olasılık ve İstatistik - erzurum atartürk Üniversitesi olasılık ve raslantı değişkenleri...

8/9/2019 Olasılık Ve İstatistik - Erzurum Atartürk Üniversitesi Olasılık Ve Raslantı Değişkenleri Ders NotlarI

1/96

CXWSXSI K]K]ÚXDÚ@ĶQCX_Ķ]C_Ķ

EjksĻjĻd vc Xkstjk`tĻ Mcĕnŕdc`jcrnMkbk Akzjk Dky`kd Ķìn`4

www.ccimcrs`etjkrn.oei


2/96

Ñ@_ÑW]úrdìc dky`kdjkrĻ` kzjĻĕĻ hnz ñĕrctni úycjcrn`n dc`mn mcrs ètjkrĻiĻzĻ múzlú` hnr aerikttk

bkzĻrjkykrkd ñĕrcònyc suìk yñ`ú`mc ietnvc cticdtcmnr. Kòkd mcrs ètjkrĻ`Ļ` hnr dntkpdkmkr mctkyjĻ vc dkpskijĻ bkjc lcjicsn nìn` kyrĻok ñzc`jn hnr ìkjĻŕik lcrcdicdtcmnr. HuìkjĻŕik nsc hnr súrcon lcrcdtnricdtcmnr.Őu k`mkdn bkjnyjc , kikìjkmĻĕĻiĻz ñĕrcòn`n`rkbktjĻdjk tkdnp cmchnjcocĕn hnr ykrmĻioĻ dky`kd ejuŕturikdtĻr. MejkyĻsĻyjk hu mcrs ètjkrĻ`Ļ`hnr mcrs dntkhĻ ycrn`c lcìcicycocĕn vc mcrsn` ñĕrctni úycsn tkrkaĻ`mk` tkvsnyc cmnjc` ycrjn vcykhkòĻ dky`kdjkrjk hcrkhcr múŕú`újmúĕú`mc dktdĻsĻ`Ļ` kikìjk`mĻĕĻ yñ`mc ejkokĕĻ lñzñ`ú`mc huju`murujkrkd hu ètjkrmk` akymkjk`ĻjikjĻmĻr.

Hu mcrs ètjkrĻ`mk‚Vrehkhnjnty, Xk`mei Qkrnkhjcs k`m _teobkstno Vreocsscs hy Ktbk`ksneus Vkpeujns’, ‚Vrehkhnjnty, Xk`mei Vreocsscs, k`m Cstniktne` ]bcery aer C`ln`ccrs hy Bc`ry _tkrd k`m Geb` \. \eems’, ‚Jcoturc @etcs e` Vrehkhnjnty ]bcery k`m Xk`mei Vreocsscs hy Gck` \kjrk`m Mcpkrtic`t ea Cjcotrnokj C`ln`ccrn`l k`m Oeiputcr

_oncòcs S`nvcrsnty ea Okjnaer`nk Hcrdcjcy, OK >=


3/96

MCX_ĶÌCX ĶĐĶ

;. EjksĻjĻdk. EjksĻjĻd `cmnr5h. EjksĻjĻĕĻ` djksnd, hkĕĻj arcdk`s vc kdsnyeiktnd tk`ĻijkrĻ

8. Dúicjcr vc Ejkyjkrk. Dúic Ķŕjcijcrnh. Hnjcŕnd ejksĻjĻdo. Deŕujju ejksĻjĻdm. Hkyc„s tcercinc. HkĕĻisĻzjĻd

?. ]cdrkr Cmnjc` Mc`cicjcrk. Hnrjcŕtnrnjinŕ mc`cyjcrh. Ìedju mc`cicjcr

=. Xkstjk`tĻ Mcĕnŕdc`jcrn (]cd Xkstjk`tĻ Mcĕnŕdc`n)k. Hcdjc`mnd mcĕcr vc vkryk`sh. EjksĻjĻd yeĕu`jud vc mkĕĻjĻi ae`dsnye`jkrĻ, Lkussnk` vc mnĕcr yeĕu`jud

ae`dsnye`jkrĻo. Deŕujju MkĕĻjĻijkrm. Hkyc„s tcercin (yeĕu`jud vc mkĕĻjĻi ae`dsnye`jkrĻ`k lc`nŕjctni)

3. ]cd Hnr Xkstjk`tĻ Mcĕnŕdc`n`n` Ae`dsnye`jkrĻk. Xkstjk`tĻ mcĕnŕdc`n l(x)h. l(x)„ n` mkĕĻjĻi ae`dsnye`jkrĻo. Ieic`tjcrm. Dkrkdtcrnstnd ae`dsnye`jkr

1. Ķdn Xkstjk`tĻ Mcĕnŕdc`nk. Hnrjcŕnd mkĕĻjĻi vc yeĕu`jud ae`dsnye`jkrĻh. Ķdn rkstjk`tĻ mcĕnŕdc`n`c knt ae`dsnye`jkr

n. Ķdn rkstjk`tĻ mcĕnŕdc`n`n` tcd hnr ae`dsnye`unn. Ķdn rkstjk`tĻ mcĕnŕdc`n`n` ndn ae`dsnye`u

nnn. Hnrjcŕnd ieic`tjcr, hcdjc`mnd mcĕcrnv. Deŕujju Ķstktnstndjcr


4/96

;. EjksĻjĻĕk Lnrnŕ

EjksĻjĻd tcernsn hcjnrsnzjnĕn` iktciktndscj ejkrkd iemcjjcìcsn njc njlnjc`nr. Cvrc`mclcrìcdjc ŕc` hútú` ejkyjkrĻ` ‚LĶWJĶ’ hnr mcĕnŕdc`n` ae`dsnye`jkrĻ ejmuĕuù múŕú`úrscd, humcĕnŕdc` vc ae`dsnye`jkrĻ` hnjnìcsn/tk`Ļijk`khnjicsn bkjn`mc ‚hcjnrsnzjndjcr’ ertkmk` dkjdkr.]khnndn hu mcĕnŕdc`n` vc ae`dsnye`jkrĻ` ‚DC_Ķ@’ ejkrkd hnjnìcsn prktndtc iúidú` mcĕnjmnr.Huù` nìn` ‚EJK_NJND ]CEXĶ_Ķ’ ‚]CDXKXJK@K@’ ejkyjkrĻ` ertkjkik mkvrk`ĻŕjkrĻ`kemkdjk`Ļr.

‚]CDXKXJK@K@’ ejkyjkr nìn` lñzjcijcr ykpĻjmĻĕĻ`mk vc ñzcjjndjc lñzjcijcrn` skyĻsĻkrtĻrĻjmĻĕĻ`mk hcjjn ertkjkikjkrĻ` skhnt mcĕcrjcrc ykdjkŕtĻĕĻ vc hu ertkjkik mcĕcrn` mkbkse`rkdn ejkyjkrĻ` dktĻjiksĻyjk mkbn mcĕnŕicic c ĕnjnin`mc ejmuĕu lñzjcijcìn ŕtnr.]cdrkrjk`k` ejkyjkrk ñr`cd ejkrkd4 cjcdtre` ykyĻ`ĻiĻ, tcjcae` krkikjkrĻ, rkmkr kjlĻjkiksĻ,dkjntc de`trejú, snstci ìñdicsn, ykzĻ-turk ktĻjiksĻ vh. ejkyjkrĻ vcrchnjnrnz. Hksntìc k`jktikd lcrcdnrsc, hu tcern`n` kikoĻ ejkyjkrĻ` ertkjkik mkvrk`ĻŕjkrĻ`Ļ ejkyjkrĻ` ejksĻjĻdjkrĻ on`sn`mc` tk`Ļijkikd vc dcstnricdtnr.

;.;. EjksĻjĻĕĻ` Iúbc`mnsjndtcdn SylujkikjkrĻ

EjksĻjĻd iúbc`mnsjnd kjk`Ļ`mk hnrìed uylujkik kjk`Ļ`k skbnptnr. Cjcdtrnd-Cjcdtre`ndIúbc`mnsjnĕn kjk`Ļ`k lnrc` uylujkikjkrĻ`mk` hkzĻjkrĻ

Cjcdtrnd mcvrcjcrmc ĻsĻj lúrújtúWkyĻa rkmkr vc rkmye sn`ykjjcrn`n` kjlĻjkìksĻCàeriksye` ]cernsnBkhcrjcŕic _nstci ]kskrĻiĻ_nstcijcrn` Lúvc`njnrjnĕn_nstci/Onbkz Bktk/HkŕkrĻsĻzjĻd Erk`jkrĻ vc EjksĻjĻdjkrĻBkhcrjcŕic K ĕjkrĻ, ñr7 Kĕ ]rkanĕn

ŕcdjn`mc sĻrkjk`khnjnr.;.8. Ìc ŕntjn EjksĻjĻd ]k`ĻijkrĻ

EjksĻjĻĕĻ` tk`ĻiĻ lc`cj ejkrkd ? tcicj ykdjkŕĻijk ykpĻjĻr.


5/96

Hu muruimk lñrújmúĕú lnhn tepjki = tk`c ejksĻ seùì vkr. Cĕcr hnz bcr ndn pkrk`Ļ`mkykzĻ lcjicsn ejksĻjĻĕĻ`Ļ ‚^^’ krĻyerskd, hu muruimk hu ejkyĻ` ejksĻjĻĕĻ ;/= ejur. Cĕcr k` kzhnr pkrk`Ļ` ^ lcjicsn ejksĻjĻĕĻ`Ļ krĻyerskd, hu muruimk ^^,]^ vc ^] seùìjkrĻ`mk` bcrhnrnhnzni krkmĻĕĻiĻz seùìjkrĻmĻr. MejkyĻsĻyjk hu ejkyĻ` ejksĻjĻĕĻ hu dcz ?/= ejur.

Djksnd tcern ndn kìĻmk` yctcrsnzmnr7;. Cŕntŕk`sk skbnp ejikyk` ejkyjkr nìn` meĕru seùì vcricz.Icscjk ktĻjk` ndn zkrĻ` tepjkiĻ`Ļ` ? ejiksĻ ejksĻjĻĕĻ`Ļ krkykjĻi. Hu muruimk iúidú`

tepjkijkr {8,?,=,3,1,,;6,;;,;8} ejkhnjnr. ^k`n ;; akrdjĻ seùì iúidú`múr. MejkyĻsĻyjktepjkiĻ` ? ejiksĻ ejksĻjĻĕĻ ;/;; mnr mnychnjnrnz. Kòkd tkhnndn hu seùì ^K@JNŐ„tĻr. Ìú`dútepjkiĻ` 8 ejiksĻ vc ? ejiksĻ ky`Ļ ŕk`sk skbnp mcĕnjmnr. ûdkrĻmkdn lnhn hnr tkhje ykpkrkdmeĕru seùou` ;/;2 ejmu ĕu ìed rkbktjĻdjk lñrújchnjnr.

8. _kyĻjkikykokd dkmkr ìed ejksĻ seùou ejk` muruijkrmk ykdjkŕĻijkrk hkŕvurikmk`ìñzúi úrctcicz.

;.8.8. HkĕĻj Arcdk`s ]k`ĻiĻ

Hnr K ejkyĻ`Ļ` ejksĻjĻĕĻ`Ļ hkĕĻj arcdk`s yejuyjk hcjnrjcicd hnr mc`cicyn ` dcrctcdrkrjkikdtk` lcìcr. Hu ` mc`cic cs`ksĻ`mk K ejkyĻ`Ļ` lñrújmúĕú muruijkr skyĻjĻr vc hu `K ejkrkd dkymcmnjnr vc K ejkyĻ`Ļ` ejksĻjĻĕĻ

V(K)9 K̀ /` ejkrkd vcrnjnr.

` K:` ejmuĕu`mk` V(K) 6 vc ; krksĻ`mkmĻr (6:V(K) :;). Hu tcernyjc njlnjn prehjcijcrmc` hnrtk`csn bnì hnr zkik` se`suz skyĻmk mc`cic ykpkikykokd ejikiĻzmĻr. Hu muruimk V(K)„ yĻse`ju skyĻmkdn mc`cicjcrmc` dcstnricd muruiu`mk dkjĻrĻz. Mkbk se`rkmk hu erk`Ļ`Ļ` `se`suz ejmuĕu muruimk hnr jnint mcĕcrn`c ykdjkŕtĻĕĻ`Ļ skvuùruz. ^k`n

V(K) 9 jni (`K /`) ejkrkd vcrnjnr.` ∐

Hu prehjcic rk ĕic` hk ĕĻj arcdk`s ykdjkŕĻiĻ, ejksĻjĻd tcernsn`n lcrìcd mú`ykmkdn ejkyjkrkuylujkikmk ñ`cijn hnr ycrc skbnptnr.

;.8.?. Kdsnyeiktnd ]k`Ļi

Ìe ĕu iemcr` ejksĻjĻd dntkhĻ`mk dujjk`Ļjk` tk`Ļi hu tk`ĻimĻr. Kòkd hu tk`ĻiĻykpikmk` ñòc tcicj hnrdkì dkvrki úzcrn`mc murikiĻz lcrcdicdtcmnr.


6/96

Ķidk`sĻz Ejky7 EjiksĻ iúidú` ejikyk` ejkymĻr. Dúic tcernsn`mc he ŕ dúicyc tcdkhújcmcr.

Ñr`cd7;. ^kzĻ-]urk ktik mc`cyn

℡9{^,]} Ñr`cd uzkyĻK9{^} ^kzĻ lcjicsn EJK^N

8. 8 dcrc ^kzĻ-]urk ktik mc`cyn℡9{^^,^],]^,]]} Ñr`cd uzkyĻ

K9{^],]^} Hnr ykzĻ vc hnr turk lcjicsn EJK^N?. Hnr mnrc`on` lúoú`ú ñjìic mc`cyn℡9{V7V≣6} ^k`n 6„mk` húyúd hútú` skyĻjkr ñr`cd uzkyĻiĻzĻ ejuŕturuyer.

=. Cjkstnd hnr ìkrpĻŕikmk `údjccr hnr pkrìkoĻĕĻ`, ìkrpĻŕik se`rksĻ kyrĻjĻŕ kìĻsĻ`Ļ lñzjcijcicmc`cyn

℡9{β7 -φ ≭ β ≭ φ} Ñr`cd uzkyĻH9{-φ /=≭ β ≭ φ /8} KyrĻjik kìĻsĻ`Ļ` -φ /= vcφ /8 krksĻ`mk ejiksĻ ejkyĻ.

Dúic Ķŕjcijcrn

p∂ K 7 p, K dúicsn`n` hnr cjcik`ĻmĻr.K⊈H 7 K dúicsn H dúicsn`n` kjt dúicsnmnr (K dúicsn`n` hútú` cjcik`jkrĻ H

dúicsn`n` mc cjcik`ĻmĻr).K9H 7 K„`Ļ` bcr cjcik`Ļ H„`n` vc H„`n` bcr cjcik`Ļ mk K„`Ļ` mk cjcik`Ļ nsc K vc

H dúicjcrn cŕnttnr.

p∂ K, K⊈H, K9H nakmcjcrn`n` dkrŕĻtkrĻ sĻrksĻyjk p∎ K, K⊍ H, K≦H ŕcdjn`mc ykzĻjĻr.∃ 7 Heŕ dúic.℡ 7 Cvrc`scj Dúic (Ñr`cd SzkyĻ).@et7 Bcrbk`ln hnr K dúicsn nìn∃̀ ⊈ K⊈℡ nakmcsn ykzĻjkhnjnr.


7/96

LĶXĶŐ

K vc H bcrbk`ln ndn dúic ejsu` K∧H njc lñstcrnjc`K vc H dúicjcrn`n` hnrjcŕnin K„ykyk mkH„yc hkĕjĻ cjcik`jkr dúicsnmnr.

Hnrjcŕni7 K∧H9{x7x K yk mk xH}

Ñr`cd7 K9{;,8,?} , H9{?,=,2} 90 K∧H9{;,8,?,=,2}K∮H njc lñstcrnjc` K vc H `n` krkdcsntn (dcsnŕnin) K vc H dúicjcrn`n` bcr ndnsn`c

hnrmc` hkĕjĻ cjcik`jkrĻ` dúicsnmnrDcsnŕni7 K∮H9{x7x K vc x H}

Ñr`cd7 K9{;,8,?} , H9{?,3} , K H9{?}Cĕcr K∮H 9∃ nsc yk`n K vc H `n` ertkd cjcik`Ļ yedsk K vc H yc ‚K^XND’ dúicjcr

mc`nr.

K vc H krksĻ`mkdn kyrĻi yk mk KZH hnìnin`mc lñstcrnjc` H dúicsn`n` K yk hkĕjĻ ejupH yc hkĕjĻ ejikyk` cjcik`jkrĻ` dúicsnmnr.

