0/2 0/2
@
B
0/= 8/= 8/=
0/=
N •@N
0/>
@]
8/>
N •BN
8/>
B]
0/>
]
]
•
•
]nj preb`bflfty tn`t yeu iujss herrjhtly fs T SH W 7 T S@] W @] W + T SB N W 7 8/> + 8/ 8 /> 7 8/=.
(0)
Trebljg 2.0.: Xelutfek ]nj TS− |N W N W fs tnj preb`bflfty tn`t ` pjrsek wne n`s NFQ tjsts kji`tfvj aer tnj tnj mfsj mfsj`s `sj. j. ]n ]nfs fs fs rjajr rjajrrj rjm m te `s ` a`ls a`lsjj-kj kji` i`tf tfv vj rjsu rjsult lt.. ]n ]njj h`sj h`sj wnjrj wnjrj ` pjrsek wne mejs ket n`vj NFQ but tjsts pesftfvj aer tnj mfsj`sj, fs h`lljm ` a`lsj-pesftfvj rjsult `km n`s preb`bflfty TS+ |N h W. Xfkh Xfkhjj tnj tnj tjst tjst fs herr herrjh jhtt 66% ea tnj tfgj, T S−|N W 7 T S+|N h W 7 4.40. 40.
(0)
Kew tnj preb`bflfty tn`t ` pjrsek wne n`s tjstjm pesftfvj aer NFQ `htu`lly n`s tnj mfsj`sj fs T S+, S+, N W T S+, S+, N W T SN |+W 7 7 . (2) TS+W T S+, S+, N W + T S+, N h W Yj h`k usj B`yjs‖ aergul` te jv`lu`tj tnjsj oefkt preb`bflftfjs. T S+|N W T SN W T S+|N W T SN W + T S+|N h W T SN h W (4. (4.66)(4. 66)(4.4442) 7 (4. (4.66)(4. 66)(4.4442) + (4. (4.40)(4. 40)(4.666>) 7 4.406=. 406=.
T SN |+W 7
(8)
]nus, jvjk tneuin tnj tjst fs herrjht 66% ea tnj tfgj, tnj preb`bflfty tn`t ` r`kmeg r`kmeg pjrsek pjrsek wne tjsts tjsts pesftfv pesftfvjj `htu`l `htu`lly ly n`s NFQ fs ljss ljss tn`k tn`k 4.42. 4.42. ]nj rj`sek tnfs preb`bflfty fs se lew fs tn`t tnj ` prferf preb`bflfty tn`t ` pjrsek n`s NFQ fs vjry sg`ll. 89
Trebljg 2.0.04 Xelutfek (`) Yj wfsn te ckew wn`t tnj preb`bflfty tn`t wj ffkm ke ieem pnetemfemjs fk k p`frs ea mfemjs. ]jstfki j`hn p`fr ea mfemjs fs `k fkmjpjkmjkt trf`l suhn tn`t wftn preb`bflfty p, betn mfemjs ea ` p`fr `rj b`m. Areg Trebljg 2.0.1, wj h`k j`sfly h`lhul`tj p. p 7 T Sbetn mfemjs `rj mjajhtfvjW 7 T SM0 M2 W 7 1/2:.
(0)
]nj preb`bflfty ea _ k , tnj preb`bflfty ea zjre `hhjpt`blj mfemjs eut ea k p`frs ea mfemjs fs pk bjh`usj ek j`hn tjst ea ` p`fr ea mfemjs, betn gust bj mjajhtfvj. k
T S_ k W 7
p 7 p k 7
f70
1 2:
k
(2)
(b) @ketnjr w`y te pnr`sj tnfs qujstfek fs te `sc new g`ky p`frs gust wj tjst uktfl TS_ k W ≭ 4.40. Xfkhj TS_ k W 7 (1/2:)k , wj rjqufrj k
1 2:
≭ 4.40
⇒
lk 4.40 k ≪ 7 8.28. lk 1/2:
(8)
Xfkhj k gust bj `k fktjijr, k 7 = p`frs gust bj tjstjm.
