Probabilistic Arithmetic by Robert Charles Williamson, B.E. (Q.I.T.

316 Pages·2006·5.07 MB·English

Probabilistic Arithmetic
by
RobCehratr Wlielsl iBa.mE(s.Qo .nI,M. .TE.n)g(,.Q Slcd.)
A thessiusb mittefdo rt hed egreoefD octoorfP hilosophy,
DepartmeonfEt l ectrEincgailn eerUinnigv,e rsoifQt uye ensland,
Augus1t9,8 9.
Statemenotf O riginiatly
candikdnaotwel'aesndb dge el tihemefa ,t eprriaels einntt heids
thbee osftt h e
To
theisosir sia glei,xn caesap ctk nowilnte hdtege exadtn t,dh mea tehrainsao blte en
submietitteihdnwe, hr o olrie n p arta,d efgoarrtet eho irsa ny outnhievtrey .r si
RobeWritl liamson.
Abstract
Thitsh edseivse tlhoiepd seo afp robabairliitshtmTiehcte i cia.sti om replace
arithompeertaitcoi nno unmsb weirtashr h imttecio peraotnir oannsdv oarmi ables.
Speccailfiwleay r,ie n terienns utmeedrm iectahloo fcd asl cucloaontlviuntgoi fo ns
probadbiisltirtiyTb huet iloonnsg.- tebream b tlgoeho aanlrd alinesd p ortmoo b ­
lemssu cahst hdee termoifnt hadeti isotnr iobftu htei ornor oatnsda olomgrf ea bic
(
equatuisoinanslgg orwihtihcmhsb ehedanev vee lfootprhe dede termciansies.t ic
)
To thiisnt hetinhsde ,ws eis su rvane uym boefpr r eviporuosploys ed methods
focra lcucolnavtoilnugat nirdoe npsr espernotbiandbiigsl tirtiyab nuedtx iaomnisn e
thediecrft eWse.d evesloomnpee rwe sfuolsrto smo eft hemseet hotdhLsea guerre
(
transformh isatnodmg erttahh,meob duutl timfianttdeh leuymn suitWaeb le.
)
fintdh atd etthoaeinh l osw rtdhiaerny c onvoleuqtuiaotanir coean lsc ualraet ed
o
secontdota hrdeyi fficualrtiidseuistne odg e pendencies.
Wherna ndvoamr iaapbplereaser p eaitnae ned xlpyr eisitsns io poton s stiob le
determidnies ttrhioebft uhtei ooenvx epsrrsaeilbolypn a irawpipslei ocfta htei on
conlvuotrieolna tWiepo rnosp.aom esteh foopdra rtoivaelrlcyto hmpiirsno gb ilne m
thfeo romfd ependbeonucnTydh se.as ree boonut nhddeis s troifbafu nutcitoino n
ofra ndvoamr iwahbelonen stl hyme a rigndali striobtfu hvteai roinaasrb el eksn own.
Theayrb ea soentd h Fer ecbhoeutnf dojsro idnits trfinubcuttiioonns .
Wed eveelffiocpi ent mneutmheforodirsc acla ltchuedlseaept einndge ncy bounds
ansdh ohwo twh ecyab nee xteinnadn eudm boewfra yFsu.r thewresm hoohrwoe w
theayrr ee lattote hd"e e xtepnrsiinoconiff p ulzseze"ytt h eowrhyi aclhl tohwes
calciuoolnffa utncotiffou znvzsay r i.aW betl hepusrso vipdreo baa biinltiesrtpirce ­
tatoiffo unz vzayr iaWbela elssst.ou dyl itmhibete ihnagvo ifo udterhp ee ndency
bounTdhsi.s tshhueos wesf uolfin netsesr vali nas roimsteih tmueatTtihiceo ns.
limirteisnaugll pstro o vaig deensel raawll aorfng uem bfeorfrsu zziya bvIlaner­s.
tertrieolnawsihtianhp u sm boefor t hiedrea asra el dsios cussed.
A numboefpr o ntteifarluliyat rfefuaolfsr u truersee aarriecd he ntaintfidhe ed
possaiplbicpltieos o f probaarbiitlhiwmshetitiicinchcc ,lm uadnea geomfe nntu ­
a n
meruincc eritna ianrtityni tfieclislayilsg teaennmctdseh e osftra unddyso yms tems,
are discussseodl.u otfrWi ahonindla oslbmtgr eaetiqhcue a tions wiasy still a long
offt,h neo toifdo enp endbeonucndyde sv eliontp hetidhs e wsoiusal pdp etaob reo f
indepeinndteenTrtheb eso tu.na druses effoudrle termrionbiunsogtfi n nedsesp en­
denacses umpotniceoa nns :d etthereram niognfpe eo ssriebslwuehl etnnso thiisn g
knoawbno tuhtje o idnetp endsetrnucoceft as u erotefr andvorami ables.
