CHAPTER Let "a" denote an acceptable power supply Let "f","m","c" denote a supply with a functional, minor, or cosmetic error, respectively.

Transcription

1 CHAPTER Sco - -. L "a", "b" do a par abov, blow h spccao S aaa, aab, aba, abb, baa, bab, bba, bbb { } -. L "" do a b rror L "o" do a b o rror "o" dos okay, o, o, oo, o, oo, oo, ooo, S o, oo, oo, ooo, oo, ooo, ooo, oooo -. L "a" do a accpabl powr supply L "","m","c" do a supply wh a ucoal, mor, or cosmc rror, rspcvly. S { a,, m, c} -. S,,,...} s o ogav grs { -. I oly h umbr o racks wh rrors s o rs, h S {,,,..., } -6. A vcor wh hr compos ca dscrb h hr dgs o h ammr. Each dg ca b S,,..., 999,,,...,9. Th S s a sampl spac o possbl hr dg grs, { } -7. S s h sampl spac o possbl wo dg grs. -8. L a ordrd par o umbrs, such as do h rspos o h rs ad scod quso. Th, S,,..., cosss o h ordrd pars { } S {,,,...,} S {,,,...,} -9. ppb. -. mllscods -. S {.,.,., K.} -. s small, m mdum, l larg; S {s, m, l, ss, sm, sl,.} S {,,,..., - mllscods. -. } auomac rasmsso sadard rasmsso wh ar whou ar wh ar whou ar rd blu black wh rd blu black wh rd blu black wh rd blu black wh -

2 -. PRESS CAVITY mmory 8 dsk sorag -7. c coc, b busy, S {c, bc, bbc, bbbc, bbbbc, } S s, s, s, S, FS, FFS, FFFA -8. { } -9 a. b. -

3 c. d... a. -

4 b. c. d.. -

5 -. a S ogav grs rom o h largs gr ha ca b dsplayd by h scal. L rprs wgh. A s h v ha > B s h v ha C s h v ha 8 < S {,,,, } b S c < or {,,, } d or {,,,, } S A C would coa h valus o such ha: 8 Thus A C would coa h valus o such ha: < 8 or {,,,, 7} g h B would coa h valus o such ha >. Thror, B C would b h mpy s. Thy hav o oucoms commo or B C s h v 8 <. Thror, A B C s h v 8 or {8, 9,, } -. a A B b A C B c C -

6 d. A B C. I h vs ar muually clusv, h A B s qual o zro. Thror, h procss would o produc produc pars wh cm ad Y cm. Th procss would o b succssul -. L "d" dod a dsord b ad l "o" do a b ha s o dsord. dddd, dodd, oddd, oodd, dddo, dodo, oddo, oodo, a S ddod, dood, odod, oood, ddoo, dooo, odoo, oooo b No, or ampl A A { dddd, dddo, ddod, ddoo} c A dddd, dodd, dddo, dodo ddod, dood ddoo, dooo d oddd, oodd, oddo, oodo, A odod, oood, odoo, oooo A A A A { dddd} A A A A { dddd, dodd, dddo, oddd, ddod, oodd, ddoo} -6

7 -. L "d" do a dcv calculaor ad l "a" do a accpabl calculaor a a S { ddd, add, dda, ada, dad, aad, daa, aaa} b A { ddd, dda, dad, daa} c B { ddd, dda, add, ada} d A B { ddd, dda} B C { ddd, dda, add, ada, dad, aad} A B 7, A, A B 9-7. a. A B, B, A B 9 b. Surac Edg E G G Surac Edg E E G E E G G E E G E G G G E E E G E G G E E G E G G -8. A B, B, A B 8-9. a A { 7.} b B {.} c A B {. < < 7.} d A B { > }. a {ab, ac, ad, bc, bd, cd, ba, ca, da, cb, db, dc} b {ab, ac, ad, a, a, ag, bc, bd, b, b, bg, cd, c, c, cg,, g, g, ba, ca, da, a, a, ga, cb, db, b, b, gb, dc, c, c, gc,, g, g} c L d dcv, g good; S {gg, gd, dg, dd} d L d dcv, g good; S {gd, dg, gg}. L g do a good board, m a board wh mor dcs, ad j a board wh major dcs. a. S {gg, gm, gj, mg, mm, mj, jg, jm, jj} b S{gg,gm,gj,mg,mm,mj,jg,jm} -7

8 -.a. Th sampl spac coas all pos h posv -Y pla. b A c B d B A B A -8

9 - a b c d -9

10 Sco - -. All oucoms ar qually lkly a PA / b PB / c PA' / d PA B PA B P -. a PA. b PB.8 c PA'.6 d PA B PA B. -6. a S {,,,,, 6} b /6 c /6 d /6-7. a S {,,,,,6,7,8} b /8 c 6/8-8.., 6-9. a...7 b a / b / -. a. b.7 -. Toal possbl: 6, Oly 8 vald, Pvald 8 / 6 / 8 -. dgs bw ad 9, so h probably o ay hr umbrs s /**; lrs A o Z, so h probably o ay hr umbrs s /6*6*6; Th probably your lcs pla s chos s h / */ a ** b */ // -. a PA 86/.86 b PB 79/.79 c PA' /. d PA B 7/.7 PA B 796/.9 PA B 79/.8-6. L A cll surac sh; B cll lgh a PA 8/.8 b PB 9/.9 c PA'.8.8 d PA B 8/.8 PA B.9 PA B.98 -

11 -7. a PA /. b PB 77/.77 c PA'..7 d PA B /. PA B 8/.8 PA B 9/.9-8. a Bcaus E ad E' ar muually clusv vs ad E E S PS P E E PE PE'. Thror, PE' - PE b Bcaus S ad ar muually clusv vs wh S S PS PS P. Thror, P c Now, B A A B ad h vs A ad A B ar muually clusv. Thror, PB PA P A B. Bcaus P A B, PB PA. Sco a PA' - PA.7 b P A B PA PB - P A B c P A B P A B PB. Thror, P A B. -.. d PA P A B P A B. Thror, P A B. -.. P A B ' - P A B -..6 P A B PA' PB - P A B A B C A B C -. a P PA PB PC, bcaus h vs ar muually clusv. Thror, P....9 b P A B C, bcaus A B C c P A B, bcaus A B d P, bcaus A B C A B C A C B C A B C P -[ PA PB PC] I A,B,C ar muually clusv, h P A B C PA PB PC...., whch grar ha. Thror, PA, PB,ad PC cao qual h gv valus. -. a 7/.7 b /.9 c No, P A B -. a /7 6 b 7 7 c d /7 -. a 7/9 7/9 b 7/9 -. a Pusasacory -/ / b Pboh crra sasacory 7/.9, No -6. a 77---/7.988 b 66/7.989 c /7 6/7.98 d 9/7 /7.96 Sco - -

12 -7. a PA 86/ b PB 79/ P A B 7/ 7 c P AB P B 79/ 79 d P BA -8.a.8 b.9 c 8/9.889 d 8/ /8.976 /. P A B P A -9. a /7 b / 7/ 86/ a / b /8 c / -6. Nd daa rom Tabl - o pag a PA... P A B..7 b PA B. P B.7 c PB.7 P A B..7 d PB A. 7 P B. PA B..7. PA B a / b 9/99 c /9/99.8 d I h chps ar rplacd, h probably would b /. -6. a PA / b P BA /9 c P A B PA PB/A / /9. d P A B PA PB - P A B 9-6. A rs s local, B scod s local, C hrd s local a PA B C //9/8.6 b PA B C //9/ a /99.8 b //99.8 c 9/9/ a /98.6 b /98.8 c a Pgas lak /7.8 b Plcrc alur gas lak /7/87/.6 c Pgas lak lcrc alur /7/7/

13 -68. No, B A, h PA/B B A PA B PB PB PB -69. A C B Sco a PA B PABPB b PA B PA BPB 6... PA PA B PA B PABPB PAB PB L F do h v ha a cocor als. L W do h v ha a cocor s w. PF PFWPW PFW PW L F do h v ha a roll coas a law. L C do h v ha a roll s coo. P F P FC P C P FC P C -7. a PA. b PA'.97 c PB A. d PB A' P A B P BAPA... P A B' P B' APA.6..8 g PB P BAPA P BA'PA'

