Conceptos de Estadística

 

@BYJ][DL J. MJP]SJC[@JLHK MJP]SJC[@JLHKP P MK NSK@[KH@JBP 2.2 2. 2 @lh`kptls @lh`kptls mk Kstbmæstj`b Kstbmæstj`b y su `dbsjnj`b`jð `dbsjn j`b`jðh h  Bhtks mk  mk  bhbdjzbr dls  dls  `lh`kptls y db `dbsjnj`b`jðh mk db kstbmæstj`b, ks `lhvkhjkhtk khtkhmkr dbs sjgujkhtks mknjhj`jlhks câsj`bs1

Mbtls. Plh dls ek`els y dls hñfkrls quk sk rkñhkh, sk bhbdjzbh y sk rksufkh rksu fkh pbrb su prkskhtb`jðh k jhtkrprktb`jðh. Dls mbtls pukmkh skr @ubdjtbtjvls l @ubhtjtbtjvls. Mbtls @ubdjtbtjvls slh bqukddls quk rksplhmkh b db prkguhtb `ubd, sk pukmkh kxprksbr `lh `brb`tkrks hufêrj`ls l bdnbhufêrj`ls, y slh ktjquktbs l hlfcrks bsjghbmls b uh btrjcutl. Mbtls @ubhtjtbtjvls slh bqukddls quk rksplhmkh b db prkguhtb `uâhtl l `uâhtls, sjkfprk skrâh hufêrj`ls plrquk b mjnkrkh`jb mk dls bhtkrjlrks, kstls sj sk pukmkh `ubhtjnj`br. Kdkfkhtl. Plh dbs khtjmbmks b`kr`b mk dbs `ubdks sk rkñhkh dls mbtls. ^brjbcdk. Ks uhb `brb`tkræstj`b mk jhtkrês mk dls kdkfkhtls. Lcskrvb`jðh. Ks kd `lhauhtl mk fkmj`jlhks rkuhjml pbrb mktkrfjhbml kdkfkhtl. Ylcdb`jðh.   Ks kd `lhauhtl mk tlmls dls kdkfkhtls quk slh lcaktl mkd kstumjl Ylcdb`jðh. kstbmæstj`l. Fukstrb   ks uh suc`lhauhtl, kxtrbæml mk db plcdb`jðh (fkmjbhtk tê`hj`bs mk Fukstrb fukstrkl), `uyl kstumjl sjrvk pbrb jhnkrjr `brb`tkræstj`bs mk tlmb db plcdb`jðh.

Kh db sjgujkhtk sk pukmkh kakfpdjnj`br dls bhtkrjlrks `lh`kptls.   Kd kstumjl `lhstb mk 09 kdkfkhtls. 

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  Plcrk `bmb uhl mk kstls 09 kdkfkhtls sk rk`bcl jhnlrfb`jðh mk 0 vbrjbcdks

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`ubhtjtbtjvbs (kmbm, kstbturb y hñfkrl mk ebcjtbhtks kh `bsb), y 3 vbrjbcdks

 

`ubdjtbtjvls (sk`tlr mlhmk vjvkh, quk ks uh mbtl bdnbhufêrj`l y ks`ukdb mk

kgrksl mk hjvkd fkmjl supkrjlr, quk ks uh mbtl hufêrj`l)   Ylr fkh`jlhbr sldl bdguhb, db lcskrvb`jðh pbrb kd kdkfkhtl ddbfbml Nbcjldb

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]BCDB1   Dls sjgujkhtks mbtls `lrrksplhmkh b uh grupl mk bdufhls mk db `brrkrbs ]BCDB1 mk Dj`kh`jbturb kh Bmfjhjstrb`jðh, mkd Jhstjtutl ]k`hldðgj`l mk @m. Auârkz.

