1)) and E(Cn=C0)=EnYi=1(0+KXk=1k(Mk+1)Xik)=[E(C1=C0)]n: E[(C1=C0)2]=KXk=1pk(0+k(Mk+1))2 E[(Cn=C0)2]=fE[(C1=C0)2]gn: (11) Itfollowsthat where1=(c1=c0)=e(c1=c0)=[pkk=1pk2k(mk+1)2?(pkk=1pkk(mk+1))2]1=2 Var(Cn=C0)=fE[(C1=C0)2]gn?f[E(C1=C0)]2gn=fE(C1=C0)g2n[(1+21)n?1](12) verylargeorverysmallreturnsoccurafteraperiodofplay. isthecoecientofvariationforasingleplay. dependent,identicallydistributedrandomvariables.takinglogsinequation(1)forthe optimalstrategyn=,wecanwrite Someinsightintothevariationcanbeobtainedfromthecentrallimittheoremforin- Thestandarddeviationcanbequitelargeincomparisontotheexpectationsothatoften [0+PKk=1pkk(Mk+1)] -alargesamplelog-normalrepresentationwherezn!n(0;1)indistributionasn!1. Here 2=KXk=1pkln2(0+k(Mk+1))?[KXk=1pkln(0+k(Mk+1))]2: Cn=C0en(p)+pnZn 4

5 4AnExample. Toillustratethestrategycalculationanditsexpectedreturnforaknownbiasedmodelwith asmallnumberofcells(k=4),considerbetsongreen,therstdozen,seconddozen, andthirddozen(seeforexamplescarne(1961)page365).specically,wecombinegreen outcomesf0;00gintoacellwithindexk=1,outcomesf1;2;:::;12gintoacellwithindex intoacellwithindexk=4. k=2,outcomesf13;14;:::;24gintoacellwithindexk=3,andoutcomesf25;26;:::;36g $2:00foreachofthedozens. (2=38;12=38;12=38;12=38)Tforacompletelybalancedwheel.Table1givestheoptimal strategycalculationinthiscase.table1.computation. Thepayoamountsfora$1:00betareM1=$17:00forgreen,withM2=M3=M4= Letthetruecellfrequenciesbep=(3=38;14=38;12=38;9=38)Tascomparedto k13/381754/38:= /323:=.983/95 214/38242/38:= /209:=.908/95 312/38236/38:=0.9581/95:=.853/95 49/38227/38:=0.71 pkmkpk(mk+1) 0[k]k weobtainfromequation(11) Itisinterestingtonotethat3=3=95>0despitep3(M3+1)=36=38<1. TocalculatetheexpectedreturnforstartingcapitalofC0=$1000aftern=100games y:pkj=1(1?(mj+1)?1<0,0=0[3]=81=95 y0 [(3 With 38)(395)2182+(14 E(C1=C0)=81 38)(895)232+(12 EC100=$1000[EC1=C0]100:=$7;607:50: 95+(338)(3 38)(395)232?(3 95)18+(14 38)(8 38)(395)18+(14 95)3+(12 95)3:=1: wehavethecoecientofvariation andthestandarddeviationfromequation(12)is 1=(C1=C0)=E(C1=C0):=0: =(C1=C0):=0: )(8 95)3+(12 38)(395)3)]1=2 Notetheextremevariability. (C100)=(EC100)[(1+21)100?1]1=2:=22;139:21: 5

6 5TheCaseofUnknown(p1;p2;:::;pK). Wenowconsiderthestatisticalproblemofdeterminingastrategy[n]=(1;2;:::;n)(K+1)n fornconsecutivegambles,tomaximize,forunknownxedp,theexpectedlogreturn ThomasKurtzmentionedbyEthier(1982),weconsidertheBayesstrategyfortheDirichlet priorjointdensityfp(p)= Asameansofderivingaclassofinterestingstrategies,andnotingthesuggestionby EX[n]jpln(Cn([n];X[n])): byraiaandschlaifer(1961)(seeforexamplewilks(1962)ordegroot(1970)).the convenientpriorforthemultinomialmodelandistheconjugatepriorgenerallyrecommended posteriordistributionalsobelongstothisdirichletclassofdistributions. where0<pk<1,pkj=1pj=1,k>0,+=pkj=1j.thisisanaturalandmathematically?(1)?(2)?(k)p1?1?(+) 1p2?1 2pK?1 ABayessolutionwillminimizetheBayesriskormaximizetheexpectedlogreturnaveragedwithrespecttotheDirichletprior: TondtheBayesstrategy,rewriteequation(13)as EPEX[n?1]jPEXnjX[n?1];Pfln(Cn?1)+ln(n0+KXk=1nk(Mk+1)Xnk)g EPEX[n]jPln(Cn([n];X[n])): (13) K where =EX[n?1]EPjX[n?1]fln(Cn?1)+KXk=1pkln(n0+nk(Mk+1))g istheposteriorexpectationofpkbasedonn?1observationswithsk[n?1]=pn?1 =EX[n?1]fln(Cn?1)+KXk=1^pnkln(n0+nk(Mk+1))g forthestartingstrategy(i=1). byequation(9)withpkreplacedbythebayesestimates^pikfori=1;2;:::;nwheresk[0]=0 ItfollowsthattheithcolumnofaBayesstrategy^[n]=(^1;^2;:::;^n)(K+1)nisgiven ^pnk=e(pkjx[n?1])=k+sk[n?1] ++n?1 i=1xik. 6

