Order Statistics. 1 n. Example Four measurements are taken on a random variable, x, which take on values.

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1 Oe Sascs e be couous epee.v. wh sbuo a es (. We eoe K be he oee aom vaable whee < < K < a because he ae couous we ca goe equal sg. m ma ( K ( K The pobabl es uco o a ae easl ou: e be a couous.v. ha has values om a aom sample o se. Reoeg he s om smalles o lages < < < (Noe: o wo s ae equal ecep wh pobabl eo sce s couous Dee a.v. such ha has values. The s calle he h ORDER STATISTIC. Somemes a ae esgae ma a m especvel. Eample ou measuemes ae ake o a aom vaable whch ake o values The oee sample woul be.6 <. <. <.6 The aom vaable o smalles oe sasc a s value s.6. Smlal he seco oe sasc s.. ec Cose a sample se aw om a populao wh a p ( a D ( a. The lages eleme he sample s ee b he ollowg; ( K ma( G K The { a} { G a} { ma( K a} { a a K a} { a} { a} K{ a} a ( a [ ( a ] ( a [ ( a ] ( a a Smlal he smalles eleme s m( K. The

2 { } { } [ ] { } [ ] [ ] [ ] [ ] [ ] a a a a a a m K K Coseg he emeae case.e. le us ocus o a geeal oe sasc ha s bewee wo small value a he hee possble eves. { } oe sasc. he pobabl es uco o he s whee. s ( a values ae ( a values o o ( h > < K Now he pobabl o he s eve s { } { } [ ] [ ] [ ] ha ae less values < < < < < K K a he pobabl o he seco eve s { } { } [ ] [ ] [ ] [ ] [ ] [ ] [ ] > > > > > > > > ae values K K Howeve hee ae was ha hs ca occu paog o hee goups. all he pobabl o he h eve s { } [ ] [ ]!!! I cab easl be see ha o o hs epesso euces o

3 [ ] [ ] [ ] beoe. as a!!! Now we ca a epesso o he o sbuo o The oe sasc wll ake o values < < < o some pemuao ( o ( K Sce o a pemuao ( o ( K K ; ; ; We see o < < < (... (... (... ( ; ;... N K K We see o < <.. ( (! ; ; Dve b a le! K < < Ths equao s smpl eplae (see Ross b agug ha oe o he veco > <... o equal > <... s ecessa a suce o > <... o equal oe o he! pemuaos o > <.... As he pobabl (es ha > <... equals a gve pemuao o > <... s us ( ( ( he equao ollows.

4 Eample ou measuemes ae ake o aom vaable havg p e >. ( 5. a ( 5.. (H: Keep m ha wll be less ha o equal o 5. ol all he obsevaos ae less ha o equal o 5.. e > (. a ( a s we ee pobabl ha he oh measueme ake s [ 5.] e e ( e. 87 b To solve hs we ee he sbuo o h oe sasc 5. { 5.} ( [ ] e [ e ] 5. { 5.} e ( e 5. ( e ( e [ ] ( e (.8 Eample: Suppose ha ma eas o obsevao have come ha he aual mamum loo e ( ee o a cea ve has a p < < (Noe: I s ulkel ha loo es woul be escbe b ahg as smple as a uom p The Am Cops o Egees ae plag o bul a levee alog a cea poo o he ve a he wa o bul hgh eough so ha hee s ol a % chace ha he seco wos loo he e eas wll ovelow he embakme. How hgh shoul he levee be? (We assume ha hee wll ol be oe poeal loo pe ea. Soluo: e h be he ese hegh. I ee he loo es o he e eas wha we eque o h s ha ( > h. As a sag po oce ha o <<

5 Theeoe! 7!! a h s he soluo o he egal equao (( h we make he subsuo u he equao smples o ( h > h ( u (. u u h h - Seg he gh ha se equal o. a he solvg o h b al a eo gves h9. ee as he ese hegh.. Eample: e be a aom sample om e >. Dee Y. Y ( he pobabl es uco o he smalles epoeal oe sasc. ake om e > Y Y [ ] e e e e [ e ] [ e ] e - s oe sasc e Eample: Suppose he legh o me mues ha ou have o wa a bak elles wow s uoml sbue ove he eval (. I ou go o he bak ou mes ug e moh wha s he pobabl ha ou seco loges wa wll be less ha 5 mues? Raom sample Oe Seco loges wa