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Mknjhj`jðh mk db Kstbmæstj`b Db kh`j`dlpkmjb Crjtâhj`b mknjhk db kstbmæstj`b `lfl db `jkh`jb kh`brgbmb mk rk`ldk`tbr, bhbdjzbr, prkskhtbr k jhtkrprktbr mbtls. Kd nbflsl mj``jlhbrjl Jhgdks Wlrm Sknkrkh`k mknjhk db kstbmæstj`b `lfl uh ârkb mk db fbtkfâtj`b bpdj`bmb lrjkhtbmb b db rk`ldk``jðh k jhtkrprktb`jðh mk mbtls `ubhtjtbtjvls y bd usl mk db tklræb mk db prlcbcjdjmbm pbrb `bd`udbr dls pbrâfktrls mk uhb plcdb`jðh. [hb mknjhj`jðh fâs `lfpdktb skræb db sjgujkhtk1 brtk y `jkh`jb mk rkuhjr, bhbdjzbr, prkskhtbr k jhtkrprktbr mbtls pbrb db tlfb mk mk`jsjlhks. @lfl sk pukmk bprk`jbr, kstb ñdtjfb mknjhj`jðh bgrkgb uh `lh`kptl fuy jfplrtbhtk `lfl dl ks db tlfb mk mk`jsjlhks.

@dbsjnj`b`jðh mk db Kstbmæstj`b Db kstbmæstj`b sk `dbsjnj`br kh mls grbhmks rbfbs1   

Kstbmæstj`b Kstbmæ stj`b mks`rjp tjvb l mkmu`tj vb. 

 

Kstbmæstj`b Kstbmæ stj`b jhn krkh`jbd krkh`jbd l j hmu`t jvb. 

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Db Kstbmæstj`b Kstbmæstj`b Mks`rjp Mks`rjp tjvb l Mkmu`tj Mkmu`tj vb vb sk  sk kfpdkb sjfpdkfkhtk pbrb rksufjr mk nlrfb hufêrj`b l grânj`b uh `lhauhtl mk mbtls. Pk rkstrjhgk b mks`rjcjr dls mbtls quk sk bhbdjzbh. Pj bpdj`bfls dbs ekrrbfjkhtbs lnrk`jmbs plr db kstbmæstj`b mks`rjptjvb b uhb fukstrb, sldl hls djfjtbrkfls b mks`rjcjr dls mbtls kh`lhtrbmls kh mj`eb fukstrb, hl sk plmrâ gkhkrbdjzbr db jhnlrfb`jðh eb`jb db plcdb`jðh.

 

Db Kstbmæstj`b Jhnkrkh`jbd pkrfjtk Jhnkrkh`jbd  pkrfjtk rkbdjzbr `lh`dusjlhks l jhnkrkh`jbs, cbsâhmlsk kh dls mbtls sjfpdjnj`bmls y bhbdjzbmls mk uhb fukstrb eb`jb db plcdb`jðh l uhjvkrsl. Ylr kakfpdl, b pbrtjr mk uhb fukstrb rkprkskhtbtjvb tlfbmb b dls ebcjtbhtks mk uhb `jumbm, sk plmrâ jhnkrjr db vltb`jðh mk tlmls dls `jumbmbhls quk `ufpdbh dls rkqujsjtls `lh uh krrlr mk bprlxjfb`jðh.  

2. 2.3 3 Sk`l Sk`lpj pjdb`j db`jðh ðh mk m k mbtls. mbtls . Fêtlmls Fê tlmls mk rk`ldk``jðh mk mbtls kh db jhvkstjgb`jðh `ubhtjt btjvb [hb vkz mknjhjml b`kr`b mk quk tkfb tkfb sk qujkrk rkbdjzbr kd kakr`j`jl kstbmæstj`l, ks hk`ksbrjl mknjhjr dbs tê`hj`bs mk rk`ldk``jðh mk mbtls pbrb `lhstrujr dls jhstrufkhtls quk hls pkrfjtbh lctkhkrdls mk db rkbdjmbm. [h jhstrufkhtl mk rk`ldk``jðh mk mbtls ks  `ubdqujkr rk`ursl mk quk sk vbdk kd jhvkstjgbmlr pbrb b`kr`brsk b dls nkhðfkhls y kxtrbkr mk kddls jhnlrfb`jðh. kddls jhnlrfb`jðh.   Mkhtrl mk `bmb jhstrufkhtl pukmkh mjstjhgujrsk mls bspk`tls1

 

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Db nlrfb1 sk rknjkrk b dbs tê`hj`bs quk utjdjzbfls pbrb db tbrkb mk bprlxjfb`jðh b db rkbdjmbm (lcskrvb`jðh, khtrkvjstb).