Akrd7 KZH9{x7xK vc x∎H}

Hurkmk KZH vc H `n` kyrĻd ejmuĕu`k yk`n (KZH)∮H96 ejmuĕu`k mnddkt cmn`nz.Ñr`cd7 K9{;,8,?}, H9{?,3} 90 KZH 9{;,8} , HZK9{3}

(KZH)∮H9{;,8}∮{?,3}9∃ (HZK)∮K9{3}∮{;,8,?}9∃ Ko hnìnin`mc lñstcrnjc` K„`Ļ` skjt túijcyc`n K yk hkĕjĻ ejikyk` vc ñr`cd uzkyĻ`k hkĕjĻ

ejk` cjcik`jkrĻ` dúicsnmnr.

]úijcyc`7 Ko 9{x7x ℡ vc x∎K}

Ñr`cd 7 K9{?,3},℡9{;,?,3,


8/96

]krkjĻ Kjk`7 KZH ]krkjĻ Kjk`7 Ko Őcdnj 8. Akrd vc túijcyc` dúicjcrn

DÚIC ĶŐ JCI DSXKJJKXN

;) ]k`Ļijkik DurkjĻ7 K∧K9K , K∮K9K

8) Hnrjndtcjnd 7(K∧H)∧O9K∧(H∧O), (K∮H)∮O 9K∮ (H∮O)

?) Mcĕnŕic 7 K∧H9H∧K, K∮H9H∮K

=)MkĕĻjik 7 K∧

(H∮O)9(K∧

H)∮ (K∧

O), K∮ (H∧

O)9(K∮H)∧

(K∮O)3) Ñzmcŕjnd 7K∧∃∃∃∃9K , K∮∃∃∃∃9∃∃∃∃ , K∧℡9℡ , K∮℡9K

1) ]úijcic 7 K∧Ko9℡ , K∮Ko9∃∃∃∃ , (Ko)o9K , ℡ o9∃∃∃∃, ∃∃∃∃o9℡


9/96

Hu èdtkmk` ntnhkrc` K∧H ycrn`c K+H vc K∮H ycrn`c KH ètksyeùù mcĕnŕnijn ejkrkddujjk`kokĕĻz.Ñr`cd7


10/96

VTHY 9VT{?}Y + VT{=}Y+………………………….+VT{>}Y

VTKHY9VT{?}Y + VT{=}Y+………………………….+VT{1}Y

Cĕcr VT{;}Y9…………….9VT{;8}Y9;/;8 90 VTKY9;/;8, VTHY9


11/96

HĶJC ŐĶD, DEŐSJJS, ]EVJKI EJK_NJNDJKX QC HK ĐNI_NWJND

Hnjcŕnd ejksĻjĻd7Ķdn K vc H ejkyĻ`Ļ` hnrjndtc lcrìcdjcŕic ejksĻjĻĕĻ`Ļ nakmc cmcr.VTK∮HY vcyk VTKHYŕcdjn`mc lñstcrnjnr. KŕkĕĻmkdn K, H vc O ejkyjkrĻ`Ļ cjc kjkjĻi.K7 {Crzurui mk bcrbk`ln hnr lú`mc sĻokdjĻĕĻ` ;3 eO„`n` úzcrn`mc ejiksĻ}H7 { Crzurui mk bcrbk`ln hnr lú`mc ; oi/i 8 úzcrn ykĕĻŕ ejiksĻ}O7 {K vc H ejkyjkrĻ`Ļ` bcr ndnsn`n` mc ky`Ļ lú`mc lcrìcdjcŕicsn}^k`n O 9 K∮HvcykO 9 KHejsu`.

O„`n` ejksĻjĻĕĻ`k K vc H„`n` ‚Hnjcŕnd ejksĻjĻĕĻ’ vcyk ‚Ertkd ejksĻjĻĕĻ’ mc`nr.

VTOY 9 VTK∮HY 9 VTKHY

Hu dkvrki ndnmc` akzjk ejky nìn` mc lcìcrjnmnr. Ñr`cĕn` bcrbk`ln C,A,L ejkyjkrĻ`Ļ`hnjcŕnd ejksĻjĻĕĻ VTBY 9 VTCALY ejkrkd vcrnjnr.

EjksĻjĻĕĻ` arcdk`s tk`ĻiĻ`mk` bkrcdctjc,

VTKY 9 `K / `, VTHY 9 `H / ` ejikd úzcrc,

VTKHY 9 VTK∮HY 9 K̀H / ` ejur._ñz deùsu K vc H ejkyjkrĻ hnrhnrjcrn`mc` hkĕĻisĻz nscjcr, ertkd ejksĻjĻdjkrĻ hksnt hnr

ìkrpĻi bkjn`n kjĻr7

VTKHY 9 VTKY VTHY

_ñz deùsu K vc H ejkyjkrĻ hnrhnrjcrn`mc` hkĕĻisĻz mcĕnj nscjcr, ertkd ejksĻjĻdjkrĻdeŕujju ejksĻjĻdjkrĻ ykrmĻiĻyjk cjmc cmnjchnjnr7

VTKHY 9 VTK|HY VTHYVTKHY 9 VTH|KY VTKY

Deŕujju ejksĻjĻd7Hnr K ejkyĻ`Ļ`, H ejkyĻ ejmudtk` se`rk ejik ejksĻjĻĕĻ`Ļ nakmc cmcr.ûdkrĻmk vcrnjc` K̀H /` K vc `KH /` H erk`jkrĻ`kŕnimn yc`nmc` hkdkjĻi. Hu erk`jkr K vc H

ejkyjkrĻ`Ļ` KH ejkyĻ`Ļ` lcrìcdjcŕicsn krksĻ`mkdn hkĕĻj arcdk`sjkrmĻr. Hu erk`jkr sĻrksĻyjk,sĻokdjĻĕĻ` ;3 eO„`n` úzcrn`mc ejmuĕu lú`jcrmc ykĕĻŕĻ` mk ; oi/i 8„`n` úzcrn`c ìĻdtĻĕĻ lú`jcrn`vc ykĕĻŕĻ` ; oi/i 8„`n` úzcrn`c ìĻdtĻĕĻ lú`jcrmc sĻokdjĻĕĻ` mk ;3eO„`n` úzcrn`mc ejmuĕulú`jcrn` ejksĻjĻĕĻ`Ļ vcricdtcmnr.

Sakd hnr iktciktndscj múzc`jcicyjc,


12/96

Ñr`cd7

^k`mkdn ŕcdnjmc lñstcrnjc` hnr ndnjnbkhcrjcŕic snstcin`mc 6 vc ; scihejjcrn`n`njctnjmnĕn hnr skyĻskj bkhcrjcŕicykpĻjikdtkmĻr. Hu snstcimc ‚^’ kjĻ`k`scihejú, ‚R’ nsc lñ`mcrnjc` scihejúlñstcricdtcmnr.

Őcdnj 1.Ķdnjn bkhcrjcŕic snstcin

Hu snstcin` ñr`cd uzkyĻ ejk` ‚℡’

℡9{(x,y)7 x9 6 vcyk ;, y9 6 vcyk ;}9{(6,6),(6,;),(;,6),(;,;)}

R ejkyĻ R 9 {(x,6),(x,;)} njc vcrnjicdtcmnr.

^ ejkyĻ ^ 9 {(6,y),(;,y)} njc vcrnjicdtcmnr.VT (x,y) 9 (n,g)Y 9 VT x9n Y VT y9g | x9n Y, (n 9 6 , ; vc g 9 6 , ;)

Hu snstcimc bkhcrjcŕic snstcin`mcdn lúrújtúmc` mejkyĻ lñ`mcrnjc` 6 hkzc` ; hkzc` 6ejkrkd kjĻ`khnjicdtcmnr. Ky`Ļ zkik` mk lñ`mcrnjc` ;„mc hkzc` 6 hkzc` ; ejkrkdkjĻ`khnjicdtcmnr.

^kpĻjk` ñjìúijcr `ctnocsn`mc kŕkĕĻmkdn ejksĻjĻdjkr hnjn`icdtcmnr.

VTy9;|x9;Y96.> 4 VTy9;|x96Y96.;VTy96|x9;Y96.; 4 VTy96|x96Y96.>

Ky`Ļ zkik`mk bkhcrjcŕic snstcin`n` tkskrĻiĻ`mk` lcjc` hnr ñzcjjndjc 6 úrctnjicejksĻjĻĕĻ ; úrctnjic ejksĻjĻĕĻ`k cŕnt ejup

VTx96Y9VTx9;Y 9 ;/8„mnr.

HkzĻ hnjcŕnd ejksĻjĻdjkrk hkdkjĻi

VTx96,y96Y 9 VTy96|x96Y VTx96Y96,>* 6,39 6,=3VTx96,y9;Y 9 VTy9;|x96Y VTx96Y96,;* 6,39 6,63


13/96

VTHY9VTH|K; Y.VTK; Y + VTH|K8Y.VTK8Y +………………….+ VTH|K`Y.VTK`Y

Hkzc` VTHY, cŕntjnĕn` skĕ tkrkaĻ`mkdn bcskpjkik`Ļ` meĕksĻ lcrcĕn ertkjkik ejksĻjĻd ejkrkd mkkmjk`mĻrĻjĻr.

Ķspkt7Kn‒jcr kyrĻd ejmuĕu`mk` KnK g9∃ , n≦ g ndc` vc ( ) vc H℡9H9H Kn HKn⊈Kn mnr. MejkyĻsĻyjk (HKn)(HK g)9∃ , bcr n≦ g nìn`.Kdsnyei ?„ú dujjk`Ļrskd (` ejkyk lc`nŕjctnjcrcd)4

VTHY9VTS` n9;HKnY9VTHK; Y+VTHK8Y+……………………+ VTHK`Y9 VTH|K; Y.VTK; Y + VTH|K8Y.VTK8Y +………………….+ VTH|K`Y.VTK`Y

Ñr`cd7Hnr ñòcdn ñr`cdjc vcrnjc` ndnjn bkhcrjcŕic snstcin nìn` VTy96Y (6 kjĻìk) vc VTy9;Y (;

kjĻìk) ejksĻjĻdjkrĻ`Ļ bcskpjkykjĻi.VTy96Y 9 VTy96|x96Y VTx96Y + VTy96|x9;Y VTx9;Y

9 6,> . 6,3 + 6,; . 6,3 96,3VTy9;Y 9 VTy9;|x96Y VTx96Y + VTy9;|x9;Y VTx9;Y

9 6,> . 6,3 + 6,; . 6,3 96,3VTy9;Y n kyrĻok {y96}∧ {y9; }9℡ ejmuĕu`mk` mnddkt cmcrcd,VTy96Y+VTy9;Y9; mc` {Kdsnyei 8} bcskpjkykhnjnrnz.

HkĕĻisĻzjĻd7Hnr H ejkyĻ`Ļ` ertkyk ìĻdiksĻ K ejkyĻ`Ļ` ertkyk ìĻdiksĻ úzcrn`mc bnìhnrctdn ykpiĻyersk, H ejkyĻ njc K ejkyĻndn hkĕĻisĻz ejkymĻr mc`nr. HkĕĻisĻzjĻd, ejksĻjĻĕĻèjmudìk ñ`cijn dkvrkijkrĻ`mk` hnrnmnr. KyrĻ ñr`cd uzkyĻ`mk tk`ĻijĻ K vc H ejkyjkrĻ`k ykj`Ļzvc ykj`Ļz

VTKHY9VTKY.VTHY nsc ‚HKĐNI_NW EJK^JKX’ mĻr mc`nr.Lc`cj nakmcsnyjc VTKHY9VTH|KY.VTKY9VTK|HY.VTHY ejmuĕu`mk`, hkĕĻisĻz ejkyjkr nìn`VTH|KY9VTHYVTK|HY9VTKY mnr.MejkyĻsĻyjk hkĕĻisĻzjĻd tk`ĻiĻ K, H ejkyjkrĻ`Ļ` hkĕĻisĻz ejmuĕu murui nìn` K ejkyĻ`Ļ`

H úzcrn`mc vc H ejkyĻ`Ļ` mk K úzcrn`mc bnìhnr ctdnsn ejikmĻĕĻ`Ļ sñyjcicdtcmnr. Hu tk`Ļi mktcicj dkvrkijkrĻiĻzjk ky`Ļ meĕrujtumkmĻr

? akrdjĻ K,H,O ejkyĻ nìn` (V(K)≦6, V(H)≦6, V(O)≦6 ejikd dkymĻyjk) hkĕĻisĻz ejikŕkrtĻŕñyjc vcrnjchnjnr7 VTKHOY9VTKY VTHY VTOYVTKHY9VTKY VTHY , VTHOY9 VTHY VTOY , VTKOY9VTKY VTOY

? ejkyĻ` hkĕĻisĻz ejiksĻ nìn` skmcoc VTKHOY9VTKY VTHY VTOY„`n` yctcrjn ejikmĻĕĻ`k vcbcr ndnjn`n` mc hkĕĻisĻz ejiksĻ lcrcdtnĕn`c mnddkt cmn`nz.


14/96

HK^C_ ]CEXCI Ķ7EjksĻjĻd tcernsn nìn`mc nòcjc`c` hnr ejky ejkrkd H ejkyĻ`k deŕujju hnr K ejkyĻ (yk`n H

ejkyĻ`Ļ` hnjn`mnĕn bkjmc K ejkyĻ) nìn` ejksĻjĻd mcĕcrn, K ejkyĻ`k deŕujju ejkrkd H ejkyĻ (yk`n KejkyĻ hnjn`mnĕn bkjmcdn H ejkyĻ) nìn` ejksĻjĻd mcĕcrn`mc` akrdjĻmĻr. Kòkd hu ndn hnrhnrn`c tcrsdeŕujjujud krksĻ`mk ìed hcjnrjn hnr njnŕdn vkrmĻr vc hu njnŕdnyc (njd kìĻdjkyk` nstktnstndìnĶ̀lnjnz]beiks Hkycs kmĻ`k ktac`) Hkycs ]cercin mc`njicdtcmnr.

Hkycs tcercin`n` nakmc cmnjnŕn7Hkycs tcercin hnr stedkstnd súrcì sĻrksĻ`mk ertkyk ìĻdk`hnr rkstlcjc ejky K njc hnr mnĕcr rkstlcjc ejky H (cĕcr H nìn` dkyhejikiĻŕ ejksĻjĻd vkrsk) nìn`deŕujju ejksĻjĻdjkrĻ vc ikrgn`kj ejksĻjĻdjkrĻ krksĻ`mkdn njnŕdnmnr, yk`n

Hkycs tcercin aeriújú nìn`mc huju`k` bcr hnr tcrnic ñzcj nsnijcr vcrnjicdtcmnr7• V(K) tcrnin`c K nìn` ñ`scj ejksĻjĻd vcyk ikrgn`kj ejksĻjĻd kmĻ vcrnjnr. Hu ñ`scjmnr,

ìú`dú H ejkyĻ bkddĻ`mk ñòcmc` bcrbk`ln hnr hnjlnyn nìcricicdtcmnr.• V(K|H) tcrnin vcrnjinŕ H nìn` K„`Ļ` deŕujju ejksĻjĻĕĻ kmĻ`Ļ kjĻr.•

V(H|K) tcrnin vcrnjinŕ K nìn` H„`n` deŕujju ejksĻjĻĕĻ kmĻ`Ļ tkŕĻr.• V(H) tcrnin H ejkyĻ nìn` 'ñ`scj' ejksĻjĻdtĻr vcyk H„`n` ikrgn`kj ejksĻjĻĕĻmĻr vciktciktndscj rejú èrikjnzc cmc` hnr skhnttnr.

Hu ŕcdnjmcdn Hkycs tcercin`n, akzjk iktciktndscj ejikmk`, sczlnyc mkyk`krkdŕñyjckìĻdjkykhnjnrnz7 Hkycs tcercin cĕcr H lñzjcijcìn ŕ nsc, K lñzjcin bkddĻ`mkdn n`k`ìjkrĻ` `cŕcdnjmc lúòcjjcŕtnrnjchnjcocĕn`n ertkyk ìĻdkrtĻr.

