Mathematics For Management

 

3/>1/>/::

 

3/>1/>1/>) Z`e tftbo tmie mh imhutes (oet y 9 tftbo tmie) t`bt Fweh's bssmstbht, Nrmthey Qwmct, heejs tf afipoete b stbak fc aofsmhgs ms > + : 9 05. Mt `bs twf sfoutmfhs4  x  9

QFOSMHG OMHEB^ ERUBZMFHQ Z`e key tf sfovmhg bh equbtmfh ms tf percfri ibt`eibtmabo fperbtmfhs fh nft` smjes fc t`e equbtmfh t`bt smipomcy mt nut jf hft a`bhge t`e set fc  sfoutmfhs tf mt . Mc yfu abh percfri ibt`eibtmabo fperbtmfhs fh nft` smjes fc bh equbtmfh t`bt msfobte boo teris mhvfovmhg t`e vbrmbnoe fh fhe smje fc t`e equbtmfh, t`e equbtmfh ms usubooy ebsy tf sfove. Zf sfove omhebr equbtmfhs, use t`e cfoofwmhg ruoes4 ^uoe 24 Bjjmhg t`e sbie qubhtmty tf nft` smjes fc bh equbtmfh jfes hft a`bhge t`e set fc sfoutmfhs tf t`bt equbtmfh. ^uoe >4 Iuotmpoymhg fr jmvmjmhg nft` smjes fc bh equbtmfh ny t`e sbie hfhzerf huiner jfes hft a`bhge t`e set fc sfoutmfhs tf t`bt equbtmfh.

CMHJMHG Z@E N^EBKESEH \FMHZ ^eaboo t`bt t`e ifht`oy afst fc fperbtmhg @fccibh's @bppytbmo Set Aomhma ms > + : p - 1. Hfw grbp` QZN's prfcmt bs b cuhatmfh fc prmae. @ere b 9 -2 (m.e., >b 9 ->) bhj n 9 :. Qmhae b ? b 9 -:/-> 9 $>, bhj mt wmoo jearebse w`eh p 8 >. Z`erecfre, cfr  p 9 >, t`e pbrbnfob wmoo bssuie bs suie mts ibxmiui vboue. W`eh  p 9 >, c(p) 9 ->> + :(>) - 1 9 2. Z`erecfre, t`e pfmht > (>, 2) ms fh t`e pbrbnfob. Mc p 9 2, t`eh c(2) 9 -2  + :(2) - 1 9 .  ]fu abh vermcy t`bt x > - 3 x  +  + = 9 < `bs t`e uhmque sfoutmfh  x  9  9 1, w`erebs t`e qubjrbtma equbtmfh x > + 1 x  +  + 3 9 < `bs hf rebo rffts.

IUOZM\O]MHG NMHFIMBOQ  ]fu iby reieiner crfi `mg` sa`ffo bogenrb t`e CFMO iet`fj. Afhsmjer t`e cuhatmfh c(p) 9 (1 - p)(2 - p). Fcteh we heej tf iuotmpoy teris t eris mhvfovmhg b vbrmbnoe tf represeht b cuhatmfh bs b pfoyhfimbo. We wmoo hfw oebrh tf iuotmpoy expressmfhs mhvfovmhg twf teris (bh expressmfh t`bt afhtbmhs twf teris ms abooej b nmhfimbo). Z`us (1 - p) bhj (2 - p) bre nmhfimbos. Zf express c(p) bs b pfoyhfimbo, yfu heej tf iuotmpoy twf  nmhfimbos. ]fu iby reieiner crfi `mg` sa`ffo bogenrb t`e CFMO iet`fj cfr iuotmpoymhg nmhfimbos4 Cmrst, Futer, Mhher, Obst.

EPE^AMQEQ (2) W`bt ms t`e jegree fc t`e cfoofwmhg pfoyhfimbo4 x 5 - : x 1 + > x >6  + >. (>) Grbp` t`e aurve c(x) 9 x > - 3 x  +  + > 9  x > - 5 x  + (:) Mt afsts $ 9 -1 × -1 9 =. W`eh t`e expfheht ms b pfsmtmve crbatmfh ‚ sby, b/n ‚ t`eh xb/n 9 n x b . We rebj t`ms expressmfh bs t`e nt` rfft fc  x  rbmsej  rbmsej tf t`e b pfwer. Cfr exbipoe, w`eh b 9 2 bhj  n 9 1, t`ms expressmfh ymeojs t`e aune rfft fc x . Cfr exbipoe, =1/> 9 > = 1 9 >7. ]fu wmoo sffh oebrh `fw ` fw tf wfrk wmt` hegbtmve expfhehts.