14 -7. L R do h v ha a produc hbs surac roughss. L N,A, ad W do h vs ha h blads ar w, avrag, ad wor, rspcvly. Th, PR PR NPN PR APA PR WPW L B do h v ha a glass braks. L L do h v ha larg packagg s usd. PB PB LPL PB L'PL' L U do h v ha h usr has mproprly ollowd sallao srucos. L C do h v ha h comg call s a compla. L P do h v ha h comg call s a rqus o purchas mor producs. L R do h v ha h comg call s a rqus or ormao. a PU CPC.7.. b PP RPR a b L A do a v ha h rs par slcd has cssv shrkag. L B do h v ha h scod par slcd has cssv shrkag. a PB P BAPA P BA'PA' // //. b L C do h v ha h hrd chp slcd has cssv shrkag. P C P C A B P A B P C A B' P A B' P C A' B P A' B P C A' B' P A' B' L A ad B do h vs ha h rs ad scod chps slcd ar dcv, rspcvly. a PB PB APA PB A'PA' 9/99/ /998/. b L C do h v ha h hrd chp slcd s dcv. P A B C P C A B P A B P C A B P B A P A Sco Bcaus P AB PA, h vs ar o dpd. -8. PA' - PA.7 ad P AB ' - P AB.7 Thror, A' ad B ar dpd vs. -8. P A B 7/, PA 86/, PB 77/. Th, P A B PAPB, so A ad B ar o dpd. -

15 -8. P A B 8/, PA 8/, PB 9/. Th, P A B PAPB, so A ad B ar o dpd. -8. a P A B /, PA /, PB 7/, Th P A B PAPB, hror, A ad B ar o dpd. b PB A PA B/PA /// I A ad B ar muually clusv, h P A B ad PAPB.. Thror, A ad B ar o dpd I s usul o work o o hs rcss wh car o llusra h laws o probably. L H do h v ha h h sampl coas hgh lvls o coamao. a PH ' H ' H ' H ' H ' PH ' PH ' PH ' PH ' PH ' ' by dpdc. Also, P H 9.. Thror, h aswr s ' ' ' ' b A H H H H H ' ' ' ' A H H H H H ' ' ' ' A H H H H H ' ' ' ' A H H H H H ' ' ' ' A H H H H H Th rqusd probably s h probably o h uo A A A A A ad hs vs ar muually clusv. Also, by dpdc P A Thror, h aswr s c L B do h v ha o sampl coas hgh lvls o coamao. Th rqusd probably s PB' - PB. From par a, PB' L A do h v ha h h b s a o. a By dpdc PA A... A PA PA... PA. 976 b By dpdc, ' ' ' ' ' P A A... A PA PA... PA c. 976 c Th probably o h ollowg squc s ' ' ' ' ' P A A A A A A6 A7 A8 A9 A, by dpdc. Th umbr o squcs cossg o v ""'s, ad v ""'s s!!!. Th aswr s L A do h v ha a sampl s producd cavy o o h mold. a By dpdc, P A A A A A. 8 b L B b h v ha all v sampls ar producd cavy. Bcaus h B's ar muually clusv, PB B... B8 PB PB... PB 8 From par a., P B. Thror, h aswr s c By dpdc, ' 7 P A A A A A. Th umbr o squcs 8 8 whch our ou o v sampls ar rom cavy o s. Thror, h aswr s

16 -9. L A do h uppr dvcs uco. L B do h lowr dvcs uco. PA PB PA B..87. Thror, h probably ha h crcu opras PA B PA PB PA B [-..][-..][-..] L A do h v ha h h radback s succssul. By dpdc, ' ' ' ' ' ' PA A A PA PA PA a P BA /99 ad Sco -7 P B P B A P A P B A' P A' / 99/ / 999/ / Thror, A ad B ar o dpd. b A ad B ar dpd. -9. Bcaus, P AB PB P A B P BA PA, P A B P B.7. P B A.8 P A. -9. L F do a raudul usr ad l T do a usr ha orgas calls rom wo or mor mropola aras a day. Th, P T F P F.. P F T. P T F P F P T F' P F' backup ma-sorag..7 l > yrs l > yrs l < yrs l < yrs a PB. b P AB.9 c P AB' d P A B P ABPB.9..7 P A B' P AB'PB' PA P A B P A B' g h P A' B P B.. P B A'.769 P A' B P B P A' B' P B'

17 -97. L G do a produc ha rcvd a good rvw. L H, M, ad P do producs ha wr hgh, modra, ad poor prormrs, rspcvly. a PG PGHPH PGMPM PGPPP b Usg h rsul rom par a., PGHPH 9.. PHG. 68 PG. 6 c PG ' HPH.. PHG '. PG ' a PDPD GPGPD G PG b PG D PG D /PD PD GPG/PD.99.99/ a PS b PCh S..897/ Sco Couous: a, c, d,, h, ; Dscr: b,, ad g Supplmal Ercss -. L D do h v ha h prmary alur mod s yp ad l A do h v ha a board passs h s. Th sampl spac s S { AAD, ', AD ', AD ', AD ', AD ' }. -. a / b / c 6/ d A B 9-7

18 -. a PA 9/.9 b PA B /. c PA B 9 9 /.99 d PA B 8/.8 PA B PA B/PB.8 -. L A do h v ha h h ordr s shppd o m. a By dpdc, P A A A P A P A P A b L ' B A A A ' B A A A ' B A A A Th, bcaus h B's ar muually clusv, PB B B PB PB PB 9... c L ' ' B A A A ' ' B A A A ' ' B A A A ' ' ' B A A A Bcaus h B's ar muually clusv, P B B B B P B P B P B P B a No, PE E E b No, E E s o c PE E E PE PE PE PE E - PE E - PE E PE E E / d PE E E / PE E PE PE PE E / PE E E PE E E

19 -7. L A do h v ha h h bol slcd s o orqud o h propr lm. a Th, P A A A A P A P A A A A A A P A A A P A A A A P A A P A b L B do h v ha a las o o h slcd bols ar o proprly orqud. Thus, B' s h v ha all bols ar proprly orqud. Th, PB - PB' L A,B do h v ha h rs, scod poro o h crcu opras. Th, PA PB ad P A B PA PB A by lpho, A wbs; PA.9, PA.9; By dpdc PA A PA PA - PA A PPossss L D do h v ha a coar s corrcly lld ad l H do h v ha a coar s lld udr hgh-spd oprao. Th, a PD PDHPH PD H'PH' b P D H P H.. P H D. 88 P D.7 -. a PE T D b PE D PE PD PE D D dcv copy a PD b PD c L A rprs h v ha h wo ms NOT spcd ar o dcv. Th, PA7/77/ Th ool als ay compo als. L F do h v ha h ool als. Th, PF'. 9 by dpdc ad PF a b P E rou P rou.8. P rou E. 9 P E

20 -6. a By dpdc,. 79. b L A do h vs ha h mach s dl a h m o your h rqus. Usg dpdc, h rqusd probably s ' ' ' ' ' P A A A A A or A A A AA or A A AA A or A AA A A or AA A A A c As par b,h probably o o h vs s P A A A A ' A ' A A.. A A ' A ' or.8 or ' A A A A A A A ' A A ' A ' or or A A ' So o g h probably o a las, add aswr pars a. ad b. o h abov o oba rqusd probably. Thror h aswr s L A do h v ha h h washr slcd s hckr ha arg. 9 8 a b /8.6 A A A A A ' ' A A ' A or or A A c Th rqusd probably ca b wr rms o whhr or o h rs ad scod washr slcd ar hckr ha h arg. Tha s, ' ' ' ' P A P AAAorAAAorAAAorAAA ' ' PA AA PA A PA AA PA A ' ' ' ' ' PA A' A PA A PA AA PA A ' ' PA AA PA A PA PA A A PA A PA ' ' ' ' ' ' ' ' PA AA PA A PA PA AA PA A PA a I washrs ar slcd, h h probably hy ar all lss ha h arg s probably all slcd washrs ar lss ha arg /. /9/9. /9/98/8.8 Thror, h aswr s b Th v E ha o or mor washrs s hckr ha arg s h complm o h v ha all ar lss ha arg. Thror, PE quals o mus h probably par a. Thror,. ' A A A ' ' A A A A A ' or or A A ' A ' A A ' A A A A ' A or -

21 -9. a b c d 68 6 P A B. 9 6 P A B P A' B.88 9 P A' B' 9.7. P AB PA B 6 / PB / 9 PBA PB A 6 / PA 8 / 9 -. L E do a rad rror ad l S,O,P do skwd, o-cr, ad propr algms, rspcvly. Th, a PE PE S PS PE O P O PE P PP b PS E P ES P S... P E L A do h v ha h h row opras. Th, PA. 98, PA , PA. 98, PA. 98. Th probably h crcu dos o opra s ' ' ' ' 7 P A P A P A P A a b P or mor provdd../..67 Md-Epadg Ercss -. L E do a rad rror ad l S, O, B, P do skwd, o-cr, boh, ad propr algms, rspcvly. PE PE SPS PE OPO PE BPB PE PPP L do h umbr o washrs slcd. a Th probably ha all ar lss ha h arg s., by dpdc Thror, b Th rqusd probably s h complm o h probably rqusd par a. Thror, -

