COMPLEX ANALYSIS AND PROBABILITY DISTRIBUTION

Transcription

1 OMPLEX ANALYSIS AND POBABILITY DISTIBUTION

2 ONTENTS omple Fuctos Ad Deretto omple Itegrto Power Seres Epso O omple Fucto Sgle dom Vrbles Probblt Dstrbutos

3 TEXT BOOKS Erw Kresg Advced Egeerg Mmtcs Joh Wle & Sos Publshers th Edto4. B. S. Grewl Hgher Egeerg Mmtcs Kh Publshers 4 d Edto

4 EFEENES hurchll.v. d Brow J.W omple Vrbles d Applctos Tt Mc Grw-Hll 8 th Edto. A. K. Kpoor omple Vrbles Prcples d Problem Sessos World Scetc Publshers st Edto. Murr Spegel Joh Schller Probblt d Sttstcs Schum s Outle Seres 3 rd Edto.

5 omple Fuctos Ad Deretto

6 Dervtve o comple ucto drecto. o deped ot does vlue Its ests ] [ lm or ' v u

7 choose choose lm or. o vlues ll or ble deret s ucto tht Show : E ' ' '

8 lm ] [ lm ] [ lm : method Aor * * '

9 ble. deret owhere s so drecto o depeds lmt The ] [ lm lm slope o thrugh le log ple. comple where ble deret ot s ucto tht Show : E ' m m m m m

10 ANALYTI FUNTION drecto. t o depede s tht such everwhere ltc s Provded ] [ lm ] [ lm ] [ lm. t ecept everwhere ltc s ucto tht Show : E ' '

11 uch-em relto A ucto =u+v s deretble d ltc re must be prtculr coecto betwee u d v reltos em - uch - d ] [ lm mgr s suppose ] [ lm rel s suppose ] [ lm ] [ lm u v v u v u v v u u L v u v v u u L v u v u L v u v u L

12 ] [ qutrt secod b qutrt outh ] [ u v v u v u ucto? ltc s ple comple o dom whch I : E u v v u ot o combto cots o octo ltc mples stsed. re reltos em - uch ltc s I d s o cojugte comple d * * * * * * *

13 dmeso. two equto s Lplce' o solutos re d ucto or result sme stsed re reltos em - uch I v u v v v u u u v v u orthogol ˆ ˆ d ˆ ˆ re l respectve curves two g correspod vectors orml costt d coctt curves o mles two For u u u u v u v u v u j v v v j u u u v u

14 OMPLEX INTEGATION

15 Sgulrtes d eros o comple ucto. t order o pole hs d cotg od eghborho some pots ll t ltc s teger postve s : pole sgulrt Isolted g g g sgulrt essetl lmt stses te o drecto rom 4 s o pole s 3. th greter order o pole s te s. th less order o pole s umber comple ero - o te s d ltc s ] [ lm s t order o pole hs tht or deto lterte A * *

16 pole smple s sgulrt ech } sh sh ] cosh [ { lm } cosh ] sh [ { lm rule s Hosptl' l' Usg teger s ep ] ep[ ep ep ep whe sgulrt hs ep ep ep ep cosh sh th d t order o poles ucto o es sgulrt Fd : E

17 . pproched s whch rom drecto t o depede d ests lm but ed udeterm o vlue mkes Sgulrt : es sgulrt emove. t sgulrt removble hs so w t o depede s lm... 5! 3!... 5! 3! ed udeterm lm : t sgulrt removble s s tht Show : E Sol

18 where t o tht b gve s t t o behvor The b b b t sgulrt essetl hs! ep t 3 order o pole hs t ltc s t ltc s set ep d o t t behvor Fd : E 3

19 o order o pole lso s ero. smple clle d s. order o ero clle d s d teger postve s d I g g. t sgulrt removble hs so w t o depede s lm... 5! 3!... 5! 3! ed udeterm lm : t sgulrt removble s s tht Show : E Sol

20 . t shg d strtg crcle log o tegrl comple Evlute : E. t o depede s result clculted The s cos cos s b clculted lso s tegrl The * * s s cos cos cos s s cos s cos cos s s cos dt dt t t t t d dt t t tdt t tdt t dt t t d t v t u v u t dt d t dt d t t t t