^UOEQ CF^ JEBOMHG WMZ@ EP\FHEHZQ  Bs ofhg bs x  ms  ms hft equbo tf zerf, yfu abh use t`e cfoofwmhg ruoes cfr expfhehts4

\fwer Cuhatmfh JECMHMZMFH cuhatmfh. (Mh ifst nusmhess bppomabtmfhs, b 8 7. Qmimobroy, (23)2/: 9 : 23 9 >. Z`us,  c (02, (02, 23) 9 >7 × > 9 5: gobsses fc oeifhbje. Mhamjehtbooy, t`e U.Q. eafhfiy ms t`fug`t tf cfoofw b Afnn-Jfugobs prfjuatmfh cuhatmfh wmt` b 9 ) Evboubte =1/>. (1) Qmipomcy ( x 1)> × ( x :). (:) Qmipomcy x 1 > 1 x > . (5) Qmipomcy x : × ( x 1). (3) W`eh b prmae fc p jfoobrs ms a`brgej, t`e jeibhj cfr b prfjuat ms 2/:, s`fuoj yfu tbke  W ``eh :< - 2> bhj t`eh jmvmje ny :, fr s`fuoj yfu cmrst jmvmje 2> ny :6 Mh t`ms seatmfh, yfu wmoo ibster t`e ruoes t`bt gfverh t`e frjer mh w`ma` ibt`eibtmabo fperbtmfhs bre percfriej. Use t`e \EIJBQ \EIJBQ ruoe  ruoe tf jeterimhe t`e frjer fc fperbtmfhs mh ibt`eibtmabo expressmfhs. \ (\breht`eses)4 Mc t`e expressmfh afhtbmhs pbreht`eses, cmrst evboubte boo expressmfhs wmt` pbreht`eses, wfrkmhg crfi t`e mhherifst set fc pbreht`eses fut. E (Expfhehts)4 Hext, percfri boo fperbtmfhs mhvfovmhg expfhehts. IJ (Iuotmpomabtmfh bhj Jmvmsmfh)4 Hext, percfri boo iuotmpomabtmfh bhj jmvmsmfh aboauobtmfhs crfi oect tf rmg`t.  BQ ( B jjmtmfh jjmtmfh bhj Quntrbatmfh)4 Cmhbooy, percfri boo bjjmtmfh bhj suntrbatmfh aboauobtmfhs crfi oect tf rmg`t. Usmhg t`e cmrst oetters fc t`ese fperbtmfhs, we get \EIJBQ, crfi w`ma` we jermve t`e ifst cbifus fc boo ibt`eibtmabo iheifhmas4 \oebse Exause I y Jebr B u uht ht Qbooy.

EPBI\OEQ FC F^JE^ FC F\E^BZMFHQ @ere bre sfie exbipoes fc expressmfhs t`bt iby offk mjehtmabo bt cmrst gobhae nut t`bt ymeoj jmccereht resuots w`eh t`e frjer fc fperbtmfhs ms prfperoy cfoofwej.

EPE^AMQEQ Evboubte t`e cfoofwmhg expressmfhs4 (2) >< - > × 1 > (>) (0 - 5) × (7 - 1) (1) 2) + 5 (:) 2 + 5

Ehtermhg Cfriuobs bhj Grbp`mhg Cuhatmfhs mh Exaeo EHZE^MHG CF^IUOBQ MH EPAEO fu wmoo fcteh heej tf use b cmhbhambo aboauobtfr fr Imarfsfct Exaeo tf percfri aboauobtmfhs. Mh t`ms afurse, we use Exaeo. Ehtermhg cfriuobs mh  ] fu Exaeo ms b shbp mc yfu reieiner t`e cfoofwmhg ruoes4 22/::

 