7 6LargeSampleOptimality. givenby Intheorem1weshowitattainstheexpectedlogreturn(p)fortheoptimalstrategywhen FortheBayesstrategy,considerthelimitingaverageexpectedlogreturnoverngambles pisknown. Write n!1ex[n]jp1nln(cn(^[n];x[n]): andconsiderthenthterminthesumontheright =1nln(C0)+1nnXi=1EX[n]jpln(^i0+KXk=1^ik(Mk+1)Xik) n=ex[n]jpln(^n0+kxk=1^nk(mk+1)xnk) (14) =EX[n?1]jpEXnjX[n?1];pln(^n0+KXk=1^nk(Mk+1)Xnk) =EX[n?1]jpKXk=1pkln(^n0+^nk(Mk+1)): (15) Lemma1Forxedp,Bayesestimator^pnandstrategy^[n]=(^1;^2;:::;^n)(K+1)nwe Torstshow havekxk=1pkln(^pnk(mk+1))kxk=1pkln(^n0+^nk(mk+1))kxk=1pkln(mk+1): Proof.Therightinequalityholdssince(^n0+^nk(Mk+1))(Mk+1). n!(p)asn!1weprovethefollowing Fortheleftinequality, =rxi=1pjiln(^pnk(mk+1))+kx since^n0^pnk(mk+1)fork>rwhererisdenedinequation(8)withprobabilities^pnk. Thiscompletestheproof. KXk=1pkln(^pk(Mk+1)) k=r+1pkln(^n0) Nextweprove 7

8 asn!1. Lemma2For0<pk<1,k=1;2;:::;KandtheBayesestimates^pnkwehave thedenitionof^pnkwehave Choose",0<"<pkanddenethesetAnk=f!:Sk[n?1](n?1)(pk?")g.Using Proof.Byadditivity,itsucestoproveEln(^pnk)!ln(pk)asn!1. EX[n]jpKXk=1pkln(^pnk(Mk+1))!KXk=1pkln(pk(Mk+1)) and 0<k=(++(n?1))(k+Sk[n?1])=(++(n?1))=^pnk<1 ln ++(n?1)!p(ank)eln(^pnk)iank0: usingfeller(1950page140,(3.6)).itfollowsasn!1that Forsn=b(n?1)(pk?")cthegreatestintegernotexceeding(n?1)(pk?"),wehave n?1 sn!psn k(1?pk)n?1?snp(ank) sn!psn k(1?pk)n?1?sn (n?sn)pk npk?sn! (16) using(16). OnthecomplementsetAcnkwehaveSk[n?1]>(n?1)(pk?")sothat ln 1^pnk=Sk[n?1]+k ++(n?1)!p(ank)!0andeln(^pnk)iank!0 k vergencetheoremtoobtain fornsucientlylarge.thusln(^pnk)isboundedonacnkandweapplythedominatedcon- pk?" 1++=(n?1)(pk?")=2 n?1++>(n?1)(pk?")+k Eln(^pnk)IAcnk!ln(pk) since^pnkconvergesalmostsurelytopkbythestronglawoflargenumbersandiacnk!1 tonishtheproofoflemma2. almostsurely.finally Nowusingthelemmasweconclude Eln(^pnk)=Eln(^pnk)IAnk+Eln(^pnk)IAcnk!0+ln(pk) 8

9 Theorem1Forxedpk,0<pk<1andBayesstrategy^[n]wehave n!1,where (^pn),usingequations(9)and(10),convergesalmostsurelytoanoptimalstrategyforp Proof.Since^pnkconvergesalmostsurelytopk,itfollowsthataBayesstrategy^n= nandaredenedbyequations(15)and(10),and n!1ex[n]jp1nln(cn(^[n];x[n])=(p): lim n!(p)as known almostsurelysinceitisalsoacontinuousfunction.ifwedenotetherandomfunction asn!1sincetheequationsarecontinuousinp.thusthefunction hn=kxk=1pkln(^n0+^nk(mk+1))!(p) ^n!(p) gnhng.sincegn!g=pkk=1pkln(pk(mk+1)),andbylemma2,egn!g,wecan applythetheoremofpratt(1960)toconclude gn=pkk=1pkln(^pnk(mk+1))andtheconstantg=pkk=1pkln(mk+1),thenbylemma1, appliedtoequation(14). 7PerformanceoftheBayesStrategy. (p)usingn?1ln(c0)!0andtoeplitzlemma(seeforexampleash(1972page270)) Forthesecondpartofthetheorem,theaverageexpectedlogreturnalsoconvergesto n=ehn!(p). forallcellnumbers(k=).ifisselectedtobelarge,thebayesstrategyobservesfor quiteafewgameswithoutbetting.ifthewheelisfavorablybiased,eventuallythebayes Forsinglenumberbets,symmetryconsiderationssuggestusingpriorparametersthesame estimates^pnkwilldiscoverthisaftermanygamesandbettingwillbegin.ifasmallis astheinitialcapital.sampleaveragesandsamplestandarddeviations used,chanceuctuationsinthecountsleadtoearlybettingonunfavorablecellsresultingin capitalnearzero. ideaofperformance.table2givesresultsfornplaysusingasthepriorparameterandc0 BecauseofthecomplexityofCnwhen[n]dependsonX[n?1],wesimulatetogetsome forr=10;000samplesaregivenwherecn(j)isthecapitalattheendofnplaysforthejth samplereplication.abiasedwheelwithp1=1=30,p2=:::=p38=29=1110wasused.this Wilson(1965). degreeofbiasinthewheelisconsistentwiththeestimatesforwheelimbalancediscussedby Cn=RXj=1Cn(j)=R;S(Cn)=[RXj=1(Cn(j)?Cn)2=(R?1)]1=2 9