6 T ( T T T ( [ ] [ ] ( [ 5]!!!..... T T 5 5 T 5 8 T Eample: Two epee aom vaables a have a Gaussa D wh eo mea a vaace.e. he commo D s e < <. π The lage o he wo obseve values s eoe b Z. E(Z. Soluo: I hs case usg he epesso o he es uco o a oe ssasc wh! [ ] (!(! a he D o he lages o he wo aom vaables a s ( ( ( Z We ca check o see Z ( has u aea. Thus Z ( ( ( Z [ ] ( Recall ha ( (. Thus we ca epess he above egal as

7 he e Z sce o all. The epece value calculao s hus Z Z E The egao s a ego o whee as show skech below. Iechagg he oe o he egao we have [ ] π π π π e e e e e Z E whee we have use a subsuo he egal o a he ac ha he egal o ep ep

8 o ( s π. Skech o he ego o egao. The o D o he oe sascs Y Y Y s equal o T Y KY ( K! ( ( ( < < < ohewse Equao (5.9. ca be eplae b og ha he oee veco (Y Y Y s equal o ( a ol ( equals oe o he! pemuaos o (. The D ha ( equals oe o hese pemuaos s us he pouc (b epeece ( ( a hee ae! such pemuaos. Thus we oba (. Summag. e be a couous.v. wh pobabl es uco (. I a aom sample o se s aw om ( he magal p o he h oe sasc s! [ ] [ ] ( (!! A he eemes hs euces o ma [ ] [ ] m Eample: Thee ucks beakow a aom alog a oa o legh. pobabl ha o ucks ae wh a sace o each ohe whee. Soluo:

9 e be sace alog he oa o beakow o ucks a. Whe we pu hem acceleag oe we ge ( ( a ( - oee sascs. We wa { > ( }.e. he hghes oe sasc ( s a leas ( awa om he seco ec. Now hs pobabl s { > ( } ove all ps such ha > Bu sce we have epee.v. uoml sbue { } { } { } { } { } { } specc values o a Bu we mus accou all possble pemuaos Ths s o a specc aageme o. To accou o all possble aagemes: { > }! bu > { }! { > } { } { } { } > a smlal o he ohes To eeme he lms o egao cose he ollowg age o vaes om o age o - vaes om o -

10 { } [ ] e 6 6 e 6 6 > The Ceal-m Theoem Cose a sequece o epee.v. wh especve eses (. We kow om he law o lage umbes ha he vaace o s small o lage. Hece (Tchebche equal he es s coceae ea s mea. Howeve he law sas ohg abou he acual shape o hs es. I us ou ha as ceases es o a omal cuve egaless o he shape o he eses. Ths s he essece o he ceal-lm heoem. Raom Vaables o Couous Tpe We assume ha he.v. ae o couous pe a epee wh { } E η We om he sum Is mea η a vaace ae gve b vaes om o -

11 η η η ( a s es b ( The ceal-lm heoem sas ha ue cea geeal coos ( appoaches a omal cuve as ceases: ( η ~ e π ( I s popel scale (e.g. wh so ha he lm o he esulg vaace s e he he above becomes a equal o pove: (a (b o some > ( < C cosa (5 These coos ae o he mos geeal o he val o he heoem. Howeve he cove a we age o applcaos. We oe ha ( s sase > C > a hs s ceal he case he gve.v. have equal vaaces. Coo ( s sase all eses ( ae eo o > C. The poo o he oegog heoem s somewha legh a wll be ome. The ceal-lm heoem ca be sae epeel o a pobablsc coseaos. I s meel a pope o covoluos volvg a lage umbe o posve ucos a s use ohe aeas e.g. he eemao o he oupu o lea ssems cascae. I he eses ( ae easoabl coceae ea η he he omal cuve ( s a close appomao o ( eve o moeae values o. Ths s llusae he ollowg eample. Eample: The.v. ae uoml sbue he (T eval as beol gue. I s eas o see ha E { } wh T s T he es o s a agle as he gue. Cleal { } T E T s 6

12 Hece he coespog omal cuve [see (] s gve b gue ( T T e T π he he es o cosss o hee paabolc peces show gue. We ow have T E{ } s wh omal esmae T T e π (.5T T Eve o such small values o he wo ses ( ae emakabl close. T T T T T T T ( ( T T e π ( T T T e π (.5T (a (b (c Demosao gues