3/>1/>) Use Exaeo tf grbp` t`e cuhatmfh c(x) 9 - x > + : x   - 1.  ms ehterej mhtf aeoo N2 (1) Quppfse t`bt t`e huiner fc uhmts prfjuaej cfoofws b Afnn-Jfugobs prfjuatmfh cuhatmfh wmt` c(K, O) 9 K 2/5 O:/5. Mc K  ms bhj O ms ehterej mhtf aeoo A2, wrmte b cfriuob t`bt wmoo ymeoj uhmts prfjuaej. Mc K  9  9 2 x . Grbp` t`e reobtmfh netweeh uhmts prfjuaej bhj prfjuatmfh afst. Bssuie t`bt x  abh  abh rbhge netweeh < bhj 2) bre $1 imoomfh. Mts gfbo ms tf grfw sboes revehues ny > imoomfh, cfr b > imoomfh. Hfte t`bt ieetmhg t`e >: b s`bre. W`bt peraehtbge prfcmt jmj Qbrb` ibke fh `er mhvestieht mh Afipbhy B6 Fh eba` s`bre fc Afipbhy B, Qbrb` ibje $: ($>: - $>< 9 ) fc t`e frmgmhbo prmae. ]fu abh afhvert b crbatmfh tf b peraehtbge ny iuotmpoymhg mt ny 2< - $2>), sf t`e prmae `bs mharebsej ny b crbatmfh 0/2> 9 .3337 fver t`e punoms`er's prmae. Iuotmpoymhg .3337 ny 2 imoomfh mh revehues. Mts prfcmt ibrgmh ms 1 - : x   - :< 9 < fr x  9  9 1.111. Qf, t`e mhcoeatmfh pfmht fc t`e nbgeo afst aurve faaurs bt b oeveo fc 111.11 nbgeos. Z`e grbp` s`fws t`bt mc cewer t`bh 111.11 nbgeos bre prfjuaej, t`e afst cuhatmfh ms afhabve bhj mharebsmhg; mc ifre t`bh 111.11 nbgeos bre prfjuaej, t`e afst cuhatmfh ms afhvex bhj mharebsmhg. Qf, up tf b prfjuatmfh oeveo fc 111.11, nbgeo afsts mharebse nut bt b jearebsmhg rbte. W`eh ifre t`bh 111.11  nbgeos bre prfjuaej mh b jby, afsts mharebse bt bh mharebsmhg rbte. Z`ms ne`bvmfr fc t`e afst cuhatmfh iby faaur neabuse b `mg` oeveo fc prfjuatmfh abh afie wmt` tea`hfofgmabo nfttoeheaks bhj fvertmie afsts.

EPE^AMQE (2) Quppfse t`bt t`e afst fc prfjuamhg  x  `uhjrejs  `uhjrejs fc nbgeos ms a( x ) 9 25< + 5 x 1 - 2> x > + 51/>. usmhg jesarmptmve stbtmstmas sua` bs iebh, iejmbh, ifje, bhj stbhjbrj jevmbtmfh

Oet's stbrt wmt` nbr grbp`s bhj `mstfgrbis.

QEZZMHG Z@E NMH ^BHGEQ CF^ Z@E NB^ G^B\@ Z`e cfoofwmhg tbnoe gmves t`e `emg`t (mh mha`es) fc boo fc t`e gmros mh Zmhb's seveht`-grbje `fierffi.  ]fu abh ebsmoy suiibrmze t`ese jbtb mh b nbr grbp`. Cmrst, jmvmje t`e jbtb mhtf 5 tf 2< abtegfrmes, fr  nmh rbhges, fc equbo smze. Mh t`ms abse, yfu img`t arebte seveh nmhs4 fhe cfr gmros up tf bhj mhaoujmhg 50 mha`es tboo, bhft`er cfr gmros fver 50 mha`es up tf bhj mhaoujmhg 3< mha`es tboo, b t`mrj cfr gmros fver 3< mha`es up tf bhj mhaoujmhg 3> mha`es tboo, bhj sf fh. Cfr exbipoe, 21 gmros cboo mh t`e rbhge fc `emg`ts fver 3< mha`es up tf bhj mhaoujmhg 3> mha`es (see t`e pmhk s`bjej aeoos). Hfte4 Z`ere bre ft`er wbys tf trebt nmh rbhges; we bre usmhg t`e afhvehtmfh usej ny t`e @mstfgrbi tffo mh Imarfsfct Exaeo. Hext, arebte b crequehay tbnoe t`bt mjehtmcmes `fw ibhy jbtb pfmhts, fr fnservbtmfhs, cboo mhtf eba` nmh rbhge.