10 (seeforexamplemarsaglia(1972)).adierentstartingseedintegera06=0wasusedto thejthrandomnumberon(0;1)givenbyxj=aj=232.theperiodis232=4=1;073;741;824 computeeachentry. Themultiplicativerandomnumbergeneratoraj=(aj?169069)mod232wasusedwith (S(Cn))n=100 =1 p=(1=30;29=1110;:::;29=1110);mk=$35;c0=$1000;r=10;000. Cn 20.12Table2.BayesStrategySimulation ( )( ) ( )( )(272.36)(1.15)(0.04) (906.92)(36.39)(0.00)(0.00)(0.000) (0.51)(0.00)(0.00) (0.00) (613.25)( )( )(296.89)(54.02) (213.69)(424.02)(915.12)( )( )(832.58) (29.13)(85.02)(323.98)( )( )( ) (5.88) (1.48)(10.55)(67.40)(227.77)( )( ) (0.00)(0.20)(3.34)(30.05)(230.72)( ) usinglogarithmicscaleclassintervalsandpositivecountsasgivenintable3. Figure1givesthehistogramof10;000valuesofCnfor=100,n=5000,C0=$1000 (0.00)(0.00)(0.00)(0.18)(5.44)(174.80) Interval[26;27)[27;28)[28;29)[29;210)[210;211)[211;212)[212;213)[213;214)[214;215) Interval[215;216)[216;217)[217;218)[218;219)[219;220)[220;221)[221;222)[225;1) Count Alog-normaldistributionissuggestedasinthecaseofknownp Table3.Countsfor10,000valuesofCn

11 knownwecalculate andtocomparetheseresultswiththoseforp=(1=30;29=1110;:::;29=1110)tassumed Figure1.Countsof10;000valuesofCnwith=100,n=5000,C0=$1000. E(C1=C0)=(174=175+(1=30)(1=175)36):=1: =(174=175;1=175;0;:::;0)T ^2 2^5 2^8 2^11 2^14 2^17 2^20 2^23

12 Table4givescorrespondingexpectationsandstandarddeviations. 1=[(1=30)(1=175)2362?((1=30)(1=175)36)2]1=2 Table4.OptimalStrategyValues ne(c1=c0) ECn (Cn) :=0: : Inadditiontoresultsforabiasedwheel,itisofinteresttoseehowtheBayesstrategy performsintheequiprobablecasep=(1=38;1=38;:::;1=38)whenthereisnofavorablebet (Mk=$35).Table5givesCnand(S(Cn))forR=10;000simulationsforsome(;n) values. (S(Cn))n=1000 = Cnpk=1=38,R=10;000. Table5.BayesSimulation (460.16)(570.81)(113.28) (148.20)(294.36)(349.26) (15.15)(61.25)(204.69) $5:714)canbebet.InrealitythereisusuallyaminimumbetandCnkmustbean 8PracticalConsiderations. TheBayesstrategyassumesthatanarbitraryfractionofthecapital(e.g.$10001=175= integermultipleofthisminimumbet.thisrestrictionshoulddiminishtheexponentialrate (0.00)(0.12)(7.39) possibletobringacomputerintosomecasinosalthoughtoday'spalmtopportablesare betoneachplayaswellasamaximumbetlimitationfurthercomplicatesimplementingthe strategy. alargeinitialcapitalc0.inaddition,arequirementtobetatleastfourtimestheminimum ofcapitalincreaseforafavorablybiasedwheel.theeectcanbereducedbystartingwith Anotherdicultyisthecomputationrequiredtodeterminebets.Itmaybenotbe 12