Trebljg 2.0.00 Xelutfek ]nj st`rtfki pefkt fs te mr`w ` trjj ea tnj jxpjrfgjkt. Yj mjffkj tnj jvjkts Y tn`t tnj pl`kt fs w`tjrjm, L tn`t tnj pl`kt lfvjs, `km M tn`t tnj pl`kt mfjs. ]nj trjj mf`ir`g fs 4.> Y 4.9 4.2 4.8 Y h 4.0 4.6
=8
L
•
YL
4.:1
M
•
YM
4.0=
L
•
M
•
Y L
4.48
Y M
4.29
h
h
Ft aellews tn`t (`) TSLW 7 TSY LW + TSY h LW 7 4.:1 + 4.48 7 4.:6. (b) h T SY MW 4.29 29 h T SY |MW 7 7 7 . T SMW 4.0= + 4.29 =0
(0)
(h) TSM|Y h W 7 4.6. Fk fkaerg`l hekvjrs`tfek, ft h`k bj hekausfki te mfstfkiufsn bjtwjjk TS M|Y h W `km TSY h |MW3 newjvjr, tnjy `rj sfgplj ekhj yeu mr`w tnj trjj.
Trebljg 2.0.02 Xelutfek ]nj jxpjrfgjkt jkms `s seek `s ` ffsn fs h`uint. ]nj trjj rjsjgbljs H p 0
0− p
H 0
h
H p 2
0− p
H 2
h
H p 8
0− p
H 8 ... h
Areg tnj trjj, TSH 0 W 7 p `km TSH 2 W 7 (0 − p) p. Afk`lly, ` ffsn fs h`uint ek tnj ktn h`st fa ke ffsn wjrj h`uint ek tnj prjvfeus k − 0 h`sts. ]nus, T SH k W 7 (0 − p)k 0 p. −
(0)
Trebljg 2.2.0 Xelutfek ]jhnkfh`lly, ` iugb`ll g`hnfkj n`s ` ffkftj kugbjr ea iugb`lls, but tnj prebljg mjshrfptfek gemjls tnj mr`wfki ea iugb`lls `s s`gplfki areg tnj g`hnfkj wftneut rjpl`hjgjkt. ]nfs fs ` rj`sek`blj gemjl wnjk tnj g`hnfkj n`s ` vjry l`rij iugb`ll h`p`hfty `km wj n`vj ke ckewljmij bjaerjn`km ea
==
new g`ky iugb`lls ea j`hn heler `rj fk tnj g`hnfkj. Zkmjr tnfs gemjl, tnj rjqujstjm preb`bflfty fs ifvjk by tnj gultfkegf`l preb`bflfty 2
2
2
2
>! 0 T SR2 V 2 I2 B2 W 7 2!2!2!2! = >! 7 04 ≎ 4.48>:. =
0 =
0 =
0 =
(0)
Trebljg 2.2.2 Xelutfek Fk tnfs gemjl ea ` st`rburst p`hc`ij, tnj pfjhjs fk ` p`hc`ij `rj helljhtjm by s`gplfki wftneut rjpl`hjgjkt areg ` if`kt helljhtfek ea st`rburst pfjhjs. (`) J`hn pfjhj fs “bjrry er hnjrry‗ wftn preb`bflfty p 7 0/2. ]nj preb`bflfty ea ekly bjrry er hnjrry pfjhjs fs p02 7 0/=461. (b) J`hn pfjhj fs “ket hnjrry‗ wftn preb`bflfty 8/=. ]nj preb`bflfty `ll 02 pfjhjs `rj “ket pfkc‗ fs (8/=)02 7 4.4809. (h) Aer f 7 0, 2, . . . , 1, ljt H f mjketj tnj jvjkt tn`t `ll 02 pfjhjs `rj fl`ver f. Xfkhj j`hn pfjhj fs fl`ver f wftn preb`bflfty 0/=, TSH f W 7 (0/=)02 . Xfkhj H f `km H o `rj gutu`lly jxhlusfvj, =
TSA 0 W 7 TSH 0 ∯ H 2 ∯ ¹ ¹ ¹ ∯ H = W 7
TSH f W 7 = TSH 0 W 7 (0/=)00 .