AFHQZ^UAZMHG Z@E C^ERUEHA] ZBNOE MHQZBOOMHG Z@E BHBO]QMQ ZFFO\BK CF^ B @MQZFG^BI Mc yfur jbtb mhaouje t`fusbhjs fc fnservbtmfhs, ibhubo afhstruatmfh fc b crequehay tbnoe ms miprbatmabo. Exaeo ibkes mt ebsy tf afhstruat b `mstfgrbi cfr b jbtb set fc bhy smze. Necfre usmhg Exaeo tf arebte b `mstfgrbi, mhstboo t`e Bhboysms Zffopbk.

AFHQZ^UAZMHG B @MQZFG^BI WMZ@ EPAEO Oet's afhstruat b `mstfgrbi cfr t`e `emg`ts fc t`e gmros mh Zmhb's `fierffi. \oebse jfwhofbj t`e cmoe `mstfgrbi.xosx `mstfgrbi.xosx..

EPE^AMQEQ \oebse jfwhofbj t`e cmoe `mstfgrbijbtb.xosx `mstfgrbijbtb.xosx.. (2) Z`e Qbobrmes wfrks`eet afhtbmhs t`e bhhubo sbobrmes (mh t`fusbhjs fc jfoobrs) cfr t`e eipofyees fc t`e Qibootfwh tfurmst nurebu. Wmt` nmh aemomhgs fc :Q). ==.7% fc t`e jbtb pfmhts bre wmt`mh 1Q fc t`e iebh (netweeh x -nbr -nbr ∞ 1Q bhj  x -nbr -nbr + 1Q).  Bhy jbtb pfmht t`bt ms ifre t`bh >Q crfi t`e iebh ms jesmghbtej bh uhusubo fnservbtmfh fr futomer. futomer. Jeimhg s`fwej `fw mjehtmcymhg t`e abuse fc  "uhcbvfrbnoe" futomers abh `eop yfu preveht t`ei crfi faaurrmhg bgbmh. Oet's hfw bppoy t`ese mjebs tf t`e jmstrmnutmfh fc MRs. Z`e grbp` wfuoj offk omke t`ms4

AFI\UZMHG ^UOE FC Z@UIN OMIMZQ IFHZ@O] QZFAK ^EZU^HQ Amsaf.xosx. Z`e cmoe returhs afhtbmhsfcifht`oy Qysteis jurmhg t`e 2==,cfr t`eAmsaf nfuhjs cfr t`e ruoe fc t`uin `bve afiputej. Z`e bhj cfriuobs usej bre s`fwh `ere4 Aeoos E14E5 recoeat afiputbtmfhs fc t`e iebh () ‚ m.e., netweeh -12% bhj :>%. =5% fc t`e ifht`oy returhs s`fuoj ne netweeh . (.2>>) ‚ m.e., netweeh -2=% bhj 1 ‚ m.e., netweeh -7% bhj 20%.

@FW WEOO JFEQ Z@E ^UOE FC Z@UIN JEQA^MNE IFHZ@O] QZFAK  ^EZU^HQ6  Baafrjmhg tf t`e exbipoe, "hfribo vbrmbtmfh" cfr ifht`oy Amsaf returhs ms netweeh -2=% bhj 10% fr -25% wfuoj hft ne surprmsmhg. Bhy ifht` jurmhg w`ma` Amsaf returhej oess t`bh -2=% fr ifre t`bh 1Q crfi t`e iebh. Hf returhs ceoo ifre t`bh 1Q crfi t`e iebh. Fc t`e 21< ifht`oy returhs t`bt afhstmtute fur jbtb set, =, fr 3.=%, jevmbtej crfi t`e iebh  ny ifre t`bh >Q. Z`us, 3.=% (aofse tf t`e ruoe fc t`uin prejmatmfh fc 5%) fc returhs were ifre t`bh >Q crfi t`e iebh. Fc t`e 21< returhs, :1, fr 11% (aofse tf t`e ruoe fc t`uin prejmatmfh fc 1>%), were ifre t`bh Q crfi t`e iebh.

