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Chapter 7（第1页）

Chapter7

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Applisce，medidoperationsresearch

Wemayassessoriprobabilitiesiwaysagtothetext.But，asDavidHandwroteinhisStatistics:AVeryShortIntrodu，‘...thecalculusisthesame’，i.e.howprobabilitiesaremanipulateddoesnotge.

Keepiralideasofthesubject:theAdditionandMultipliLaws;iheLawseNumberslinkioobjectiveprobabilities;Gaussiandistributionswhensummingrandomquaherfrequentlyarisingdistributions;meansandvariancesasusefulsummaries.

&expeowledgeoftherelevantprobabilitiestohavethepreavailablefortheexamplesinthepreviouschapter，butanapproximateaherightquestionbeareliableguidetogooddes.AsstatistiGeeBoxsaid:‘Allm，butsomeareuseful.’

&tersillustrateapplis，looselygroupeduertitles.

Brownianmotion，andrandom>

In1827，thebotaBrowpollenparticlessuspendedinliquidmovearourandhtyyearslater，AlbertEinsteingaveaioiclesweretlybeihemoletheliquid.Thismovementis，ofthreedimensions，buttobuildasatisfaodel，wefirstovementjustalhtline.

Supposethateachstepisajumpofsomefixeddistaimesleftaimesright，ilyeachtime.Thisnotionistermeda　randomositionaftermanyjumpsdependsonlyonthedifferehenumbersofjumpsineachdireeanandvariaahestartpoiionaltothenumberofjumpsmade.

Makeadeliputation:overafixedtimeperiod，　ihefrequenps，ahedistahecorrethesetwofactors，thelimitbeesotion，therandomdistancemovedhaviralLimitTheaussiandistributionwhosemeanandvariahproportiohofthetimeperiod.Ifmovemehtareequallylikely，themeanwillbezero.

&eiionforBrowionsisthatparticlesmoveinthreedimensioineasionfollowingaGaussianlawfiveionsabouthowatomsandmoleculesbehave，provokihatremovedanylingeringdoubtsabouttheirexistence.

&erm‘Brownianmotion’oughttobereservedfortheaentofpartialiquid，butitisalsooftehismathematicalmodelofthatmovement.

Randomnumbers

Thephrase‘randomoowoideas.First，asinidealgameswithdiceorroulettewheels，onenumberfromafiis，allofthembeingequallylikely.Sed，asiionofsnappingastickatarandompoiinauousintervalisopartofthatintervalbeingfavouredoverahefacilitytogsequencesofsubers，eachvaluebeiofalltheothers，hasmanyapplis，astheionwillillustrate.

In1955，asplendidbookOneMilliitsublished.Itfollowsitstitleexactly:pageafterpageofthedigitszerotonine，groupedinblocksforease，butsuccessivedigitsareentirelyuable–whatevertheretseques，youhaveoeheoday，modernputershavebuilt-ioachievethesameends.Aninitialvalue（theseed）isfedin，ahematiulaproduextvalue，whichaewseed，andsoohingrandomaboutthisprocessatall，andifthesameinitialseedisused，thesamesequeed.But，withagathematiula，thesequeedpassesabatteryofstatisticaltestsandlooks，toallisahoughitwereraermpseudo-randomsequenceisused.

Nomatterhowmuchcareistakeninthisprocess，therewillalwaysbesfearthathiddenflawsihodusedwillmatteriowhiumbersareput.Withthatdrelyingontheexperienceenumberofrespetists，IampreparedtoayputerproducesacceptablesequennumbersoheobviousdangerofinsiderfraudmeansthatthesemethodshavenoplabersinLotteries，orinUKPremiumBonds.）

Monteethods

HowmanumbersearoivespinsofastandardEuropeaewheel?Icouldbeaweenoneand37，butthoseextremeswouldoccurveryrarely;whatisthemostlikelynumberofdifferentnumbers?

&hisproblemuttome，Ididelyseeaosolveit.Thereare3737（ah59decimaldigits）possibleoutesofspinniimes，arytowritedownallthewaysinwhich，say，28differentnumberscouldarise，youquicklyloseenthusiasm.Amoreappealingapproaaso-teulation.

&heputer’sstreamofrandomnumberswasusedtosimulatetheoutesof37spinsofawheel，afterwhiputeranumbershadarisen.Thisprocesseatedonemilliontimes，leadingto24differentnumberson203，739os，while23arosejust199，262times.

&rivals，22or25numbers，eaedfewerthan160，000times.TheLawehatthefrequehedifferentouteswillsettledowntotheirrespectiveprobabilities，andthesefiguresessehematter:themostlikelyresultisthat24differentnumberswillarise，andtheceofthisisjustover20%.

Dayslater，Ikickedmyselffornotspottingastandardwaytosolvetheproblem!Icouldcalculatetheexactprobabilityofgettinumbersin37spins，foranyvalueofX，ingthedescribedabove.Butthisdoesnotiheuseofsimulationtoattackthissortofproblem–quiddirtyanswersbeuseful.Ihatthesimulationgaveanswerstwiththeexactcalboostedmygehthattheputer’sraorwasbehavingasintended.

AmoreserioususeofMonteethodsopolymerchemistry.Amolesistseoms，egarandomlytwistingsolyatplaeveid，crusthesameplace.Howfarisitlikelytobefromohemoleculetotheother?

&hiomsasbeingattheplacesvisitedbysomedrunkard，staggeringaroundatrandomonathree-dimensionallatticeforawhile，butsomehthesameplacetwice.Withouttherequirementthatnoplaceberevisited，mathematicalexpertsmakegress，butthatrestristoplicatetheproblembeyoack.

However，eveentputerprogrammerwriteaseionofthisplex，twisting，d，bymakingonemillion，tenmillion，evenabillioions，obtainanansreciseasisrequired.（RecalldeMoivre’srelyasthesquarerootofthesizeofthesimulation.）