f70
Trebljg 2.2.8 Xelutfek (`) Ljt Bf , Lf , Ef `km H f mjketj tnj jvjkts tn`t tnj ftn pfjhj fs Bjrry, Ljgek, Er`kij, `km Hnjrry rjspjhtfvjly. Ljt A 0 mjketj tnj jvjkt tn`t `ll tnrjj pfjhjs mr`w `rj tnj s`gj fl`ver. ]nus, A 0 7 { X 0 X 2 X 8 , L0 L2 L8 , E0 E2 E8 , H 0 H 2 H 8 } T SA 0 W 7 T SX 0 X 2 X 8 W + T SL0 L2 L8 W + T SE0 E2 E8 W + T SH 0 H 2 H 8 W =:
(0) (2)
Trebljg 2.2.= Xelutfek (`) Xfkhj tnjrj `rj ekly tnrjj pfjhjs ea j`hn fl`ver, wj h`kket mr`w aeur pfjhjs ea `ll tnj s`gj fl`ver. Njkhj TSA 0 W 7 4. (b) Ljt Mf mjketj tnj jvjkt tn`t tnj ftn pfjhj fs ` mfffljrjkt fl`ver areg `ll tnj prfer pfjhjs. Ljt X f mjketj tnj jvjkt tn`t pfjhj f fs tnj s`gj fl`ver `s ` prjvfeus pfjhj. @ trjj aer tnfs jxpjrfgjkt fs rjl`tfvjly sfgplj bjh`usj wj step tnj jxpjrfgjkt `s seek `s wj mr`w ` pfjhj tn`t fs tnj s`gj `s ` prjvfeus pfjhj. ]nj trjj fs 2/00
0
M0
6/00
X 2
M2
=/04
1/04
X 8
M8
1/6
8/6
X =
M=
Ketj tn`t?
• TSM0 W 7 0 bjh`usj tnj ffrst pfjhj fs “mfffljrjkt‗ sfkhj tnjrj n`vjk‖t bjjk `ky prfer pfjhjs. • Aer tnj sjhekm pfjhj, tnjrj `rj 00 pfjhjs ljat `km 6 ea tnesj pfjhjs `rj mfffljrjkt areg tnj ffrst pfjhj mr`wk. • Ifvjk tnj ffrst twe pfjhjs `rj mfffljrjkt, tnjrj `rj 2 helers, j`hn wftn 8 pfjhjs (1 pfjhjs) eut ea 04 rjg`fkfki pfjhjs tn`t `rj ` mfffljrjkt fl`ver areg tnj ffrst twe pfjhjs. ]nus TS M8 |M2 M0 W 7 1/04. • Afk`lly, ifvjk tnj ffrst tnrjj pfjhjs `rj mfffljrjkt fl`vers, tnjrj `rj 8 pfjhjs rjg`fkfki tn`t `rj ` mfffljrjkt fl`ver areg tnj pfjhjs prjvfeusly pfhcjm. ]nus TSM= |M8 M2 M0 W 7 8/6. Ft aellews tn`t tnj tnrjj pfjhjs `rj mfffljrjkt wftn preb`bflfty T SM0 M2 M8 M= W 7 0 =9
6 00
1 8 6 7 . 04 6 ::
(0)
@k `ltjrk`tj `ppre`hn te tnfs prebljg fs te `ssugj tn`t j`hn pfjhj fs mfstfkiufsn`blj, s`y by kugbjrfki tnj pfjhjs 0, 2, 8 fk j`hn fl`ver. Fk `mmftfek, wj mjffkj tnj euthegj ea tnj jxpjrfgjkt te bj ` =-pjrgut`tfek ea tnj 02 mfstfkiufsn`blj pfjhjs. Zkmjr tnfs gemjl, tnjrj `rj (02) = 7 02! >! jqu`lly lfcjly euthegjs fk tnj s`gplj sp`hj. ]nj kugbjr ea euthegjs fk wnfhn `ll aeur pfjhjs `rj mfffljrjkt fs k = 7 02 ¹ 6 ¹ 1 ¹ 8 sfkhj tnjrj `rj 02 hnefhjs aer tnj ffrst pfjhj mr`wk, 6 hnefhjs aer tnj sjhekm pfjhj areg tnj tnrjj rjg`fkfki fl`vers, 1 hnefhjs aer tnj tnfrm pfjhj `km tnrjj hnefhjs aer tnj l`st pfjhj. Xfkhj `ll euthegjs `rj jqu`lly lfcjly, 02 ¹ 6 ¹ 1 ¹ 8 6 k= T SA = W 7 7 7 (02)= 02 ¹ 00 ¹ 04 ¹ 6 ::