13 betting.apracticalapproximationmightinitiallyobservethewheelforalongperiod withoutbetting,ifthisispermitted,andthenuseaxedstrategyforpestimatedfrom isafelonyinnevada. theinitialcounts-a\wheelclocking"approach. unobtrusive.accordingtobarnhart(1992),theuseofanelectronicdevicetoaidgambling isnoguaranteethatthecasinowillnotchangethewheeliftherearelargewinnings. ones.evenifafavorablewheelisfoundsuchasdescribedbywilson(1965page33),there TheBayesstrategiesthatdowellintable2havealongperiodinitiallywithverylittle (e.g.food,lodging,parking,etc.).includingsuchanoverheadcostwouldchangethebayes solutionandmakefavorablereturnsevenmoredicult. Amajordicultyisndingawheelwithsucientfavorablebiasandavoidinggaed 9Conclusions. Iftruefrequenciesareknownaccuratelyandafavorablebiasexists,theoptimalstrategy Wedidnotincludeaxedoverheadcosttoobservethewheelwhennobetsareplaced earningpossibilitiesusingpriorparametervaluesintherange200500arequite frequenciesareunknown,anexponentialincreasealsooccursonfavorablybiasedwheelsbut interestingfor2000ormoreplays.however,becauseofextremevariation,largelossesas wellaslargewinningsarepossible. aconsiderablenumberoftrialsarerequired(seetable2).despiteallthedicultiesthe expectedreturnincreasesexponentiallywiththenumberofgamesplayed(seetable4).when Acknowledgements. References ThanksgotoStewartEthierforhelpfulcorrectionsandsimplicationstoanearlierdraft. ThanksalsogotoMarkFinkelstein,RichardJohnson,TomLeonard,BinYu,andDavid Blackwellforcommentsandsuggestionsforimprovement. [1]Ash,RobertB.(1972)RealAnalysisandProbability.SanDiego:AcademicPress,Inc. [2]Barnhart,R.T.(1992)BeatingtheWheel.WinningStrategiesatRoulette.NewYork: [4]Ethier,S.N.(1982)Testingforfavorablenumbersonaroulettewheel.Journalofthe [3]Breiman,L.(1961)Optimalgamblingsystemsforfavorablegames.Proceedingsofthe FourthBerkeleySymposiumonMathematicalStatisticsandProbabilityI65-78.UniversityofCaliforniaPress LyleStuart. AmericanStatisticalAssociation77,

14 [5]Feller,William(1950)AnIntroductiontoProbabilityTheoryandItsApplications,Vol [6]Finkelstein,MarkandWhitley,Robert(1981)Optimalstrategiesforrepeated [7]Kelley,J.L.(1956)Anewinterpretationofinformationrate.BellSystemTechnicalJournal36, games.advancesinappliedprobability13, I.NewYork:JohnWiley&Sons. [8]Mangasarian,O.L.(1969)NonlinearProgramming.NewYork:McGrawHill. [9]Marsaglia,George(1972)Thestructureoflinearcongruentalsequences,inApplicationsofNumberTheorytoNumericalAnalysis.EditedbyS.K.Zaremba.NewYork: AcademicPress, [10]Pratt,JohnW.(1960)Oninterchanginglimitsandintegrals.AnnalsofMathematical [11]Raia,H.andSchlaifer,R.(1961)AppliedStatisticalDecisionTheory.DivisionofResearch,GraduateSchoolofBusinessAdministration,HarvardUniversity,Boston. [12]Rockafellar,R.T.(1970)ConvexAnalysis.Princeton:PrincetonUniversityPress. [13]Scarne,John(1961)Scarne'sCompleteGuidetoGambling.NewYork:Simon&Schuster. York:NorthHolland. Statistics31, [14]Srivastava,M.SandKhatri,C.G.(1979)AnIntroductiontoMultivariateStatistics.New [15]Wilks,S.S.(1962)MathematicalStatistics.NewYork:JohnWiley&Sons. [16]Wilson,AllanN.(1965)TheCasinoGambler'sGuide.NewYork:Harper&Row. 14

( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( )

( ) = ( ) = {,,, } β ( ), < 1 ( ) + ( ) = ( ) + ( ) { } ( ) = ( ) = {,,, } ( ) β ( ), < 1 ( ) + ( ) = ( ) + ( ) max, ( ) [ ( )] + ( ) [ ( )], [ ( )] [ ( )] = =, ( ) = ( ) = 0 ( ) = ( ) ( ) ( ) =, ( ), ( ) =, ( ), ( ). ln ( ) = ln ( ). + 1 ( ) = ( ) Ω[ (

More information

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?

36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous? 36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this

More information

Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

More information

Pain-Free Injections

Pain-Free Injections T N S Bk P-F Ij I P Bk DV Dv Iv D D - I 8 8 9 1 D v C U C Y N D j I W U v v I k I W j k T k z D M I v 965 1 M Tk C P 1 2 v T I C G v j V T D v v A I W S DD G v v S D @ v DV M I USA! j DV - v - j I v v

More information

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010 MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic

More information

Lecture 2 Binomial and Poisson Probability Distributions

Lecture 2 Binomial and Poisson Probability Distributions Lecture 2 Binomial and Poisson Probability Distributions Binomial Probability Distribution l Consider a situation where there are only two possible outcomes (a Bernoulli trial) H Example: u flipping a

More information

SteganographyinaVideoConferencingSystem? AndreasWestfeld1andGrittaWolf2 2InstituteforOperatingSystems,DatabasesandComputerNetworks 1InstituteforTheoreticalComputerScience DresdenUniversityofTechnology

More information

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2 . Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the

More information

Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

Average rate of change of y = f(x) with respect to x as x changes from a to a + h: L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,

More information

Math 115 Spring 2011 Written Homework 5 Solutions

Math 115 Spring 2011 Written Homework 5 Solutions . Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

More information

Manual for SOA Exam MLC.