Hkycs tcercin`n` nspkt cmnjnŕn7Hu tcercin nspkt cticd nìn` deŕujju ejksĻjĻd tk`ĻiĻ`mk`ykrkrjk`ĻjĻr. H ejkyĻ hnjn`nrsc K ejkyĻ`Ļ` ejksĻjĻĕĻŕñyjc vcrnjnr7

Ky`Ļŕcdnjmc K ejkyĻ vcrnjinŕ nsc H ejkyĻ`Ļ` ejksĻjĻĕĻŕumur7

Hu ndn mc`djci yc`nmc` múzc`jc`np hnrjcŕtnrnjnrsc,


15/96

VTHY 0 6 vc VTKnY≦ 6, n nìn`

@E]7V TK g|HY„yc K g ‒`n` H de ŕuju`k hk ĕ jĻ ‚se`rkmk`’ ejksĻjĻ ĕ Ļ mc`nr. VTH|K gY„yc nsc H„`n`

K g„yc de ŕuju`k hk ĕ jĻ ‚ñòcmc`’ ejksĻjĻ ĕ Ļ mc`nr.

Lc`cjmc prktndtc ‚se`rkmk`’ ejksĻjĻdjkr lñzjcijcr seùou`mk bcskpjk`Ļr,‚ñòcmc`’ ejksĻjĻdjkr nsc lcìin ŕ ñjìúijcrc mkyk`Ļjkrkd dcstnrnjnr (tkbin` cmnjnr.)

Ñr`cd7Ķdn tkhkd mejusu hnsdúvn múŕú`újsú`. ]khkd ; nìn`mc ;6 tk`c ìndejktkjĻ hnsdúvn vc ?6tk`c skmc hnsdúvn huju`muĕu dkhuj cmnjsn`. ]khkd 8 nìn`mc nsc bcr ndn tnp hnsdúvnmc` 86ŕcrtk`c ejmuĕu hnjn`sn`. Cvn` dúìúd ìeouĕu hnr tkhkĕĻ rkstlcjc scìnp hu tkhkdtk` rkstlcjc hnrhnsdúvn scìnp kjsĻ`. Ìeouĕu` hnr tkhkĕĻ mnĕcrn`c vc hnr tnp hnsdúvnyn mnĕcrn`c tcronb cticdtcejmuĕu`k mknr cjninzmc bnìhnr lñstcrlc hujuìkikdtkmĻr. Ìeouĕu` scìtnĕn hnsdúvn`n` skmcejmuĕu lñrújsú`. Ìeou ĕu` hu skmc hnsdúvnyn tkhkd ;„mc` scìinŕ ejiksĻ`Ļ` ejksĻjĻĕĻ`Ļ` `cejkokĕĻ prehjcin hurkmk nòcjcìcdtcmnr.

_czln njc tkhkd ;„mc skmc hnsdúvn skyĻsĻ`Ļ` ìndejktkjĻ hnsdúvn skyĻsĻ`k lñrc mkbk akzjkejmuĕu lñz ñ`ú`c kjĻ`krkd nòcjc`c` ejksĻjĻĕĻ` %36„mc` mkbk akzjk ejkokĕĻ bcic` kjlĻjk`Ļr.Hu seruyk ocvkp Hkycs tcercin dujjk`Ļjkrkd dcsn` ejkrkd vcrnjchnjnr.

Ñòc seruyu mcĕnŕtnrnp Hkycs tcercin uylujk`khnjcocdŕcdjc sedikd lcrcdicdtcmnr7Ìeou ĕu` hnr skmc hnsdúvn scìinŕ ejmuĕu hnjnìcdtcmnr. E bkjmc hu deŕujjk hnrjndtc tkhkd;„mc` scìni ykpiksĻ ejksĻjĻĕĻ `c ejkokdtĻr5

Hñyjcoc Hkycs tcercin aeriújú`c uyikd nìn`, K ejkyĻ ìeouĕu` tkhkd ;„mc` scìniykpiksĻ4 H ejkyĻ nsc ìeouĕu` hnr skmc hnsdúvn scìicsn ejsu`.Ķstc`njc` ejksĻjĻd hñyjcoc V(K|H)ejkokdtĻr vc huù bcskpjkikd nìnŕ̀u ejksĻjĻdjkrĻ` hujuìksĻ lcrcdnr7

• V(K) vcyk bnìhnr mnĕcr hnjln ejikmk` ìeouĕu` tkhkd ;„mc` scìni ykpiksĻ ejksĻjĻĕĻ4Ķdn tkhkd krksĻ`mk tcronb ejikyĻp scìnin` cŕnt ejksĻjĻĕĻ ejmuĕu dkhuj cmnjicdtcmnr.

• V(H) vcyk bnìhnr mnĕcr hnjln ejikmk` ìeouĕu` hnr skmc hnsdúvn scìicsn ejksĻjĻĕĻ4 Mnĕcrhnr nakmc njc hu ìeouĕu` bcr hnr tkhkdtk` hnr skmc hnsdúvn scìic ejksĻjĻĕĻmĻr. HuejksĻjĻd, ñòc bcr ndn tkhkdtk` kyrĻ kyrĻ ejkrkd scìnjc` hnr tkhkdtk` hnr skmc hnsdúvnscìic ejksĻjĻĕĻ njc hu tkhkĕĻ scìic ejksĻjĻĕĻ`Ļ` hnrhnrn`c ìkrpĻjiksĻ vc se`rk hu ndnìkrpĻiĻ` tepjkìksĻ surctnyjc cjmc cmnjnr. ]khkdjkrmk ejk` skmc hnsdúvn`n` skyĻsĻ`Ļ`tepjkik erk`tĻsĻ`mk` hnjnìcdtcmnr dn tkhkd ; mc` hnr skmc hnsdúvn scìic ejksĻjĻĕĻ


16/96

Hñyjcoc ìeouĕu` hnr skmc hnsdúvn scìnin hnjn`mnĕn`c lñrc, tkhkd ;„mc` scìni ykpiksĻ`Ļ`ejksĻjĻĕĻ % 16„mĻr vc sczlninzc lñrc scìtnĕninz %36„mc` mkbk húyúdtúr.

Ñr`cd7Hnr bkhcrjcŕic dk`kjĻ`mk V6 nbtnikjjc 6, V; nbtnikjjc ; lñ`mcrnjicdtcmnr.( V; 9;-V6) Dk`kjmkdn lúrújtú mejkyĻsĻyjkθ ejksĻjĻĕĻ njc 6, ;-θ ejksĻjĻĕĻyjk mk ; kjĻ`Ļyer.KjĻoĻmk ; lñzjcijc`mnĕn`c lñrc ; lñ`mcrnjin ŕ ejik ejksĻjĻĕĻ `cmnr5

Ocvkp7

Dk`kjĻ` ykpĻsĻ yudkrĻmk vcrnjinŕtnr. VTR9;|^9;Y ejksĻjĻĕĻ`Ļ krĻyeruz.

θ

θ θ Ñr`cd7Hnr dk`scr tcstn`c njnŕdn` kŕkĕĻmkdn se`uìjkr tk`Ļijk`Ļyer ejsu`K 7]cstc lñrc tcst cmnjc` dnŕn dk`scr.H 7Dnŕn lcrìcdtc` dk`scr.Ko7]cstc lñrc tcst cmnjc` dnŕn dk`scr mcĕnj.Ho7Dnŕn lcrìcdtc` dk`scr mcĕnj.

kŕkĕĻ kd j ` h j `ĕ ` k k kjĻi


17/96

, ,, , , ,

,

^k`n skmcoc tcstjcrn` ejmuĕuù sñyjcmnĕn dnŕnjcrn` lcrìcdtc` dk`scr ejik nbtnikjn % 2,<mĻr. Lcrn dkjk` % >;,? júd dĻsĻimk dk`scr tcŕbnsn deùjk` dnŕn dk`scr ejikykokdtĻr.MejkyĻsĻyjk ‚Ķ̂ Ķ HĶX ]C_] MCĐĶJ’

Hu seùì ŕkŕĻrtĻoĻ lnhn lñzúdsc mc dk`scr ejk` vc ejikyk`jkrĻ` skyĻsĻ krksĻ`mkdnmc`lcsnzjnd (yk`n dk`scr ejk`jkrĻ` kzjĻĕĻ) tcstn hkŕkrĻsĻz dĻjikdtkmĻr.

_KÎK AEXIÚJJCX Ķ ^crn`c deykrkd ñr`cdjcic7

Hnr terhkmkdn ùikrkjk`mĻrĻjiĻŕ tepjkr ñòc ìcdnjnyer, ùikrksĻ dkymcmnjnyer vc se`rktcdrkr terhkyk deùjuyer. Hu muruimk njd ñr`cdjcicmc ‚`’ tk`c (terhkmk ` tk`c tep ejmuĕuvkrskyĻiĻyjk) akrdjĻ scìc`cd vc ndnòn ñr`cdjcicmc yn`c ‚`’ tk`c akrdjĻ scìc`cd sñz

deùsumur. _eùì ejkrkd ‚`’ cjcik`Ļ ejk` terhkmk` ‚r’ cjcik`jĻ ‚`r‚ tk`c akrdjĻ sĻrkjkìĻŕ ñr`cdjcr cjmc cmnjchnjnr.

^crn`c deyikmk` ñr`cdjcic7

^n`c yudkrĻmk hkbscmnjc` muru` nìn` hu scacr ìcdnjc` tepjkrĻ` ycrn`c lcrn deùjikmĻĕĻ`Ļmúŕú`cjni. MejkyĻsĻyjk njd ìcdni nìn` ‚`’ tep vkrdc` ndnòn ìcdni nìn` ‚`-;’ tep úìúòúmc ‚`-8’ ŕcdjn`mc kzkjk` skyĻmk tep ejkokdtĻr.

_eùì ejkrkd7

, ≚ ∘ ; ∘ 8 ∘ ? ∘ + ; 9 ! ∘ ! Hu muruimk ejuŕkhnjcocdŕcdnjjcrn` skyĻsĻ

, 9 9 !! dkmkr ‚akrdjĻ sĻrkjkìĻŕ’ ñr`cd cjmc cmnjchnjnr.‚`’ cjcik`jĻ hnr pepújèsye`mk` ejuŕ khnjcocd ‚r’ cjcik`jĻ kjt pepújèsye`jkr7

EjksĻjĻdtk ejuŕkhnjcocd tcicj serujkrmk` hnrnmnr. Ñr`cĕn` ;„mc` =„c dkmkr


18/96

;,; 8,; ?,; =,; ^crn`c deykrkd ñr`cdjcicmc nsc;,8 8,8 ?,8 =,8 =x=9;1 lrup cjmc cmnjnr.;,? 8,? ?,? =,? ;,= 8,= ?,= =,=

Hnzc 1 akrdjĻ lrup vcrc` mnznjni nìn` lc`cj hnr aeriúj cjmc cmnjnr. Hu mnznjnic` „n`r„jndeihn`ksyeù mc`nr vc scihejú njc lñstcrnjnr.

‚r’ tk`c cjcik`mk` eju ŕk` pepújksye` nìn` r! tk`c akrdjĻ mnznjni sñz deùsumur.MejkyĻsĻyjk kjt pepújèsyeù nìn` .r! tk`c akrdjĻ sĻrkjkikmk` hkbscmchnjnrnz.Hu muruimk,

.r!9 9 ! 9 !!. ! ≚ ≚ yc ky`Ļ zkik`mk Hnèi dktskyĻsĻ mc`nr.

Qc 9 !! ! 9 !!. ! ≚ 9 ^k`n

HCX@ESJJN MC@CICJCX Ķ _eùou ‚V’ ejksĻjĻĕĻ njc hkŕkrĻ{ΰ; 9s} vc q9;-

p ejksĻjĻĕĻ njc hkŕkrĻsĻzjĻd{ΰ; 9a} ejsu` vc tcd hnr mc`cic nìcrc` hnr mc`cicyn cjc kjkjĻi. Humuruimk ñr`cd uzkyĻ℡9{s,a}ejur. Hu mc`cyn tcdrkrjkmĻĕĻiĻzĻ vkrskykjĻi hu muruimk yc`nñr`cd uzkyĻ℡89℡x℡ ejur vc℡8 9{ss, sa, as, aa}. Hu uzky 889= tk`c ejky nìcrnr℡x℡ ìkrpĻiĻ‚DKX]CW^C@ ÌKXVNI’ ejkrkd kmjk`mĻrĻjĻr. Cĕcr ` tk`c ‚hkĕĻisĻz’ mc`cic ykpkrskd

ñr`cd uzkyĻ

℡̀9℡x℡x℡x℡x…………………x℡, ejur vc hu ñr`cd uzky 8` cjcik` nìcrnr.

` ‟ dcrc

9 7 9 8


19/96

ŕ , ∘ ŕ ≚ 4 , ]k`ĻijkmĻĕĻiĻz hu ‚h(d4`,p)’ yc ‚ Hnèi Dkùù’ mc`nr. ‚`’ mc`cic ykpkrkd ‚d’ hkŕkrĻ cjmcctic nbtnikjn krkik ejkyĻ`k mk ‚ Hnèi Mc`cicjcrn’ mc`nr.

Ñr`cd7 _k`nycmc ;6 hkĕĻisĻz hnt úrctc` kjĻìkdtkmĻr. BktkjĻ hnt kjĻìk ejksĻjĻĕĻ ( 6 lñ`mcrnjnp

;kjĻìksĻ ykmk tki tcrsn ) 6,66; mnr. _k`nycmc ‚C` Kz’ hnr bktk ejik ejksĻjĻĕĻ `cmnr.Ocvkp7

℀ ; ∘ ℀ì ℀

9 ;∘ 6,66;6,>>> 9 ;∘ 6,>>> ≃ 6,6;

Hnèi dkùù njc njlnjn aeriújjcr7

;) ‚d’ ykmk mkbk kz hkŕkrĻ ejik nbtnikjn

9 4 ,9 8) C` kz ‚d’ hkŕkrĻ ejik nbtnikjn

9 4 ,9 9 ; ∘ ?) C` kz ‚g’ c` akzjk ‚d’ hkŕkrĻ ejik nbtnikjn

9 4 ,9 Ñr`cd7

℡9{;,8,?,=} , K9{;,8} , H9{8,?} , O9{?,=}

9 | | Cŕntjnĕn kokhk skĕjk`Ļr iĻ5 E`k hkdkjĻi


20/96

XK_]JK@]N MCĐĶŐ DC@JCX Ķ Xkstlcjc mcĕnŕdc`7 Mcĕcrn mc`cymc` mc`cyc mcĕnŕc` seùìjkr njc hcjnrtnjc` mcĕnŕdc`c

mc`nr. Ñr`cĕn`, Hnr etepkrdtk skkt 6>.66„mk pkrd bkjn`mcdn krkìjkrĻ` skyĻsĻ, bnjcsnz hnr zkrĻ`ktĻjĻŕĻ`mk úst yúzc lcjcocd skyĻ hnrcr rkstlcjc mcĕnŕdc` mcĕcrnmnr.Xkstjk`tĻ mcĕnŕdc`n dkvrkiĻ erngn`kj ñr`cd (ejksĻjĻd) uzkyĻ`Ļ` ycrn`c ejkyjkrĻ` rkdki

dúicjcrn`mc` ejuŕturmuĕu hnr uzkyĻ dujjkìkiĻzĻ yk`n rccj(X) cdsc`mc ìkjĻŕkhnjicinzniúidú` dĻjkokdtĻr.

Ñr`cd uzkyĻ ‚℡’ ejk` hnr B mc`cyn`n cjc kjkjĻi.℡ ñr`cd uzkyĻ`Ļ` cjcik`jkrĻ (ΰ) Bmc`cyn`n` rkstlcjc seùìjkrĻmĻr. Cĕcr bcr hnr cjcik` (ΰ) nìn` rccj hnr skyĻ dujjk`Ļrskdΰ„jcr vcrccj(X) cdsc` krksĻ`mk hnr dkrŕĻjĻd hujik durkjĻ hujuruz. Hu durkjĻ R(ΰ) njc lñstcrcjni. Hñyjchcjnrjn dĻsĻtjkikjkrk skbnp durkjk ‚XK_JK@]N MCĐĶŐDC@Ķ’ mnyeruz. MejkyĻsĻyjk ‚R(.)’ ykmk dĻskok ‚R’ rkstlcjc mcĕnŕdc`n℡ meic`n`mc ìkjĻŕk` vc rccj cdsc`mc mcĕcrjcr kjkhnjc` hnrae`dsnye`mur.

Őcdnjmcdn kìĻdjk`mĻĕĻ lnhn hnr cŕjcŕtnric seùou`mk R rccj cdsc`mc mcĕcrjcr kjikdtkmĻr.Ñzcjjndjc4

R 7 rksjk`tĻ mcĕnŕdc`n x7 rccj skyĻjkrmkdn dkrŕĻjĻĕĻñzcj hnr ejkyk dkrŕĻjĻd lcjicdtcmnr vc hnz hu ejkyk hnr ejksĻjĻd mcĕcrn ktkikd nstcrnz. HuejksĻjĻd, mnr vc hu ejksĻjĻĕk ‚EJK_NJND MKĐNJNI

AE@D Ķ̂E@S (VMA)’ mcrnz

R(.)