(2)
(h) ]nj sjhekm gemjl ea mfstfkiufsn`blj st`rburst pfjhjs g`cjs ft j`sfjr te selvj tnfs l`st qujstfek. Fk tnfs h`sj, ljt tnj euthegj ea tnj jxpjrfgjkt bj tnj 02 7 =6: hegbfk`tfeks er pfjhjs. Fk tnfs h`sj, wj `rj fikerfki = tnj ermjr fk wnfhn tnj pfjhjs wjrj sjljhtjm. Kew wj heukt tnj kugbjr ea hegbfk`tfeks fk wnfhn wj n`vj twe pfjhjs ea j`hn ea twe fl`vers. Yj h`k me tnfs wftn tnj aellewfki stjps?
0. Hneesj twe ea tnj aeur fl`vers. 2. Hneesj 2 eut ea 8 pfjhjs ea ekj ea tnj twe hnesjk fl`vers. 8. Hneesj 2 eut ea 8 pfjhjs ea tnj etnjr ea tnj twe hnesjk fl`vers. Ljt kf jqu`l tnj kugbjr ea w`ys te jxjhutj stjp f. Yj sjj tn`t k0 7
= 7 1, 2
k2 7
8 7 8, 2
k8 7
8 7 8. 2
(8)
]nj kugbjr ea pessfblj w`ys te jxjhutj tnfs sjqujkhj ea stjps fs k0 k2 k8 7 := Xfkhj `ll hegbfk`tfeks `rj jqu`lly lfcjly, T SA 2 W 7
k0 k2 k8 02 =
=>
:= 1 7 7 . =6: ::
(=)
Trebljg 2.2.: Xelutfek Xfkhj tnjrj `rj N 7 :2 jqufpreb`blj sjvjk-h`rm n`kms, j`hn preb`bflfty fs 9 oust tnj kugbjr ea n`kms ea j`hn typj mfvfmjm by N .
(`) Xfkhj tnjrj `rj 21 rjm h`rms, tnjrj `rj 21 sjvjk-h`rm n`kms wftn `ll 9 rjm h`rms. ]nj preb`bflfty ea ` sjvjk-h`rm n`km ea `ll rjm h`rms fs T SR9 W 7
21 9 :2 9
21! =:! 7 7 4.44=6. :2! 06!
(0)
(b) ]njrj `rj 02 a`hj h`rms fk ` :2 h`rm mjhc `km tnjrj `rj 02 sjvjk h`rm 9 n`kms wftn `ll a`hj h`rms. ]nj preb`bflfty ea mr`wfki ekly a`hj h`rms fs T SA W 7
02 9 :2 9
7
02! =:! 7 :.62 Ù 04 1 . :!:2! −
(2)
(h) ]njrj `rj 1 rjm a`hj h`rms (O,P,C ea mf`gekms `km nj`rts) fk ` :2 h`rm mjhc. ]nus ft fs fgpessfblj te ijt ` sjvjk-h`rm n`km ea rjm a`hj h`rms? TSR9 A W 7 4.
Trebljg 2.2.1 Xelutfek ]njrj `rj N : 7 :2 jqu`lly lfcjly ffvj-h`rm n`kms. Mfvfmfki tnj kugbjr ea : n`kms ea ` p`rtfhul`r typj by N wfll yfjlm tnj preb`bflfty ea ` n`km ea tn`t typj.
(`) ]njrj `rj 21 ffvj-h`rm n`kms ea `ll rjm h`rms. ]nus tnj preb`bflfty : ijttfki ` ffvj-h`rm n`km ea `ll rjm h`rms fs T SR: W 7
21 : :2 :
21! =9! 7 7 4.42:8. 20! :2!