II
Acknowledgements
Iw oullidtk oae c knowtlhheede glaepns d u ppooftr htfelo lowpienogwp hlcoeo n­
tribtuott hetidhs e isnvi asr iwoauyss .
Mositmo prtaInw toluyll idtk oep ubllyti hcamnykw ifAen ghawrhaohd a,s
astseimdse g reaitnnl uym erwoauyossv terhy eer asW.i thhoeuhrte alnpsd u pport
thtihse sis hwaovbueel edpn o nsosItia bmdl eee.gp rlayt feoafrlut llha es sistance
shhea psr ov.i ded
Iw oualldls iotk ote h ank:
Mys urpveiPsroorsf,oe Trso mD ownsc,o nftoirsn uupepdao srsti,s atnadn ce
encouernaatgn,efd mo bre ians go lovfae drm inisptrroabtlievmest harana ther
generoaft otrh em;
Formheerao dfd epartPmreosfnoetMrs a tD arvewnhiohz eal,pm eea dsa be­
ginngirnagd sutautdeae nnldta taesrse idis nmt yo btaiaCn PiRnsAgc holianr ship
ordteor purPshuDse t umdyi es;
Theh eaodfd epanrttP,mr eofeTsosmPo arr nfeotlrhl u,es oefd epartmental
fcailiintt ihpeers e paorfat thtiihosen s is;
Dr PDhiialm ooftn hdDe e partomfMe antth emUantiivcesor,fQs uieteyn sland,
fours eafduvlai ncddei scuasnsfdiro to hnleso oafns ompea pers;
Ana nonyrmeofueosrf te hIeen ternJaotuiroonfnAa apllp roxRiemaastoen ing,
foar quewshtiliceohtdn os ect5i.oo2fcn h ap3t;e r
ProfeBsrsiAoanrnd erosfto hnAe u strNaaltiiaUonnn iavle wrhsoisstieym ,p le
quesatsti owo hne tIhh ealrdo oaktae ndly i mirteisnwugalp sta sor ftt h e motivation
focrh ap5t;e r
DrH elMecnG illoift vhDreea pya rtomfMea ntth ematicso,fQ u Ueneinvse­rsity
lanfdoa,rd iscuosncs oinofind ienntceear nvodat lhsme art taenrds ltohaoenfa
book;
DrG uyWe sotft hDee partmEecnotn ooUmfni icvse,or fQs uieteyn sfloarn d,
discuasnsdi otnhsoe fs olmpoeaa pne rs.
Thfelo lowpienogap slsseti emdeb ys endcionmgm eonnst osm qeu estainodn s
reproiftn htepsia rp ePrrsos:fo eArsl iAcgeo g(iUnnoi voefCr asliitfyao tBr enrikae ­
leyD)rP, i eBroon is(sGnoeenrEeal le cRtersrieccLah a bNse,wY o rkD)r, JBoreeln ­
ne(rP laoA lot)P,r ofeDsisdoDiruesbr o ainsHd e nPrria (dUen ivoefTr osuiltoyu se
111
lV
IIPIr)o,sf oers J iir(gIennsf tiGiAiatnrrug lteo wffa ndte UMnaitvheermsaittiakt
FrerigbP)ur,os foeRrso bGiinl( eQsu een's OUnntia)vPr,eri roosfiVetLs.y.sG,oi rr ko
(KiUenvi vietryDs)rE ,l lReins (dUanlei rvsoifOt soyl) P,r osfoeMrs.S ambandham
(AatnltUan isvietrDyr)E l,i Sea nc(hMeazr sePirlofleeBs)es,ro trSh cohlwde izer
(UnivoefrM saistsya chAumshee)trtP,ssr t,of eEsusgoeSrne en e(tUan ivoefr sity
SydnPeryo)f,Re osssSsoh ra c(ttSearn fUonrisdvi etrDyr) ,O Sshkeayr(n Mions cow),
ProsfoeUrss hSiuom i(tUan iveorfRs oicthye asntdPe rro)f eKslsaoWures i se
(PhysiTkecahlniisBscuchnh-dee saBnrsatuanlstcW,he wsGetei rgmy,a) n.