 

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Kd `lhtkhjml1 qukmb kxprksbml kh db kspk`jnj`b`jðh mk dls mbtls quk hk`ksjtbfls `lhskgujr. Pk `lh`rktb kh uhb skrjk mk ætkfs quk hl slh ltrb `lsb quk dls jhmj`bmlrks quk pkrfjtkh fkmjr b dbs vbrjbcdks, pkrl quk bsufkh belrb db nlrfb mk prkguhtbs, puhtls b lcskrvbr, kdkfkhtls pbrb rkgjstrbr,, kt`. rkgjstrbr

Kd jhstrufkhtl sjhtktjzb kh sæ tlmb db dbclr prkvjb mk jhvkstjgb`jðh1 rksufk dls bplrtks mkd fbr`l tkðrj`l bd skdk``jlhbr mbtls quk `lrrksplhmkh b dls jhmj`bmlrks y, plr dl tbhtl, b dbs vbrjbcdks y `lh`kptls utjdjzbmls? utjdjzbmls ? pkrl tbfcjêh sjhtktjzb sjhtk tjzb kd mjskòl `lh`rktl kdkgjml pbrb kd trbcbal.  trbcbal.  Fkmjbhtk uhb bmk`ubmb `lhstru``jðh mk dls jhstrufkhtls mk rk`ldk``jðh, db jhvkstjgb`jðh bd`bhzb db hk`ksbrjb `lrrksplhmkh`jb khtrk tklræb khtrk  tklræb y ek`els.

 

  Mbtls Mb tls Yrjfbrjl Yrjfbrjls sZP Pk`uhmbrj k`uhmbrj ls.  

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Mbtls prjfbrjls1 slh bqukddls quk kd jhvkstjgbmlr lctjkhk mjrk`tbfkhtk mk db rkbdjmbm, rk`ldk`tâhmldls `lh sus prlpjls jhstrufkhtls.

 

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Mbtls sk`uhmbrjls1 slh rkgjstrls ks`rjtls quk prl`kmkh tbfcjêh mk uh `lhtb`tl `lh db prâ`tj`b, pkrl quk yb ebh sjml kdkgjmls y prl`ksbmls plr ltrls jhvkstjgbmlrks.

Dls mbtls prjfbrjls y sk`uhmbrjls hl slh mls `dbsks kskh`jbdfkhtk mjnkrkhtks mk jhnlrfb`jðh, sjhl pbrtks mk uhb fjsfb sk`ukh`jb1 tlml mbtl sk`uhmbrjl eb sjml prjfbrjl kh sus lrægkhks, y tlml mbtl prjfbrjl, b pbrtjr mkd flfkhtl kh quk kd jhvkstjgbmlr `lh`duyk su trbcbal, sk `lhvjkrtk kh mbtl sk`uhmbrjl pbrb dls mkfâs.

]ê`hj ]ê `hj `bs Mk S Sk`ldk`` k`ldk``jðh jðh Mk Mbtls Mbtls Yrjfbrjl Yrjfbrjls. s.  

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Db Lcskrvb`jðh. Lcskrvb`jðh . @lhsjstk kh kd usl sjstkfâtj`l mk hukstrls skhtjmls lrjkhtbmls b db `bptb`jðh mk db rkbdjmbm quk qukrkfls kstumjbr.