Ketj tn`t tnfs h`k bj rjwrfttjk `s 21 2: 2= 28 22 TSR: W 7 , :2 :0 :4 =6 => wnfhn snews tnj suhhjssfvj preb`bflftfjs ea rjhjfvfki ` rjm h`rm. =6
(0)
(b) ]nj aellewfki sjqujkhj ea subjxpjrfgjkts wfll ijkjr`tj `ll pessfblj “aull neusj‗ 0. Hneesj ` cfkm aer tnrjj-ea-`-cfkm. 2. Hneesj ` cfkm aer twe-ea-`-cfkm. 8. Hneesj tnrjj ea tnj aeur h`rms ea tnj tnrjj-ea-`-cfkm cfkm. =. Hneesj twe ea tnj aeur h`rms ea tnj twe-ea-`-cfkm cfkm. ]nj kugbjr ea w`ys ea pjraergfki subjxpjrfgjkt f fs k0 7
08 , 0
k2 7
02 , 0
k8 7
= , 8
k= 7
= . 2
(2)
Ketj tn`t k2 7 02 bjh`usj `atjr hneesfki ` tnrjj-ea-`-cfkm, tnjrj `rj 0 twjlevj cfkms ljat areg wnfhn te hneesj twe-ea-`-cfkm. fs
]nj preb`bflfty ea ` aull neusj fs T Saull neusjW 7
k0 k2 k8 k= :2 :
8, 9== 7 7 4.440=. 2, :6>, 614
(8)
Trebljg 2.2.9 Xelutfek ]njrj `rj 2: 7 82 mfffljrjkt bfk`ry hemjs wftn : bfts. ]nj kugbjr ea hemjs wftn jx`htly 8 zjres jqu`ls tnj kugbjr ea w`ys ea hneesfki tnj bfts fk wnfhn tnesj zjres ehhur. ]njrjaerj tnjrj `rj :8 7 04 hemjs wftn jx`htly 8 zjres.
Trebljg 2.2.> Xelutfek
Xfkhj j`hn ljttjr h`k t`cj ek `ky ekj ea tnj = pessfblj ljttjrs fk tnj `lpn`bjt, tnj kugbjr ea 8 ljttjr werms tn`t h`k bj aergjm fs = 8 7 1=. Fa wj `llew j`hn ljttjr te `ppj`r ekly ekhj tnjk wj n`vj = hnefhjs aer tnj ffrst ljttjr `km 8 hnefhjs aer tnj sjhekm `km twe hnefhjs aer tnj tnfrm ljttjr. ]njrjaerj, tnjrj `rj ` tet`l ea = ¹ 8 ¹ 2 7 2= pessfblj hemjs.
:4
Gerjevjr, fa tnj ffrst shr`thnjm bex n`s tnj g`rc, tnjk tnjrj `rj =+ c g`rcjm bexjs eut ea k − 0 rjg`fkfki bexjs. Hektfkufki tnfs `riugjkt, tnj preb`bflfty tn`t ` tfhcjt fs ` wfkkjr fs p 7
:+c =+c 8+c 2+c 0+c (c + :)!(k − :)! 7 . k k−0k−2k−8k−= c!k!
(0)
By h`rjaul hnefhj ea k `km c, wj h`k hneesj p hlesj te 4.40. Aer jx`gplj, k 6 00 0= 09 c 4 0 2 8 p 4.4496 4.402 4.404: 4.4464
(2)
@ i`gjh`rm wftn K 7 0= bexjs `km : + c 7 9 sn`mjm bexjs weulm bj quftj rj`sek`blj.
Trebljg 2.8.0 Xelutfek (`) Xfkhj tnj preb`bflfty ea ` zjre fs 4.>, wj h`k jxprjss tnj preb`bflfty ea tnj hemj werm 44000 `s 2 ehhurrjkhjs ea ` 4 `km tnrjj ehhurrjkhjs ea ` 0. ]njrjaerj T S44000W 7 (4.>)2 (4.2)8 7 4.44:02.