Iw oualldls iotk ote h atnhkie n dulogfe nEtclheee c tErnigcianlel eirbirnagr ian
MrBsa rbKaorram eanndtdyh ,Me a thematicMsr FsLr iebKiradanarso ikwain.
Iti sp lae atsoua rcken owal eddtegobDe to naKlndu thL easnLldai mep ort
fotrh dee veloopf'm lEXe anntfA\.d T X undwehri tchhti hse wsaiss pr,ae npda red
E
toJ amAelse xafnohdrie 'srl bIibb liopgrreapprhoiTcche essseow ret.ro meoo lsst
usuelf.
Finanscuipapflor roatmC ommonwPeoaslttghr RaedsueaaAtrweca hra dnad
grafnrtot mhA eu satlriRaens eGarrcahnS cthse imgser ateafcuklnloyw ledged.
FinaIlw loyul,li dtk oae c knowalnie ndtgeel dleebtctotK uaarPllo ppwehro,,
throhuigwshr ithiantsga su,g htth iemm ep orotfac nlcaeor ifet xyp osaintdi on
honeosftt hyo ught.
Publications
Mosotft hmea teirnti hatilhs e hsaibsse eonrw, i blelp, u blieslhseedwT hheer e.
flolowiisanl gio sfpt a peerist,ph uebrl issuhbemdi,ot rnt eeandrg,si ubmission,
whircehp moartte ornit ahlte o poiftc h tihse sSiosmo.ef t hepsaep erresp ort
fruthmeart eriails iwnnhocitlc iuhntd hetidhs e sis.
1.Ro beCr.Wt i lliaanmTdso omDn o wn"sP,r obaAbriiltihsamtnetidthc Die ic s ­
triboufFt uinocnto ifRo annsdV oarmi abPlreosc,e"eo dfti h1nesgl tsA STED
SympoosniS uimgP nracole ssainindtg s AppBlriicsabAtauingoeu1n,ss9t ,8 7,
11-2111989,7(.Pr atosfc hap2ti,en rc lfurudtihndeger t aaniedlx sa mpolne s
thhei stomgertahmo d).
2.R obeCr.Wt i lliaanmTdsoo mnD own"sT,hI en vearnsDdee termoifn ant
a 22 U nifoDrsimtlryi bRauntdeModam t rSitxat,i"sa tnidcP sr obability
x
Lett7e,1r 6s7,0- 1,1978(9hC.a p7te)r.
3.R obeCr.Wt i lliaanmTdso omDn o wn"sPo,rb abiAlriistthiNmcue mteirci:c al
Methfoodrs Covu n anDde pendBeonucnyda sc,ct"ee pd
Calculnatliing
o to s
fopru blictahtIeint oernna tiioJnnoa ulrn aolf A pproixmatRee asoni1ng9, 89.
(Cphta3er) .
4.R obeCr.Wt i llia"mAsEnox nt,rL eimmeTi hteo rfeomDr e pndeenBcoyu nds
of Normalised VSaaurbimlsae csoec,fp" tf Reopadrun bdloimic nIan tfoiro­n
matioSnc ienc1es9,8 (9hC.a p5t.e)r
5R.o beCr.Wt i llia"mTshLoean ow,f LNaurmgbefe orFrsu zVzaryi aubnldeesr
aG eneTrraila ngular NPorrimn cuEixnptdlreeeenr,vs "ifi sroori neo snu bmis­
sitaonFu zzyS etasn dS ystems1,9 8(9hC.a p6t).e r
6. RobWeirltl iCa".mI snotneA,rr viatlha mnePdtr iocb abAirliitshtmiect ic,"
toa ppeiantr h per oceeodfSi CnAgNs-I 8M9A CS-GAIMnMt-eGrIn ational
SympoosniC uomm puAtreirt hamneSdte ilacfl -iVdNautmienrgMi ectahlo ds,
BasOeclt,o be(rA s 1u9m8m9ao.rfcy h ap3ta enr5dsa , n add iscuosfts hieo n
relatiboentswpehreionpb abairliitshatmnieidctn itcea rrviatilhc m)e.t
7.R obeCr.Wt i lliaanmTdso omDn o wn"sPo,rb abiAlriistthimRcee tliacti:o n­
shiwpistO hht eIrd eatsob, es" u bmittott heIendt e rnatioJnoaulrn aolf A p­
proximaRteea soni1ng9,8 (9C.h a4p)t.e r
v
VI
8.R obeCr.Wt i lliaanmTdso omDn o wn"sN,u meMreitchafolod Crsa lculating
ConvoloufPt rioobnasDb iisltirtiyb tuobt esi uobnmsi,tt"oMta tehde matics
andC omputresi nS imulati1on9,8 (9Ca.hp t2e)r.