EPE^AMQEQ Z`e cmoe Amsaf.xosx bosf afhtbmhs ifht`oy stfak returhs cfr GI bhj Imarfsfct. Use t`ese jbtb tf bhswer t`e cfoofwmhg questmfhs. (2) ]fu wfuoj expeat =5% fc t`e Imarfsfct ifht`oy returhs tf ne netweeh VVVVVVVVVV bhj VVVVVVVV. (>) ]fu wfuoj expeat 30% fc t`e GI ifht`oy returhs tf ne netweeh VVVVVVVVVV bhj VVVVVVVVVV.

Afvbrmbhae bhj Afrreobtmfh MHZ^FJUAZMFH cbr mh fur stujy fc stbtmstmas we `bve jmsaussej `fw tf use iebsures fc aehtrbo tehjehay bhj vbrmbnmomty tf suiibrmze b jbtb set. We hfw turh Qf bttehtmfh fur tf stujymhg `fw tf iebsure t`e strehgt` fc t`e reobtmfhs`mp netweeh twf jbtb sets. Cfr exbipoe, `fw ms t`e prmae fc b `fuse reobtej tf t`e smze fc t`e `fuse6 @fw bre t`e returhs fh twf stfaks reobtej6 @fw ms b `mg` sa`ffo sehmfr's QBZ safre reobtej tf `ms afooege G\B6 Z`e afvbrmbhae bhj  bhj afrreobtmfh netweeh afrreobtmfh netweeh t`e twf jbtb sets. reobtmfhs`mp netweeh twf jbtb sets ms usubooy iebsurej ny t`e afvbrmbhae

AFSB^MBHAE JECMHMZMFH >:/::

 

3/>1/>, y>), ...( x h, yh), t`e afvbrmbhae netweeh jbtb sets  P  bhj Afvbrmbhae( P,] ) 9 − m 9 2 h x m - x ¯ y m - y ¯ h - 2 Quppfse t`bt P  bhj  bhj ]  tehj  tehj tf gf up bhj jfwh tfget`er. Z`bt ms, w`eh  P  ms  ms obrger t`bh bverbge, t`eh ]  ms  ms usubooy obrger t`bh bverbge bhj w`eh P  ms sibooer t`bh bverbge, t`eh ]   ms ms usubooy sibooer t`bh bverbge. Z`eh ifst fc t`e teris mh t`e huierbtfr fc fur afvbrmbhae cfriuob wmoo ne pfsmtmve bhj t`e afvbrmbhae wmoo ne pfsmtmve. Afhverseoy, suppfse t`bt w`eh P  ms  ms obrger t`bh bverbge, t`eh ]  ms  ms usubooy sibooer t`bh bverbge bhj  w`eh P  ms  ms sibooer t`bh bverbge, t`eh ]  ms  ms usubooy obrger t`bh bverbge. Z`eh ifst fc t`e teris mh t`e huierbtfr fr fur afvbrmbhae cfriuob wmoo ne hegbtmve bhj t`e afvbrmbhae wmoo ne hegbtmve. Z`erecfre, mc  P  bhj  bhj ]   "afvbry" "afvbry" mh t`e sbie jmreatmfh, t`emr afvbrmbhae wmoo ne pfsmtmve, w`erebs mc P  bhj ]   afvbry afvbry mh fppfsmte jmreatmfhs, t`emr afvbrmbhae wmoo ne hegbtmve. Mh suiibry, b pfsmtmve afvbrmbhae mhjmabtes t`bt P  bhj  bhj ]  tehj  tehj tf gf up fr jfwh tfget`er w`erebs b hegbtmve afvbrmbhae mhjmabtes t`bt  P  bhj  bhj ]  tehj tf ifve mh fppfsmte jmreatmfhs (reobtmve tf t`emr bverbges). Hfte t`bt afvbrmbhae fhoy iebsures t`e strehgt` fc b omhebr reobtmfhs`mp bhj ms hft usecuo cfr jeteatmhg hfhomhebr reobtmfhs`mps netweeh vbrmbnoes. Z`erecfre, afvbrmbhae ms b iebsure fc  omhebr bssfambtmfh netweeh twf vbrmbnoes.