(0)
(b) ]nj preb`bflfty tn`t ` hemj werm n`s jx`htly tnrjj 0‖s fs T Stnrjj 0‖sW 7
: (4.>)2 (4.2)8 7 4.4:02. 8
(2)
Trebljg 2.8.2 Xelutfek Ifvjk tn`t tnj preb`bflfty tn`t tnj Hjltfhs wfk ` sfkilj hn`gpfeksnfp fk `ky ifvjk yj`r fs 4.82, wj h`k ffkm tnj preb`bflfty tn`t tnjy wfk > str`fint KB@ hn`gpfeksnfps. T S> str`fint hn`gpfeksnfpsW 7 (4.82)> 7 4.44400. ::
(0)
]nj preb`bflfty tn`t tnjy wfk 04 tftljs fk 00 yj`rs fs T S04 tftljs fk 00 yj`rsW 7
00 (.82)04 (.1>) 7 4.444>=. 04
(2)
]nj preb`bflfty ea j`hn ea tnjsj jvjkts fs ljss tn`k 0 fk 0444! Ifvjk tn`t tnjsj jvjkts teec pl`hj fk tnj rjl`tfvjly snert ffaty yj`r nfstery ea tnj KB@, ft sneulm sjjg tn`t tnjsj preb`bflftfjs sneulm bj guhn nfinjr. Yn`t tnj gemjl evjrleecs fs tn`t tnj sjqujkhj ea 04 tftljs fk 00 yj`rs st`rtjm wnjk Bfll Russjll oefkjm tnj Hjltfhs. Fk tnj yj`rs wftn Russjll (`km ` streki suppertfki h`st) tnj preb`bflfty ea ` hn`gpfeksnfp w`s guhn nfinjr.
Trebljg 2.8.8 Xelutfek Yj ckew tn`t tnj preb`bflfty ea ` irjjk `km rjm lfint fs 9 /01, `km tn`t ea ` yjllew lfint fs 0/>. Xfkhj tnjrj `rj `lw`ys : lfints, I, V , `km R ebjy tnj gultfkegf`l preb`bflfty l`w? :! T SI 7 2, V 7 0, R 7 2W 7 2!0!2!
2
2
9 01
0 >
9 01
.
(0)
]nj preb`bflfty tn`t tnj kugbjr ea irjjk lfints jqu`ls tnj kugbjr ea rjm lfints T SI 7 RW 7 T SI 7 0, R 7 0, V 7 8W + T SI 7 2, R 7 2, V 7 0W + T SI 7 4, R 7 4, V 7 :W :! 7 0!0!8!
8
9 01
:! + 4!4!:! ≎ 4.0==6.
9 01
0 >
0 >
:
:! + 2!0!2!
2
2
9 01
9 01
0 >
(2)
Trebljg 2.8.= Xelutfek Aer tnj tj`g wftn tnj negjheurt `mv`kt`ij, ljt Y f `km Lf mjketj wnjtnjr i`gj f w`s ` wfk er ` less. Bjh`usj i`gjs 0 `km 8 `rj negj i`gjs `km i`gj 2 fs `k `w`y i`gj, tnj trjj fs :1
Y •Y 0 Y 2 0− p 2
p Y 0 0− p L 0
p
0− p
L2
p(0− p)
p Y 8 L 0− p p
Y
0
p8
Y 0 L2 L8
p2 (0− p)
L0 Y 2 Y 8
p(0− p)2
L0 Y 2 L8
(0− p)8
8 •
Y
2 8 L 0 − p 8 p L •L L p(0− p) 2
Y 0 L2 Y 8
•
• •
2
]nj preb`bflfty tn`t tnj tj`g wftn tnj negj heurt `mv`kt`ij wfks fs T SN W 7 T SY 0 Y 2 W + T SY 0 L2 Y 8 W + T SL0 Y 2 Y 8 W 7 p(0 − p) + p8 + p(0 − p)2 .
(0)
Ketj tn`t TSN W ≭ p aer 0/2 ≭ p ≭ 0. Xfkhj tnj tj`g wftn tnj negj heurt `mv`kt`ij weulm wfk ` 0 i`gj pl`yeffl wftn preb`bflfty p, tnj negj heurt tj`g fs ljss lfcjly te wfk ` tnrjj i`gj sjrfjs tn`k ` 0 i`gj pl`yeffl!