22 -. L do h umbr o ks producd. Rvu a ach dmad - Ma pro Ma pro [--] Ma pro [--]. [--] Ma Pro Mamum Pro 7.7 $ 77. a.7 $ 7 a. $ a Thror, pro s mamzd a ks. Howvr, h drc pro ovr ks s small. -6. L E do h probably ha o o h bols ar dd as corrcly orqud. Th rqusd probably s PE'. L do h umbr o bols h sampl ha ar corrc. Th, PE PE P PE P PE P PE P PE P ad P //9/8/7.87. Th rmag probably or ca b drmd rom h coug mhods Appd B-. Th,!!!!!! 6!!!! P. 696!!!!! 6!!!!!!!! P 67.! 6!!!!!!!! P. 9! 6!! -7. P //9/8/7. ad PE, PE., PE.., PE.., PE Th, PE ad PE'.69 PA ' B' P[ A' B']' PA B [ PA PB PA B] PA PB PAPB [ PA ][ PB ] PA ' PB ' -

23 -8. Th oal sampl sz s ka a kb b k a k b. k a b ka a PA, P B k a k b k a k b ad ka ka PA B k a k b k a b Th, k a b ka a k a b k a ka PAPB [ k a k b] k a b k a b PA B Sco -. o CD S-. From h mulplcao rul, h aswr s S-. From h mulplcao rul, 6 S-. From h mulplcao rul, S-. From quao S-, h aswr s! 688 S-. From h mulplcao rul ad quao S-, h aswr s!! S-6. From quao S-, 7! squcs ar possbl!!!!! S-7. a From quao S-, h umbr o sampls o sz v s 6968!! 6! b Thr ar ways o slcg o ocoormg chp ad hr ar 888 ways o slcg our coormg chps. Thror, h umbr o sampls ha coa acly o ocoormg chp s 888 c Th umbr o sampls ha coa a las o ocoormg chp s h oal umbr o sampls mus h umbr o sampls ha coa o ocoormg chps.!! Tha s - 77!!!! S-8. a I h chps ar o dr yps, h vry arragm o locaos slcd rom h rsuls a! dr layou. Thror, P 9 layous ar possbl. 7! b I h chps ar o h sam yp, h vry subs o locaos chos rom h rsuls a dr layou. Thror,! 7!! 79 layous ar possbl. -

24 7!!! squcs ar possbl. 7! b!!!!!! S-9. a squcs ar possbl. c 6! 7 squcs ar possbl. S-. a Evry arragm o 7 locaos slcd rom h comprss a dr dsg.! P dsgs ar possbl.!! dsgs ar!7! b Evry subs o 7 locaos slcd rom h comprss a w dsg. 79 possbl.!!9! c Frs h hr locaos or h rs compo ar slcd ways. Th, h our 9!!! 9 locaos or h scod compo ar slcd rom h rmag locaos 6 ways. From h mulplcao rul, h umbr o dsgs s 6 77 S-. a From h mulplcao rul, prs ar possbl b From h mulplcao rul, 8 6 ar possbl c Evry arragm o hr dgs slcd rom h dgs rsuls a possbl pr. P! 7! 7 prs ar possbl. 8 S-. a From h mulplcao rul, 6 bys ar possbl 7 b From h mulplcao rul, 8 bys ar possbl!!! 6! 8!!!!! S-. a Th oal umbr o sampls possbl s o ak has hgh vscosy s Th umbr o sampls whch acly. Thror, h probably s 8!!! 8 b Th umbr o sampls ha coa o ak wh hgh vscosy s rqusd probably s. 7. Thror, h 6!!!!!!!!! 6 c Th umbr o sampls ha m h rqurms s Thror, h probably s. 6. -

25 S-. a Th oal umbr o sampls s ocoormg par s 9. 9/.9.!. Th umbr o sampls ha rsul o! 9!!!!!!8! Thror, h rqusd probably s!!7! b Th umbr o sampls wh o ocoormg par s. ocoormg par s.. Th probably o a las o S-. a Th probably ha boh pars ar dcv s. 8 b Th oal umbr o sampls s pars s 9! 9!8!!. Thror, h probably s. 8 9!! 9. Th umbr o sampls wh wo dcv. -

26 CHAPTER Sco - -. Th rag o s {,,,...,} - Th rag o s {,,,..., } -. Th rag o s {,,,..., 99999} - Th rag o s {,,,,,},,...,9 -. Th rag o s {. Bcaus 9 pars ar coormg, a ocoormg par mus b slcd 9 slcos. } } -6 Th rag o s {,,,...,. Alhough h rag acually obad rom los ypcally mgh o cd %. -7. Th rag o s covly modld as all ogav grs. Tha s, h rag o s,,,... { } -8 Th rag o s covly modld as all ogav grs. Tha s, h rag o s,,,... { } -9. Th rag o s {,,,...,} - Th possbl oals or wo ordrs ar /8 /8 /, /8 / /8, /8 /8 /, / / /, / /8 /8, /8 /8 6/8. 6 Thror h rag o s,,,, Th rag o s {,,, K,} - Th rag o s {,,,...,} Sco - -. P. P / 6 / 6 / 6 / 6 /. / - a P. / b P.< <.7 P. P /6 / / c P > d P < P P. / / / P or / /6 / -. All probabls ar grar ha or qual o zro ad sum o o. -

27 a P /8 /8 /8 /8 /8 b P > - /8 /8 /8 /8 7/8 c P- /8 /8 /8 6/8 / d P - or /8 /8 /8 /8 / -6 All probabls ar grar ha or qual o zro ad sum o o. a P P.7 b P> -P c P<<6P.9 d P or > P PP -7. Probabls ar ogav ad sum o o. a P 9/ b P / / / c P < / 7/ / d P > -8 Probabls ar ogav ad sum o o. a P // /6 b P /[// ] 6/6 c P > P /6 d P P / / -9. P mllo., P mllo.6, P mllo. - P mllo., P mllo., P mllo P. 8 P [.98..]. P [ ].76 P umbr o wars ha pass P..8 P P..8.8 P.8. - P mllo.6, P mllo., P -. mllo. - umbr o compos ha m spccaos P... P P umbr o compos ha m spccaos P.... P P P

28 Sco -, / F / / 6 < <.. < < -6 whr P. P / 6 / 6 / 6 / 6 /. / -7., /8 /8 F /8 7 /8 < < < < < a P. 7/8 b P. c P-. < 7/8 /8 / d P > P /8 /8 whr / 8 / 8 / 8 / 8 / 8, / / F 9 / 6 / < < < < < -8 whr / / / 7 / 9 / a P <. / b P 6/ c P > P 9/ 6/ d P < P P 9/ / / / -9.,., F.7,, < < < whr P mllo., P mllo.6, P mllo. -

29 -,., F.,, < < < whr P mllo., P mllo., P mllo. -.,.8, F.,.88,, < < < < whr , ,.8.8,., -,., F.,, <.. < < whr P mllo.6, P mllo., P -. mllo. -. Th sum o h probabls s ad all probabls ar grar ha or qual o zro; pm:.,. a P b P. c P P. d P> P. - Th sum o h probabls s ad all probabls ar grar ha or qual o zro; pm:.7,., 7. a P.9 b P > 7 c P.9 d P>. P.7 -. Th sum o h probabls s ad all probabls ar grar ha or qual o zro; pm: -.,.,. a P b P.7 c P 6 P. d P<. P < P<< -

30 -6 Th sum o h probabls s ad all probabls ar grar ha or qual o zro; pm: /8., /.7, /8. a P /8 b P /.9 c P /6.9 d P>/. P / Sco - -7 Ma ad Varac µ E..... V µ Ma ad Varac µ E.. /./ / / 6./ / 6. V. µ / 6 9/ Drm E ad V or radom varabl rcs -. µ E / 8 /8 /8 /8 / 8 V µ / 8 / 8 / 8 / 8 / 8. - Drm E ad V or radom varabl rcs - µ E V µ Ma ad varac or rcs -9 µ E mllo V µ mllo.6 -

31 - Ma ad varac or rcs - µ E.... mllo V µ mllo.. -. Ma ad varac or radom varabl rcs - µ E V µ Ma ad varac or rcs - µ E mllo V.. µ mllo -. Drm whr rag s [,,,,] ad ma s 6. µ E Sco - -6 E /, V [- -]/ 8-7. E /, V [- -]/ E, V