21 emples. prevous show s d pth log e o tegrl comple Evlute : E 3 pth. deret o depeds tegrl The : cos s cos : cos s cos : 3b 3 3 ds s dt t ds s dt t dt t t t dt t t t

22 d d cotours two betwee rego ltc s ucto tht Show. wth completel les t tht smll l sucet beg dgrm Argd d cotour closed two osder : E d d d d d d d cotour o tht s cotour o drecto tke I ltc s d b bouded s re d ltc. s or curve b bouded dom closed o ucto cotuous s : orem s Morer'

23 uch s tegrl ormul d wth pot s d cotour closed o d wth ltc s I ep ep or d e d e e e I d d d d I

24 : ucto comple o dervtve o orm tegrl The ' d d d d h d h h h h h h h '! dervtve th For ] [ lm ] [ lm lm

25 POWE SEIES EXPANSION OF OMPLEX FUNTION

26 Tlor d Luret seres! sde pot s d pot o cetered rdus o crcle o d sde ltc s I!! seres geometrc s epd o les where so o d sde ltc s d d d

27 ... ep. seres Tlor s epded d t ltc s The. o d sde pot or ever t ltc s but t order o pole hs I eres Luret s ded s c be sde ll or Thus b g g p p p p p p o cetered d crcles two betwee rego ltc s! d d d g d g g b b p p

28 p p k m m - k p p m m sgulrt essetl o vlue lowest d to mpossble o resdue clled s t order o pole hs ll or but d to possble t ltc ot s I. t order o ero hve to sd s wth s term vshg - o rst or hppe m It. or ll t ltc s I

29 pole. ech t o resdue d d 3 order o pole s d order o pole s tht ver Hece. d es sgulrt bout o seres Luret Fd : E 3 8. s t o resdue 3 order o pole s ] [ set pot order o pole s ] 3! 5 4 3! [ 8 8 pot

30 How to obt resdue? t resdue ]} [! { lm lmt Tke! ] [ d d m b m d d m m m m m m m m m m m m lm lm lm d t ero - o d ltc s d t smple hs I ] [ lm pole smple For ' ' h g h g h g h g h g h g m

31 . pot t ep ucto o resdue evlute Hece. t o resdue or epresso geerl dervg bout o seres Luret g cosder B. pot t order o pole hs tht Suppose : E m e e e d d d d ] [! ep ep ] ep [ ] [ : t pole or d t order o poles ep ep 3 3

32 esdue orem m m m d I d e d e d e d I d e d e d d I m or or set t order o pole hs

33 wth poles ts t o resdues o sum s wth poles o umber te or ecept ltc d cotour closed o d wth cotuous s d j j j j

34 cotour ope log o tegrl The I cotour closed or lm lm lm ecept o sgulrt o tht eough smll chose s rg d g surroud eghbour some wth ltc s t pole smple hs I d e e d I d d d d I

35 Itegrls o susodl uctos d d d F s cos crcle ut ep set s cos or cos cos Evlute : E b d b b I d b b b b b d b b b b d b b d d b b cos cos cos cos 4 4 4

36 ] [ ] [ lm t pole } ] [ 4 { lm ]} [! { lm t pole ] [! { lm : esdue crcle ut wth d t poles double b b b b b b b b I b b b b b b b b b b b m b b b b b b b b b b b d d m d d m b d b b b I b m m m

37 Some te tegrls d. s ero to teds log tegrl o o mmum or est both d 3. s tht s codto sucet s ero to teds o o mmum tmes rdus o semcrcle o s. rel o s whch o oe poles o umber te or ecept Im ple - hl upper ltc s : propertes ollowg hs Γ d d d d Γ Γ j j

38 rel s Evlute : E 4 d I ! s t o coece set ple - hl upper t ol t 4 order o poles s I d d d d d d d Γ Γ

39 For poles o rel s: ero s tegrl s th ster vshes I e e e e set or t pole hs or cotour closed or s deed tegrl o vlue Prcpl d d d d d d d d d d P d d d d d d d d P θ Γ

40 rel cos o vlue prcpl Fd : E m d m m d m P m d m P e d e P d e d e d e d e d e d e I d e I m m m m m m m m m cos s d s cos d d As s d ple - hl upper pole o tegrl osder

