完本神战

Chapter 7（第1页）

Chapter7

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。）

Orsupposeyouwaheareaularlyshapedleaf。Drawareglearouhehepositionsenumberofpoirandomwithiangle。Yourestimateultiplyihewhlebytheproportionofpointsthatfallwithintheleaf’sboundaries。

Asafinalillustrativeappli，supposePaulishopiuparolfillingstatioallsfourpumps，theminimumviableherewillberoomforuptoeightothercarstowaitinaqueue;eachextrapumpremovestaces，soifheihemaximumofeightpumps，therewouldbe。Toworkouthoswillmaximizehisprofits，heulationsofwhatealledfour，five，six，sevepumps。