R(ΰ)X

K

℡

K

.


21/96

Hu muruimk ktĻjk` ndn pkrk`Ļ` mk ykzĻ ejik ejksĻjĻĕĻ V(x96), hnrn`n` turk hnrn`n` ykzĻejik ejksĻjĻĕĻ V(x9;), vc bcr ndnsn`n` mc ykzĻ ejik ejksĻjĻĕĻ V(x98), njc nakmc cmcrnz.

Cĕcr ‚ejksĻjĻd mkĕĻjĻi ae`dsnyeùù (VMA)’ ìnzicyc ìkjĻŕĻrskdŕñyjc hnr lrkand cjmccmcrnz.

Xkstjk`tĻ mcĕnŕdc`n` tk`ĻiĻ`mk csks ejk` ñr`cd uzkyĻ`Ļ` bcr hnr ñĕcsn dkrŕĻjĻĕĻ`mk XccjskyĻ cdsc`n`mc (X) hnr dkrŕĻjĻd hujkhnjicsnmnr.

Ñr`cd7 _Ļ`Ļatkdn ñĕrcònjcr nìnÈ ; 7 Heyu ;36oi„mc` dúìúd ejk`jkr

E87 Heyu ;1>oi„mc` dúìúd ejk`jkr ŕcdjn`mc tk`ĻijkìĻŕ ejkyjkrĻ cjc kjkjĻi.E?7 Heyu ;16oi„mc` húyúd ejk`jkr

R7 Hnr ñĕrcòn`n` heyuù` uzu`juĕu nsc

R(E ; )9K; 9{x | x : ;36} V(R : ;36) Hu ejkyjkr mcĕnŕdc`R(E8)9K89{x | x : ;1>} V(R : ;1>) ejksĻjĻdjkrmĻr.R(E?)9K?9{x | x 0 ;16} V(R 0 ;16)

Xkstjk`tĻ mcĕnŕdc`n rccj cdsc`mc kjmĻĕĻ mcĕcrjcrc lñrc ‚súrcdsnz’ vc ‚súrcdjn’ rkstjk`tĻmcĕnŕdc`n ejikd úzcrc ndnyc kyrĻjĻrjkr.

Cĕcr R rkstjk`tĻ mcĕnŕdc`n`n` X mcdn mcĕcr dúicsn skyĻjkhnjnr ejkrkd se`suz hnr dúic nsc

A(x)9V(R≭x)

;

6,


22/96

Hnr ìed muruimk bcr hnr ejksĻjĻd nìn` tcd tcd ejksĻjĻd ae`dsnyeù VT.Y ‒yn ykzikd lkrnp ejkhnjnr.Huù` nìn` VMA (ejksĻjĻd mkĕĻjĻi ae`dsnyeùù) dujjkìkyk nbtnykì muykrĻz.

VMA 7 Ax(x)9VT R≭ x Yŕcdjn`mc tk`Ļijk`Ļr. @E]7 Xkstjk`tĻ mc ĕ n ŕdc`jcrn`n` húyúd ‚R’ ( ^,W vs. ejkhnjnr ), hu rkstjk`tĻ mc ĕ n ŕdc`n`n`kjmĻ ĕ Ļ mc ĕ crjcrn` nsc dúìúd ‚x’ njc lñstcrnjmn ĕ n`c mnddkt cmn` hu muruimk A x(y) ykzikiĻz ètksye`mk bcrbk l̀n hnr prehjci eju ŕturikz. Ax(y)9VTR ≭ yY ejur.

EjksĻjĻd MkĕĻjĻi Ae`dsnyeùù` Ax(x)„n` ñzcjjndjcrn7

;) Ax(∐)9; Ax(-∐)96

8) 6≭ Ax(x)≭;

?) x; ≭ x8 ↘ Ax(x; )≭ Ax(x8)=) V(x; : R ≭ x8) 9 Ax(x8) - Ax(x; )≣ 6

Ñr`cd7Hnr hnjlnskykrĻ` bkaĻzksĻ`mkdn hnt mnznsn`n nòcjcycjni. Cĕcr hnt ‚;’ nsc R9; vc ‚6’ nsc

R96 mĻr. Hntn` 6 ejik ejksĻjĻĕĻ`Ļ` ‚q’, ; ejik ejksĻjĻĕĻ`Ļ` mk ‚;-q’ ejmuĕuù vkrskykjĻi.Hu muruimk℡9{6,;} ejur. Hu ejksĻjĻĕĻ` mkĕĻjĻi ae`dsnyeù Ax(x)„n bcskpjkykjĻin) x : 6 7 Hu muruimk {R≭ x}9ǰ mnr vc Ax(x)96nn) 6≭ x : ;7 Hu muruimk {R≭ x}9q mur.nnn) x≣ ; 7 Hu muruimk {R≭ x}9℡ ejur yk`n V(R≭;)9;

Ax(x)

;


23/96

Ñr`cd7

Hnr rkstjk`tĻ mcĕnŕdc`n kŕkĕĻmkdn ejksĻjĻd mkĕĻjĻi ae`dsnyeù`k skbnptnr.

6 , ∘∐ ∘; Ax(x) 9 + , ∘ ; : : ; ; , ; ≭ : ∐

k) R 9;/=

h) R 0 ?/=

Ñòc hu ejksĻjĻd mkĕĻjĻi ae`dsnyeùù ìnzcjni

Ax(x)

;

6,3

x-; 6 ; 8

‚k’ ŕĻddĻ`Ļ ìñzicd nìn` 9 jni↘ ∘ ,6 : ^k`n ejksĻjĻd mkĕĻjĻi ae`dsnyeùù` x èdtksĻ`mkdn sejmk` jnintn`n tk`ĻijkykjĻi.Hurkmkŕuù nmmnk cmnyeruz V{R9x}9 A(x)-A(x-)

Ķ spkt7

∘ cjc kjkjĻi


24/96

EJK_NJND ÊĐS@JSD AE@D_Ķ̂E@S (pma)

EjksĻjĻd yeĕu`jud ae`dsnyeù.Ñzcjjndjcrn7

∐

∘ ∐ ∘ ∐ a(x)„n` yeruiu7

∘ ^ctcrnòc dúìúd hnr x mcĕcrn nìn`

∘ mejkyĻsĻyjk dúìúdx nìn ̀VTx:R ≭ x+ xY9A(x) x cĕcr x↘6„k lnmcrsc

VTR9xY9a(x)x↘6 ejur.

HCDJC@MĶD MCĐCX


25/96

R ∐

∐

Dkrcscj hcdjc`mnd mcĕcr cĕcr rkstlcjc mcĕnŕdc`ninz hnr mnrc`ì úzcrn`mcdn lcrnjnin nakmccmnyersk mnrcòn` úzcrn`mc bkrok`k` lúì hu dkrcscj hcdjc`mnd mcĕcrjc erk`tĻjĻmĻr.

Icrdczn ieic`t mnyc tk`Ļijkykok ĕĻiĻz ieic`tjcrmc rkstlcjc mcĕnŕdc`mc` hcdjc`mndmcĕcrn` ìĻdkrtĻjmĻdtk` se`rk cjmc cmnjc` ieic`tjcrmnr.

R R x x

`9; nìn` icrdczn ieic`tn` ‚6’ ejkok ĕĻ kìĻdtĻr.`98 nìn` nsc ejmudìk ñ`cijnmnr. Hu`k ‚vkryk`s’ mc`njnr.

x x x∐

∐

∘ ∘ 8 ∘ 8 ∘ 8

∘

mcĕcrn mc stk`mkrt skpik ejkrkd kmjk`mĻrĻjĻr.Ñr`cd7

k) Őcdnjmc vcrnjc` ejksĻjĻd yeĕu`judae`dsnyeùù` lcìcrjn hnr ‚pma’ ejiksĻ nìn` ‚k’

`c ejikjĻ5h) CTxYo) 8 m)

Ìñzúi7 k’ ŕĻddĻ`Ļ` ìñzúiú nìn` ejksĻjĻd

a(x)

k


26/96

9 ∐

∐

9 ;= 9

| ;

3 ; 9

∘

∐

∐ ;= 9;| 9 ;; ; 3 ∘ ; 9 ; =;

⇘ 9 ; =; ∘ 9 ; =; ∘ ;; ⇘ 9 = 9 ⇘ 9 . Ñr`cd7

KŕkĕĻmk ejksĻjĻd mkĕĻjĻi ae`dsnyeù vcrnjc` R rkstjk`tĻ mcĕnŕdc`n`n` ejksĻjĻd yeĕu`judae`dsnyeùù ìnzn`nz.

ĕ MejkyĻsĻyjk ejksĻjĻd mkĕĻjĻiae`dsnyeùù` túrcvn hnzc ejksĻjĻdyeĕu`jud ae`dsnyeùù vcrnr.

Ocvkp7

A(x)

;

x8

a(x)


27/96

Mnĕcr hnr tk`Ļimk rcot (mndmñrtlc`) ae`dsnyeù dujjk`Ļjkrkd vcrnjnr.

j i↘∐

Őcdnjmc k↘∐ lnttnĕn`mc mndmñrtlc` lc`nŕjnĕnkzkjkrkd 6„k lnmcr vc yúdscdjnĕn krtkrkd∐„klnmcr.

R9R n èdtksĻ`mkdn súrcdsnzjn ĕ n` túrcvn7

Cjninzmc hnrni hkskikd ae`dsnyeù ejsu`.

h r i hkskikd ae`ds yeù ; ≣ 66 6

_ejmk lñstcrmnĕninz ae`dsnye` S(x-xn) yk`n hnrnihkskikd ae`dsnyeùù` x n dkmkr dkyiĻŕ ŕcdjnmnr.(Őcdnj h)

_kĕmkdnŕcdnjmc (ŕcdnj o) nscŕcdnj h `n` x↘6 lnmcrdc` hnr

krcot(kx)

k

x

-;/8k ;/8k

S(x)

;

x6

Őcdnj kS(x-xn)

;

x6 xn

Őcdnj h

A(x)

;


28/96

| ≚ 9 j i∁↘ ;∁ ∘ ∁

| 9 ∘ . ^k`n x 9 xn èdtksĻ`mkdn hnr súrcdsnzjnĕn` túrcvn x 9 xn èdtksĻ`mk tk`ĻijĻ vc yúdscdjnĕn súrcdsnzjnĕn njc meĕruerk`tĻjĻ ejk` hnrν ae`dsnyeù njc vcrnjnr.

9 ≭ 9 jmuĕuù b tĻrj y r d y`Ļ ŕcd jmc ŕcdj `mc y z h j r z.

M j yĻsĻyj . Ñr`cd7

↘VMA nsc

_Ļd Dujjk`Ļjk` EjksĻjĻd ê ĕ u`jud Ae`dsnye`jkrĻ7

;) S`naeri pma (múzlú` mkĕĻjĻi)

a(x)

;/ x

xxn- x/8 xn+ x/8

a(x)

;/8

x6

A(x)

;

;/8

x6

;a(x)


29/96

Hu ejksĻjĻd yeĕu`jud ae`dsnyeùù` VMA„sn`n ìnzcocd ejurskd

9 6 ∘ ∘ ; ≣

ĕ κ + 8 . 9 ∘ ;8 8) Cxpe`k`snycj pma (Ústscj mk ĕ ĻjĻi)(³06)

; Ertkjkik mcĕcr 9 ³Qkryk`sϏ89³ 8

Ústscj mkĕĻjĻiĻ` ejksĻjĻd yeĕu`judae`dsnyeùù` VMA„sn`n ìnzcrscd

A(x)

;

x6 k h

a(x)

;/³ -

x6

∘


30/96

?) @erikj pma (Lkussnk` mk ĕ ĻjĻi)

; 8

@(³,Ϗ; 8)

@(³,Ϗ88)

³

Ertkjkik mcĕcr CTxY9³Qkryk`s 9Ϗ8 Hu ejksĻjĻd yeĕu`jud ae`dsnyeùù` VMA„n nsc

;8;+ cr 8 ; 8 Hu n`tclrkjn kjikd zer ejmuĕu nìn` ‚cra’ ae`dsnyeù`mk` ykdjkŕĻi ykpkokĕĻz.

cr ≚ ;8


31/96

3) ≭ cr 1)

0 ∘ cr KŕkĕĻmkdnŕcdnjmc, deyu ikvn ertkjkikmk` hnr stk`mkrt skpik mkbk dúìúdtúr. Hnr `erikjmkĕĻjĻi nìn` hu (deyu ikvn) cĕrn`n` kjtĻ`mk dkjk` kjk`, tepjki kjk`Ļ` %12'n`n dkpskr.Ertkjkikmk` ndn stk`mkrt skpik krkjĻĕĻ`mk `edtkjkr nìn` cĕrn`n` kjtĻ`mk dkjk` kjk` (kìĻd, ertkvc deyu ikvn kjk`) tepjki kjk`Ļ` %>>.


32/96

98(6,;>;)

96,?2811x6,?28 9 83,62≂ 83 dnŕn 86 njc =6 krksĻ`mk `et kjĻr.

=) Xkyjcnlb mk ĕ ĻjĻi (k 0 6) vc (≣̀ 6 )

Ertkjkik mcĕcr ³9 Qkryk`s

Hu pma bkhcrjcŕicmclcondicjn (akmn`l) dk`kjĻ` iemcjjc`icsn`mc dujjk`ĻjĻr.

a(x)

,

xx


33/96


34/96


35/96

a(x)

V(R96)

x

A(x)

;

x6 ; 8 ?

‚`’ c ĕcr ìed húyúdsc

ĕ .

1) Vensse` mk ĕ ĻjĻiĻ (k06 ) 7

Vensse` mkĕĻjĻiĻ, ejksĻjĻd durkiĻ vc nstktnstnd hnjni dejjkrĻ`mk hnr kyrĻd ejksĻjĻd mkĕĻjĻiĻ ejup,hcjjn hnr skhnt hnrni zkik` krkjĻĕĻ`mk icymk`k lcjic skyĻsĻ`Ļ` ejksĻjĻĕĻ`Ļ nakmc cmcr. Huzkik` krkjĻĕĻ`mk ertkjkik ejky icymk`k lcjic skyĻsĻ`Ļ` hnjn`mnĕn vc bcrbk`ln hnr ejkyjk eùbcic` tkdnp cmc` ejky krksĻ`mkdn zkik` akrdĻ`Ļ`, ñòcdn zkik` akrdjkrĻ`mk` hkĕĻisĻzejuŕtuĕu dkhuj cmnjnr.

Vensse` mkĕĻjĻiĻ ìed dcrc hcjnrjn skhnt zkik` krkjĻĕĻ hnrnijcrn huju`k` prehjcijcrcuylujkìkdjk hcrkhcr, mnĕcr hnrniscj krkjĻdjĻ prehjcijcrc mc (yk`n hnrni uzkdjĻd, kjk` vcykbkoni nìcrc` prehjcijcrc mc) hkŕkrĻ njc uylujk`khnjnr


36/96

! ;

Őnimn hu`jkrĻ` `ksĻj cjmc cmnjmnĕn`c hkdkjĻi. Ķ spkt7

⇘ ;

⇘ ≤ ; ≧ ; . 9

⇘ 9 ! ∘ ! ! ; ∘ 9 ;! ∘ ; ∘ 8………… ∘ ;. ;∘

9 ;! ; ∘ 9 ! ; ∘ ]kyjer scrnsn`mc`

⇘ ;! ⇘ ! .

Ñr`cd7Hnr hnjlnskykrmk ;6 hn` tk`c cjcdtre`nd cjcik` vkrmĻr. Bcr hnr cjcik` mnĕcrn`mc`hkĕĻisĻz hnr yĻjmk hezujik ejksĻjĻĕĻ (p9;6-=) ejkrkd vcrnjnyer. HnjlnskykrĻ` yĻjse`u`mkìkjĻŕĻyer ejik ejksĻjĻĕĻ `cmnr5 Hnr cjcik` hezujmuĕu`mk hnjlnskykrĻ`mk hezujmuĕu dkhujcmnjnyer.Ocvkp7


37/96

9 7 hnrni zkik`mkdn ejkyjkrĻ` skyĻsĻύ7krkjĻĕĻ` uzu`juĕu (t , t +ύ )

4 , + 9 ! ]rkand iemcjjcicjcrn`mc ìedìk dujjk`Ļjk` ye ĕ u`jud nakmcsn

Ñr`cd7 Őcdjn`mc vcrnjc` hnr VMA nìn` pma yn huju`uz.