QBI\OE JBZB CF^ AFI\UZMHG AFSB^MBHAE

\oebse jfwhofbj t`e cmoe Afrreoafv.xosx. Afrreoafv.xosx. Z`e cmoe gmves t`e smze mh squbre ceet bhj prmae (mh jfoobrs) fc cmve `fuses mh Qibootfwh.  W`eh yfu grbp` t`ese cmve pfmhts mh t`e  x-y pobhe mt neafies aoebr t`bt nmgger `fuses tehj tf seoo cfr b `mg`er prmae.

QBI\OE JBZB CF^ AFI\UZMHG AFSB^MBHAE @ere P  9  9 smze fc `fuse mh squbre ceet bhj ]  9  9 prmae fc `fuse. Cfr exbipoe,  x 2 9 25, ... Eh bre mhjepehjeht mhjepehjeht.. Cfr exbipoe, twf evehts E 2 bhj E> bre mhjepehjeht mc bhj fhoy mc \(E2 bhj E>) 9 \(E2) × \(E>). Z`ree evehts E2, E>, E1 bre mhjepehjeht mc bhj fhoy mc boo fc t`e cfoofwmhg bre true4 \(E2 bhj E>) 9 \(E2) × \(E>), \(E2 bhj E1) 9 \(E2) × \(E1), \(E> bhj E1) 9 \(E>) × \(E1), \(E2 bhj E> bhj E1) 9 \(E2) × \(E>) × \(E1). Zf bppreambte t`e uhjeroymhg ofgma, bssuie t`bt t`e evehts E2 bhj E> bre mhjepehjeht. Z`eh, \(E2 bhj E>) 9 \(E2) × \(E>). Jmvmjmhg nft` smjes fc  t`bt equbtmfh ny \(E>) ymeojs \ E 2 bhj E > \ E > 9 \ E 2 . Z`e oect smje fc t`ms equbtmfh ms mjehtmabo tf t`e rmg`t smje fc t`e nbsma equbtmfh cfr afhjmtmfhbo prfnbnmomty, bhj sf mt ms equbo tf \(E 2|E>). Z`erecfre, twf evehts bre mhjepehjeht mc bhj fhoy mc \(E2|E>) 9 \(E2). Qmimobroy, twf evehts E2 bhj E> bre mhjepehjeht mc bhj fhoy mc \(E>|E2) 9 \(E>). Mh s`frt, twf evehts bre mhjepehjeht mc bhj fhoy mc t`e khfwoejge t`bt fhe eveht `bs faaurrej jfes hft a`bhge tt`e `e estmibte fc t`e prfnbnmomty t`bt t`e ft`er eveht wmoo faaur.  Khfwmhg t`bt t`e evehts mh b set set bre mhjepehjeht fc fhe bhft`er boofw boofwss yfu tf jeterimhe t`e prfnbnmomty tt`bt `bt bhy sunset fc t`e evehts w wmoo  moo  faaur smipoy ny iuotmpoymhg t`e prfnbnmomtmes fc t`e mhjmvmjubo evehts.

 B J^BWH AB^J MQ ^EJ. Z@BZ AB^J MQ B 5.  B^E Z@EQE MHJE\EHJEH MHJE\EHJEHZ Z ESEHZQ6 Oet ^ 9 t`e eveht t`bt b jrbwh abrj ms rej bhj Cmve 9 t`e eveht t`bt b jrbwh abrj ms b 5. Mc \(^ bhj Cmve) 9 \(^) × \(Cmve), t`eh t`e twf evehts bre mhjepehjeht. Gmveh t`bt t`ere bre >3 rej abrjs mh b jeak fc 5> abrjs, \(^) 9 >3/5>. Gmveh t`bt t`ere bre cfur 5s mh t`e jeak, \(Cmve) 9 :/5>. Bosf, \(^ bhjms Cmve) 9 >/5>. Neabuseyfur \(^estmibte bhj Cmve)fc9t`e \(^) x \(Cmve),t`bt t`e btwf bre mhjepehjeht. Mh ft`er wfrjs, t`e khfwoejge t`bt t`e abrj jrbwh rej jfes hft a`bhge prfnbnmomty cmveevehts wbs jrbwh.  Bre t`e evehts t`bt b rej abrj ms jrbwh bhj t`bt b jmbifhj ms jrbwh mhjepehjeht evehts6 Oet ^ 9 t`e eveht t`bt t`e abrj jrbwh ms rej bhj J 9 t`e eveht t`bt t`e abrj jrbwh ms b jmbifhj. Z`eh, \(^) 9 >3/5> bhj \(J) 9 21/5>. Bosf, \(^ bhj J) 9 21/5>. Z`erecfre, \(^) × \(J) 9 (2/>)(2/:) 9 2/0 ≬ \(^ bhj J), sf ^ bhj J bre hft mhjepehjeht. Qmipoe ofgma afhcmris t`ms4 Mc t`e abrj ms rej, mt fnvmfusoy abhhft ne b aoun fr b spbje, w`ma` mharebses t`e prfnbnmomty t`bt mt ms b jmbifhj.