Manual for SOA Exam MLC. Chapter 4. Life Insurance. Extract from: Arcones Manual for the SOA Exam MLC. Fall 2009 Edition. available at http://www.actexmadriver.com/ 1/14 Level benefit insurance in the continuous case In this chapter,

More information

ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME

ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME Alexey Chuprunov Kazan State University, Russia István Fazekas University of Debrecen, Hungary 2012 Kolchin s generalized allocation scheme A law of

More information

From Binomial Trees to the Black-Scholes Option Pricing Formulas

From Binomial Trees to the Black-Scholes Option Pricing Formulas Lecture 4 From Binomial Trees to the Black-Scholes Option Pricing Formulas In this lecture, we will extend the example in Lecture 2 to a general setting of binomial trees, as an important model for a single

More information

Lecture 13: Martingales

Lecture 13: Martingales Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

More information

Taylor and Maclaurin Series

Taylor and Maclaurin Series Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

More information

Compound Inequalities. AND/OR Problems

Compound Inequalities. AND/OR Problems Compound Inequalities AND/OR Problems There are two types of compound inequalities. They are conjunction problems and disjunction problems. These compound inequalities will sometimes appear as two simple

More information

Matrix-Chain Multiplication

Matrix-Chain Multiplication Matrix-Chain Multiplication Let A be an n by m matrix, let B be an m by p matrix, then C = AB is an n by p matrix. C = AB can be computed in O(nmp) time, using traditional matrix multiplication. Suppose

More information

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +... 6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

More information

Wealsopresentaperformancemodelanduseittoanalyzeouralgorithms.Wendthatasymp- 1.1.Dataparallelism.Highlyparallel,localmemorycomputerarchitectures

Wealsopresentaperformancemodelanduseittoanalyzeouralgorithms.Wendthatasymp- 1.1.Dataparallelism.Highlyparallel,localmemorycomputerarchitectures Machine,adistributed-memorySIMDmachinewhoseprogrammingmodelconceptuallysuppliesone Choleskyfactorizationofasparsematrix.OurexperimentalimplementationsareontheConnection processorperdataelement.incontrasttospecial-purposealgorithmsinwhichthematrixstructure

More information

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

More information

I. Pointwise convergence

I. Pointwise convergence MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.

More information

Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates

Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates Cash Flow Financial transactions and investment opportunities are described by cash flows they generate. Cash flow: payment

More information

AP Calculus AB 2011 Scoring Guidelines

AP Calculus AB 2011 Scoring Guidelines AP Calculus AB Scoring Guidelines The College Board The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 9, the

More information

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

F = ma. F = G m 1m 2 R 2

F = ma. F = G m 1m 2 R 2 Newton s Laws The ideal models of a particle or point mass constrained to move along the x-axis, or the motion of a projectile or satellite, have been studied from Newton s second law (1) F = ma. In the

More information

Practice with Proofs

Practice with Proofs Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

More information

Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

More information

2.3. Proposition. There exists an enumeration operator W a such that χ A = W a (A ) for all A N.

2.3. Proposition. There exists an enumeration operator W a such that χ A = W a (A ) for all A N. THE ω-enumeration DEGREES IVAN N. SOSKOV Abstract. In the present paper we initiate the study of the partial ordering of the ω-enumeration degrees. This ordering is a semi-lattice which extends the semi-lattice

More information

Algorithms and Methods for Distributed Storage Networks 9 Analysis of DHT Christian Schindelhauer

Algorithms and Methods for Distributed Storage Networks 9 Analysis of DHT Christian Schindelhauer Algorithms and Methods for 9 Analysis of DHT Institut für Informatik Wintersemester 2007/08 Distributed Hash-Table (DHT) Hash table does not work efficiently for inserting and deleting Distributed Hash-Table

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis

Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)! Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

More information

RNA-seq. Quantification and Differential Expression. Genomics: Lecture #12

RNA-seq. Quantification and Differential Expression. Genomics: Lecture #12 (2) Quantification and Differential Expression Institut für Medizinische Genetik und Humangenetik Charité Universitätsmedizin Berlin Genomics: Lecture #12 Today (2) Gene Expression per Sources of bias,

More information

100. In general, we can define this as if b x = a then x = log b

100. In general, we can define this as if b x = a then x = log b Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,

More information

Chapter 4: Nominal and Effective Interest Rates

Chapter 4: Nominal and Effective Interest Rates Chapter 4: Nominal and Effective Interest Rates Session 9-10-11 Dr Abdelaziz Berrado 1 Topics to Be Covered in Today s Lecture Section 4.1: Nominal and Effective Interest Rates statements Section 4.2:

More information

Principles of Annuities

Principles of Annuities Principles of Annuities The Mathematical Foundation of Retirement Planning Knut Larsen Brigus Learning Inc. Friday, June 5, 2015 Learning Objectives for this Presentation 1. Know the structure of the course