9.R obeCr.Wt i lliamsQouno,t oi"feT nhTNtewo or mRaaln dVoamr i:a bAlnes
HistoSrtiucdaypl,r "e pianr1 a9t8(i9To.rn at,ch heeis s toofar ys ipmrpolbe­
lerme lattope rdo babairlieittshitmicc) .
RobeCr.Wt i llia"mTshDoein s,cT r-ectoen jTurgaanmts,efi"o np r reptairoan.
10.
Tob es ubmittoItn oefrmda tioSnc ineces1,9 8(9E.x tetnhTde-s c onjugate
trans-fsoerseme ct6i.ao4nnc d h ap5t)e.r
Contents
1 ProbaibiltsicA rithm-eTthiecV erIyd ea 1
Motiv.at.i.on. . 1
1.1
1.2O utloifRn ees u.l ts 3
12..1Di ffeMreetnhtf oodCrsa lcuCloantvionlgau ntdi oLtnahs-e
guerre TMreatnhs.ofdo. r.m. ... 3
. 0 • • • • • • • •• ••
12..2T he DepeBnoduenandncsNdy u merMiectahloo fdC sa lcu-
latTihnegm 3
. . . . . . . . . . . . . . . 0 • • • •• •• •• •• •
12..3P recuMruslotrDissip,cl oieve easrn,Rd e tliaonwsihFtiuhpz sz y
Sets. . ...... . ...... . 4
. . . . 0 • • • • •
12..4T hIen vearnsde DeteorfaRm ainndaMnoatmt rix 4
12..5A LimiRteisnfugolD rte pendBeonucnyd s 5
1.T3h eSstirsu c.t u..r e 5
. .
1.N4o tatCioonnvaeln. t ions 6
2 NumeriMceatlh ofdosCr a lculCaotnivnogl uotfiP ornosb ability
Ditsributions 8
2.I1n trodAuicamtn iAdon na,l yMteitchaold s 8
21..H1i stMoortyi,v aantOdiu otnl. i ne 9
2.21 E.xact AnRaesluyl.tt is.c al. . . 10
2.1S.p3e ciFaorlmiuslfeaoder Convolutions 12
2..14I nteTgrraanls f.o rms 13
. .
2.1T.h5He - funMcettihoond 15
2.2Mi scelNluamneeroMiuecsta hlfo odCrsa lcuCloantvionlgu ti1o6n s
2.2N.o1r mAaplp roxiamnaodtt ihPoeanrrs a metDeirsitsreid1b 6u tions
2.2M.e2t hBoadssoe ndM oments 18
2.2N.o3n -LTiranneosarfrm ations 20
VB
VBl
2.2D.i4r Seacmtp olifn Dtgeh nes fiotSriu emasn Ddi fferCeonnc-e
volu.t io.n s... .. .. .. . 21
2.2T.h5Se k inner/MAectkhroody d 22
2.2S.p6l BianseMe edt ho.d.s . . 23
2.3La guTerrarnes afnoOdrt mhsOe rrt honTorramnaslf orms 24
2.3T.h1Le a guerre MTertahn.osd.f .o. r.m .. . .2 5
2.3D.i2s triobfPu rtoidoaunncsQdt u so tiuesnittnhsgLe a guerre
anMde lTlriann sf.o r.m s. . . . .. ..2 7 . . . . . . . . . . . .
2.3D.i3s triobfPu rtoidoaunnc Qdtu so tiCeanltcsuw liattthhe ed
LaguTerraroners Dmfi re.c t.l y . . . . .2 8. . . . . . . . . . . .
2.3D.si4rt ibuotfiPo rnosd auncQdtu so tiuesnittnhsgLe a guerre
TransforLmo gatanhrdmi ic/ExpTornaoenrsnmftaitoaiflo n
Varia.b le.s . . . . . . . . 3.2 . . . . . . . . . . . . . . . . .
2.3O.t5h Teyrp oefLsa grueTe rransafnoRdre mlsa Rteesdu lt4s0.