Ks uhb tê`hj`b bhtjgub1 b trbvês mk sus skhtjmls, kd skhtjmls,  kd elfcrk `bptb db rkbdjmbm quk dl rlmkb, quk dukgl lrgbhjzb jhtkdk`tubdfkhtk. Murbhtk jhhufkrbcdks lcskrvb`jlhks sjstkfâtj`bfkhtk rkpktjmbs. Kd usl mk hukstrls skhtjmls ks uhb nukhtk jhbgltbcdk mk mbtls quk, tbhtl pbrb db b`tjvjmbm `jkhtænj`b `lfl pbrb db vjmb prâ`tj`b, rksudtb mk jhkstjfbcdk vbdlr. jhkstjfbcdk  vbdlr.    

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Db Khtrkvjstb. Khtrkvjstb. @lhsjstk kh uhb jhtkrb``jðh uhb jhtkrb``jðh khtrk mls pkrslhbs, kh db `ubd kd jhvkstjgbmlr

nlrfudb

mktkrfjhbmbs

prkguhtbs

rkdbtjvbs

bd

tkfb

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jhvkstjgb`jðh, fjkhtrbs quk kd jhvkstjgbml prlplr`jlhb vkrcbdfkhtk l plr ks`rjtl db jhnlrfb`jðh quk dk ks sldj`jtbmb. Kxjstkh bmkfâs ltrls  ltrls  prl`kmjfjkhtls mk rk`ldk``jðh mk mbtls prjfbrjls, khtrk dls quk njgurbh kd ddbfbml `ukstjlhbrjl ddbfbml `ukstjlhbrjl mk butl- bpdj`b`jðh, dls tkst, dls mjbgrbfbs dls mjbgrbfbs sl`jl fêtrj`ls, dbs ks`bdbs y mjnkrkh`jbdks skfâhtj`ls, kt`. sjh kfcbrgl, tlmls tjkhkh su lrjgkh, kh ñdtjfb jhstbh`jb, kh dbs mls prjh`jpbdks tê`hj`bs fkh`jlhbmbs.

 

 

2.0 2. 0 Mjst Mjstrjc rjcu`j u`jðh ðh mk nrk`ukh`jbs nrk`ukh`j bs. Mkspuês mk `lhtbr `lh db djstb mk mbtls  (tbd y `lfl nukrlh rk`ldk`tbmls), ks `lhvkhjkhtk y hk`ksbrjl lrmkhbr kstls mk b`ukrml b db vbrjbcdk quk sk vbyb b bhbdjzbr. B kstk lrmkhbfjkhtl sk dk ddbfb austbfkhtk lrmkhb`jðh.  Bmkfâs mk db `lhvkhjkh`jb `lhvkhjkh`jb mk lrmkhbr dls mbtls, sk prkskhtb ltrb `lhvkh `lhvkhjkh`jb jkh`jb `lfl dl ks kd bgrupbfjkhtl mk kstls mk b`ukrml b su vbdlr, ks mk`jr, `rkbr `dbsks l grupls mk `jkrtl tbfbòl, pbrb bsæ `lhtbcjdjzbr `ubhtls mbtls qukmbh bgrupbmls kh `bmb `dbsk y nb`jdjtbr db jhtkrprktb`jðh mk dls mbtls, b kstk kakr`j`jl sk dk ddbfb Mjstrjcu`jðh mk nrk`ukh`jbs.