Trebljg 2.8.: Xelutfek
(`) ]njrj `rj 8 ireup 0 cfhcjrs `km 1 ireup 2 cfhcjrs. Zsfki If te mjketj tn`t ` ireup f cfhcjr w`s hnesjk, wj n`vj T SI0 W 7 0/8,
T SI2 W 7 2/8.
(0)
Fk `mmftfek, tnj prebljg st`tjgjkt tjlls us tn`t T SC |I0 W 7 0/2,
T SC |I2 W 7 0/8.
(2)
Hegbfkfki tnjsj a`hts usfki tnj L`w ea ]et`l Treb`bflfty yfjlms T SC W 7 T SC |I0 W T SI0 W + T SC |I2 W T SI2 W 7 (0/2)(0/8) + (0/8)(2/8) 7 9/0>. :9
(8)
Trebljg 2.=.0 Xelutfek Areg tnj prebljg st`tjgjkt, wj h`k hekhlumj tn`t tnj mjvfhj hegpekjkts `rj hekffiurjm fk tnj aellewfki w`y. Y 0
Y 8
Y 2
Y :
Y =
Y 1
]e ffkm tnj preb`bflfty tn`t tnj mjvfhj wercs, wj rjpl`hj sjrfjs mjvfhjs 0, 2, `km 8, `km p`r`lljl mjvfhjs : `km 1 j`hn wftn ` sfkilj mjvfhj l`bjljm wftn tnj preb`bflfty tn`t ft wercs. Fk p`rtfhul`r, T SY 0 Y 2 Y 8 W 7 (0 − q )8 , T SY : ∯ Y 1 W 7 0 − T SY :h Y 1h W 7 0 − q 2 .
(0) (2)
]nfs yfjlms ` hegpesftj mjvfhj ea tnj aerg 8
(0-q)
2
0-q
0-q
]nj preb`bflfty TSY W tn`t tnj twe mjvfhjs fk p`r`lljl werc fs 0 gfkus tnj preb`bflfty tn`t kjftnjr wercs?
T SY W 7 0 − q (0 − (0 − q )8 ).
(8)
Afk`lly, aer tnj mjvfhj te werc, betn hegpesftj mjvfhj fk sjrfjs gust werc. ]nus, tnj preb`bflfty tnj mjvfhj wercs fs T SY W 7 S0 − q (0 − (0 − q )8 )WS0 − q 2 W.
14
(=)
Xfkhj j`hn ea tnj =4 bft tr`ksgfssfeks fs `k fkmjpjkmjkt trf`l, tnj oefkt preb`bflfty ea h herrjht bfts, m mjljtfeks, `km j jr`surjs n`s tnj gultfkegf`l preb`bflfty T SH 7 h, M 7 m, J 7 mW 7
=4! h m j θ ν h!m!j!
4
h + m + j 7 =43 h,m,j ≪ 4, etnjrwfsj.
(2)
Trebljg 2.=.= Xelutfek Areg tnj st`tjgjkt ea Trebljg 2.=.0, tnj hekffiur`tfek ea mjvfhj hegpekjkts fs Y 0
Y 2
Y 8
Y =
Y :
Y 1
By syggjtry, ketj tn`t tnj rjlf`bflfty ea tnj systjg fs tnj s`gj wnjtnjr wj rjpl`hj hegpekjkt 0, hegpekjkt 2, er hegpekjkt 8. Xfgfl`rly, tnj rjlf`bflfty fs tnj s`gj wnjtnjr wj rjpl`hj hegpekjkt : er hegpekjkt 1. ]nus wj heksfmjr tnj aellewfki h`sjs? F Rjpl`hj hegpekjkt 0 Fk tnfs h`sj q T SY 0 Y 2 Y 8 W 7 (0 − )(0 − q )2 , 2 T SY = W 7 0 − q, T SY : ∯ Y 1 W 7 0 − q 2 .