More information

2. Discrete random variables

2. Discrete random variables 2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be

More information

Math 431 An Introduction to Probability. Final Exam Solutions

Math 431 An Introduction to Probability. Final Exam Solutions Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

More information

Transient Voltage Suppressor SMBJ5.0 - SMBJ440CA

Transient Voltage Suppressor SMBJ5.0 - SMBJ440CA Features: Glass passivated junction Low incremental surge resistance, excellent clamping capability 600W peak pulse power capability with a 10/1,000μs waveform, repetition rate (duty cycle): 0.01% Very

More information

Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

More information

1 Interest rates, and risk-free investments

1 Interest rates, and risk-free investments Interest rates, and risk-free investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 ($) in an account that offers a fixed (never to change over time)

More information

Introduced by Stuart Kauffman (ca. 1986) as a tunable family of fitness landscapes.

Introduced by Stuart Kauffman (ca. 1986) as a tunable family of fitness landscapes. 68 Part II. Combinatorial Models can require a number of spin flips that is exponential in N (A. Haken et al. ca. 1989), and that one can in fact embed arbitrary computations in the dynamics (Orponen 1995).

More information

Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

More information

Sequences and Series

Sequences and Series Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite

More information

10.2 Series and Convergence

10.2 Series and Convergence 10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

More information

Solutions of Equations in One Variable. Fixed-Point Iteration II

Solutions of Equations in One Variable. Fixed-Point Iteration II Solutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

Wales & West Utilities Ltd

Wales & West Utilities Ltd Wales & West Utilities Ltd Wales & West House Spooner Close Coedkernew Newport NP10 8FZ Registered in England and Wales: No. 5046791 Indicative Distribution Transportation Charges from 1 April 2015 1.

More information

Modeling and Analysis of Information Technology Systems

Modeling and Analysis of Information Technology Systems Modeling and Analysis of Information Technology Systems Dr. János Sztrik University of Debrecen, Faculty of Informatics Reviewers: Dr. József Bíró Doctor of the Hungarian Academy of Sciences, Full Professor

More information

SHORETEL GENERAL SERVICES ADMINSTRATION Federal Acquisition Service Authorized Federal Supply Schedule Price List - June 2014

SHORETEL GENERAL SERVICES ADMINSTRATION Federal Acquisition Service Authorized Federal Supply Schedule Price List - June 2014 SHORETEL GENERAL SERVICES ADMINSTRATION Federal Acquisition Service Authorized Federal Supply Schedule Price List - June 2014 Contract# GS-35F-0085U Contract Term: 11-9-2012 through 11-8-2017 Federal ID#:

More information

EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

More information

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.

Critical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima. Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =

More information

Key Concepts and Skills

Key Concepts and Skills McGraw-Hill/Irwin Copyright 2014 by the McGraw-Hill Companies, Inc. All rights reserved. Key Concepts and Skills Be able to compute: The future value of an investment made today The present value of cash

More information

The Demand for Life Insurance: An Application of the Economics of Uncertainty: A Comment

The Demand for Life Insurance: An Application of the Economics of Uncertainty: A Comment THE JOlJKNAL OF FINANCE VOL. XXXVII, NO 5 UECEMREK 1982 The Demand for Life Insurance: An Application of the Economics of Uncertainty: A Comment NICHOLAS ECONOMIDES* IN HIS THEORETICAL STUDY on the demand

More information

Universal Algorithm for Trading in Stock Market Based on the Method of Calibration

Universal Algorithm for Trading in Stock Market Based on the Method of Calibration Universal Algorithm for Trading in Stock Market Based on the Method of Calibration Vladimir V yugin Institute for Information Transmission Problems, Russian Academy of Sciences, Bol shoi Karetnyi per.

More information

Sample Solutions for Assignment 2.

Sample Solutions for Assignment 2. AMath 383, Autumn 01 Sample Solutions for Assignment. Reading: Chs. -3. 1. Exercise 4 of Chapter. Please note that there is a typo in the formula in part c: An exponent of 1 is missing. It should say 4

More information

Financial Mathematics for Actuaries. Chapter 1 Interest Accumulation and Time Value of Money

Financial Mathematics for Actuaries. Chapter 1 Interest Accumulation and Time Value of Money Financial Mathematics for Actuaries Chapter 1 Interest Accumulation and Time Value of Money 1 Learning Objectives 1. Basic principles in calculation of interest accumulation 2. Simple and compound interest

More information

東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文

東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文 東 海 大 學 資 訊 工 程 研 究 所 碩 士 論 文 指 導 教 授 楊 朝 棟 博 士 以 網 路 功 能 虛 擬 化 實 作 網 路 即 時 流 量 監 控 服 務 研 究 生 楊 曜 佑 中 華 民 國 一 零 四 年 五 月 摘 要 與 的 概 念 一 同 發 展 的, 是 指 利 用 虛 擬 化 的 技 術, 將 現 有 的 網 路 硬 體 設 備, 利 用 軟 體 來 取 代 其

More information

Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014

Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014 Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, steen@cs.vu.nl Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction

More information

Manual for SOA Exam MLC.