Mktkrfjhb`jðh mk Hñfkrl mk @dbsks. Db prjfkr prkguhtb quk surgk bd qukrkr `dbsjnj`br dls mbtls ks kh `ubhtls grupls sk vb b `dbsjnj`br dls mbtls, puks cjkh, buhquk eby `jkrtbs nlrfudbs quk brrlabh kd hufkrl jmðhkl pbrb bgrupbr dls mbtls, db vkrmbm quk db `bhtjmbm mk `dbsks db mktkrfjhb db pkrslhb quk eb`k kd kstumjl mk b`ukrml b db hk`ksjmbm mk dl quk qujkrb bhbdjzbr, pbrb bdgujkh db hk`ksjmbm plmræb skr ñhj`bfkhtk 3 `dbsks, dls mbtls pkqukòls y dls mbtls grbhmks, l 0 `dbsks, l 29, kt`., y ks tbfcjêh jfplrtbhtk `lfkhtbr quk hl hk`ksbrjbfkhtk kd tbfbòl mk kstbs tjkhkh quk skr jgubd. Dls butlrks mk djcrls sugjkrkh quk skb khtrk ; y ; l 39 `dbsks mkpkhmjkhml mk dls mbtls, pkrl mkck prkmlfjhbr db hk`ksjmbm mkd kstumjl. Ybrb njhks mkd prkskhtk tkxtl, kh `bmb kakfpdl sk skòbdbrb kd hufkrl mk `dbsks, y kstbs skrâh mk jgubd tbfbòl.

@bd`u @b d`u dl mkd Bh` el mk @dbsk. [hb vkz tlfbmb db mk`jsjðh mkd hñfkrl mk `dbsks.sk prl`kmk bd `bd`udbr kd bh`el l tbfbòl mk `dbsk mk db sjgujkhtk fbhkrb1 mk db lrmkhb`jðh mk dls mbtls sk tlfb kd mbtl fbylr y sk rkstb kd mbtl fkhlr, b kstb mjnkrkh`jb sk dk ddbfb rbhgl. B`tl skgujml mk mjvjmk kd mkd rbhgl khtrk kd hufkrl mk `dbsks kstbcdk`jmbs. Kd rksudtbml ks kd bh`el mk `dbsk, kd `ubd, sj ks hk`ksbrjl, sk rkmlhmkb b db `jnrb fâs bdtb pbrb gbrbhtjzbr quk kd mbtl fâs grbhmk qukmk `lhtkfpdbml kh db ñdtjfb `dbsk `lhstrujmb. @bmb `dbsk tjkhk Djfjtk Pupkrjlr, Djfjtk Jhnkrjlr y Fbr`b mk @dbsk.

 

Dls dæfjtks mk `dbsk mkckrâh `lhtkhkr dbs fjsfbs `jnrbs mkspuês mkd puhtl quk dls mbtls quk sk kstêh bgrupbhml, dl bhtkrjlr mbrâ `lfl rksudtbml uhb `dbsjnj`b`jðh fbs prk`jsb.

@lhstru``jðh mk dbs `dbsks. Kd dæfjtk jhnkrjlr mk db prjfkr `dbsk ks kd mbtl fâs cbal, kd fjsfl quk sk utjdjzl pbrb `bd`udbr kd bh`el mk `dbsk. Kstk dæfjtk y kd mk tlmbs dbs `dbsks, ks `krrbml, ks mk`jr, kd vbdlr kstâ jh`dujml kh db `dbsk. Hl bsæ kd dæfjtk supkrjlr quk ks bcjkrtl, y buhquk bpbrk`k kh db `dbsk, kstk vbdlr hl pkrtkhk`k b db fjsfb.  Bd dæfjtk jhnkrjlr sk dk sufb kd bh`el mk `dbsk pbrb kh`lhtrbr kh`lhtrbr kd kd dæfjtk supkrjlr, kstk dæfjtk supkrjlr skrâ kd dæfjtk jhnkrjlr mk db sjgujkhtk `dbsk y bsæ su`ksjvbfkhtk ebstb `lfpdktbr dbs `dbsks rkqukrjmbs.

Mjstrjcu` jðh mk Nrk`ukh`jbs. Nrk`ukh`jbs. Db tbrkb sjgujkhtk ks mjstrjcujr dls mbtls mk b`ukrml b su vbdlr kh db `dbsk quk dk `lrrksplhmb. Db `bhtjmbm mk mbtls quk bgrupk `bmb `dbsk sk dk `lhl`k `lfl Nrk Nrk`ukh`jb `ukh`jb Bcs ldutb m k @dbsk, quk, sjkhml kstb dbs fâs utjdjzbmb, `bmb vkz quk sk mjgb nrk`ukh`jb sk kstbrâ eb`jkhml rknkrkh`jb b kstb.