32 -9. /Y, Y, 6, 7, 8, 9. 9 E / EY. 7 mm 9. V mm - E cods h pcd umbr o lrs s V cods h varac s Y, Y,,,..., E mm,.8 V mm - Th rag o Y s,,,...,, E 9/. EY //.../ [ ] E.. V 8., VY 8. 6., Y.6 c c E c ce -, c cµ c µ V c cv - s a dscr radom varabl. s dscr bcaus s h umbr o lds ou o 8 ha has a rror. Howvr, s o uorm bcaus P P. -7

33 Sco A bomal dsrbuo s basd o dpd rals wh wo oucoms ad a cosa probably o succss o ach ral. a rasoabl b dpdc assumpo o rasoabl c Th probably ha h scod compo als dpds o h alur m o h rs compo. Th bomal dsrbuo s o rasoabl. d o dpd rals wh cosa probably probably o a corrc aswr o cosa. rasoabl g probably o dg a dc o cosa. h h lls ar dpd wh a cosa probably o a udrll, h h bomal dsrbuo or h umbr packags udrlld s rasoabl. bcaus o h burss, ach ral ha cosss o sdg a b s o dpd j o dpd rals wh cosa probably a a. E p. b. Valus ad ar h las lkly, h rm valus P b P c P d P <

34 -8 Bomal,..9 prob o E p.. a Th valu o ha appars o b mos lkly s.. b Th valu o ha appars o b las lkly s. -9. a P. b P c P 9. 9 d P < ad p.. F..87 < < < < whr

35 -6. ad p..9 F < < < < whr L do h umbr o dcv crcus. Th, has a bomal dsrbuo wh ad -6. a p.. Th, P P b P P..999 c P d E. V L do h umbr o ms h l s occupd. Th, has a bomal dsrbuo wh ad p. a. P b. P P c. E. -6. a, p /., sc E p. b c P P

36 -66 E.. V µ a s bomal wh ad p. P >. P P b s bomal wh ad p. P > P 9 [ ] [ ]. 897 c L Y do h umbr o ms cds h v sampls. Th, Y s bomal wh ad p.9 rom par b. P Y P Y [.9.8 ]. 6 Th probably s.6 ha a las o sampl rom h v wll coa mor ha o dcv L do h passgrs wh cks ha do o show up or h lgh. Th, s bomal wh ad p.. a P b P P > P L do h umbr o dcv compos amog hos sockd. a. b. P P c. P.98 -

37 -69. L do h umbr o qusos aswrd corrcly. Th, s bomal wh ad p.. a P b P < L do h umbr o morgs h lgh s gr. a b P P c P > P.6.7 Sco a. P... b. P c. P d. P P P P > P E. /p gvg p. a. P... b. P c. P... 8 P P P P d P > P

38 -7. L do h umbr o rals o oba h rs succssul algm. Th s a gomrc radom varabl wh p.8 a P P P P P P b P P [ P P P c ] [ ] [ ] L do h umbr o popl who carry h g. Th s a gav bomal radom varabl wh r ad p. a P P < [ P P ].. E r / p /. b L do h umbr o calls dd o oba a coco. Th, s a gomrc radom varabl wh p. 9 9 a P P > P [ P P P P b ] [ ] c E /. -76 L do h umbr o morgs dd o oba a gr lgh. Th s a gomrc radom varabl wh p.. a P -... b By dpdc,.8.7. Also, P >.7-77 p., r 8 a. P E b. µ days 9 c Ma umbr o days ul all 8 compurs al. Now w us p µ days or 7. yars 8 E Y L Y do h umbr o sampls dd o cd Ercs -66. Th Y has a gomrc dsrbuo wh p.69. a PY b Y s a gomrc radom varabl wh p.897 rom Ercs -66. PY c EY /

39 -79. L do h umbr o rals o oba h rs succss. a E /. b Bcaus o h lack o mmory propry, h pcd valu s sll. p r -8 Ngav bomal radom varabl: ; p, r. Wh r, hs rducs o ; p, r p - p, whch s h pd o a gomrc radom varabl. Also, E r/p ad V [rp]/p rduc o E /p ad V p/p, rspcvly. r p r -8. a E / b P c P9 9 d P Th mos lkly valu or should b ar µ. By ryg svral cass, h mos lkly valu s L do h umbr o amps dd o oba a calbrao ha coorms o spccaos. Th, s gomrc wh p.6. P P P P L do h umbr o lls dd o dc hr udrwgh packags. Th s a gav bomal radom varabl wh p. ad r. a E /. b V [.999/. ] 997. Thror, L do h umbr o rasacos ul all compurs hav ald. Th, s gav bomal radom varabl wh p -8 ad r. a E 8 b V [ -8 ]/

40 -8 L do a gomrc radom varabl wh paramr p. L q -p. [ ] [ ] [ ] [ ] p q p p p p p q p q pq p q q q p q q q p q p q p q q q p p p p p V p p p q p q dq d p q dq d p q p p p E p p q q dq d p dq d p dq d p p p p p p Sco has a hyprgomr dsrbuo N,, K c a P b., h sampl sz s oly 6 P c P d. 8. N K p E N N p p V -

41 -87. a b c 6 6 P P P P 6 6 / / / P 6 P d E /.8 V..86/ N, ad K P... -6

42 -89., / 6, F /, 9 /,, < < < < whr ,., 6 6.,. -9 L do h umbr o uaccpabl washrs h sampl o. a. b. c. d. P P P 7 7 P E / 7 7!!6! 7!!6!.!7! 9!6! 7!!6! / L do h umbr o m who carry h markr o h mal chromosom or a crasd rsk or hgh blood prssur. N8, K a P b 6 9 8! 6!!9! 8!!79! 9!!. P > P [ P P ] P 6 8! 6!!! 8!!79!!!.76 P > P [.76.].8-7

43 -9. L do h umbr o cards h sampl ha ar dcv. a b P P P P P!!!!!!.6.96 P P!!!!!! P.7.9.6!!.7!! -9. L do h umbr o blads h sampl ha ar dull. a P P P 8 8 P P 8!!! 8!!! 8!!.9 8!!.769 b L Y do h umbr o days dd o rplac h assmbly. PY c O h rs day, O h scod day, P P !!! 8!!! 6!!.8 8!!!!7! 8!!! O h hrd day, P.9 rom par a. Thror, PY !!.968 8!7! -9 L do h cou o h umbrs h sa's sampl ha mach hos h playr's sampl. Th, has a hyprgomrc dsrbuo wh N, 6, ad K 6. a b c P P 6 P ! 6!! d L Y do h umbr o wks dd o mach all s umbrs. Th, Y has a gomrc dsrbuo wh p,88,8 ad EY /p,88,8 wks. Ths s mor ha 78 curs! -8

44 -9. a For Ercs -86, h populao corrco s 96/99. For Ercs -87, h populao corrco s 6/9. Bcaus h populao corrco or Ercs -86 s closr o o, h bomal appromao o h dsrbuo o should b br Ercs -86. b Assumg has a bomal dsrbuo wh ad p., P P Th rsuls rom h bomal appromao ar clos o h probabls obad Ercs -86. c Assum has a bomal dsrbuo wh ad p.. Cosquly, P ad P ar h sam as compud par b. o hs rcs. Ths bomal appromao s o as clos o h ru aswr as h rsuls obad par b. o hs rcs. -96 a. From Ercs -9, s appromaly bomal wh ad p / /7. P P populao corrco s /9.86 b From Ercs -9, s appromaly bomal wh ad p / /8 P P populao corrco s / Sco a P. 8! b P P P P!!. 8 c P 9.! 8 d P !. -98 a P b P. 99!!.. c P. 7! d P ! -9

45 λ -99. P.. Thror, λ l Cosquly, E V a L do h umbr o calls o hour. Th, s a Posso radom varabl wh λ. P 78..! b P.!!! c L Y do h umbr o calls wo hours. Th, Y s a Posso radom varabl wh λ. PY. 6! d L W do h umbr o calls mus. Th W s a Posso radom varabl wh λ. PW 7.! -. a L do h umbr o laws o squar mr o cloh. Th, s a Posso radom varabl.. λ P.! wh.. b L Y do h umbr o laws squar mrs o cloh. Th, Y s a Posso radom varabl wh. λ P Y. 679! c L W do h umbr o laws squar mrs o cloh. Th, W s a Posso radom varabl wh λ. P W. d P Y P Y P Y P Y.6 - a E λ. rrors pr s ara..... b. P. 9989!! 99.89% o s aras -. a L do h umbr o cracks mls o hghway. Th, s a Posso radom varabl wh λ. P. b L Y do h umbr o cracks a hal ml o hghway. Th, Y s a Posso radom varabl wh λ. P Y P Y. 6 c Th assumpos o a Posso procss rqur ha h probably o a cou s cosa or all rvals. I h probably o a cou dpds o rac load ad h load vars, h h assumpos o a Posso procss ar o vald. Spara Posso radom varabls mgh b appropra or h havy ad lgh load scos o h hghway. -