13 Covegece Coceps The oegog scusso ca be omulae ems o lms. I he ollowg we shall gve a smple eplaao o lms volvg.v. These coceps wll also be eee he eo o cou eeao a egao o sochasc pocesses ( As we kow a sequece o umbes es o a lm gve > we ca a ege such ha < o eve > ( Cose ow he sequece K K ( o.v. We have o each epemeal oucome ζ a sequece o umbes ( ζ ( ζ K ( ζ K ( a ( epeses a aml o such sequeces; hece we ca o loge use ( o ee covegece o he ee aml. I happes ha ( coveges o eve ζ he we sa ha ( coveges evewhee (e. The lm o each sequece wll epe o ζ ; ha s o he s geeal a.v. The above covegece moe s oo escve. I ma cases ( oes o covege o eve ζ bu mgh covege a weake sese. Covegece almos evewhee (a.e. (o wh obabl : We sa ha he sequece coveges o wh pobabl he se o oucomes ζ such ha lm ( ζ ( ζ o ( has pobabl equal o. Ths s we he om { } o (5 Covegece he mea-squae sese (m.s.. The sequece es o he m.s. sese E { } o (6 Covegece pobabl (p (o Sochasc Covegece o Covegece > ha - s lage ha he gve Measue. We ow om he pobabl { } umbe >. Ths s a sequece o umbes epeg o. I coveges o eo o eve { > } o (7 The we sa ha he sequece es o pobabl. Covegece sbuo(. We eoe b ( a ( he sbuo ucos o he.v. a especvel. I (8 o eve po whch ( s couous he we sa ha es o sbuo (see ceal-lm heoem. Cauch ceo. Suppose ha es o some sese. I geeal he lm s o kow a oe o es o covegece oe mus use a ceo ha avos. As he case o oa umbes such a ceo s he esece o he lm (Cauch

14 o a a m > m I he above lm ess a.e. o he m.s. sese o p he coveges he coespog sese. o eample gve > we ca such ha E { m } < o > a a m > he he sequece coveges he m.s. sese. I (8 oe ess o he covegece o m (- (. Compaso o vaous covegece moes. The smples om o covegece s (6. Ths volves a sgle sequece o umbes. I ( we have a aml o sequeces oe o each ζ. I (7 we also have a aml o sequeces oe o each. all (8 we eal wh covegece o a sequece o ucos; we ow have a sequece o umbes ( o eve eal. The vaous covegece moes ae elae as g.. Each po o hs gue epeses a pacula sequece. We see o eample ha coveges he m.s. sese he coveges also pobabl. Iee applg (5-58 o he.v. - we oba wh a { } { } E > a.e. m.s. p No covegece g. I he m.s. sese he he umeao above es o eo. Hece o a e he le-ha se mus also e o eo; ha s pobabl. The covese s o ue. A sequece mgh covege pobabl bu o a m.s. sese. The easo s ha { > }

15 s small hs oes o mpl ha { } E s also small because - mgh ake lage values wh small pobabl. Howeve oe ca show whou much cul ha he eses ( o he.v. ae eo o lage o > c a a > he p covegece a m.s. covegece ae equvale. We shall all comme o he elaoshp bewee a.e. a p covegece. Tha he s mples he seco s eas o pove. B a coueeample oe ca show ha he covese s o ue. We choose sea o gve a aïve eplaao o he eece bewee hese wo covegece moes. I g. we ζ ζ ζ ζ ζ ζ ζ ζ (a (b g. have ploe - o vaous oucomes ζ as a uco o. To smpl he awg we use a couous cuve sea o scee values. Covegece pobabl meas ha o a specc > ol a small peceage o hese cuves wll have oaes eceeg (g a. I s o couse possble ha o oe o he above cuves wll ema less ha o eve >. Covegece wh pobabl howeve emas ha mos cuves wll ema below o eve > (g. b. The aw o age Numbes The law o lage umbes s evelope as a coolla o he DeMove-aplace heoem was ecosee a geeale ove a peo o wo ceues b he mos amous ames pobabl heo. Wh he ecepo o Boels heoem he ollowg vesos ca be eve om Tchebches equal.