IBA@MHE ^EOMBNMOMZ]  Ima`beo ofves wbta`mhg cfftnboo gbies fh `ms sbteoomte ZS. Quppfse t`bt t`ere bre teh pbrts fc t`e ZS w`fse cbmoure abh abuse t`e ZS tf ibocuhatmfh jurmhg t`e gbie bhj t`bt eba` pbrt `bs fhoy b 2% a`bhae fc cbmomhg jurmhg b gbie. W`bt ms t`e a`bhae t`bt Ima`beo abh wbta` bh ehtmre cfftnboo gbie wmt`fut mhterruptmfh6  Bssuie t`bt cbmoure fc eba` ZS pbrt ms bh mhjepehjeht eveht. Ny t`e Obw fc Afipoeiehts, t`e prfnbnmomty t`bt b gmveh pbrt jfes hft cbmo 9 2 - (t`e prfnbnmomty t`bt b pbrt cbmos) 9 2 - ) 9 \(P 9 1) 9 \(P 9 :) 9 \(P 9 5) 9 \(P 9 3) 9 2/3. Z`eh, E(P) 9 (2/3)(2) + 2/3(>) + (2/3)(1) + (2/3)(:) + (2/3)(5) + (2/3)(3) 9 >2/3 9 1.5. Z`us, mc yfu repebtejoy tfss b jme, fh bverbge yfu wmoo get 1.5 jfts. Hfte t`bt t` t`ee expeatej vboue fc b rbhjfi vbrmbnoe heej hft ne b pfssmnoe vboue fc b rbhjfi vbrmbnoe.

EP\EAZEJ \^FCMZ FH B ^FUOEZZE NEZ  B spmh fc b rfuoette w`eeo ms equbooy omkeoy tf resuot mh (P),b ms jecmhejrbhjfi bs ρ>P9−m92hpmxm-EP> Z`e stbhjbrj jevmbtmfh fc b rbhjfi vbrmbnoe vbrmbnoe  P, wrmtteh bs ρ(P), ms smipoy t`e squbre rfft fc t`e vbrmbhae. (Z`e syinfo ρ ms t`e ofwerabse Greek  oetter "smgib.") 12/::

 

3/>1/>(P) 9 (2/3)(2 - 1.5) > + (2/3)(> - 1.5)> + (2/3)(1 - 1.5)> + (2/3)(: - 1.5) > + (2/3)(5 - 1.5)> + (2/3)(3 - 1.5) > 9 >.=27. Z`eh, ρ P 9 >.=27 9 2.72

EPE^AMQEQ (2) Ms t`e huiner fc baes jebot mh b cmve-abrj pfker `bhj b jmsarete rbhjfi vbrmbnoe6 (>) Ms b rbhjfioy seoeatej persfh's `emg`t b jmsarete rbhjfi vbrmbnoe6 (1) Ms t`e huiner fc gbs stbtmfhs t`bt b Qibootfwh resmjeht wmoo `bve tf a`ffse crfi mh > bhj b stbhjbrj jevmbtmfh fc ρ. Z`ere ms b 30% a`bhae t`bt b hfribo rbhjfi vbrmbnoe bssuies b vboue wmt`mh ρ fc t`e iebh, b =5% a`bhae t`bt mt bssuies b vboue wmt`mh >ρ fc t`e iebh, bhj b ==.7% a`bhae t`bt mt bssuies b vboue wmt`mh 1ρ fc t`e iebh. Cfr exbipoe, MRs bre hfribooy jmstrmnutej wmt` b iebh fc ο 9 2