46 - a. E λ. alurs pr sampls. L Y h umbr o alurs pr day E Y E E λ. alurs pr day. b.l W h umbr o alurs parcpas, ow λ. ad P W a L do h umbr o laws squar o plasc pal. Th, s a Posso radom varabl wh... λ P. 66 b L Y do h umbr o cars wh o laws, P Y c L W do h umbr o cars wh surac laws. Bcaus h umbr o laws has a Posso dsrbuo, h occurrcs o surac laws cars ar dpd vs wh cosa probably. From par a., h probably a car coas surac laws s Cosquly, W s bomal wh ad p.9. P W P W P W a L do h alurs 8 hours. Th, has a Posso dsrbuo wh λ.6. P.6.8 b L Y do h umbr o alur hours. Th, Y has a Posso dsrbuo wh λ.8. P Y P Y 8. 8 Supplmal Ercss -7. L do h umbr o os h sampl ha do o coorm o pury rqurms. Th, has a hyprgomrc dsrbuo wh N,, ad K.!! P P.7!! -8 L do h umbr o calls ha ar aswrd scods or lss. Th, s a bomal radom varabl wh p a P b P 6 P6 P7 P8 P9 P c E

47 -9. L Y do h umbr o calls dd o oba a aswr lss ha scods. a P Y b EY /p /.7 / - L W do h umbr o calls dd o oba wo aswrs lss ha scods. Th, W has a gav bomal dsrbuo wh p.7. a PW b EW r/p /.7 8/ -. a L do h umbr o mssags s o hour.. 7 b L Y do h umbr o mssags s. hours. Th, Y s a Posso radom varabl wh λ P Y! c L W do h umbr o mssags s o-hal hour. Th, W s a Posso radom varabl wh λ.. P W < P W P W s a gav bomal wh r ad p. E r / p /. rquss P! -. Possoλ., Possoλ P Y!!!.98 - L do h umbr o dvduals ha rcovr o wk. Assum h dvduals ar dpd. Th, s a bomal radom varabl wh ad p.. P P a. P, P., P., P., P.6 P6., P7.8, P8., P9., P.6 b. P., P.., P., P..6, P. P..8, P., P.., P.6-6 L do h umbr o assmbls dd o oba dcvs. Th, s a gav bomal radom varabl wh p. ad r. a E r/p. b V *.99/. 9 ad.9-7. I assmbls ar chckd, h l do h umbr o dcv assmbls. I P.9, h P.. Now,..99 l.99 l. 99 P ad.99.. Thror, l. l Ths would rqur 99. -

48 -8 Rqur. Thror, c. Thror, c L do h umbr o producs ha al durg h warray prod. Assum h us ar dpd. Th, s a bomal radom varabl wh ad p a P. b E. c P > P a P...6 b P > c P.7 < <....7 d E V L do h umbr o bols h sampl rom supplr ad l Y do h umbr o bols h sampl rom supplr. Th, s a hyprgomrc radom varabl wh N,, ad K. Also, Y s a hyprgomrc radom varabl wh N,, ad K 7. a P or Y P PY b P[ ad Y or Y ad ] 9 - L do h umbr o rrors a scor. Th, s a Posso radom varabl wh λ.768. a P> P b L Y do h umbr o scors ul a rror s oud. Th, Y s a gomrc radom varabl ad P P P EY /p.8 -

49 -. L do h umbr o ordrs placd a wk a cy o 8, popl. Th s a Posso radom varabl wh λ.8. a P P [ /!] b L Y do h umbr o ordrs wks. Th, Y s a Posso radom varabl wh λ, ad PY> - PY - - /! - /! - [ ] a. hyprgomrc radom varabl wh N,, ad K E.7.E E.97.E E E E 7. 8 E.9.E b

50 -7. L do h umbr o os h sampl ha cd h mosur co. Th s a bomal radom varabl wh. W ar o drm p. I P.9, h P.. Th whch rsuls p.79. p p., gvg lpl., -8 L do a rval o m hours ad l do h umbr o mssags ha arrv m. Th, s a Posso radom varabl wh λ. Th, P.9 ad -.9, rsulg. hours.6 scods -9. a L do h umbr o laws pals. Th, s a Posso radom varabl wh λ.. P b L Y do h umbr o laws o pal, h PY PY L W do h umbr o pals ha d o b spcd bor a law s oud. Th W s a gomrc radom varabl wh p.98 ad EW /.98. pals.. c P Y P Y. 98 L V do h umbr o pals wh or mor laws. Th V s a bomal radom varabl wh ad p.98 P V Md Epadg Ercss -. L ollow a hyprgomrc dsrbuo wh paramrs K,, ad N. To solv hs problm, w ca d h gral pcao: E k k k P Usg h rlaoshps K N K N K K K ad N N N w ca subsu o E K : -

51 E k k P ] [ k j k k k Z E N K N j K N j K j N K N N K N K K N K N K Now, Z s also a hyprgomrc radom varabl wh paramrs, N, ad K. To d h ma o, E, s k : E N K Z E N K ] [ I w l p K/N, h E p. I ordr o d h varac o usg h ormula V E [E}, h E mus b oud. Subsug k o E k w g [ ] ] [ N K N K E Z E N K Z E N K Z E N K E Thror, V N K N K N K N K N K N K I w l p K/N, h varac rducs o V p p N N -. Show ha usg a sum. p p To bg,, by do o a sum hs ca b rwr as p p p p p p p p p p -6

52 - [ ] 6 6 ]... [ a b a b a b a b a a b b a b a a a b b b a b a b a b a b a b V a b a b a b a b a b a b a b a b a a b b a b a b b a a E b a b a b a a b b a - L do h umbr o ocoormg producs h sampl. Th, s appromaly bomal wh p. ad s o b drmd. I, h.9 P. P. Now, P p p p. Cosquly,, ad. p 9. l l. p. Thror, s rqurd - I h lo sz s small, % o h lo mgh b suc o dc ocoormg produc. For ampl, h lo sz s, h a sampl o sz o has a probably o oly. o dcg a ocoormg produc a lo ha s % ocoormg. I h lo sz s larg, % o h lo mgh b a largr sampl sz ha s praccal or cssary. For ampl, h lo sz s, h a sampl o s rqurd. Furhrmor, h bomal appromao o h hyprgomrc dsrbuo ca b usd o show h ollowg. I % o h lo o sz s ocoormg, h h probably o zro ocoormg produc h sampl s appromaly 7. Usg a sampl o, h sam probably s sll oly.9. Th sampl o sz mgh b much largr ha s dd. -7

53 - L do h umbr o pals wh laws. Th, s a bomal radom varabl wh ad p s. h probably o o or mor laws a pal. Tha s, p.9. P < P P 99 p p p p 97 p p. P P p p 96 P p p 98 P -6 L do h umbr o rolls producd. Rvu a ach dmad.... ma pro ma pro.. [..-] ma pro.. [..-]. [..-] ma pro.. [..-]. [..-]. [..- ]. -. Pro Ma. pro. $ a.7 $ a $ a L do h umbr o accpabl compos. Th, has a bomal dsrbuo wh p.98 ad s o b drmd such ha P. 9. P Thror, compos ar dd. -8

54 CHAPTER Sco - -. a P < d b P < <. d. 88 c P d d P < d. 987 P d a P < d.. Th, l.. b P d.. Th, l.9. - a P < d. 7, bcaus or < b, P >. d bcaus or > c P < < d d P <. d P >. P <. d. 8 d

55 - a P < d, bcaus or <. Ths ca also b obad rom h ac ha s a probably dsy uco or <. b P d. 6 c P < P. From par b., P. 6. Thror, P < d P8 < < d. 8 8 P < d. 9. Th, l a P <., by symmry. b P. <. d c P... d.. d P <. P < or >.... P <. d.... Th, a P > d. b P < < d. c P < d. 6 / d P < d.. / Th, 9., ad l