Nrk`ukh`jb Skdbtjvb. Ks db nrk`ukh`jb mk `dbsk kh rkdb`jðh bd tltbd mk mbtls mbt ls `lh quk sk `ukhtb. Pk lctjkhk mjvjmjkhml db nrk`ukh`jb khtrk kd hñfkrl mk mbtls.

Nrk`ukh`jb B`ufudbmb. Ks db nrk`ukh`jb mk tlmls dls vbdlrks fkhlrks b `bmb uhl mk dls dæfjtks supkrjlrks. Pk lctjkhk sjtuâhmlsk kh tlmls y `bmb uhl mk dls djfjtks supkrjlrks y eb`krsk db prkguhtb ‖¼@uâhtls vbdlrks slh fkhlrks b kstk<

 

  Kakfpdl   Kakfpdl @âd`ud @â d`ud l mkd bh`el mk `dbsk ]lfbhml `lfl cbsk b dls mbtls prlplr`jlhbmls kh db tbcdb 2.2, db bgrupb`jðh mk dls mbtls rknkrkhtk b db kstbturb kh ; `dbsks, mk jgubd tbfbòl skræb db sjgujkhtk1 Mbtl Fbylr 6 2.50 Mbtl Fkhlr 6 2.;3 Hufkrl mk `dbsks mkskbml1 ;

∓   ℈       

 

 

   .  ∓.    9.9>3  

Db lpkrb`jðh mb `lfl rksudtbml mk 0 mægjtls mkspuês mkd puhtl mk`jfbd, y dls mbtls kh kstumjl sldl tjkhkh 3, kstl lcdjgb b rkmlhmkbr eb`jb brrjcb kd rksudtbml b 3 mægjtls @ ≅ 9.97. 

@lhstru``j ðh mk dls jhtkrvbdls mk `dbsk

Db prjfkr `dbsk tkhmrâ `lfl dæfjtk jhnkrjlr kd mbtl fkhlr 2.;3 y kd dæfjtk supkrjlr skrâ 2.;: (2.;3 + 9.97). Kd dæfjtk jhnkrjlr mk db skguhmb `dbsk skrâ 2.;:, bd `ubd sk dk sufbrb mk hukvl kd bh`el mk `dbsk pbrb lctkhkr kd dæfjtk supkrjlr mk db skguhmb y bsæ su`ksjvbfkhtk. Hðtksk quk mkd dbml mkd dæfjtk jhnkrjlr mk `bmb `dbsk bpbrk`k `lr`ektk —_— quk qujkrk mk`jr quk ks djfjtk `krrbml, fjkhtrbs quk mkd dbml mkd dæfjtk supkrjlr bpbrk`k kd sæfcldl —)‖ quk sjghjnj`b djfjtk bcjkrtl, quk sjghjnj`b quk kd k d dæfjtk jhnkrjlr sj pkrtkhk`k

b db `dbsk, fjkhtrbs quk kd dæfjtk supkrjlr hl, kstk pkrtkhk`k b db `dbsk sjgujkhtk.

 

  @âd`udl mk dbs fbr`b mk `dbsk Db fbhkrb mk `bd`udbr db fbr`b mk `dbsk `lrrksplhmjkhtk ks db sjgujkhtk1

   3 

Kh mlhmk1 F` jj  6    6

Fbr`b mk `dbsk mk db `dbsk

DPj 6 Djfjtk

supkrjlr mk db `dbsk

DJj 6 Djfjtk

jhnkrjlr mk db `dbsk

 

j 

j

j

Kakfpdjnj`bhml kd `âd`udl mk db fbr`b mk `dbsk mk db prjfkr `dbsk, qukmbræb bsæ1

2.;3+2.3 ;  :  2.;;;

 

Mk fbhkrb sjfjdbr sk `bd`udbh dbs fbr`bs mk `dbsk mk dbs mkfâs `dbsks.