16 Ceal m Theoem (coue Oe o he mos mpoa heoems apple pobabl a sascs s he Ceal m Theoem. We saw om he pevous eample ha he.v. coveges sbuo o he saa omal.v. Ths pheomeo s shae b a lage class o.v. Theoem e be epee a ecall sbue.v. wh E [ ] V ( <. Dee Y as Y whee The Y coveges sbuo o a saa omal.v. as. We ca show hs s ue b eg Z whe e E Z Z The M.G.. o Z M Z E[ Z ]! Now Y A he MG o Y s [ ] ( Z M M Z k! whee k E [ Z ] Takg lm o M ( as s akg log o M ( log Epag hs usg log a leg log k! M (

17 k! Yel!! log log k k M whee succeeg ems clue ec. so we eglec hghe oe. lm o lm log 6 log e M M k M Whch s he MG o aom saa omal.v. We ca coclue Y o saa omal. Noes: We sa ha he.v. s asmpocall omal. Also hee ae ohe oms o he heoem whch allow o o be ecall sbue a ca be epee. Ths s wha makes he heoem mos valuable!! The pobabl sbuo asg om lookg o ma epee values o (sample mea o a e sample se selece om same pop s calle he SAMING DISTRIBUTION o. The paccal mpoace o he C..T. s ha o lage he samplg sbuo o ca be appomae b a omal sbuo.v. omal saa Z b Z b b Eample Samples o se wee aw om a pop havg he p...

18 e > elsewhee The sample mea was compue o each sample. The elave equec hsogam o hese values o samples o se 5 s show gue 7.. gues 7. a 7. show smla esuls o samples o se 5 a especvel. Alhough all he elave equec hsogams have a so o bell shape oce ha he eec owa a smmec omal cuve s bee o he lage. A smooh cuve aw hough he ba gaph o gue 7. woul be eal ecal o a omal es uco wh a mea o a a vaace o (. The ceal lm heoem poves a ve useul esul o sascal eece o we ow kow o ol ha has mea a vaace he populao has mea a vaace bu also ha he pobabl sbuo o s appomael omal. o eample suppose we wsh o a eval (ab such ha ( a b. 95 Ths pobabl s equvale o a b.95 o cosas a. Because has appomael a saa omal sbuo. om Table he Appe we kow ha a hece (.96 Z o a.96 b.96 a.96 b.96

19 gue gue gue

20 Eample 7.6 The acue seghs o a cea pe o glass aveage (housas o pous pe squae ch a have a saa evao o. (a Wha s he pobabl ha he aveage acue segh o peces o hs glass ecees.5? (b a eval ha clues he aveage acue segh o peces o hs glass wh pobabl.95. Soluo: (a The aveage segh has appomael omal sbuo wh a saa evao. Thus.5 >.5 > s appomael equal o.5.5 Z > Z >.. ( Z > om able o Nomal Vaaes. The pobabl o seeg a aveage value (o moe ha.5 us above he populao mea hs case s ve small. (b We have see ha o a omall sbue. I hs poblem a Appomael 95% o sample mea acue seghs o samples o se shoul le bewee.6 a.. Eample. A cea mache use o ll boles wh lqu has bee obseve ove a log peo o me a he vaace he amous o ll s ou o be appomael ouce. oweve he mea ouces o ll epes o a ausme ha ma chage om a o a o opeao o opeao. I 5 obsevaos o ouces o ll spese ae o be ake o a gve a (all wh he same mache seg he pobabl ha he sample mea wll be wh. ouce o he ue populao mea o ha seg. Soluo: We wll assume 5 s lage eough o he sample mea o have appomael a omal sbuo. The

21 (. [. (.] Sce has appomael a saa omal sbuo he above pobabl s appomael 7 [.5 Z.5]. 866 Eample Acheveme es scoes om all hgh seos a cea sae have a mea a vaace o 6 a 6 especvel. A specc hgh school class o sues ha a mea scoe o 58. Is hee evece o sugges ha hs hgh school s eo? (Calculae he pobabl ha he sample mea s a mos 58 whe. Soluo: e eoe he mea o a aom sample o scoes om a populao wh 6 a We kow om. We wa o appomae Theoem 7. ha ( s appomael a saa omal vaable whch we eoe b Z. Hece 58 6 ( 58 Z 6 Z.5.6 om able o Nomal Vaaes he Appe o he e. Because hs pobabl s so small s ulkel ha he specc class o ees ca be egae as a aom sample om a populao wh 6 a 6. Thee s evece o sugges ha hs class coul be se ase as eo. We kow ha a bomall sbue.v. Y ca be epesse as a sum o epee Beoull aom vaable. Y whee wh p wh ( p (Y coul epese umbe o successes.