56 .. -7 a P >.d... b P >.9.d. Th, 99.6 ad a P < 7.8.d b P < 7.8 or > 7. P < 7.8 P > 7. bcaus h wo vs ar muually clusv. Th rsul s c P 7.7 < < 7..d a P <. or >.7 P <. P >.7 bcaus h wo vs ar muually clusv. Th, P <. ad P >.7 d....7 b I h probably dsy uco s crd a. mrs, h or. < <.8 ad all rods wll m spccaos. -. Bcaus h gral d s o chagd whhr or o ay o h dpos ad ar cludd h gral, all h probabls lsd ar qual. Sco - -. a P<.8 P.8 bcaus s a couous radom varabl. Th, P<.8 F b P >. P c P < F d P 6 F 6 > -. a P <.8 P.8 F.8 bcaus s a couous radom varabl. Th, F b P >. P c P < - d P < < P < F F.7.. -

57 -. Now, or < ad F or <. Th, F,, > -. Now, / 8 or < < ad F or <. Th, F d 8, < 9, < 6, d Now, or < ad F or <., Th, F, > / -6. Now, or < ad F / / d / d / or <., Th, F /, > P> - P - F -/. -7. Now,. or 7.6 < < 7. ad F.d. 9. or 7.6 < < 7.. Th,, < 7.6 F. 9., 7.6 < 7., 7. P > 7 P 7 F7.. bcaus s a couous radom varabl

58 -8, > -9. < < < 9.,., -. < < <..,., d F or < <. Th, < < F,,., Sco d E.. d V d E d d V -

59 -... d E.6... d d V d E d d V d E d d V -7. a l 6 d E l d d V b. Avrag cos pr par $.*9.9 $.7-8. d E -6

60 -9. a E d. -. a Usg grao by pars wh u ad E d dv d, w oba. Now,. V d. Usg h grao by pars wh u. dv ad, w oba V.. d. From h do o E h gral abov s rcogzd o qual. Thror, V.... b P >. d. 679 E V. Thror,.d. V.887..d. 8. b Clarly, crg h procss a h cr o h spccaos rsuls h gras proporo o cabls wh spccaos. P 9 < < P < <.d.. Sco - -. a E../.,.. V., ad.... b P <..d

61 -. a E -/, V /, ad.77 b P < < d.. Thror, should qual a. or 9.7 < <.. E. 9.7/.,. 9.7 V.8, ad.. b F.d or 9.7 < <.. Thror, 9.7, < 9.7 F 99., 9.7 <.,. c P <. F a Th dsrbuo o s or.9 < <.. Now,, <.9 F 9.,.9 <.,. b P >. P. F.. c I P >.9, h F.9 ad F.. Thror, ad d E..9/. ad V. 8-8

62 .. - E.8m.. V.8 m b P < d /.7 /.7 / d c... F d /.7 d /.7... or. < <.. Thror,.., F /.7.,, -6. or < <7 7 <.. <.. a P > 7.d b P < 6.d c E 6. scods 7 V.8 scods -7. a Th dsrbuo o s or. < <.. Thror,, <. F.,. <.,. P >. F. [..]. b c I P >., h F. ad F.9. Thror, -..9 ad.. d E../. µm ad.. 6 V 8. µ m -9

63 -8. a P >.d.. Sco -6 b P >.9 ad P >. d.. Now,.-.9 ad c E / ad V a PZ<..968 b PZ< c PZ> d PZ >. pz <..98 P. < Z <.76 PZ<.76 PZ > a P < Z < PZ < PZ > b P < Z < PZ < [ PZ < ].9 c P < Z < PZ < [ PZ < ].997 d PZ > PZ <. P < Z < PZ < PZ < a PZ <.8.9 b PZ <. c I PZ > z., h PZ < z.9 ad z.8 d I PZ > z.9, h PZ < z. ad z.8 P. < Z < z PZ < z PZ <. PZ < z.79. Thror, PZ < z ad z. -. a Bcaus o h symmry o h ormal dsrbuo, h ara ach al o h dsrbuo mus qual.. Thror h valu Tabl II ha corrspods o.97 s.96. Thus, z.96. b Fd h valu Tabl II corrspodg o.99. z.8. c Fd h valu Tabl II corrspodg o.8. z. d Fd h valu Tabl II corrspodg o z.. -

64 -. a P < PZ < / PZ <..99 b P > 9 P < 9 PZ < 9/ PZ < c P6 < < P < Z < P < Z < PZ < PZ < ].9. d P < < P < Z < P < Z < PZ < PZ <. P < < 8 P < 8 P < P 8 Z < P Z < PZ < PZ < a P > P Z >.. Thror, ad. b P > P Z > P Z <.9. Thror, P Z <. ad.6. Cosquly, 6.7. c P < < P < Z < P Z < P Z <. P Z <.. Thror, P Z <. ad.. Cosquly, d P < < P/ < Z < /.9. Thror, /.96 ad.9 P < < P/ < Z < /.99. Thror, /.8 ad.6 -

65 -. a P < P Z < PZ <..99 b P > P Z > PZ >. PZ < c P < < 7 P < Z < P. < Z <. PZ <. PZ < d P < < 9 P < Z < P.7 < Z < PZ < PZ <.7].88 8 P < < 8 P < Z < P.7 < Z <.7 PZ <.7 PZ < a P > P Z >.. Thror,. b P > P Z >.9. Thror, P Z <. Thror,.6, ad.6. c P < < 9 P < Z <.. Thror, PZ < PZ <. whr PZ <.8. Thus PZ <.6. Cosquly,.6 ad 6.. -

66 d P < < P < Z <.9. Thror, P Z < PZ <..9 ad P Z <.8.9. Cosquly, P Z <.8. Bcaus a probably ca o b grar ha o, hr s o soluo or. I ac, P < P. < Z.696. Thror, v s s o y h probably rqusd cao qual.9. P < < P < Z < P < Z <.99 Thror, /.8 ad a P < 6 P Z < PZ < b P8 < < 9 P < Z < P < Z < PZ < PZ <.9 6 c P > P Z >.9. 6 Thror,.6 ad a P < P Z < PZ < b P < P Z < PZ <..6.6% ar scrappd -

67 a P >.6 P Z >. PZ >. PZ < b P.7 < <.6 P < Z <.. P.6 < Z <.6 PZ <.6 PZ < c P < P Z <.9.. Thror,..8 ad a P < PZ <. PZ <.. b P <. P Z < PZ <.. ad. 6. P >.6 P Z > PZ >.7.. Thror, h proporo o cas scrappd s..7., or.% c P. < <..99. Thror, P < Z <.99.. Cosquly, P Z <.99 ad Th lms ar., a P < P Z < PZ < b P > 6 P Z > PZ > - PZ < c P < P Z < Thror,. ad 7 -

68 µ -. a I P >.999, h P Z > Thror, µ.9 ad µ.9.. µ b I P >.999, h P Z > Thror, µ -.9 ad µ a P >. P Z >. PZ > b P. < <. P < Z <.. P < Z < PZ < PZ <.77. c P >.9, h P Z >.9... Thror,..8 ad a P > 7 P Z > P Z < b P < 8 P Z < P Z <..88 c,, bys*8 bs/by 8,, bs 8,, bs. scods 6, bs/sc -

69 a P > 9. P < P Z > P Z <.. PZ > PZ < PZ < PZ < Thror, h aswr s.866. b Th procss ma should b s a h cr o h spccaos; ha s, a µ c P89.7 < < 9. P < Z <.. P < Z <.997. Th yld s * % a P89.7 < < 9. P < Z <.. P < Z <.997. P b L Y rprs h umbr o cass ou o h sampl o ha ar bw 89.7 ad 9. ml. Th Y ollows a bomal dsrbuo wh ad p.997. Thus, EY 9.97 or a P < < 8 P < Z < P. < Z < - PZ < PZ <... b P >.. Thror, P Z >. ad Thror, 6. hours.8. -6

70 7-8. a P < P Z < 6 PZ <... 7 b P >.9. Thror, P Z >.9 ad 6 6 Cosquly, c P > 7 P Z > P Z >. 6 Phr lasrs oprag ar 7 hours / / a P >.6 P Z >. PZ >. -PZ < b P. < <.6 P < Z <.. P. < Z < c P. < <.6 P < Z <.6.6 P < Z <..6.6 Thror, P Z <.997. Thror,.8 ad a P > P Z >. PZ >.7 b I P <.999, h P Z <.999. Thror, /.9 ad /.9.. µ c I P <.999, h P Z < µ Thror,.9 ad µ.. -7