Kd rksudtbml mk bgrupbr dls mbtls mk db tbcbd mk dls mbtls bhtkrjlrks sjgujkhml dls `rjtkrjls y `âd`udls bhtkrjlrfkhtk kstbcdk`jmls skræb kd sjgujkhtk1  sjgujkhtk1  

MJP]SJC[@JLH MK NSK@[KH@JBP Hl. mk `dbsk 2 3 0 = ; ]ltbdks

Jhtkrvbdl mk @dbsk _2.;3, 2.;:) _2.;:, 2.>>) _2.>>, 2.70) _2.70, 2.59) _2.59, 2.57)

Fbr`b mk @dbsk 2.;;; 2.>3; 2. >3; 2.>:; 2.7>; 2.50;

Nrk`ukh`jb  Bcsl  Bc sldu du tb 3 23 7 ; = 09

Nrk`ukh`jb Nrk`ukh`jb Skdbtjvb  B` ufud  B`uf ud bmb 3÷09 6 9.97 3 23÷096 9.=9 2= 7÷096 9.30 32 ;÷096 9.27 3> =÷096 9.20 09 2.99

 

  Db sufb mk dbs nrk`ukh`jbs mkck skr jgubd bd hñfkrl tltbd mk mbtls kh kd kstumjl. Db sufb mk dbs nrk`ukh`jbs rkdbtjvbs mkck skr jgubd 2 l bprlxjfbrsk `ubhml dbs `jnrbs slh rkmlhmkbmbs. Db nrk`ukh`jb mk db ñdtjfb `dbsk mkck skr jgubd bd hñfkrl tltbd mk mbtls.

2. 2.0. 0.2 2 Yldæglh Yldæglh ls mk nrk`ukh` nrk`ukh`jb, jb, ejstlgrbfbs ejstl grbfbs y laj vbs. Db tbcdb mk mjstrjcu`jðh mk nrk`ukh`jbs ks ñtjd kh db prkskhtb`jðh mkd rksufkh mk mbtls, pkrl bmkfâs sjrvk pbrb prkskhtbr dls mbtls pkrl belrb mk fbhkrb grbnj`b Kxjstk uhb grbh `bhtjmbm mk grânj`ls, pkrl bquæ sldl sk fkh`jlhbrbh 0 mk kddls.

[h ejstlgrbfb  ejstlgrbfb  ks uh `lhauhtl mk rk`tâhgudls `lh cbsks kh kd kak elrjzlhtbd, `khtrls kh dbs fbr`bs mk `dbsk, dlhgjtumks jgubdks b dls bh`els mk `dbsk y ârkbs prlplr`jlhbdks b dbs nrk`ukh`jbs mk `dbsk

[h Yldæglhl mk Nrk`ukh`jb ks uh grbnj`l mk trlzls l skgfkhtls mk db nrk`ukh`jb mk `dbsk `lh rkdb`jðh b db fbr`b mk `dbsk.

Db Lajvb Lajvb   ks uh pldæglhl mk nrk`ukh`jbs b`ufudbmbs y sk grbnj`b rkdb`jlhbhml db nrk`ukh`jb b`ufudbmb mk `bmb `dbsk `lh kd dæfjtk supkrjlr mk `bmb `dbsk.

 

EJP]LGSBFB MJP]SJC[@JLH MK BD[FLP YLS KP]B][SB 2= 23 29 5

Bdufhls > = 3 9 2.;3-2.;:

2.;:-2.>>

2.>>-2.70

2.70-2.59

2.59-2.57

Kstbturb kh Fts.

 

YLDJGLHL MK NSK@[KH@JB MK KP]B][SBP KP]B][SBP MK BD[FHLP 2= 23 29 5

bdufhls > = 3 9 2,;;;

2,>3;

2,>:;

Kstbturb kh fts.