Ocvkp7

a(x)

A(x) 9V(R≭x)

;

1/2

3 / 2

= / 2

8/2

-; 6 ; 8


38/96

;

= + = +

∘ ; + 8= +

; tepjkiĻ` n`tclrkjn n`tclrkjjcrn` tepjkiĻ`k cŕnttnr. MejkyĻsĻyjk

+ ;= 9 ;= + ; 9 ;=.; 9 ;=

= 9;= 9 ;=.; 9 ;= ∘ ; ; ; 9 ;.; 9 ;

8= 9 ;= ∘ 8 ;=.; 9 ;= ; ;

_eùìjkrĻ` bcpsn`n tepjkrskd seùou` ; ejiksĻ lcrcdnr.

+ + + ; Cĕcr A(x)9V(-;≭ x≭ ; ) serujiu ŕ ejskymĻ ocvkp ;/= + ;/= + ;/2 9 3/2 ejkokdtĻ.

Ñr`cd7k) pma lrkanĕn vcrnjc` ae`dsnyeù` VMA ‒sn`n ìnzn`nzh) V(=:x≭2,3) mcĕcrn`n hujuùz.


39/96

Ocvkp7

k) Bcr hnr krkjĻdnìn` kyrĻ kyrĻ bcskpjkik ykpkrskd

; ≭ x :? KrkjĻĕĻ nìn` A(x)„n ykzkokd ejurskd.

6,8 ∘ ;

6 ì ∘∐ ; 9 6 . MejkyĻsĻyjk skmcoc ndn`on n`tclrkj dkjĻr. 9 6,8 ∘ ; 9 6,88 ∘

6,88 ∘ ∘ ;8 ∘ ; 96 88∘ ;8 ŕ 8ŕ

a(x)

6,=

6,8

x6 ; 8 ? = 3 1 < 2 >


40/96

3≭x:< KrkjĻĕĻ nìnÀ(x)

9 9 6 +6,8 ∘ ;+ 6+ 6,8 ∘ 3

9 6,=+ 6,8 9 6,1 . KrkjĻĕĻ nìn` A(x)

9 9 6,2 + 6,89 6,2 +6,8 . >≭ x:∐ KrkjĻĕĻ nìn` A(x)

9 9 6,2 + 6,89 6,2 +6,8 9 ; .

Bcr krkjĻd nìn` bcskpjkik ykptĻd hu`jkrĻ ìnzcjni ìnznjc` lrkand hnzmc` nstc`c` VMA ejkokdtĻr.

A(x)

;

6,2

6,1

6 =


41/96

9 6,=+ 6,8,96,3^k mk

V( = : x≭ 2,3) 9 A(2,3) - A(=) 9 6,> - 6,= 9 6,3 ejur.

DEŐSJJS Ê ĐS@JSDJKX vc MKĐNJNIJKX

‚℡’ Cvrc`scj dúicsn`c knt hútú` ΰ cjcik`jkrĻ`Ļ nìcrc` hnr O ejkyĻ`Ļ vc hu℡ dúicsn`n`cjcik`jkrĻ`Ļ rccj cdsc`c ktkyk` hnr R rkstjk`tĻ mcĕnŕdc`n cjc kjkjĻi. Hu muruimk yn`c hucvrc`scj dúic`n` kjt dúicsn ejk` hnr H dúicsn tk`ĻijkykjĻi{ ΰ 7 R (ΰ )≭ x } , {ΰ 7ΰ H }

H ejkyĻ`k hkĕjĻŕkrtjĻ mkĕĻjĻi

≚ , ŕ . HurkmkVTR ≭ x,HY ………………….{ R≭ x } vc H ejkyĻ`Ļ` hnjcŕnd ejksĻjĻĕĻmĻr. Cĕcr x↘∐ {R ≭∐ } dcsn` ejkyk mc`d ejur vcA(∐ ,H)9; ejur.

A(x|H) @erikj hnr mkĕĻjĻi ae`dsnyeùù` hútú` ñzcjjndjcrn`c skbnptnr.Deŕujju yeĕu`jud ae`dsnyeù mk ky`Ļŕcdnjmc

≚ . Ñr`cd7

H ejkyĻ`ĻH ≚ R ≭ ;6} ŕcdjn`mc tk`ĻijkykjĻi vc A(x|H)„yn bcskpjkykjĻi.n) x≣;6 nìn`{R≭;6} ejkyĻ{R≭x} ejkyĻ`Ļ` hnr kjt dúicsnmnr. MejkyĻsĻyjkVR ≭ ;6,R ≭ x 9 VR ≭ ;6 9 A ;6 ejur. AxH V R ≭ x R ≭

;6V R ≭ ;6 9 ;

nn) x≭ ;6 nìn`{R≭ x} ejkyĻ{R≭ ;6} ejkyĻ`Ļ` kjt dúicsnmnr mejkyĻsĻyjkVTR≭ ;6, R ≭ xY9VTR≭ xY9ATxY

AH V R ≭ xA


42/96

A(x) A(x|H)

; ;

6,3

x x6 ;6 86 6 ;6

HĶX XK_JK@]N MCĐĶŐDC@Ķ@Ķ@ AE@D_Ķ̂E@JKXNIúbc`mnsjnd uylujkikjkrĻ`mk djksnd hnr prehjci4 hnr snstcin` lnrnŕn`c uylujk`k` lnrnŕ nìn`ìĻdĻŕĻ` bcskpjk`iksĻmĻr. Hñyjcsn hnr snstcin` lnrnŕn hnr rkstjk`tĻ mcĕnŕdc`n nsc ìĻdĻŕĻ mklc`cjjndjc hnr rkstjk`tĻ mcĕnŕdc`n ejkokdtĻr. Hu muruimk cĕcr lnrnŕ rkstjk`tĻ mcĕnŕdc`jcrn nìn`VMA vcyk pma nakmcjcrn mc hnjn`nyersk snstcin` ìĻdĻŕĻ`mkdn rkstjk`tĻ mcĕnŕdc`n nìn` VMA vcykpma bcskpjk`khnjnr.Hu muruiu hnr ñr`cdjc kìĻdjkykjĻi.Hnr mnrc`ì (X) cjcik`Ļ úzcrn`mc` kdk` kdĻi (N) húyúdjúĕú V njc nakmc cmnjc` indtkrmk hnr lúì

ejuŕturikdtkmĻr.

≚ Hurkmk V vcyk V(.) hkzc` mc V(N) nakmcsn njctk`Ļijk`k` vc ae`dsnye` ejkrkd kmjk`mĻrĻjk` húyúdjúdbcr N mcĕcrn nìn` ejuŕk` lúoú hcjnrjcr.Hu muruimk, cĕcr N hnr rkstjk`tĻ mcĕnŕdc`n ejkrkdtk`Ļijk`iĻ ŕ nsc, V9N8X ŕcdjn`mc tk`ĻijĻ húyúdjúdyc`n hnr rkstjk`tĻ mcĕnŕdc`n ejuŕturikdtkmĻr. E zkik`

ìñzújicsn lcrcdc` seru7Qcrnjc` hnr durkjĻ (ae`dsnye`) vc pma„na x(x) tk`Ļijk`k` R rkstjk`tĻ mcĕnŕdc`n nìn`, yc`n^9l(x) rkstjk`tĻ mcĕnŕdc`n` pma n `c ejkokdtĻr5

X

N


43/96

]cd hnr rkstjk`tĻ mcĕnŕdc`n`n` (ae`dsnye`u)7

R(6,;) krkjĻĕĻ`mk tk`ĻijĻ u`naeri hnr rkstjk`tĻ mcĕnŕdc`n nakmc cmnjsn`. Qc ^98R+? yc`nrkstjk`tĻ mcĕnŕdc`n nìn` Ay(y)95

≭ y 8R+ ? ≭ y} 9 R ≭ ;/8y∘ ?} ej o dtĻr. MejkyĻsĻyjkAy 9 Ay ∘?8 na mcsnyjc dej yjĻdj cjmc cmnjchnjnr. Hurkmk` bkrcdctjc,^„`n` pma„n nìn`7a y 9 mmyAy 9 mmyAy ∘?8 ejur.

9 ∘ ?8 ⇘ 8 9 9 ∘ ?

8 9 .

9 .;8 ⇘ 9 ∘?8 .

;8 .

Lc`cjjcic^9kR+h ŕcdjn`mc pma„n a x(x)ejk` súrcdjn hnr R rkstjk`tĻ mcĕnŕdc`nk 0 6 nìn`

^ ≭ y} 9 R+ h ≭ y} 9 R ≭ y ∘ h ⇘ h h

a x(x) a y(y)

;6,3

6 ; ? = 3


44/96

V R : y ∘h 9 V R ≭ y ∘ h Hnzni nìn` lcrcdjn mcĕnj kik meĕru

≣ ∘ 9 0 ∘ .Nvc NNcŕntjndjcrn`mc`

≭ ∘ + ≣ ∘ 9 ; ⇘

≣ ∘ 9 ; ∘ ≭ ∘ ⇘ 9 ;∘ 9 ≦ 6 ejur.Hnjcŕdc ae`dsnye`7 a(x) vc l(x) vcrnjsn` a(l(x)) nakmcsn ael(x) ejkrkd lñstcrnjnr. Qc hu`k hnjcŕdc ae`dsnye` kmĻvcrnjnr.

. Icscjk7 9 + ; 9 ? + ; 9 ? +; + ; 9 1? + ; 9 8? + ;.? .

Jn`ccr Ejikyk` Hnr ae`dsnye` ñr`cĕn7VMA„n Ax(x) ejkrkd tk`ĻijĻ súrcdjn hnr rkstjk`tĻ mcĕnŕdc`n nìn`̂ 9R 8 ejsu`.

≭ } 9 ≭ } 9 ∘ ≭ ≭ } Hu muruimk ≭ } 9 9 ∘ ∘ . . ĕ. y 0 6 nìn4̀

9 9 ;8 ∘ ∘;8 ∘


45/96

y l(x)9^

y+my

my

a x(x)

mx; mx8 mx? xmx; +x; x; x8- mx8 x8 x? mx?+x?

| | 9 : ≭ + + ∘ : ≭ + : ≭ +

⇘ + + 9

9 + + 9 9 9 ; 9 9 9 ; . ⇘ 9 + +

↘9 ≦6

y


46/96

Őcdjn`mc tk`Ļijk`k` dúicjcrn lñz ñ`ú`c kjkjĻi Bcr ndn muruimk mk pma tk`ĻiĻ`mk` V T CnY 9a x (xn).|mxn |ejkrkd ykzĻjĻr._eùì ejkrkd

.| | 9 .| x| . Qc hu cŕntjnĕn` bcr ndn tkrkaĻ`Ļ |my| njc hñjcrscd,

9 . 9 . .

Hu muruimk y9l(x) nakmcsn`n` dñdjcrn nìn` my/mxn9l„(x) kjkrkd

9 | | ŕcdj `mc lc`cj h r `ds y ` cjmc c xn9 xn(y) , l„(xn)≦ 6Hu cjmc cmnjc` nakmc l(x) nakmcsn`n` hnrìed dñdú ejmuĕu`mk` hu tkrz prehjcijcrn` ìñzúiú`údejkyjkŕtĻrk` hnr yñ`tcimnr. Cĕcr vcrnjc` hnr y nìn` ^9l(x) nakmcsn`n` dñdú yedsk a y(l)96ejkokdtĻr.Ñr`cd7

R7T6,;Y u`naeri mkĕĻjĻik skbnp ejsu`. ^98R+?ŕcdjn`mc tk`Ļijk`k` ae`dsnye` nìnà y(y)95l(x) 9 8x+? 8x + ? 9 y↘ x 9 (y-?)/8

l„(x)98

∘ ?8 8 9 ∘ ?8 8 ejur. ^9l(x) n` hcdjc`mnd mcĕcrn vc stk`mkrt skpiksĻ

9

9 9 ∘ ejmuĕuù hnjnyeru


47/96

8

8 st `m rt s i yĻ y z rs d + 8 ∘ ∘ ∘ 8

∘ jmuĕuù ĕrcì ŕt d.M j yĻsĻyj jur. Ñr`cd7 pma„n bcskpjk`kokd ae`dsnye` uylujkikjkrĻ`mk s ` túrú hnr ae`dsnye` djksnd hnruylujkikmĻr. Hurkmk R rkstjk`tĻ mcĕnŕdc`n`n`

;8 , ∘ 6 ĕ . ŕcdjn`mc u`naeri mkĕĻjmĻĕĻ dkhuj cmnjcocdtnr.Ìñzúi N7 (Djksnd ykdjkŕĻijk ìñzcrscd) ñòc hu ae`dsnyeù` ŕcdjn`n ìnzcjni.

y9sn`x

y

sn`-; (y) φ-sn`-; (y) x


48/96

9 ;8 ;;∘ + ;8 ;

; ∘ 9 ; ;

; ∘ 6 ; r jĻĕĻ `.

h) -; ≭ y≭ 6 nìn ̀Hc`zcr nŕjcijcr ykpĻjĻrsk seùìŕu bkjn kjĻr.

; ;; ∘ ;6 ĕ Ìñzúi NN7 (lc`cj ìñzúi) ^9sn`(x) nìn` pma 9 5

;8 , ∘ 6 ĕ Hurkmk l(x) ae`dsnyeù l(x) 9 sn`(x) ejkrkd tk`Ļijk`kokdtĻr. Hu muruimk^-sn`(x)96 mc`djcin`n` dñdjcrn nìn`.

k)

6 ∘

KyrĻok

oes . Lc`cj aeriúj uykrĻòk l„(x) nakmcsn`n` dñdjcrmcdn mcĕcrn bcskpjkìkjĻmĻr. Huù` nìn`4

9 oes; 9 oes

oes; 9 oes ∘ oes.oes s `.s ` ∘oes . ‚6’ ejmuĕuù hnjnyeruz tkhn. @et7

sn`(Ѹ )9 y ↘ Ѹ 9sn`-; (y)oes(sn`-; (y))9oes(Ѹ )9 ;∘


49/96

9 9 ; ∘ _e` ejkrkd

; ; ∘ + ∘ ; ∘ 9 ; ;;∘ 9 ; ;;∘ ≭ ≭ ; h) y : 6 ndc` mc hc`zcrŕcdnjmc nŕjcijcr tcdrkrjk`kokdtĻr.Ñr`cd7

y98x+; ejkrkd vcrnjinŕ ejsu`. a x(x) a y(y)

y98x+;

8/?;/?

x y

-8 6 ; -? 6 ?

+

y 9 - ? nìn` a x(-8) 9 6y 9 ? nìn` a x(;)98/?a y(y) 9 (;/8) (8/?) 9 ;/? a y(?) 9 ;/?


50/96

Ñr`cd7a x(x)↘@(6,;) yk`n ³96 vcϏ89; ejkrkd vcrnjnyer.

≭ m yc t `ĻijĻ jsu` jmuĕuù b tĻrj y r d ]k`ĻijĻ ejmuĕu krkjĻĕk lñrc ñòc lrkanĕn`nìnzcrscd yk`mkdnŕcdnj ejuŕur.Ìñzúic lcìcjni7V{ ^≭ y } 9 V{ R≭ y }↘ x 0 6 9 Ax(y)V{ ^≭ y } 9 V{ R≭ 6 } 9 ;/8 ejur. Lkusnkèjmuĕu`mk` mejkyĻ.Őnimn Ax(y) vc a(y) yn ìnzjni.

HĶJC ŐĶD MKĐNJNI QC Ê ĐS@JSD AE@D_Ķ̂E@JKXN

Hnr ejksĻjĻd uzkyĻ`mk hnrmc` ìed rkstjk`tĻ mcĕnŕdc`n tk`Ļijkikd iúidú`múr. Ñr`c ĕn`7{ R≭ x ,^ ≭ y } 9 { R≭ x } ∮ { ^ ≭ y } dúicsn ΰ ℡ ejkyjkrĻ`mk` ejuŕsu` vc R(ΰ)≭x,

^( ΰ)≭y ejsu`. Hu muruimk { R≭ x , ^ ≭ y } ejkyĻ njc tk`ĻijĻ èdtkjkr x„ vc y„ múzjcijcrn`mckjttk vcrnjc ŕ̀cdnjmcdn tkrkjĻ hñjlcyc mc`d lcjnr.H lñ i ĕ j l jŕ lú j jk kd ì ji ŕ K` kd ` l ` j i k h `jk

a x(y)

yAx(y) a(y)

;6,3

6,3

y y

h ` k` jk ( ∐ ∐) 9 6 jk kd


51/96

vc hu`mk` mejkyĻ Axy(-∐,-∐) 9 6 ejkokdtĻr.nn) Ñtc yk`mk` { R≭ ∐ } vc { ^ ≭ ∐ } dúicjcrn mc dcsn` ejkyjkrĻ nakmc cttndjcrn nìn`

{ R≭ x , ^ ≭ ∐ } 9 { R≭ x } vc{ R≭ ∐, ^ ≭ y } 9 { ^ ≭ y } ejkokĕĻ`mkÀxy(x,∐) 9 Ax(x)Axy(∐,y) 9 Ay(y) ejur.

nnn) Cĕcr Axy(x,y) súrcdjn vc túrcvn kjĻ`khnjc` hnr ae`dsnye` ejkrkd tk`ĻijkìĻŕsk hnjc ŕndejksĻjĻd ye ĕ u`jud ae`dsnyeù

, , ℀.