71 Sco a E. 8, V..6 8 ad a 7 8 Th, P 7 P Z P Z b P7 < 9 P < Z P. < Z P < b E, V..9 9 ad. Th, P < P Z < P Z <. 8 8 < < P < Z < P.67 < Z <.67 c P L do h umbr o dcv chps h lo. Th, E., V a P > P Z > P Z >. P Z b P < < P < Z < P < Z < P Z.6 P Z < L do h umbr o dcv lcrcal cocors. Th, E /., V..9.. a P.9.78 b. P P Z < P Z < Th appromao s smallr ha h bomal. I s o sasacory sc p<. c Th, E /, V..8. P.8.77 P P Z < P Z <..6 Normal appromao s ow closr o h bomal; howvr, s sll o sasacory sc p s o >. -8

72 -6. L do h umbr o orgal compos ha al durg h usul l o h produc. Th, s a bomal radom varabl wh p. ad. Also, E. ad V P P Z P Z. P Z < L do h umbr o parcls cm o dus. Th, s a Posso radom varabl wh λ,. Also, E λ, V, P >, P Z >,, P Z >. -67 L do h umbr o rrors o a wb s. Th, s a bomal radom varabl wh p. ad. Also, E. ad V..9.7 P P Z P Z.8 P Z < L do h umbr o parcls cm o dus. Th, s a Posso radom varabl wh λ,. Also, E λ, V a, P, P Z, P Z <, P Z 9,9, b. P < 9,9 P Z < P Z <. 87, c. I P >., h, P Z >..,, Thror,. ad, -69 L do h umbr o hs o a wb s. Th, s a Posso radom varabl wh a o ma, pr day. E λ, ad V, a,, P, P Z <, P Z P Z Epcd valu o hs days wh mor ha, hs pr day s.8*68. days pr yar -9

73 -69 b. L Y do h umbr o days pr yar wh ovr, hs o a wb s. Th, Y s a bomal radom varabl wh 6 ad p.8. EY 8. ad VY P Y 8. > P Z P Z. P Z <. -7 E. ad V..8 6 a P P Z P Z P7 P Z P.98 Z b c I P >., h P Z >.. 6 Thror,. ad s h umbr o mor rrors o a s par o pags o. s a Posso radom varabl wh a ma o. pr pag a. Th umbr o rrors pr pag s a radom varabl bcaus wll b dr or ach pag... b. P. 67! P P.67. Th ma umbr o pags wh o or mor rrors s. pags c. L Y b h umbr o pags wh rrors. P Y > P Z P Z. P Z <

74 Sco -9 λ -7. a P λ d b P d. 8 c P d. 867 d P < < d. 7 P d. ad.6-7. I E, h λ.... a P >. d b P >.. c P > d P <. d. 9 ad L do h m ul h rs cou. Th, s a poal radom varabl wh λ cous pr mu. a P >. d. 679 /6. b / P < d. 8 6 c P < <. 7 / a E /λ /. mus b V /λ /.,. c P < d. 9,

75 -76. Th m o alur hours or a lasr a cyomry mach s modld by a poal dsrbuo wh a P >,. d b P <,. d c P, < <, d L do h m ul h rs call. Th, s poal ad λ calls/mu. E a P > d. b Th probably o a las o call a -mu rval quals o mus h probably o zro calls a -mu rval ad ha s P >. P > /.. Thror, h aswr s Alravly, h rqusd probably s qual o P <.866. c / P < < /. / d P <.9 ad P <. 9. Thror,. mus L b h l o rgulaor. Th, s a poal radom varabl wh λ / E / 6 a Bcaus h Posso procss rom whch h poal dsrbuo s drvd s mmorylss, hs probably s 6 P < 6 / 6 / 6 d b Bcaus h alur ms ar mmorylss, h ma m ul h alur s E 6 yars. -

76 -79. L do h m o alur hours o as a prsoal compur. Th, s a poal radom varabl ad λ / E.... a P >,. d. 98, 7,,... b P < 7,. d L do h m ul a mssag s rcvd. Th, s a poal radom varabl ad λ / E /. a P > / / d. 679 b Th sam as par a. c E hours. 7, -8. L do h m ul h arrval o a a. Th, s a poal radom varabl wh λ / E. arrvals/ mu... a P > 6. d b P <. d a P >. d. ad. mus. b P <.9 mpls ha P >.. Thror, hs aswr s h sam as par a... c P <. ad 6.9 mus. -8. L do h dsac bw major cracks. Th, s a poal radom varabl wh λ / E. cracks/ml... a P >. d. b L Y do h umbr o cracks mls o hghway. Bcaus h dsac bw cracks s poal, Y s a Posso radom varabl wh λ. cracks pr mls. PY. 77! c / λ mls. -

77 a P < <. d. 9. b P > By dpdc o h rvals a Posso procss, h aswr s Alravly, h aswr s P >.. Th probably dos dpd o whhr or o h lghs o hghway ar coscuv. c By h mmorylss propry, hs aswr s P >. rom par b. -8. L do h lm o a assmbly. Th, s a poal radom varabl wh λ / E / alurs pr hour. a P < / /. d. / b P > /. 86 c From h mmorylss propry o h poal, hs aswr s h sam as par a., P < a L U do h umbr o assmbls ou o ha al bor hours. By h mmorylss propry o a Posso procss, U has a bomal dsrbuo wh ad p. rom Ercs -8a. Th, P U P U b L V do h umbr o assmbls ou o ha al bor 8 hours. Th, V s a bomal radom varabl wh ad p P < 8, whr dos h lm o a assmbly. 8 Now, P < 8 / / d Thror, PV L do h umbr o calls hours. Bcaus h m bw calls s a poal radom varabl, h umbr o calls hours s a Posso radom varabl. Now, h ma m bw calls s. hours ad λ /. calls pr hour 6 calls hours P P !!!! -88. L Y do h umbr o arrvals o hour. I h m bw arrvals s poal, h h cou o arrvals s a Posso radom varabl ad λ arrval pr hour. PY > P Y. 899!!!! -

78 -89. a From Ercs -88, PY >.899. L W do h umbr o o-hour rvals ou o ha coa mor ha arrvals. By h mmorylss propry o a Posso procss, W s a bomal radom varabl wh ad p.899. PW b L do h m bw arrvals. Th, s a poal radom varabl wh λ arrvals pr hour. P >. ad P > d.. Thror,. hours. -9. L do h umbr o calls mus. Bcaus h m bw calls s a poal radom varabl, s a Posso radom varabl wh λ / E. calls pr mu calls pr mus. a P > [ ] P. 8!!!! b P. 979! c L Y do h m bw calls mus. Th, P Y. ad P Y mus d. Thror,. ad a From Ercs -9, PY >..y dy.y 6. b Bcaus h calls ar a Posso procss, h umbr o calls dsjo rvals ar dpd. From Ercs -9 par b., h probably o o calls o-hal hour s Thror, h aswr s [ ] 6.. Alravly, h aswr s h probably o o calls wo hours. From par a. o hs rcs, hs s. c Bcaus a Posso procss s mmorylss, probabls do o dpd o whhr or o rvals ar coscuv. Thror, pars a. ad b. hav h sam aswr. -9. a / θ / θ P > θ d. 679 θ θ / θ b P > θ. θ / θ c P > θ. 98 θ d Th rsuls do o dpd o θ. θ 6. -

79 -9. s a poal radom varabl wh λ. laws pr mr. a E / λ mrs... b P >. d. c No, s Ercs -9 par c. d P <.9. Th, P < ad..... Thror, P > 8. d 8 8 /.9 Th dsac bw succssv laws s hr lss ha 8 mrs or o. Th dsacs ar dpd ad P > 8.9. L Y do h umbr o laws ul h dsac cds 8 mrs. Th, Y s a gomrc radom varabl wh p.9. a PY b EY /.9.9. λ -9. E λ d. Us grao by pars wh u ad dv. Th, E λ λ d λ λ / λ λ λ V λ λ d. Us grao by pars wh u ad λ λ dv λ λ. Th, λ λ λ λ λ λ λ V λ λ d Th las gral s s o b zro rom h do o E. Thror, V. λ d Sco a Th m ul h h call s a Erlag radom varabl wh λ calls pr mu ad r. b E / mus. V /. mus. c Bcaus a Posso procss s mmorylss, h ma m s /. mus or scods -6