Manual for SOA Exam MLC. Chapter 5 Life annuities Extract from: Arcones Manual for the SOA Exam MLC Fall 2009 Edition available at http://wwwactexmadrivercom/ 1/70 Due n year deferred annuity Definition 1 A due n year deferred

More information

Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations

Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations 56 Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Florin-Cătălin ENACHE

More information

CONTINUOUS COUNTERPARTS OF POISSON AND BINOMIAL DISTRIBUTIONS AND THEIR PROPERTIES

CONTINUOUS COUNTERPARTS OF POISSON AND BINOMIAL DISTRIBUTIONS AND THEIR PROPERTIES Annales Univ. Sci. Budapest., Sect. Comp. 39 213 137 147 CONTINUOUS COUNTERPARTS OF POISSON AND BINOMIAL DISTRIBUTIONS AND THEIR PROPERTIES Andrii Ilienko Kiev, Ukraine Dedicated to the 7 th anniversary

More information

Essential Topic: Continuous cash flows

Essential Topic: Continuous cash flows Essential Topic: Continuous cash flows Chapters 2 and 3 The Mathematics of Finance: A Deterministic Approach by S. J. Garrett CONTENTS PAGE MATERIAL Continuous payment streams Example Continuously paid

More information

Math 202-0 Quizzes Winter 2009

Math 202-0 Quizzes Winter 2009 Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile

More information

IN recent years, advertising has become a major commercial

IN recent years, advertising has become a major commercial IEEE/ACM TRANSACTIONS ON NETWORKING On Modeling Product Advertisement in Large Scale Online Social Networks Yongkun Li, Bridge Qiao Zhao, and John C.S. Lui Fellow, IEEE, ACM Abstract We consider the following

More information

A Laboratory Information. Management System for the Molecular Biology Lab

A Laboratory Information. Management System for the Molecular Biology Lab A Laboratory Information L I M S Management System for the Molecular Biology Lab This Document Overview Why LIMS? LIMS overview Why LIMS? Current uses LIMS software Design differences LIMS software LIMS

More information

ÖVNINGSUPPGIFTER I SAMMANHANGSFRIA SPRÅK. 15 april 2003. Master Edition

ÖVNINGSUPPGIFTER I SAMMANHANGSFRIA SPRÅK. 15 april 2003. Master Edition ÖVNINGSUPPGIFTER I SAMMANHANGSFRIA SPRÅK 5 april 23 Master Edition CONTEXT FREE LANGUAGES & PUSH-DOWN AUTOMATA CONTEXT-FREE GRAMMARS, CFG Problems Sudkamp Problem. (3.2.) Which language generates the grammar

More information

Administrative Professionals Staffing Practices by Susan Malanowski

Administrative Professionals Staffing Practices by Susan Malanowski Administrative Professionals Staffing Practices by Susan Malanowski Introduction This paper provides benchmark data on the relationships of the number of administrative professionals to the number of total

More information

Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, and Discrete Changes

Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, and Discrete Changes Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, Discrete Changes JunXuJ.ScottLong Indiana University August 22, 2005 The paper provides technical details on

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

2008 AP Calculus AB Multiple Choice Exam

2008 AP Calculus AB Multiple Choice Exam 008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus

More information

Online Appendix Bid Preference Programs and Participation in Highway Procurement Auctions

Online Appendix Bid Preference Programs and Participation in Highway Procurement Auctions Online Appendix Bid Preference Programs and Participation in Highway Procurement Auctions Elena Krasnokutskaya Department of Economics, University of Pennsylvania Katja Seim Department of Business & Public

More information

Chapter 9 Lecture Notes: Acids, Bases and Equilibrium

Chapter 9 Lecture Notes: Acids, Bases and Equilibrium Chapter 9 Lecture Notes: Acids, Bases and Equilibrium Educational Goals 1. Given a chemical equation, write the law of mass action. 2. Given the equilibrium constant (K eq ) for a reaction, predict whether

More information

ELEMENTARY PROBLEMS AND SOLUTIONS

ELEMENTARY PROBLEMS AND SOLUTIONS Edited by A. P. HILLMAN Assistant Editors GLORIA C. PADILLA CHARLES R. WALL Send all communications regarding ELEMENTARY PROBLEMS SOLUTIONS to PROFESSOR A. P. HILLMAN; 709 Solano Dr., S.E.,; Albuquerque,

More information

To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.

To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes. INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number

More information

Estimation following a phase II/III clinical trial

Estimation following a phase II/III clinical trial Estimation following a phase II/III clinical trial Sue Todd Department of Mathematics and Statistics University of Reading joint work with Peter Kimani and Nigel Stallard Warwick Medical School Cambridge

More information

Chapter Two. THE TIME VALUE OF MONEY Conventions & Definitions

Chapter Two. THE TIME VALUE OF MONEY Conventions & Definitions Chapter Two THE TIME VALUE OF MONEY Conventions & Definitions Introduction Now, we are going to learn one of the most important topics in finance, that is, the time value of money. Note that almost every

More information

Parameter Estimation for Black-Scholes Equation. Peter Gross Advisor: Dr. Jialing Dai. Final Report URA Spring 2006

Parameter Estimation for Black-Scholes Equation. Peter Gross Advisor: Dr. Jialing Dai. Final Report URA Spring 2006 Parameter Estimation for Black-Scholes Equation Peter Gross Advisor: Dr. Jialing Dai Final Report URA Spring 2006 Abstract The Black-Scholes equation is a hallmark of mathematical finance, and any study

More information

Manual for SOA Exam MLC.