MejkyĻsĻyjk a xy(x,y) mx my 9VT x : R≭ x + mx , y : ^≭ y + myY ejkokĕĻ nìn` túi x vc ymcĕcrjcrn nìn` a xy(x,y)≣ 6 ejur.

nnnn) ûdkrĻmkdn cŕntjnĕn n`tclrkj kjkrkd nakmc cmcrscd

, , cŕntjnĕn cjmc cmnjnr. Hu muruimk Axy(x,y) ae`dsnyeù `clktna ejikyk` hnr ae`dsnyeù` n`tclrkjnejkrkd tk`Ļijk`mĻĕĻ`k lñrc kzkjk` dkrkdtcrmc ejkikz. ^k`n, ( x; , y; ) vc ( x 8 , y 8 ) ndn tk`cskyĻ ìnatn ejikd úzcrc x; ≭ x 8 , y; ≭ y 8 ŕkrtĻ mk skĕjk`ĻrskAxy(x; , y; )≭ Axy( x8 , y8 ) ejikjĻmĻr.KyrĻok hnr súrcdsnzjnd èdtksĻ`mk Axy(x,y) skĕmkdn vc ústtcdn mcĕcrjcrn kjĻr. ^k`n

, j i↘

↘ , .

]úi hu ñzcjjndjcrn ñzctjcrscd N) Axy(∐,∐)9; , Axy(x, -∐) 9 Axy(-∐,y) 9 Axy(-∐,-∐) 96

Axy(∐,y) 9 Ay(y) 4 Axy(x,∐) 9 Ax(x) NN) x; ≭ x8 , y; ≭ y8 nsc Axy(x; , y; )≭ Axy( x8 , y8 ) 9 j i↘↘ , . { x; : R ≭ x8 , y; : ^ ≭ y8} dúicsn`n` cjcik`jkrĻ`Ļ lñz ñ`ú`c kjkjĻi.

Axy(x8 ,y8) 9 {;}+{8}+{?}+{=}ejur. Hnz hu tepjkiĻ` bcr hnr

y„(x,y)

;y8

VT R ≭ ^ ≭ Y 9 A ( ) A ( ) A ( ) + A ( ) j


52/96

VTx; :R ≭ x8 , y; :^ ≭ y8 Y 9 A xy(x8 ,y8) - A xy(x8 ,y; ) - A xy(x; ,y8) + A xy(x; ,y; ) ejur. Ax(x) vcAy(y) ae`dsnye`jkrĻ hnr hnjcŕnd mkĕĻjĻi ae`dsnyeù`mk` cjmc cmnjnrsc hu`jkr

ikrgn`kj mk ĕ ĻjĻi ae`dsnye`jkrĻ ejkrkd kmjk`mĻrĻjĻrjkr. ^k`n7

,∐ , ∐, ,

Ikrgn`kj yeĕu`jud ae`dsnye`jkrĻ nsc

ejkokdjkrĻ ì `yud rĻm dcŕntjndjcrn` túrcvn kjĻ`mĻĕĻ`mk,

,∐ , ∐, , Hurkmk` bkrcdctjc hkzĻ seùìjkr7 N) a (x,y)≣ 6 , túi x vc y mcĕcrjcrn nìn`

, ; NNN) a xy(x , y) hnr ejksĻjĻĕĻ nakmc cticzdc`7a xy(x , y)mymx 9 VT x : R≭ x+mx , y : ^ ≭ y+my Y hnr ejksĻjĻd nakmc cmcr.HKĐNI_NW XK_JK@]N MCĐĶŐDC@JCXĶ

R vc ^ lnhn ndn rkstjk`tĻ mcĕnŕdc`n, bcr x,y mcĕcr ndnjnsn nìn` { R≭x } vc{ ^≭y } ejkyjkrĻhnrhnrn`mc` hkĕĻisĻz nscjcr dc`mnjcrn mc hkĕĻisĻz ejkrkd kmjk`mĻrĻjĻrjkr. Mkbk ñòcdntk`Ļijkrmk` K vc H lnhn ndn ejky VTKHY 9 VTKYVTHYŕkrtĻ skĕjkìksĻ bkjn`mc hnrhnrjcrn`mc`hkĕĻisĻz ejkrkd tk`ĻijkìĻŕjkrmĻ. Hurkmk

KH 9 { R≭ }∮{ ^≭ } k`ĻiĻ

jk kd Ļ


53/96

, ejkokdtĻr.Hu seùìjkr ĻŕĻĕĻ`mk túrcv kjĻ`krkd

cjmc cm j rd hur m ` HkĕĻisĻz ejkyjkr nìn`ŕkrtjĻ ejksĻjĻd yeĕu`jud ae`dsnye`jkrĻ`Ļ` ikrgn`kj ejksĻjĻdyeĕu`jud ae`dsnye`jkrĻ`Ļ vcrcocĕn seùouù cjmc cmcrnz. KyrĻok mkbk ñòc cjmc cttnĕninz hnrseùìtk`…7VTx; :R ≭ x8 , y; :^ ≭ y8 Y 9 A xy(x8 ,y8) - A xy(x8 ,y; ) - A xy(x; ,y8) + A xy(x; ,y; )VTx; :R ≭ x8 , y; :^ ≭ y8 Y 9 A x(x8).A y(y8) - A x(x8).A y(y; ) - A x(x; ).A y(y8) + A x(x; ).A y(y; )

9 A x(x8) T A y(y8) - A y(y; ) Y - A x(x; ) T A y(y8) - A y(y; ) Y9 T A x(x8)- A x(x; ) Y T A y(y8) - A y(y; ) Y

VTx; :R ≭ x8 , y; :^ ≭ y8 Y9 VTx; :R ≭ x8Y.VT y; :^ ≭ y8 Y seùou cjmc cmnjnr.Ñr`cd7

KŕkĕĻmk vcrnjc` a xy ae`dsnyeù nìn` R vc ^ rkstjk`tĻ mcĕnŕdc`jcrn`n` hkĕĻisĻz ejmuĕuùlñstcrn`nz.

, ; 8 ; 8 ; 8 ejkokĕĻ nìn` R vc ^ hkĕĻisĻz ejkokdjkrmĻr.Ñr`cd7

, , 6 ; 6 ;6 , m ĕcr muruij rŕcdjn`mc tk`ĻijkìĻŕ ejsu`.Ocvkp7

N) K95

, ; 90

E kik` ; 6


54/96

E kik` ;8 , 6 ;6 ,m ĕcr muruij r

Hc`zcr ŕcd jmc + ;8 , 6 ;6 ,m ĕcr muruij r NNN)Axy(x,y) 9 5Axy(x,y) 9 VT R≭x,^≭y Y tk`ĻiĻ ykpĻjĻrsk sñz de`usu ejksĻjĻĕĻ bcskpjkikd nìn` akrdjĻ muruijkrĻlñz ú`ú`c kjikd lcrcdnr. Hu`k lñrc ,

k) x≣ ; vc y≣ ; ejiksĻ muruiu`mk…4

, ;

h) 6 : x ≭ ; , y ≣ ; ejiksĻ muruiu`mk…4

9 + 9 ‸8 ‸ ‸ ‸

;8 ‸ ;8 , ;8 . o) 6 : y≭ ; , x ≣ ; ejiksĻ muruiu`mk…4

y„

(x,y)

;

x„;

y„

(x,y)

;

x„;

y

y


55/96

9 +

9 + 8 ejkrkd hujuùr.

:6 ,y ` `bcr mcĕcr vcy 6 , x `bcr mcĕcr ejiksĻ muruiu`mk…4Axy(x,y) 9 6 ejur.

a) VT R+ ≭̂ ; Y 95 Ķakmcsn`n` `c k`jkik lcjmnĕn`c hkdkjĻi.

R+^9; me ĕrusuù` ústú R+^0; hñjlcsn`n hcjnrtnyer

R+^9; me ĕrusu

R+^9; me ĕrusuù` kjtĻ R+^:; hñjlcsn`nhcjnrtnyer

; 9 ‸. 9 + ‸ ; ∘ ;

y„

;

x„;

y„

;(x,y)

; x„

y„

;

x„;

( x,y)

ĶDĶXK JK@]N MCĐĶŐDC@Ķ@Ķ@ ]CD AE@DĶ̂E@S T W9l(R,^)Y


56/96

ĶDĶ XK_JK@]N MCĐĶŐDC@Ķ@Ķ@ ]CD AE@D_ĶE@S T W9l(R, )Y

Ìe ĕu iúbc`mnsjnd uylujkikjkrĻ`mk hnr W rkstjk`tĻ mcĕnŕdc`n hcjjn hnr krkjĻdtk R vc ^lnhn ndn (bkttk mkbk ìed) rkstjk`tĻ mcĕnŕdc`n`c hkĕjĻ ejkrkd nakmc cmnjchnjnr. Icscjk7

- Hnr duvvctjc`mnrnon lnrnŕn`mcdn W sn`ykjn, R lnhn hnr nŕkrct vc hu`mk` hkĕĻisĻz ejkrkdtk`Ļijk`k` ^ lúrújtú on`sn`mc` W9R+^ ŕcdjn`mc tk`Ļijk`khnjnr.

- Ķdn nŕkrctn ìkrpk` hnr snstcimc, R hu ìkrpĻoĻ`Ļ` hnr lnrnŕn`mcdn, ^ mc hu ìkrpĻoĻ`Ļ` mnĕcrlnrnŕn`mcdn sn`ykj ejikd úzcrc W9R ŕ̂cdjn`mc tk`ĻijĻ ìĻdĻŕ sn`ykjn W„`n` pman `ksĻjtk`Ļijk`khnjnr5

- R vc ^ meĕrujtujkrĻ`mk hkĕĻisĻz bkrcdct cmc` hnr pkrìkoĻd nìn` R vc ^ njc hucdsc`jcrmcdn bkrcdctn nakmc cmc` rkstjk`tĻ mcĕnŕdc`jcrn tk`Ļijk`Ļrsk W9(R8+^ 8);/8 ŕcdjn`mctk`ĻijĻ vc tepjki ycr mcĕnŕtnricyn nakmc cmc` W rkstjk`tĻ mcĕnŕdc`n`n` pman `ksĻjbcskpjk`khnjnr5

Lc`cj ejkrkd W9l(R,^) ŕcdjn`mc tk`Ļijk`kokĕĻiĻz hu túr prehjcijcrn` ìñzúiú mkbkñòc ìñzmúĕúiúz ^9l(R) ŕcdjn`mc tk`ĻijĻ prehjcijcrn ìñzúiú`mc` akrdjĻ mcĕnjmnr. Hurkmkñyjc hnr Oz ìñzúi dúicsn hujuìkjĻmĻr dn huù` nìn`

{ W≭ z },{ (R,^) ∂ Oz } ejkyjkrĻ ky`Ļ muruiu nakmc cticjnmnr. Hu muruimk

,. ∂ ejkokdtĻr. Hu túr prehjcijcrn ìñzúiú ^ 9 l(R) ŕcdjn`mc tk`Ļijk`k` prehjcijcrn` ìñzúiú`mc`mkbk dkrikŕĻdtĻr. Kòkdn) ykrmĻioĻ cdsc` , nn) dkrkdtcrnstnd ae`dsnye`jkrĻ` dujjk`ĻiĻ njc huprehjcijcr ìñzújchnjnrjcr. Ñòc hkzĻ ñr`cdjcrn nòcjcycjni.Ñr`cd7

W9R^ ŕcdjn`mc R vc ^ rkstjk`tĻ mcĕnŕdc`jcrn on`sn`mc` tk`ĻijĻ W rkstjk`tĻ mcĕnŕdc`nnìn` ìñzúi dúicsn O z yn cjmc cmn`.A z(z) vc a z(z) 95

y9z/x vcyk x9z/y

y9z/x vcyk x9z/y

{ W≭ z } njc nakmc cmnjc` ìñzúi dúicsn

y

x

∐ ∐ j


57/96

, ∘ ∘∐, ∐, ∘ , ejur. Hurkmk` bkrcdctjc,

9 95 serusuù `ksĻjocv j rĻz5 ûdkrĻmk cjmc cmnjc` Az(z) cŕntjnĕn nìn`,, ≚ ,hcj rs z `tclr j ` t `Ļij yĻ y

, ∘ ∘∐, ∐, ∘ , . 9 ∘ ∘∐ + ∐ ∘ 9 ∘ 90 9 ; ∘ ; 9 ; , ; ,

; , ejkrkd hujuùr.

Ñzcj hnr murui ejkrkd R vc ^ hkĕĻisĻz vc hc`zcr ejikd úzcrc

9 ≚ + ejkrkd tk`ĻijkìĻŕ ejsu`.Humuruim, ; 0 . ; + ; + 90 9 ;∘j i .

y


58/96

Ìñzúi dúicsn y9z-x c ĕrnsn`n` seju`mk dkjk`hñjlc ejkokĕĻ`k lñrc

9 90 9 ∘ ∘ ∘∐ ejur.

Hur m , ≚ ejkrkdt `Ļij `mĻĕĻ`Ļ d huj cm y Hurkmk Az(z) ae`dsnyeùù cjmc cticd nìn` , mcĕcr ` bcs j i d lcrcd r. z i `

, 0

∘ , seùoucjmc cm j r. Ìe ĕu muruimk R vc ^ hnrhnrn`mc` hkĕĻisĻz ndn rkstjk`tĻ mcĕnŕdc`n ejkrkd tk`ĻijĻ

ejkhnjnrjcr. E zkik`

, ŕcdj `mc t `ĻijĻ hu m y`Ļ z jc ae`dsnye`jkrĻ`Ļ` de`vejúsyeù ejkrkd mk hnjn`nr.Lñrújmúĕú lnhn hu tnp prehjcijcrn tk`Ļijkrmk` ykrkrjk`krkd ìñzicd bcr zkik` nìn`

yy9z-x vcykx9z-y

x

a x(x) a y(y)


59/96

;

;/8x y

; 8

h) a x(x) ae`dsnyeùù` y cdsc`n`c lñrc snictrnsn`n kjĻp lrkanĕn` pkrkictrcsn`n

mcĕnŕcjni vc lrkanĕn z dkmkr ñtcjcycjni. Hnz huù`jk a x(z-y) yn cjmc cmcrnz.a x(z-y)

;

y

z-; z

o) z : 6 nìn` a x(z-y) njc a y(y) ìkdĻŕk` dĻsiĻ`k hkdkjĻi

a x(z-y) ;

;/8 a y(y)

z-; z 8

m) 6 : z : ; nìn`

a x(z-y) ;


60/96

9 ;.;8 ;8 ;8

a) 8 : z : ? nìn`

;.;8 ;8 ? ∘ 8 .

l) z 0 ? a z(z) ìkdĻŕk` kjk` ejikmĻĕĻ nìn` 6 ejur.6 4…………………z:6

6,3z 4………………....6:z:;

a z(z) 9 6,3 4…………………;:z:8 cjmc cmnjnr.

(?-z)/8 4……………8:z:?

6 4 ………………....z0?

a z(z) `n` ejksĻjĻd yeĕu`jud ae`dsnyeùù ìnzcrscd

VT z:8 Y 90 ?/=VT z:? Y 90;

cjmc cmcrnz.

; a x(z-y)

;/8 a y(y)

z-; z 8

; a x(z-y)a y(y)

;/8

z-; 8 z

a z(z)

¹

Ķ D Ķ XK_JK@]N MCĐĶŐ DC@ Ķ @ Ķ @ Ķ D Ķ AE@D_ Ķ Ê@S7


61/96

Q9l(x,y) vc \9b(x,y) tnpn`mcdn prehjcijcrmnr

R + Q

Hurkmk { kn } mcĕcrjcrndkzk`oĻ nakmc ctsn`jcr.

^ + \+

^udkrĻmkdnŕcdnj njc hcjnrjn hnr bkhcrjcŕic snstcin ( stcrye tcicj-hk`tjĻ snstci eju ŕuiulnhn ) nakmc cmnjicdtcmnr. Qcyk { kn } mcĕcrjcrn ; ejkrkd scìnjmndjcrn`mc, hcjnrjn hnr bkhcrjcŕicsnstcin`mc Q vc \ sn`ykjjcrn ( sĻrksĻyjk R vc ^ sn`ykjjcrn`n` akrdĻ vc tepjkijkrĻ ) lñ`mcrnjc`sn`ykjn tcspnt cticdtc dujjk`ĻjĻr.