41 Itegrl o multvlued uctos opposte. d equl ot re to jog D d AB les two log vlues ts d multvlue s tegrd The. d let We otg. t s pot brch Sgle s such uctos d Multvlue L or : E 3 d I ! lm ] [! 3 lm d t pole tegrl cotour to o cotrbut o mke crcles two d s tegrd d d d d e

42 D le log AB le log d d 8 3 d d e e e d d e e d d d d d d D B A D AB

43 SINGLE ANDOM VAIABLES

44 Bsc ocepts A epermet s process b whch observto or mesuremet s obted. Epermet: ecord ge Epermet: Toss de Epermet: ecord opo es o Epermet: Toss two cos

45 A smple evet s outcome tht s observed o sgle repetto o epermet. The bsc elemet to whch probblt s ppled. Oe d ol oe smple evet c occur whe epermet s perormed. A smple evet s deoted b E wth subscrpt.

46 Ech smple evet wll be ssged probblt mesurg how ote t occurs. The set o ll smple evets o epermet s clled smple spce S.

47 Emple The de toss: Smple evets: Smple spce: 3 4 E E E 3 E 4 S ={E E E 3 E 4 E 5 E 6 } S E E 3 E 5 5 E 5 E E 4 E 6 6 E 6

48 A evet s collecto o oe or more smple evets. The de toss: A: odd umber B: umber > A ={E E 3 E 5 } E A E E 3 E 4 E 5 E 6 B S B ={E 3 E 4 E 5 E 6 }

49 Two evets re mutull eclusve whe oe evet occurs or cot d vce vers. Epermet: Toss de A: observe odd umber B: observe umber greter th : observe 6 D: observe 3 Mutull Eclusve Not Mutull Eclusve B d? B d D?

50 The probblt o evet A mesures how ote we thk A wll occur. We wrte PA. Suppose tht epermet s perormed tmes. The reltve requec or evet A s Number o tmes A occurs I we let get tel lrge P A lm

51 PA must be betwee d. I evet A c ever occur PA =. I evet A lws occurs whe epermet s perormed PA =. The sum o probbltes or ll smple evets S equls. The probblt o evet A s oud b ddg probbltes o ll smple evets coted A.

52 dg Probbltes Probbltes c be oud usg Estmtes rom emprcl studes ommo sese estmtes bsed o equll lkel evets. Emples: Toss r co. PHed = % o U.S. populto hs red hr. Select perso t rdom. Ped hr =.

53 Emple Toss r co twce. Wht s probblt o observg t lest oe hed? st o d o E PE H HH H T HT H TH T T TT Pt lest hed = PE + PE + PE 3 = = 34

54 Emple A bowl cots three M&Ms oe red oe blue d oe gree. A chld selects two M&Ms t rdom. Wht s probblt tht t lest oe s red? st M&M d M&M E PE m m m m m m m m m B G B BG GB G Pt lest red = PB + PB+ PG + PG = 46 = 3

55 outg ules I smple evets epermet re equll lkel ou c clculte P A A N umber o totl umber smple evets A o smple evets You c use coutg rules to d A d N.

56 The m ule I epermet s perormed two stges wth m ws to ccomplsh rst stge d ws to ccomplsh secod stge re re m ws to ccomplsh epermet. Ths rule s esl eteded to k stges wth umber o ws equl to 3 k Emple: Toss two cos. The totl umber o smple evets s: = 4

57 Emples Emple: Toss three cos. The totl umber o smple evets s: = 8 Emple: Toss two dce. The totl umber o smple evets s: 6 6 = 36 Emple: Two M&Ms re drw rom dsh cotg two red d two blue cdes. The totl umber o smple evets s: 4 3 =

58 Permuttos The umber o ws ou c rrge dstct objects tkg m r t tme s P r! r! where!... d!. Emple: How m 3-dgt lock combtos c we mke rom umbers 3 d 4? The order o choce s mportt! 4 4! P 4 3 3! 4

59 ombtos The umber o dstct combtos o dstct objects tht c be ormed tkg m r t tme s r! r! r! Emple: Three members o 5-perso commttee must be chose to orm subcommttee. How m deret subcommttees could be ormed? The order o choce s ot mportt! 3!5 5! 3!