80 -97. L Y do h umbr o calls o mu. Th, Y s a Posso radom varabl wh λ calls pr mu. a PY. 7! b PY > - P Y. 87.!!! L W do h umbr o o mu rvals ou o ha coa mor ha calls. Bcaus h calls ar a Posso procss, W s a bomal radom varabl wh ad p.87. Thror, PW L do h pouds o maral o oba parcls. Th, has a Erlag dsrbuo wh r ad λ.. r a E pouds. λ. b V,. ad, 87. pouds. -99 L do h m bw alurs o a lasr. s poal wh a ma o,. a. Epcd m ul h scod alur E r / λ /., hours b. No o alurs hours E N k P N.6767 k! k - L do h m ul mssags arrv a a od. Th, has a Erlag dsrbuo wh r ad λ mssags pr mu. a E / /6 mu scods. bv 8 / mu / scod ad. 7 mu.7 scods. c L Y do h umbr o mssags ha arrv scods. Th, Y s a Posso radom varabl wh λ mssags pr mu mssags pr scods. P Y P Y.9 [ ]!!!!! d L Y do h umbr o mssags ha arrv scods. Th, Y s a Posso radom varabl wh λ. mssags pr scods. P Y P Y

81 -. L do h umbr o bs ul v rrors occur. Th, has a Erlag dsrbuo wh r ad λ rror pr b. r a E bs. λ r b V ad 67 bs. λ c L Y do h umbr o rrors bs. Th, Y s a Posso radom varabl wh λ / rror pr b rror pr bs. P Y P Y. [ ] 8!!! - λ /. r a E r / λ /. mus b m -. m. m cl Y b h umbr o calls bor scods λ.* P Y > P.7.!!! [ ] a L do h umbr o cusomrs ha arrv mus. Th, s a Posso radom varabl wh λ. arrvals pr mu arrvals pr mus. P > P. [ ] 9!!!! b L Y do h umbr o cusomrs ha arrv mus. Th, Y s a Posso radom varabl wh λ arrvals pr mus. P P. [ ] 87!!!!! -. L do h m days ul h ourh problm. Th, has a Erlag dsrbuo wh r ad λ/ problm pr day. a E days. b L Y do h umbr o problms days. Th, Y s a Posso radom varabl wh λproblms pr days. P Y <. -. a Γ 6! [ ]!!!! b Γ Γ Γ π / Γ Γ / 6 c Γ π.67 r -6 Γ r d. Us grao by pars wh u r ad dv -. Th, Γ r r r r d r Γ r. -8

82 r r λ r y λ λ y dy -7 ; λ, r d d. L y λ, h h gral s λ Γ r. From h Γ r do o Γr, hs gral s rcogzd o qual. -8. I s a ch-squar radom varabl, h s a spcal cas o a gamma radom varabl. r 7 / r 7 / Now, E 7 ad V. λ / λ / Sco β. ad δ hours E Γ!, V Γ.. [ Γ. ] a P < F. 989 b P > F.. -. L do lm o a barg. β ad δ hours 8.8 a > 8 F 8. 7 b P E Γ Γ.. Γ. π hours c L Y do h umbr o bargs ou o ha las a las 8 hours. Th, Y s a bomal radom varabl wh ad p.7. P Y a. E δγ 9Γ / 9Γ / hours b. V β δ Γ β δ β [ Γ ] 9 Γ 9 [ Γ ] c. < F. 76 P 8.6 hours -9

83 -. L do h lm. a E δγ δγ δ 6. Th δ. Now,. P >. 7 b P < L do h lm a E 7Γ 6. b V 7 Γ 7 [ Γ.] 7 7.π, c P > a.β, δ E Γ Γ.. Γ. π.. hours b. V Γ [ Γ ] Γ [ Γ.] 6. c. P < F I s a Wbull radom varabl wh β ad δ, h dsrbuo o s h poal dsrbuo wh λ.... or > Th ma o s E /λ. or > -

84 Sco - -7 s a logormal dsrbuo wh θ ad ω 9 a. W l P < P < P W < l Φ Φ..9 b. Fd h valu or whch P.9 W l P P P W < l Φ.9 l c. µ E θ ω / 9 / θ V ω ω. -8 a. s a logormal dsrbuo wh θ- ad ω 9 W P < < P < < Pl < W < l l l Φ Φ Φ.97 Φ.7.6 W l b. P < P P W < l Φ. l c. µ E V θ ω θ ω ω / 9 / ,,.87 -

85 -9 a. s a logormal dsrbuo wh θ ad ω W l P < P < P W < l Φ Φ..986 b. P < < P < > P > l l Φ Φ l Φ Φ.66 Φ Φ /.7 c. Th produc has dgradd ovr h rs hours, so h probably o lasg aohr hours s vry low.. - s a logormal dsrbuo wh θ. ad ω a W l. P > P > P W > l Φ Φ W l. b. P P P W < l Φ. l...6 scods c. µ E V θ ω θ ω ω /. /

86 - Fd h valus o θad ω gv ha E ad V 8, l θ ω / θ ω ω 8 y θ ω ad h y ad 8 y y y y Squar y ad subsu o 8 y y 9. Subsu y o ad solv or. 9. θ l..8 ad ω l9.. - a. Fd h valus o θad ω gv ha E ad, l θ ω / θ ω ω y θ ω ad h y y y y y ad Squar y ad subsu o y y Subsu y o ad solv or 7. 6 θ l ad ω l. 69 b. W l 8.6 P > P > P W > l Φ.686 Φ W l 8.6 c. P > P > P W > l Φ..686 l hours.686 -

87 - L ~Nµ,, h Y ollows a logormal dsrbuo wh ma µ ad varac. By do, F Y y PY y P log y µ < y P < log y F log y Φ. Sc Y ad ~ Nµ,, w ca show ha Y Y log y y Fally, Y y log yµ log Y y F y log y F y y y y π. Supplmal Ercss. - a P <.. d.. 6 b P >. d.. 7. c P. < <.. d F. d. F,,, < <. Th, -6 E. d. 8 V... d 9 9 d d 9 9 -

88 -7. L do h m bw calls. Th, λ / E. calls pr mu.... a P <. d.9... b P < <. 8.. c P <.9. Th, P <. d.9. Now,. mus. -8 a Ths aswr s h sam as par a. o Ercs P <. d.9 b Ths s h probably ha hr ar o calls ovr a prod o mus. Bcaus a Posso procss s mmorylss, dos o mar whhr or o h rvals ar coscuv. P... >. d a L Y do h umbr o calls mus. Th, Y s a Posso radom varabl wh λ. P Y..!!! b L W do h m ul h h call. Th, W has a Erlag dsrbuo wh λ. ad r. EW /. mus. - L do h lm. Th λ / E / 6. < / 6 / 6. P d L W do h umbr o CPUs ha al wh h hr yars. Th, W s a bomal radom varabl wh ad p.9 rom Ercs -. Th, P W P W

89 - s a logormal dsrbuo wh θ ad ω a. W P < < P < < Pl < W > l l l Φ Φ Φ.96 Φ W l b. P < P < P W < l Φ. l c. µ E V θ ω θ ω ω / / a. Fd h valus o θad ω gv ha E ad V l θ ω / θ ω ω y θ ω ad h y ad y y y y Squar or y y y.6 ad subsu o y y y y subsu y back o ad solv or.6 θ l. ad ω l b. P W l. < P < P W < l Φ.98 Φ

90 - L do h umbr o brs vsbl a grd cll. Th, has a Posso dsrbuo ad λ brs pr cm 8, brs pr sampl. brs pr grd cll... a P P. 9.! b L W do h umbr o grd clls amd ul coa brs. I h umbr o brs hav a Posso dsrbuo, h h umbr o brs ach grd cll ar dpd. Thror, W has a gav bomal dsrbuo wh p.9. Cosquly, EW /.9. clls..9 c VW. Thror, 6. W clls L do h hgh o a pla... a P>. P Z > PZ > -. - PZ b P. < <. P < Z < P- < Z < c.p >.9 P Z >.9 ad Thror, a P >. rom par a. o Ercs - s.. b Ys, bcaus h probably o a pla growg o a hgh o. cmrs or mor whou rrgao s low. -7. L do h hckss.. a P >. P Z > PZ > b P. < <. P < Z < P -. < Z < Thror, h proporo ha do o m spccaos s P. < <... c I P <.9, h P Z >.9. Thror,.6 ad L do h do damr. I P. < <.6.997, h P < Z < P < Z < Thror,. 6 ad

91 -9. I P.- < <., h P-/. < Z < / Thror, /. ad.. Th spccaos ar rom.8 o.. - L do h l. a P < P Z < P Z < P Z. 6 d I P >.9, h PZ < hours Cosquly, ad,µ, -. I P >,.99, h PZ > Thror, 6 µ,98. µ -. ad - Th probably a produc lass mor ha hours s [ P > ], by dpdc. I [ P > ].99, h P > µ Th, P > P Z > Thror, µ ad µ,6 hours. - s a poal dsrbuo wh E 7 hours 8 a P < d. 6 7 b. > d P Thror,. 9 7 ad 7 l hours -8