DĻskok Q9l(x,y) vc \9b(x,y) ae`dsnye`jkrĻ tk`ĻijĻ ndc`, a xy(x,y) hnrjcŕnd ejksĻjĻdyeĕu`jud ae`dsnye`jkrĻ njc túrcvn kjĻ`khnjc` l(x,y) vc b(x,y) ae`dsnye`jkrĻ nìn` Avw(v,w) vca vw(v,w)95

Hu túr prehjcijcrn` ìñzúijcrn`mc

, , ∃,

,

. 9 ∃,, t `Ļij i sĻ`m ` ym j ` v9l(x,y) vc w9b(x,y) ejikd úzcrc tk`Ļijk`k`x9ϓ(v,w) vc y9ς (v,w) ae`dsnye`jkrĻ on`sn`mc`

≚ ,℀ , ⋥ ,.

⋻ ⋠℀ ⋥ ℀ i tr s `c

l(x,y)9vb(x,y)9w mñ`úŕúi iktrnsn`n` gkoehyc` iktrnsn,

⋥

+

+

Ìñzúi7 R9ϓ( )98 3


62/96

R9ϓ(v,w)98v-3w vc^9 ς (v,w)9-v+?w ejkokdtĻr. 8, ∘3, ∘; 9 ? ej r d bcs j `Ļr. Hu muruimk

9 8 ∘3∘; ? 90 9 8∜? ∘ ∘3∜∘; 9 ; , 9 ;

8∘;

8 +

, 9 ; ;8 ∘;8 8 ∘ 3 ∘ ? 90 , 9 ;8 ∘;83 ∘ 81 + ?= ejur. ^k`n hu mñ`úŕúi dercjksyeù ejikyk` ndn rkstjk`tĻ mcĕnŕdc`n`n, dercjksyeù ejk` yc`n

ndn rkstjk`tĻ mcĕnŕdc`n`c mñ`úŕtúriú ŕtúr.

Őu k`k dkmkr lñrmúĕúiúz aeriújjcr Q9l(x,y) vc \9b(x,y) ae`dsnye`jkrĻ hnrchnrae`dsnye` ejmudjkrĻ zkik` ykzĻjkhnjnrjcr. Kòkd (x; ,y; ), (x8,y8), (x?,y?)…………… (x̀,y`)lnhn `-tk`c Q9l(xn,yn) , \9b(x n ,yn) dñdú`ú` ejiksĻ muruiu`mk ìñzúi4

, 9 , , Őcdjn`mc cjmc cmnjnr. Hurkmkdn ‚n’ n`mnsn gkoehyc` iktrnsn`n`n. dñdtcmcĕcrjc`mnrnjcocĕn`n nakmc cmcr.Ñr`cd7

, ℀, tk` , 0 6

tk` , 6 ejkrkdvcr js `.

Cĕcr R vc ^ hnrhnrn`mc` hkĕĻisĻz vc hc`zcrŕcdnjmc mcĕnŕnyersk vc

, ∘ ejkrkdvcr ji ŕsc a vw(v,w)95Ìñzúi7

Ñ

^9Qsn`(w)9ς (v,w) mĻr.

∘


63/96

,

, ∘

a vw(v,w)9 v.a xyTv.oes(w) ,v.sn`(w)Y

, ;8 ∘ ;8 , 0 6,∘8 86, m ĕcr muruij r a vw(v,w)9a v(v).a w(w) ejkokdtĻr.

^k`n Q vc \ hnrhnrn`mc` hkĕĻisĻz rkstjk`tĻ mcĕnŕdc`jcrnmnr. Hurkmk Q Xkyjcnlb, \ mk 8φ krkjĻĕĻ`mk u`naeri mkĕĻjiĻŕ ndn rkstjk`tĻ mcĕnŕdc`nmnr.

DKXKD]CXĶ_] ĶD AE@D_Ķ̂E@7

Hnr R rkstjk`tĻ mcĕnŕdc`n`c knt dkrkdtcrnstnd ae`dsnye`

Őcdjn`mc tk`ĻijĻmĻr. Hu tk`Ļi c gwx tcrnin`mcdn dujjk`Ļjikyk` (-) nŕkrctn mĻŕĻ`mk

a x(x)ae`dsnyeù Aeurncr trk`saeriu ejkrkd mk kmjk`mĻrĻjkhnjnr.Dkrkdtcrnstnd ae`dsnye` hkĕĻisĻz rkstjk`tĻ mcĕnŕdc`jcrn`n` tepjkiĻ`Ļ` bcskpjk`mĻĕĻ

prehjcijcrmc sĻdìk dujjk`ĻjĻr. Mkbk ñòc lñriúŕ ejmuĕuiuz ndn rkstjk`tĻ mcĕnŕdc`n`nàe`dsnyeù ŕcdjn`mcdn uylujkikjkrĻ` hnr ñr`cĕn W9R; +R8 ŕcdjn`mc tk`ĻijĻ hnr tepjkiae`dsnyeù nìn` lcrcdjn ejk` nŕjcijcr ykpĻjkokd ejursk,

∜ ∘ Őcdjn`mc tepjki rkstjk`tĻ mcĕnŕdc`n ejksĻjĻd yeĕu`jud ae`dsnyeù vc on`sn`mc` nakmc cmnjnrjcr. ûdkrĻmkdn lnhn tk`ĻijkìĻŕ n`tclrkj nŕjcin`c de`vejúsye` kmĻ

vcrnjnr vc k`kjntnd ejkrkd bcskpjkikd bcr zkik` prktnd ejikykhnjnr.Kòkd yudkrĻmkdn lnhn tk`ĻijĻ dkrkdtcrnstnd ae`dsnye`jkr on`sn`mc` hu de`vejúsye`

nŕjcin hksnt hnr ìkrpik nŕjcin`c mñ`úŕtúrújcocdtnr. ^k`n7


64/96

; hu

` .

mcrcocmc` i ic`t vcr r.

`9; nìn` n`tclrkjn` se`uou4

; 9 . ejur. HĶJC ŐĶD DKXKD]CXĶ_] ĶD AE@D_Ķ̂E@

, ,…………. . .. 9 . ... 9

9 ;8 .. . .. ∜

Cŕntjndjcrn yudkrĻmk ykpĻjk` tk`Ļijkik`Ļ` hnjcŕnd ejksĻjĻd mkĕĻjĻi ae`dsnye`u nìn` ìedheyutju lc`nŕjctnjinŕ bkjnmnr.

HĶXJCŐĶD IEIC@]JCX W9l(x,y) vc

, . 9 Cĕcr R vc ^ hnrhnrn`mc` hkĕĻisĻzsk z9l(x) vc w9b(^) mc hkĕĻisĻzmĻr.CT l(x) b(y) Y 9 CT l(x) YCT b(^) Y ejur.

DEQKX^K@_


65/96

R vc ^ ndn rkstjk`tĻ mcĕnŕdc`n ejikd úzcrc

, ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ejkrkd tk`Ļijk`Ļr.

, ji jĻmĻr. , hur m d j ŕd d rcj sy ` d ts yĻs -;≭ ≭; krksĻ`mkmĻr.Cĕcr R vc ^ hkĕĻisĻz nscOev(x,y) 9CTxyY-CTxY.CTyY

9 CTxY.CTyY- CTxY.CTyY9 6 vc 9 6 ejur.

R vc ^ hkĕĻisĻz nsc Oev(x,y)96 , 9 6 ejur. ^k`n dercjksye` yedtur.↘ Cĕcr Oev(x,y)96 nsc bcr zkik` R vc ^ hkĕĻisĻzmĻr. Kik↘Cĕcr 9 6 nsc R vc ̂ bcr zkik` hkĕĻisĻz ejikz.

Ñr`cd7 W9kR+h^ nsc (Ϗz)8 ‒yn,Ϗx , Ϗy vc on`sn`mc` nakmc cmn`nz.Ocvkp7 CTzY9CTkx+hyY9kCTxY+hCTyY

9k³ x + h³ y Qc Ϗ8 9CT( z-³ z )8Y 9 CT{(kx+hy)-(k³ x+h³ y)}8Y

9CT{ k(x-³ x)+h(y-³ y)}8Y9k8CT( x-³ x)8Y + h8CT(y-³ y)8Y + 8khOev(xy)9 k8(Ϗx)8 + h8(Ϗy)8 + 8kh (Ϗx Ϗy)

Ñzcj ejkrkd R vc ^ hkĕĻisĻz nsck 9 h 9 ; nsc W9R+^ 90 (Ϗ )8 9 (Ϗ )8 + (Ϗ )8 ejur

]cerci7 R; , R8, R?………………R@ ejksĻjĻd mkĕĻjĻi ae`dsnye`jkrĻ a ; (x; ), a 8(x8),a?(x?) à(x`) ejkrkd tk`Ļijk`iĻŕ `’ tk`c hnrhnrn`mc` hkĕĻisĻz rkstjk`tĻ mcĕnŕdc`n


66/96

a ?(x?),…………… a (x ) ejkrkd tk Ļijk iĻŕ ‚ tk c hnrhnrn mc hkĕĻisĻz rkstjk tĻ mcĕnŕdc ntk`Ļijk`iĻ ŕ ejsu`jkr. Hu`jkr nìn`4

6 ejikdzcrc ≚ + + + t `Ļij `sĻ` Qcrnjc` hnrγ 06 mcĕcrn nìn` yctcrjn se`suz ‚`’ skyĻ nìn` (Ϗd ) mcĕcrn (Ϗd ) 0 γ _ ̀ d9;,8,……….` ŕkrtĻ`Ļ skĕjkr vc hu muruimk ≚ ……………..+ / @erikjnzc tepjkiĻ `erikj mkĕĻjĻi VMA„sn`c ykdĻ`skrj i↘ ………………..̀erikjm ĕĻjĻi ji dt tmĻr. ]cerci7 6 ; 9 6,;8……….. ejikd úzcrc tk`ĻijĻhkĕĻisĻz vc hc`zcr mkĕĻjĻiĻ ejk` R; , R8, R?………………R@ rkstjk`tĻ mcĕnŕdc`jcrn vcrnjsn` ≚ ; tk`Ļijk`Ļrsk

j i↘ 6,; ejkokdtĻr.


67/96

;) Hnr zkr ktĻjiksĻ mc`cyn`mc ejuŕkhnjcocd seùìjkr n9;,8,?,=,3,1 ejsu`. hu mc`cyjc njnŕdnjnejkrkd R9n8 rkstjk`tĻ mcĕnŕdc`n tk`Ļijk`Ļyer.

k) Hu rkstjk`tĻ mcĕnŕdc`n`c njnŕdn` ejksĻjĻd mkĕĻjĻi ae`dsnyeùù (VMA) ìnzn`nz.h) A(;8) mcĕcrn`n hujuùz.o) V(3:R≭


68/96

m)

c)

a)


69/96

l)

b)

?) Úrctnjc` hnr mnrcòn` úzcrn`mc lñstcrnjc` bktk mcĕcrn`n` úzcrn`mc ejiksĻ nbtnikjn p96.68 mnr.Hu muruimk úrctni hk`mĻ`mk` rkslcjc ;6 tk`c mnrc`ì scìtnĕninzn múŕú`cjni.

R(d)7 d tk`c mnrcòn` hezud ejiksĻ rkstjk`tĻ mcĕnŕdc`n ejsu` vc hu rkstjk`tĻ mcĕnŕdc`n4

≭ ≭ ≭

mcĕcrjcrn`n kjsĻ`.k) VTx9gY yn g9;,8,?,= nìn` bcskpjkyĻ`.h) EjksĻjĻd mkĕĻjĻi ae`dsnyeù (VMA) A(x)„n ìnzn`.


70/96

=) EjksĻjĻd yeĕu`jud ae`dsnyeù (pma) kŕkĕĻmk vcrnjc` ae`dsnye` nìn`4

k) d ‒ yĻ hujuùz.h) CTxY ‒ n hujuùz.o) Qkryk`sĻ hujuùz. ( 8)

m) EjksĻjĻd mkĕejĻi ae`dsnyeùù (VMA) hujuùz.c) V(6:R≭;) ejksĻjĻĕĻ`Ļ hujuùz.

3) KŕkĕĻmk VMA n vcrnjc` R rkstjk`tĻ mcĕnŕdc`n nìn`,

k) EjksĻjĻd yeĕu`jud ae`dsnyeùù hujuùz.

h)

CTxY n hujuùz.o) Qkryk`sĻ ( 8) hujuùz.m) V(6:R≭;) ejksĻjĻĕĻ`Ļ hujuùz.


71/96


72/96

c) V(6:x≭;)9A(;)-A(6)9

3)k)

h) o) m) V(6:x≭;)9A(;)-A(6)9 ;-6.396.3Ìñzúijú _ct-8

;)


73/96

R(d)7 d tk`c mnrc`on` hezud ejiksĻ

9

:≭

:≭

:≭

9

???888;;;66

)(

d d d d

d R

k) VTx9gY yn bcskpjkyĻ`, g96,;,8,?h) A(x) ‒n ìnzn`.o) a(x)„ n ìnzn`.

?)

^udkrĻmk R rkstjk`tĻ mcĕnŕdc`n`n` pma n vcrnjinŕtnr. Hu`k lñrc4

k) d95h) VTR98Y95


74/96

h)


75/96

h)

o)

?)k)

?=8=6

81.6

3.66?

868

88;.6

)8(3.6)8(8

)?(;.6;)(

8

?

?

6

8

8

8

?

6

+

∘∘+9

++∘

++9

∘+++++⇘9∤ ∤ ∤ ∤∤∐

∐∘

∘

∘ ∘

d d

dx x xd

xd xd x x a ν ν

l)


76/96

l)

?83.?

?8.6;3.;=3.6

?8.6(63.6=?2663.6?.6

83.6

6?

8;.6

86

?63.6

?;.6

)8(3.6;.6)8(63.6)?(;.6)(YT

88

?

?

?

6

8

?

6

8

8

9∘9+∘9

+∘+

+∘

∘+∘9

9++

∘

++

∘99

∘+++++99 ∤ ∤ ∤ ∤ ∤∐

∐∘

∘

∘ ∘

x x x x x

x x

mx x x xmxmx x xmx x x x xa R C ν ν

b)

?8.;;

?8.62.?

8>.6)Y?

;1=(6T63.6>.6

83.6

6?

?;.6

86

?8

=63.6

?;.6

)8(3.6;.6)8(63.6)?(;.6)(YT

8??=

8

6

8

8

8

8?

6

88?

?

888

9+9

++∘∘+9

9++

∘

++

∘99

∘+++++99 ∤ ∤∤∤∤∘∘

∐

∐∘

x x x x x

x x

xm x x x x x xmx x a x R C ν

>6?3183.;6

?8.;;

?83.?

?8.;; 88 9∘9

∘9Ϗ

n)

g) A(;)96 ?


77/96

g) A(;)96.?

=)

k) T Y8;;8;8;

68;;)( 8 / 68 / 8 /

6

8 / 9⇘9⇘9∘∘⇘9∐

∘⇘9⇘9 ∘∐∘∘∐

∐∘

∐

∘

∤ ∤ d d ccd dcmxdcmx x a x x

h) 8 / 8 / 8 / 8 / 8 / )(8;)()( t t x

t

x

t

ct R Vcct

ccmx x a t R V ∘∘∐∘∘∐

∘

∐

9≣⇘+9∐

∘999≣ ∤∤

o)

V(R≣t+s | R≣t)9t)V(R

t)Rs+tV(R≣

≣≣

V(R≣t+s , R≣t)9 V(R≣t+s)⇘ V(R≣t+s | R≣t)9 8 / 8 / 8 / )(

t)V(R s)tV(R s

t

st

cc

c ∘∘

+∘

99≣

+≣

m) Bcpsn Cŕnt (c-3)

Ìñzúijú _ct-?

c) V(^9?)95


78/96

c) V( 9?)95a) ax(R)95, ay(y)95l) CTRY95, CT^Y95b) 55, 88 99 y x Ϗ Ϗ

Ocvkp-;71 tk`c 6„mk` akrdjĻ ejksĻjĻĕĻ ejk` murui vkr. Hu muruijkr4 (;,;), (;,?), (8,;), (8,?), (=,;) vc (=,?).

;9∤∤ mxmy axy ejikjĻ. Hurkmkdn ejkyjkr nìn` n`tclrkj tepjkik mñ`úŕcocĕn`mc`4

∛∛ 9 ;)( xy axy ejikjĻ.

k) oT;.;+;.?+8.;+8.?+=.;+=.?Y9; 90 o9;/82h) ^:R ejk` ? ñr`cd èdtksĻ vkr.