60 Emple A bo cots s M&Ms our red d two gree. A chld selects two M&Ms t rdom. Wht s probblt tht ectl oe s red? The order o choce s ot mportt! ws!4! to 6! choose M 5 & Ms. ws gree!!! to choose M & M. 4! 4!3! 4 ws to choose red M & M. 4 =8 ws to choose red d gree M&M. P ectl oe red = 85

61 Evet eltos The uo o two evets A d B s evet tht er A or B or both occur whe epermet s perormed. We wrte A B S A B A B

62 Evet eltos The tersecto o two evets A d B s evet tht both A d B occur whe epermet s perormed. We wrte A B. S A B A B I two evets A d B re mutull eclusve PA B =.

63 Evet eltos The complemet o evet A cossts o ll outcomes o epermet tht do ot result evet A. We wrte A. A S A

64 lcultg Probbltes or Uos d omplemets There re specl rules tht wll llow ou to clculte probbltes or composte evets. The Addtve ule or Uos: For two evets A d B probblt o r uo PA B s P A B P A P B P A B A B

65 lcultg Probbltes or omplemets We kow tht or evet A: PA A = Sce er A or A must occur PA A = so tht PA A = PA+ PA = A A PA = PA

66 lcultg Probbltes or Itersectos I prevous emple we oud PA B drectl rom tble. Sometmes ths s mprctcl or mpossble. The rule or clcultg PA B depeds o de o depedet d depedet evets. Two evets A d B re sd to be depedet d ol probblt tht evet A occurs does ot chge depedg o wher or ot evet B hs occurred.

67 odtol Probbltes The probblt tht A occurs gve tht evet B hs occurred s clled codtol probblt o A gve B d s deed s P A B P A B P B P B gve

68 Deg Idepedece We c redee depedece terms o codtol probbltes: Two evets A d B re depedet d ol PA B = PA or PB A = PB Orwse re depedet. Oce ou ve decded wher or ot two evets re depedet ou c use ollowg rule to clculte r tersecto.

69 The Multplctve ule or Itersectos For two evets A d B probblt tht both A d B occur s PA B = PA PB gve tht A occurred = PAPB A I evets A d B re depedet probblt tht both A d B occur s PA B = PA PB

70 The Lw o Totl Probblt Let S S S 3... S k be mutull eclusve d ehustve evets tht s oe d ol oe must hppe. The probblt o or evet A c be wrtte s PA = PA S + PA S + + PA S k = PS PA S + PS PA S + + PS k PA S k

71 The Lw o Totl Probblt S A A S k A S S. S k PA = PA S + PA S + + PA S k = PS PA S + PS PA S + + PS k PA S k

72 Bes ule Let S S S 3... S k be mutull eclusve d ehustve evets wth pror probbltes PS PS PS k. I evet A occurs posteror probblt o S gve tht A occurred s P P S P A S S A or P S P A S...k

73 dom Vrbles A qutttve vrble s rdom vrble vlue tht t ssumes correspodg to outcome o epermet s chce or rdom evet. dom vrbles c be dscrete or cotuous. Emples: = SAT score or rdoml selected studet = umber o people room t rdoml selected tme o d = umber o upper ce o rdoml tossed de

74 POBABILITY DISTIBUTIONS

75 Probblt Dstrbutos or Dscrete dom Vrbles The probblt dstrbuto or dscrete rdom vrble resembles reltve requec dstrbutos we costructed hpter. It s grph tble or ormul tht gves possble vlues o d probblt p ssocted wth ech vlue. We must hve p d p

76 Probblt Dstrbutos Probblt dstrbutos c be used to descrbe populto just s we descrbed smples hpter. Shpe: Smmetrc skewed moud-shped Outlers: uusul or ulkel mesuremets eter d spred: me d stdrd devto. A populto me s clled m d populto stdrd devto s clled.

77 The Me d Stdrd Devto Let be dscrete rdom vrble wth probblt dstrbuto p. The me vrce d stdrd devto o re gve s Me : m p Vrce : m p Stdrd devto :

78 Emple Toss r co 3 tmes d record umber o heds. p p -m p m p.5 8 m p

79 Itroducto Dscrete rdom vrbles tke o ol te or coutbl umber o vlues. Three dscrete probblt dstrbutos serve s models or lrge umber o prctcl pplctos: The boml rdom vrble The Posso rdom vrble

80 The Boml dom Vrble M stutos rel le resemble co toss but co s ot ecessrl r so tht PH. Emple: A geetcst smples people d couts umber who hve gee lked to Alhemer s dsese. o: Hed: Tl: Perso Hs gee Does t hve gee Number o tosses: PH: = Phs gee = proporto populto who hve gee.