2.3C.o6n cluosnti hoLena sgu Terrarnes Mfeotrhmo d 43
2.4T hHei sgtroaamn Rde laMteetdh ods 43
2.4T.h1He i stoRgerparme se.n tation 43
2.4C.o2m biHniisntgo tgorC aamlsc uCloanutvteoi lons 45
2.4K.a3p lMaent'hsoo fDd si crPertoeb aDbiisltirtiyb uti5o0n s
2.4M.o4o rHei'sst oMgertahm.o d. 57
2.5I ntaelAr rvithamneEdtr irAcon ra lysis 57
2.5T.h1Se t anIdnatredAr rviatlh metic 58
2.5T.r2i pAlreixt hamneDdte imcp sQtuearn'Atsri ilteh .m et5i9c
2.5I.n3t eArrviatlh amneCdto incfi dIenntceefr rvoaml st he Metrol-
ogist's Point of .V ie.w .. .. 6.1 . . . . . . . .
2.5P.e4r mutationa-nPRdee rltaMuteretbdha .ot diso. n .. .6.4 . .
3 NumeriMceatlh ofdrosC alculDaetpienngd eBnocuyna dnsdC on-
volutions 66
3.I1n trod.uc.t.io.n. 67
3.1T.h1e Problem 68
3.21 T.hIed oefPa r obabAirliitshatmnietdcth Nieec e fdoD re pen-
denBcoyu n.d s. . . . . . . . 6.9 . . . . . . . . . . . . . . . .
3.1M.e3t hoofHd asn dtlhiEenrg r ionPr rso babAirliitshtmie7ct0 i c
3.41 O.utloiftn hReee sotfT hiCsh ap.t er 71
3.2De finiatniOdot nehrsP relim.in.ar.i.es. 71
IX
3.2D.si1t ribFuutnicotni ons 72
3.2B.i2n aOrpye rations 72
3.2Q.u3a Isniv-e.rs.e s 72
3.2T.r4i angular-norms 73
3.2C.o5p ualnads JDoitisrnitb Fuuntcitoin.o ns 73
3.2T.h6Te r iaFnugnlcet iaonndas n (Td- convolutio7n4s
T p
3.3De pendBeonucnaydn std h ePirro perties 75
3.3T.h1De e pendBeonucn.yd .s . . . 76
3.3P.o2i ntBweisste- PNoastsuoirftbe hlB eeo unds 78
3.3E.x3a mp.le..s . . . . . . . . 8.3 . . . . . .
3.3E.x4c luUssieLv oeow fae nrUd p pBeoru nds 90
3.4Nu merRiecpaerlne tsation 90
3.4D.ua1l ity 91
" "
3.4N.u2m erRiecparle seannttdah Cteai locnu olflad tbainoudnd b95
3.4N.u3m erCiaclaclu olf(TaC -toinolvnuo ti.o.ns. 99
3.4C.a4l cuolfMa otmieonnatn osdt hFeurn ctionals 103
·
3.4E.x5a mp.le..s . . .. .. .. .. .. .. 1 0.3 .
. ·
3.5In corpaonrUdap tdiantDgie npge ndIenonfrcmya tion .1 10
3.5G.e1n era.l .I de.a . .. .. .. .. .. .1 .1 0. . . .
3.5I.n2t erporfeD teapteinodInemn pclibieyCes Xd YoJ W .1 11
3.5U.s3oe fC X YoJ Wi nP robabAirliitshtCmiaeclt ciucl a.t1 i41o ns
3.5M.e4a suorfe sD ependenc.e . . . ..1 1.5 . .
3.U6s oefM ixtfuorNreo sn monoOtpoenriact ions 116
·
3.6I.n1t rodauncGdte inoeAnrp aplr o.a.ch 116
·
3.6C.o2m plicAartiisfoirnnotsgmh eo fuL osweae nrUd p pPerro b-
abiDliiistbtyu rti.o.ns. ......... .. . . .1.1.7 . . . .
3.6D.i3ffi cuIlnttireosbd yuU cseitdnh gNe u merAipcparlo ximal9t ions1
3.6N.u4m erAilcgaolr fiotSrhp mlsiD titsitnrgi ibnuPttoois ointsi ve
anNde gaPtairvtes R eacnoidmn biTnhge m. . . . .1.2.1 . . .
3.6C.o5n cluosni uotsnhoesefM i xtfuorNreo sn monoOtpoenriac-
tions 128
3.7C onclu.si.o.ns. ... . . . . 128
·
4 ReliaotnswhiitpOhst heIrd eas 131