22 Now! The aco o successes als s Y Ths s ow a sample mea. We see he o lage Y s appomael omal wh a mea o E a vaace [ ] p Y V V ( p( p p( p Because Y p(-p. Y has a omal sbuo wh a mea o p a vaace o Because bomal pobables ae cumbesome o calculae o lage we make use o omal appomao. gue gue. shows he hsogam o a bomal sbuo o a p.6. The heghs o he bas epese he especve bomal pobables. o hs sbuo he mea s p(.6 a he vaace s p(-p(.6(..8. Supempose o he bomal sbuo s a omal sbuo wh mea a vaace. 8. Noce how he omal cuve closel appomaes he bomal hsogam. o he suao splae gue suppose we wsh o ( Y b g.. eac bomal pobables ou Appe o he e. Y 5. B he

23 The value s he sum o he heghs o he bas om up o a clug. ookg a a he omal cuve gue we ca see ha he aeas he bas a a below ae bes appomae b he aea ue he cuve o he le o.5. The ea.5 s ae so ha he oal ba a s clue he aea ue coseao. Thus W epeses a omall sbue aom vaable wh a.8 (. he ( Y ( W.5 W ( Z.68 om able o Nomal Vaaes. We see ha he omal appomao o.8 s close o he eac bomal pobabl o.5. The appomao woul eve be bee wee lage. The omal appomao o he bomal sbuo woks well o eve moeael lage as log as p s o close o eo o oe. A useul ule o humb s o make sue s lage eough so ha p ± p( p les wh he eval ( beoe he omal appomao s use. Ohewse he bomal sbuo ma be so asmmec ha he smmec omal sbuo cao pove a goo appomao. Eample Slco waes comg o a mcochp pla ae spece o coomace o speccao. om a lage lo o waes ae spece. I he umbe o ocoomaces Y s o moe ha he lo s accepe. he appomae pobabl o accepace he popoo o ocoomaces he lo s p.. Soluo The umbe o ocoomaces Y has a bomal sbuo he lo ee s lage. Beoe usg he omal appomao we shoul check o see ha p( p (.(.8 p ±. ±. ±.8 s eel wh he eval ( whch s. Thus he omal appomao shoul wok well. Now he pobabl o accepg he lo s ( Y ( W.5 Whee W s omall sbue aom vaable wh p a p( p. I ollows ha W.5 ( W.5 ( Z Thee s ol a small pobabl o accepg a lo ha has % ocoomg waes.

24 Eample Caae A beleves ha she ca w a c eleco she ca poll a leas 55% o he voes pecc I. She also beleves ha abou 5% o he cs voes avo he. I voes show up o voe a pecc I wha s he pobabl ha caae A eceves a leas 55% o he voes? Soluo e eoe he umbe o voes a pecc I who voe o caae A. We mus appomae (.55 whe p he pobabl ha a aoml selece voe avos caae A s.5. I we hk o he voes a pecc I as a aom sample om he c he has a bomal sbuo wh p.5. Applg ou Theoem ollows ha (.55 Z (.5 ( Z.9. 8 om able he Appe o e. We ca make use o he ceal lm heoem o geeae omall sbue aom vaables. Recall ha s sae as ollows: e be epee a ecall sbue aom vaables wh E a V. e Y ( whee ( Y coveges sbuo o a saa omal aom vaable as. We ca geeae omal aom vaables as ollows:. Geeae s om a spece sbuo wh kow a Compue he sample mea.. Evaluae (.. The wll be saa omal aom vaable wh mea o a vaace o.. The omal aom vaable wll be gve b.. Repea seps - m mes whee m s he umbe o omal aom vaables ese. The sbuo mos oe use sep s he uom sbuo o (. Whe usg hs sbuo o evaluae.5 a. To make he algohm moe ece le.