Manual for SOA Exam MLC. Chapter 4 Life Insurance Extract from: Arcones Manual for the SOA Exam MLC Fall 2009 Edition available at http://wwwactexmadrivercom/ 1/13 Non-level payments paid at the end of the year Suppose that a

More information

General Network Analysis: Graph-theoretic. COMP572 Fall 2009

General Network Analysis: Graph-theoretic. COMP572 Fall 2009 General Network Analysis: Graph-theoretic Techniques COMP572 Fall 2009 Networks (aka Graphs) A network is a set of vertices, or nodes, and edges that connect pairs of vertices Example: a network with 5

More information

Financial Mathematics for Actuaries. Chapter 2 Annuities

Financial Mathematics for Actuaries. Chapter 2 Annuities Financial Mathematics for Actuaries Chapter 2 Annuities Learning Objectives 1. Annuity-immediate and annuity-due 2. Present and future values of annuities 3. Perpetuities and deferred annuities 4. Other

More information

Engineering Change Notice Engineer training

Engineering Change Notice Engineer training Engineering Change Notice Engineer training Rules(1/2) No phone calls in the training room I know you are busy, if the phone call is urgent, take it and I will cover what you missed at the end if necessary

More information

Inverse Functions and Logarithms

Inverse Functions and Logarithms Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a one-to-one function if it never takes on the same value twice; that

More information

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

More information

Lectures on Number Theory. Lars-Åke Lindahl

Lectures on Number Theory. Lars-Åke Lindahl Lectures on Number Theory Lars-Åke Lindahl 2002 Contents 1 Divisibility 1 2 Prime Numbers 7 3 The Linear Diophantine Equation ax+by=c 12 4 Congruences 15 5 Linear Congruences 19 6 The Chinese Remainder

More information

Black-box Performance Models for Virtualized Web. Danilo Ardagna, Mara Tanelli, Marco Lovera, Li Zhang ardagna@elet.polimi.it

Black-box Performance Models for Virtualized Web. Danilo Ardagna, Mara Tanelli, Marco Lovera, Li Zhang ardagna@elet.polimi.it Black-box Performance Models for Virtualized Web Service Applications Danilo Ardagna, Mara Tanelli, Marco Lovera, Li Zhang ardagna@elet.polimi.it Reference scenario 2 Virtualization, proposed in early

More information

BX in ( u, v) basis in two ways. On the one hand, AN = u+

BX in ( u, v) basis in two ways. On the one hand, AN = u+ 1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

More information

Factorization Theorems

Factorization Theorems Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization

More information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

x a x 2 (1 + x 2 ) n.

x a x 2 (1 + x 2 ) n. Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

More information

SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions

SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions SAT Subject Test Practice Test II: Math Level II Time 60 minutes, 50 Questions All questions in the Math Level 1 and Math Level Tests are multiple-choice questions in which you are asked to choose the

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

How To Price Garch

How To Price Garch 2011 3rd International Conference on Information and Financial Engineering IPEDR vol.12 (2011) (2011) IACSIT Press, Singapore A Study on Heston-Nandi GARCH Option Pricing Model Suk Joon Byun KAIST Business

More information

Arithmetic Coding: Introduction

Arithmetic Coding: Introduction Data Compression Arithmetic coding Arithmetic Coding: Introduction Allows using fractional parts of bits!! Used in PPM, JPEG/MPEG (as option), Bzip More time costly than Huffman, but integer implementation

More information

Pythagorean vectors and their companions. Lattice Cubes

Pythagorean vectors and their companions. Lattice Cubes Lattice Cubes Richard Parris Richard Parris (rparris@exeter.edu) received his mathematics degrees from Tufts University (B.A.) and Princeton University (Ph.D.). For more than three decades, he has lived

More information

RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES

RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES RECURSIVE ENUMERATION OF PYTHAGOREAN TRIPLES DARRYL MCCULLOUGH AND ELIZABETH WADE In [9], P. W. Wade and W. R. Wade (no relation to the second author gave a recursion formula that produces Pythagorean

More information

7. Solving Linear Inequalities and Compound Inequalities

7. Solving Linear Inequalities and Compound Inequalities 7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing

More information

Use finite approximations to estimate the area under the graph of the function. f(x) = x 3

Use finite approximations to estimate the area under the graph of the function. f(x) = x 3 5.1: 6 Use finite approximations to estimate the area under the graph of the function f(x) = x 3 between x = 0 and x = 1 using (a) a lower sum with two rectangles of equal width (b) a lower sum with four

More information