(0) (2) (8)
]nfs fgplfjs T SY 0 Y 2 Y 8 ∯ Y = W 7 0 − (0 − T SY 0 Y 2 Y 8 W)(0 − T SY = W) q 2 7 0 − (: − =q + q 2 ). 2 12
(=)
Fk tnfs h`sj, tnj preb`bflfty tnj systjg wercs fs T SY F W 7 T SY 0 Y 2 Y 8 ∯ Y = W T SY : ∯ Y 1 W q 2 7 0 − (: − =q + q 2 ) (0 − q 2 ). 2
(:)
FF Rjpl`hj hegpekjkt = Fk tnfs h`sj, T SY 0 Y 2 Y 8 W 7 (0 − q )8 , q T SY = W 7 0 − , 2 T SY : ∯ Y 1 W 7 0 − q 2 .
(1) (9) (>)
]nfs fgplfjs T SY 0 Y 2 Y 8 ∯ Y = W 7 0 − (0 − T SY 0 Y 2 Y 8 W)(0 − T SY = W) q q 7 0 − + (0 − q )8 . 2 2
(6)
Fk tnfs h`sj, tnj preb`bflfty tnj systjg wercs fs T SY F F W 7 T SY 0 Y 2 Y 8 ∯ Y = W T SY : ∯ Y 1 W q q 7 0 − + (0 − q )8 (0 − q 2 ). 2 2
(04)
FFF Rjpl`hj hegpekjkt : Fk tnfs h`sj, T SY 0 Y 2 Y 8 W 7 (0 − q )8 , T SY = W 7 0 − q, q 2 T SY : ∯ Y 1 W 7 0 − . 2
(00) (02)
(08)
]nfs fgplfjs T SY 0 Y 2 Y 8 ∯ Y = W 7 0 − (0 − T SY 0 Y 2 Y 8 W)(0 − T SY = W)
2
7 (0 − q ) 0 + q (0 − q ) . 18
(0=)
Fk tnfs h`sj, tnj preb`bflfty tnj systjg wercs fs T SY F FF W 7 T SY 0 Y 2 Y 8 ∯ Y = W T SY : ∯ Y 1 W q 2 7 (0 − q ) 0 − 0 + q (0 − q )2 . 2
(0:)
Areg tnjsj jxprjssfeks, fts n`rm te tjll wnfhn substftutfek hrj`tjs tnj gest rjlf`blj hfrhuft. Afrst, wj ebsjrvj tn`t TS Y F F W ; TSY F W fa `km ekly fa 2 q q q 0 − + (0 − q )8 ; 0 − (: − =q + q 2 ). 2 2 2
(01)
Xegj `lijbr` wfll snew tn`t TSY F F W ; TSY F W fa `km ekly fa q 2 < 2, wnfhn ehhurs aer `ll kektrfvf`l (f.j., kekzjre) v`lujs ea q . Xfgfl`r `lijbr` wfll snew tn`t TSY F F W ; TSY F FF W aer `ll v`lujs ea 4 ≭ q ≭ 0. ]nus tnj bjst pelfhy fs te rjpl`hj hegpekjkt =.
Trebljg 2.:.0 Xelutfek R`tnjr tn`k oust selvj tnj prebljg aer :4 trf`ls, wj h`k wrftj ` aukhtfek tn`t ijkjr`tjs vjhters H `km N aer `k `rbftr`ry kugbjr ea trf`ls k. ]nj hemj aer tnfs t`sc fs aukhtfek SH,NW7twehefk(k)3 H7hjfl(2*r`km(k,0))3 T70-(H/=)3 N7(r`km(k,0)< T)3
]nj ffrst lfkj premuhjs tnj k Ù 0 vjhter H suhn tn`t H(f) fkmfh`tjs wnjtnjr hefk 0 er hefk 2 fs hnesjk aer trf`l f. Kjxt, wj ijkjr`tj tnj vjhter T suhn tn`t T(f)74.9: fa H(f)703 etnjrwfsj, fa H(f)72, tnjk T(f)74.:. @s ` rjsult, N(f) fs tnj sfgul`tjm rjsult ea ` hefk flfp wftn nj`ms, herrjspekmfki te N(f)70, ehhurrfki wftn preb`bflfty T(f).
1=