25 Combao o Covegece obabl a Covegece Dsbuo Ma mes we wll be eese he lmg behavo o he pouc o quoe o seveal ucos o a se o aom vaables. The ollowg heoem whch combes covegece pobabl wh covegece sbuo apples o he quoe o wo ucos a Y. Eample Theoem Suppose ha coveges sbuo o a aom vaable a Y coveges pobabl o u. The Y coveges sbuo o. oo The poo o he heoem s beo he scope o hs e bu we llusae s useuless wh he ollowg eample. Suppose ha ae epee a ecall sbue aom vaables E a V. Dee S as wh ( ( S (. Show ha S coveges sbuo o a saa omal aom vaable. Soluo: I Eample above we showe ha S coveges pobabl o. Hece ollows om pevous Theoem pas (c a ( ha S (a hece S coveges pobabl o. We also kow om Theoem 7. ha coveges sbuo o a saa omal aom vaable. Theeoe S S coveges sbuo o a saa omal aom vaable b Theoem above Reewal Theo Relabl heo eals wh pobablsc moels ha gove behavo o ssems o compoes a pems omulao o opmum maeace polces. Some ssems ae que comple a o uesa how he wok we mus s uesa smple ssems.

26 We cose a ssem o a sgle compoe ha has a aom le legh bu ma be eplace beoe als such as a lgh bulb. We kow o a osso pocess he me bewee wo successve occueces s epoeal wh mea λ ( λ s mea umbe o occueces pe u me. A moe geeal sochasc pocess ca be obae b coseg eaval mes o be oegave.v. ecall sbue bu o ecessal epoeal. Suppose - epee ecall sbue aom vaables each wh a ( a (. Also E[ ] V ( < Now ssem opeaes b seg a ew compoe leg bu ul als he epeag he pocess. The he occuece o ees s he -sevce alue o compoes. eaval me bewee alue (- a N aom vaable equal o umbe o -sevce alues eval ( Noe: N ma ege k such ha k wh N > (.e. o alue Cose ow he pobabl sbuo o N. I s epoeal he N s osso bu s o epoeal sbuo o N ma be cul o mpossble o ge a we ee o vesgae asmpoc behavo o N as. Noe ha wh S (cose ha o be sevce le measue [ N ] [ S ] oucomes ha allow also allow Suppose a such a wa ha c some cosa Ths mples ha as a.e. lm ha compoe lase a leas o

27 Now we ca we hs pobabl < c S c N Sce om hs eomao oge o c c We have S whch s appomael omal o lage. I Φ saa omal sbuo uco Φ The c c c N c c N Φ Φ Φ Hece N ca be egae as appomael omal o lage wh mea a vaace.

28 Eample A use a elecoc ssem s eplace wh a ecal use upo alue. I s kow ha hese uses have a mea le o hous wh vaace o.5. Begg wh a ew use he ssem s o opeae o hous. a I he cos o a use s $5 he epece amou o be pa o uses ove he -hou peo. b I eplaceme uses ae sock he pobabl ha he wee all use he -hou peo. Soluo: a No sbuo o lemes o he uses s gve so asmpoc esuls ma be emploe. (Noe ha s lage as compae o. eg N eoe he umbe o eplacemes we have E ( N. We woul epec o use o eplacemes plus he oe all place he ssem. Thus he epece cos o uses s (5( $5 (b N s appomael omal wh mea a vaace (.5. Thus ( ( N N ( Z. 8 whee Z eoes a saa omal aom vaable. To mme he umbe o -sevce alues compoes oe ae eplace a age T o a alue whcheve comes s. Hee age ees o legh o me sevce a T s a cosa. Ue hs age eplaceme scheme he e-aval mes ae gve b Y m( T. Cose a age eplaceme polc whch c epeses he cos o eplacg a compoe ha has ale sevce a c he cos o eplacg a compoe (whch has o ale a age T. (Usuall c >c. To he oal eplaceme coss up o me we mus cou he oal umbe o eplacemes N a he umbe o sevce alues. e W < T T. bus ou be he eplaceme me

29 N The W eoes he oal umbe o -sevce alues. Thus he oal eplaceme cos up o me s N N C c W c N W cos oeplace he bue ou compoe plus cos o eplaceme ue oage The epece eplaceme cos pe u me s equel o ees a hs ca be appomae o obsevg ha ue he coos sae hs seco N E W E( N E( W Thus E ( C c E ( N E ( W c E ( N E ( W [ ] whch o lage becomes [ c( < T c( T ]. E( Y E( Y ae umbe o eplacemes me I ollows ha E( C { c ( T c[ ( T ]} E Y whee ( s sll he sbuo o. A opmum eplaceme polc ca be ou b choosg T o mme he epece cos pe u me