81 The Boml Epermet. The epermet cossts o detcl trls.. Ech trl results oe o two outcomes success S or lure F. 3. The probblt o success o sgle trl s p d rems costt rom trl to trl. The probblt o lure s q = p. 4. The trls re depedet. 5. We re terested umber o successes trls.

82 Boml or Not? Ver ew rel le pplctos sts se requremets ectl. Select two people rom U.S. populto d suppose tht 5% o populto hs Alhemer s gee. For rst perso p = Pgee =.5 For secod perso p Pgee =.5 eve though oe perso hs bee removed rom populto.

83 The Boml Probblt Dstrbuto For boml epermet wth trls d probblt p o success o gve trl probblt o k successes trls s P k k p k q k! k! k! p k q k or k.... ecll k! k! k! wth!... d!.

84 The Me d Stdrd Devto For boml epermet wth trls d probblt p o success o gve trl mesures o ceter d spred re: Me : m p Vrce : pq Stdrd devto : pq

85 umultve Probblt Tbles You c use cumultve probblt tbles to d probbltes or selected boml dstrbutos. Fd tble or correct vlue o. Fd colum or correct vlue o p. The row mrked k gves cumultve probblt P k = P = + + P = k

86 The Posso dom Vrble The Posso rdom vrble s model or dt tht represet umber o occurreces o speced evet gve ut o tme or spce. Emples: The umber o clls receved b swtchbord durg gve perod o tme. The umber o mche brekdows d The umber o trc ccdets t gve tersecto durg gve tme perod.

87 The Posso Probblt Dstrbuto s umber o evets tht occur perod o tme or spce durg whch verge o m such evets c be epected to occur. The probblt o k occurreces o ths evet s For vlues o k = The me d stdrd devto o Posso rdom vrble re Me: m P Stdrd devto: k k m e k! m m

88 umultve Probblt Tbles You c use cumultve probblt tbles to d probbltes or selected Posso dstrbutos. Fd colum or correct vlue o m. The row mrked k gves cumultve probblt P k = P = + + P = k

89 otuous dom Vrbles otuous rdom vrbles c ssume tel m vlues correspodg to pots o le tervl. Emples: Heghts weghts legth o le o prtculr product epermetl lbortor error

90 otuous dom Vrbles A smooth curve descrbes probblt dstrbuto o cotuous rdom vrble. The depth or dest o probblt whch vres wth m be descrbed b mmtcl ormul clled probblt dstrbuto or probblt dest ucto or rdom vrble.

91

92 otuous Probblt Dstrbutos There re m deret tpes o cotuous rdom vrbles We tr to pck model tht Fts dt well Allows us to mke best possble ereces usg dt. Oe mportt cotuous rdom vrble s orml rdom vrble.

93 The Norml Dstrbuto The ormul tht geertes orml probblt dstrbuto s: e m or e m d re populto me d stdrd devto. The shpe d locto o orml curve chges s me d stdrd devto chge.

94 The Stdrd Norml Dstrbuto To d P < < b we eed to d re uder pproprte orml curve. To smpl tbulto o se res we stdrde ech vlue o b epressg t s -score umber o stdrd devtos t les rom me m. m

95 The Stdrd Norml Dstrbuto Me = ; Stdrd devto = Whe = m = Smmetrc bout = Vlues o to let o ceter re egtve Vlues o to rght o ceter re postve Totl re uder curve s.

96 Fdg Probbltes or Geerl Norml dom Vrble To d re or orml rdom vrble wth me md stdrd devto stdrde or rescle tervl terms o. Fd pproprte re usg Tble 3. Emple: hs orml dstrbuto wth m = 5 d =. Fd P > 7. P P P

97 The Norml Appromto to Boml We c clculte boml probbltes usg The boml ormul The cumultve boml tbles Jv pplets Whe s lrge d p s ot too close to ero or oe res uder orml curve wth me p d vrce pq c be used to ppromte boml probbltes.