V(^:R)9 { }( ) { }( ) { }( ) ( )82;2?.=;.=;.8

82;?,=;,=;,8 9++9++ VVV

o) ^0R ejk` 8 ñr`cd èdtksĻ vkr.

V(^0R)9 { }( ) { }( ) ( ) 82>

?.8?.;82;

?,8?,; 9+9+ VV m) ^9R ejk` tcd ñr`cd èdtksĻ vkr.

V(^9R)9 { }( ) ( )82;;.;

82;;,; 99V

c) y9? ejk` ? ñr`cd èdtksĻ vkr.

V(^9?)9 { }( ) { }( ) { }( )828;

82;8

821

82??,=?,8?,; 9++9++ VVV

a)

9

9

9

9

mnlcr x x x

x ax=8;

,,,,

6< / =< / 8< / ;

)( 99

9

mnlcr y y

y ay ?;

,,,

6= / ?= / ;

)(

Ñr`cĕn`4

R98 nsc { }( ) { }( ) 822?,8;,8)8( 9+9 VV ax l) CTRY9? CT^Y93/8

b)23,;6 88 99 yx Ϗ Ϗ

Ocvkp 87


79/96

Ocv p 8

∛∛∘9 9

99 3

3

;6

;6

?? 6),(YT x y

y x axy xy R^ C

?) R vc ^ { }?,8,; mcĕcrjcrn`n kjk` múzlú` yeĕu`jud ae`dsnye`u`k skbnp rkstjk`tĻ mcĕnŕdc`jcrnejsu`. Hu`k lñrc4

k) Q98R+8^ 90 av(v)95h) \9R-^ 90 aw(w)95

Ocvkp ?7Hu prehjcic ndn ìñzúi ŕcdjnyjc ykdjkŕĻjkhnjnr.

k-;)R vc ^ múzlú` mkĕĻjĻik skbnp ejmudjkrĻ`mk(yk`n bcr ? mcĕcrn ejik ejksĻjĻdjkrĻ cŕntejmuĕu`mk v rkstjk`tĻ mcĕnŕdc`n`mc hu >muruiu kjik ejksĻjĻĕĻ cŕnt yk`n ;/> ejkokdtĻr.)

Hu`k lñrc 4

Ñr`cĕn`4= mcĕcrn ; dcrc kjĻ`khnjmnĕn`mc` ;/>2 mcĕcrn ? dcrc kjĻ`khnjmnĕn`mc` ?/>


80/96

Ñ`oc W9R+^ yn hujkjĻi

N) W:8

az(z)96, z:8

NN) z98 hu muruimk skmcoc z-; vc ; úst ústc lcjcocd az(z)9>;

?;*

?; 9 , z98

z9? z-; vc 8, z-8 vc ; úst ústc lcjcocd az(z)9>8

?;*

?;

?;*

?; 9+ , z9?

z9= z-; vc ?, z-8 vc 8, z-? vc ; úst ústc lcjcocd az(z)9>?

>;

>;

>; 9++ , z9=

hc`zcr ŕcdnjmc, az(z)9>8 ,z93 vc az(z)9

>; ,z91


81/96

h ŕĻddĻ hc`zcrŕcdnjmc ìñzújúr.

=) R vc ^ rkslcjc mcĕnŕdc`jcrn nìn`4

T Ymnlcr x x

x ax 8,6,,68;)( ∂∘9 vc T Ymnlcr y y y ay ;,6,,688)(

∂ ∘9 nsc,

\9R+^ njc vcrnjc` \ rkstjk`tĻ mc ĕnŕdc`n nìn` aw(w) yn huju`uz.

Ocvkp =7De`vejúsye` ejkrkd ocvkp kŕkĕĻmkdn lnhn huju`ur.

≭≭∘+∘

≭≭∘

≭≭+∘

9

mnlcr

wwww

ww

wwww

w aw

,6

?8,1;

8?

8>

8>

8;,8;

1<

;6,1;

8?8

)(?8

?8


82/96

3) R T6,8Y krkjĻĕĻ`mk ^ nsc T6,8Y krkjĻĕĻ`mk S`naeri mkĕĻjĻik skbnp rkstjk`tĻ mcĕnŕdc`jcrnejsu`. Hu`k lñrc4

yudkrĻmkdn snstcimc d mcĕcrn`c skbnp mnrc`ìjcr dkrkrjĻjĻd tcstn`mc` lcìnrnjnyerjkr vc snstciìĻdĻŕĻ`mk W03 ejursk hezud, W≭3 ejursk skĕjki ejmudjkrĻ`k dkrkr vcrnjnyer. d lnrnŕn sĻokdjĻdtejcrk`s mcĕcrn ejup d98 ejkrkd hnjn`nyer, snstcimc huù 366O mc tcst cmcrcd ìĻdĻŕk 8d ejkokdŕcdnjmc vcrnjicsn tkskrjkìĻŕ. Kòkd R vc ^ lúrújtújcrn yúzú`mc` snstci ìĻdĻŕĻ W98(d+R+^)ejuyer. Hu`k lñrc snstcic lnrc` mnrc`ìjcrn` tejcrk`s mcĕcrjcrn`n` snstci ìĻdĻŕĻ`mkdn ertkjkikmcĕcrn vc lnrc` hnr mnrcòn` hezud ejik nbtnikjn `cmnr5 (V(W03)95)@E]7 R vc ^ hkĕĻisĻz.

Ocvkp 37Ñòc R vc ^ `n` de`vejúsyeùù kjkrkd Q9R+^ yn hujikd nìn` skmcoc Q rkstjk`tĻmcĕnŕdc`n`n` hkŕjk`lĻì hntnr mcĕcrn`n 8 njc ñtcjcyn` se`rk W98\ rkstjk`tĻ mcĕnŕdc`n`n huju`.

_eùì4


83/96

2?

1=6?

21=?

?8?

;82;1;

?2=?3;821=

?;


84/96

)8 / )(cxp(8

;)8 / Yoessn`8oessn`

oessn`8sn`oesTcxp(8

;),(

8888

88888

88888

Ϗ φϏ

Ϗ β β β β

β β β β φϏ

wv

R^ ^ R

R^ ^ R wv avw

+∘9

∘+

+++∘9

8)

+∘∘

∘

∘

9 )8(;8;cxp

;8;),( 88

8888 y pxy x y x axy

ϋ Ϗ ϋ φϏ

z9kx+hyw9ox+my

k)

azw(z,w)95h) k,h,o,m `n` bk`ln deihn`ksye`u nìn` z vc w hkĕĻisĻz ejur.

oW9koR+ho^ mW9kmR+hm^

k\9koR+km^ h\9hoR+hm^

oz-kw9^(ho-km) mW-h\9R(km-ho)

km hok\ oW ^ ∘

∘9

km homW h\

hokm h\ mW R

∘

∘9

∘

∘9

),( w z^ ϓ 9 ),( w z R ρ 9

;//~ ∘∘∘

∈∈∈∈ khh

khm

wz ρρ

∘++∘+ 88;;; 88888888 kowzwk zohmwzwh zm


85/96

T Y

T Y

+∘+∘

∘+∘+

∘∘

∘

∘

9

+∘+∘

+∘++

∘∘

∘

∘

9

+∘∘∘∘∘∘∘9

))((8)8()8(;*

);(8;cxp

;8;

)(8)(8)()(;*

);(8;cxp

;8;

8*);(8cxp;8),(

888888

8888

88

888888

8888

888888

km hohm kowzkhkhwom m o z

km ho

km howzkhwomzhm kowzkhwm o z

km ho

khwkmwzhowzomzkm howv avw

ϋ ϋ ϋ

ϋ Ϗ ϋ φϏ

ϋ ϋ Ϗ ϋ φϏ

ϋ ϋ Ϗ ϋ φϏ

h) ko+hm9 ϋ (ho+km) 90 hkĕĻisĻz ̀ erikj ejur.

?) R vc ^ hkĕĻisĻz mcĕnŕdc`jcrn R~T6 ?Y vc ^~T6 =Y S`naeri mkĕĻjĻik skbnpjcr. W9R+ŕ̂cdjn`mctk`Ļijk`mĻĕĻ`k lñrc

k) 5)( 9w xρ

h) 5)( 9w yρ o) 5)( 9w zρ m) az(z)95c) CTzY95a) 58 9 zϏ

k)


86/96

9

9

∘

9∘999 ∤

∘

wocw

w

wc

gcc

gw g

cc gw

c gw

mxcw

gw

x

gw gw gw gw

gw gwx gwx x

8

?sn`*)(

8?

)8?sn`(

88?);(?

;6?

?;

?;)(

8?

8??

6

8?

8?

8?

?

ρ

ρ

h) Hc`zcr ŕcdnjmc )8(sn`)( 8 wocw gw y 9ρ

o)

T Y;;8

;

8818


87/96

h) ∤

∐

∘9 gwx

)()( khj k`t ↝

∘

λλ ;

)(


88/96

h) ∤∐∘ mwcw x ax x )()( ρ tkhjemk` gwct u +λ )(

ww ∘↘ hnzni nìn` gw

ct u t ∘

↝∘

λ λ ;)(

≣+9

∘∘

mnlcr

xcc x ax x x

,6

6,8)(8=

)(8;

3.68886.3)V(x ;8

8=

3.6

8= ∘∘∘∘∐

∘∘ +9∐

∘∘

9+9≣ ∤ ccccmxcc

x x x x

o)

=

8)8(YT

=

=6 6

8=

6

8=

∘

99

⇘99

9+9+9

∘

∘

∐ ∐∘∘

∐

∘∘

∤ ∤∤

x

x

x x x x

cvmxmu

cmv xu

mx xcmx xcmxcc x xC

2;

6;1=8

==88

====

6

= 9∐

∘

∘

9∘

∘

∘

9∘∘∘∘∐

∘

∤∤ x x x x

x c xcmxc xc xc

Ky`Ļŕcdnjmc mnĕcr tkrkatk` mk28 lcjcocd vc CTzY9

2?

m)

T Y

?1;86)8()=(

8;6

)(; 88

99+9

9∘+

∘

99

9w gw

g gw g

gww

mwm

g xC xρ

a)888


89/96

1=;;

;13

1=86

28

1==

6)8(8

)=(=;

6)(;YT

8

?

8

?

8

88

8

88

9

99+9

9

∘

+∘

99

9

Ϗ

ρ w gw

g gw g

gww

mwm

g xC x

3)R~S(6,;) , ^~S(6,;) vc R vc ^ hk ĕĻisĻz ejsu`.

W9R+^\9R-^ ejsu`. z vc w `u` hk ĕĻijĻ akdkt njnŕdnsnz ejmuĕu`u lñstcr.

8,8w z

yw z

x ∘

9

+

9 ejur. 6:z:8 , -;:w:;z+w≭8, z-w≭8 z0|w|

8;

8;

8;

8;

8;

8;

8;

8;

8;

),(~ 9∘

∘9

∘

99

z x

z y

w x

z x

w z g

ν ν

ν ν

ν ν

ν ν

),(),(~),( nn y x axyw z gw z azw 9

:≭∘≭+::∘::9

mnlcr

ww zw zw zw z azw,6


90/96


91/96

Ļ


92/96

Ļ

;) Hnr zkr ndn dcrc ktĻjĻyer.Ķjd ktĻŕtk ? lcjnrsc ndn ktĻŕ seùou`mk tepjkiĻ` < ejik nbtnikjn`cmnr5

8)

ûdkrĻmkdnŕcdnjmc lñstcrnjc` ? scihejú` (;,8 vc ?) njctnjmnĕn hnr bkhcrjcŕic dk`kjĻ`mklc`cjmc (ertkjkik mkvrk`Ļŕjkr lñz ñ`ú`c kjĻ`mĻĕĻ`mk) ?, 8 ‒ mc`, 8„ mc ;„ mc` 8 dkt mkbkakzjk sĻdjĻdtk lñ`mcrnjnyer.

VT^9;|R9;Y9;-λ, VT^98|R9;Y9λ /8, VT^9?|R9;Y9λ /8 ejkrkd vcrnjinŕ vc mnĕcr ejkhnjcocd

nbtnikjjcrjc hcrkhcrŕcdnj úzcrn`mc lñstcrnjinŕtnr. Hurmk VT^9;|R9;Y ; lñ`mcrnjmnĕn`mc ;kjĻìksĻ deŕujju nbtnikjnmnr. Hu`k lñrc, kjĻoĻ tkrkatk ^9; kjĻ`mĻĕĻ`k lñrc ; lñmcrnjinŕ ejik nbtnikjn ̀ cmnr5

?) Hnr terhkmk 8 dĻriĻzĻ vc ? hcykz tep hujuìkdtkmĻr. Hu terhkmk` hnr tep ìcdnjnp rc`lndkymcmnjnyer vc ycrn`c lcrn deyujuyer. Hu`k lñrc ykpĻjk` = ìcdni seùou`mk c` kz 8dĻriĻzĻ tep ìwdnjinŕ ejik ejksĻjĻĕĻ `cmnr5

=) Hnr ìnat zkr ;6 dcrc ktĻjĻyer. Hu zkrjkrĻ` tepjkijkrĻ`Ļ` c` kz hnr dcrc < ejik nbtnikjn `cmnr5

3) Hnr sĻ`Ļatk= crdcd vc ? dĻz ñĕrcòn vkrmĻr. Hu ñĕrcònjcr tcdcr tcdcr sĻ`kvk kjĻ`Ļyerjkr.Ñĕ ` j ` h d d h dĻŕ dj ` l ` h Ļ kjkik jk Ļ`k k kjĻìk jk ĻjĻdjk Ļ

ykmk kjtĻsĻ turk skmcoc hnrn ykzĻ) mcvki cmnyer. AkrdjĻ ejk`Ļ ktk` bkdci ejkrkd

hcjnrjc`nyer Hu`k lñrc njd turmk (yk`n njd mc`cicmc) bkdcin` hcjnrjcìcsn ejksĻjĻĕĻ


93/96

hcjnrjc nyer. Hu k lñrc njd turmk (yk n njd mc cicmc) bkdcin hcjnrjc icsn ejksĻjĻĕĻ`cmnr5 (eyuòujkrĻ` skyĻsĻ`Ļ < kjkrkd prehjcin ìñzú`.)2) K`cstczn mc dujjk`Ļjk` eteiktnd hnr `cacs múzc`jcic kylĻtĻ (H), VH ejksĻjĻĕjk hezujuyer.

Hu hezujik, nzjcic snstcin (I) hu hezudju ĕu kjlĻjkikzsk, bkstk`Ļ` ñjúiú`c schcpvcrnyer.Ķzjcic snstcin (I) mc V I ejksĻjĻĕĻyjk hezujuyer.Ķdn snstcin` hezujikjkrĻtkikic` hk ĕĻisĻz ejkyjkrmĻr. Bkstk`c`n` preacsñrú cĕcr VI 0VH nsc nzjcic onbkzĻdujjkìk`Ļ` tkikic` lcrcdsnz ejmu ĕuù skvuùyer. Hu prcañsñrc dktĻjĻriĻsĻ`Ļz. ̂ edske`k hnr ejksĻjĻd dursu kjiksĻ`Ļ iĻ tkvsnyc cmcrsn`nz5 VI 96.; vc VH96.63 ejkrkd ocvkhĻ`ĻzĻmcstcdjcynon nŕjcin lñstcrn`.

>) KŕkĕĻmkdnŕcdnjmcdn snstcimc ;8 uìjkrĻ krksĻ`k hkĕjk`k` 3Q ‒ jud hnr pnjjc ‚K’ kipújúykdĻjikd nstcìcdtcmnr. K`kbtkrjkrĻ` bcrhnrn nìn` dkpkjĻ deùimk ejik nbtnikjn ‚p’ mnr vcbcpsn hnrhnrn`mc` hkĕĻisĻz ìkjĻŕikdtkmĻr. p96.= ejkrkd jkihk`Ļ` ykìk nbtnikjn`nbcskpjkyĻ`Ļz.

Ļ

;) ^98


94/96

k) CT Y95 (Ocvkp7 ;?)h) 59 yϏ (Ocvkp7 =18< )o) V(^0-;?) (Ocvkp7 6.3)

8) ^9R 8+? vc R≂T-8,8Y krkjĻĕĻ`mk múzlú` mkĕĻjĻik skbnpsc4k) CT^Y95 (Ocvkp7 ;?/?)

h) 59

yϏ (Ocvkp7


95/96


96/96

olasılık ve İstatistik - erzurum atartürk Üniversitesi olasılık ve raslantı değişkenleri...

Documents