2,7>;

2,50;

 

LAJ^B "FKHLS R[K" KP]B][SB MK BD[FHLP 0; 09 3; 39

Bdufhls 2; 29 ; 9 2.;3-2.;:

2.;:-2.>>

2.>>-2.70

2.70-2.59

2.59-2.57

Kstbturb kh fts

 

2.= 2. = Fkmjmbs Fkmjmbs mk tkhmkh`jb `khtrbd. pbrb pbrb uh `lha `lhauht uhtl lm mk k mb mbtls tls hl bgrupbmls y mbtls bgrupbmls. bgrupbmls . Db kstbmæstj`b mks`rjptjvb kh su nuh`jðh câsj`b mk rkmu`jr mbtls, prlplhk uhb skrjk mk jhmj`bmlrks quk pkrfjtkh tkhkr uhb pkr`kp`jðh râpjmb mk dl quk l`urrk kh uh nkhðfkhl. Db prjfkrb gbfb mk jhmj`bmlrks `lrrksplhmk b dbs —Fkmjmbs mk ]khmkh`jb @khtrbd‖.

Kxjstkh vbrjls prl`kmjfjkhtls pbrb kxprksbr fbtkfâtj`bfkhtk dbs fkmjmbs mk tkhmkh`jb `khtrbd, mk dls `ubdks, dls fâs `lhl`jmls slh1 db fkmjb brjtfêtj`b, db flmb y db fkmjbhb. Dls æhmj`ks `bd`udbmls pbrb uhb plcdb`jðh sk ddbfbh pbrâfktrls, fjkhtrbs quk dls `bd`udbmls pbrb uhb fukstrb sk ddbfbh kstbmæstj`ls. 

2.=.2 2.= .2 Fk Fkmj mjb, b, Fk Fkmj mjb b pl plhm hmkrbmb. krbmb.

 

Db Fkmjb Db fkmjb, db fkmjb brjtfêtj`b y kd prlfkmjl slh kd fjsfl `lh`kptl, y kqujvbdk bd `âd`udl mkd prlfkmjl sjfpdk mk uh `lhauhtl mk mbtls. Kd `âd`udl pbrb db fkmjb mk mbtls hl bgrupbmls ks kd sjgujkhtk1

Fkmjb Fk mjb pbrb M Mbtls btls Hl Bgr Bgrupbmls upbmls Fkmjb Ylcdb`jlhbd

Kh mlhmk1

   −   6  

 

»6fkmjb plcdb`jlhbd H6 `bhtjmbm mk mbtls kh db plcdb`jðh

Fkmjb Fk mjb Fukstr bd

Kh mlhmk1

   − ̋  6  

 

6 fkmjb fukstrbd

̋

6 mbtl j  

h6 `bhtjmbm mk mbtls kh db fukstrb

Kd `âd`udl mk kstbs mls fkmjmbs ks kxb`tbfkhtk kd fjsfl, db mjnkrkh`jb ks db hlfkh`dbturb quk sk utjdjzb austbfkhtk pbrb eb`kr hltbr quk sk trbtb mk uh pbrâfktrl l uh kstbmæstj`l.

Kakfpdl1 ]lfbhml dls mbtls mk db tbcdb mk dls 09 bdufhls y suplhjkhml quk dls mbtls mk db vbrjbcdk kmbm slh mbtls fukstrbdks, db fkmjb skræb db sjgujkhtk1

 

  h609 Πxj6 =>.55

̋ . 6

6 2.;> fts.

Kd rksudtbml bhtkrjlr sjghjnj`b quk db fkmjb l kd prlfkmjl mk db kstbturb mk dls 09 bdufhls mkd djstbml ks, 2.;> fts.

Fkmjb Fk mjb Yb Ybrb rb Mb Mbtls tls Bgru Bgrupbmls pbmls @ubhml hl sk `ukhtb `lh db jhnlrfb`jðh mk db tltbdjmbm mk dls mbtls lrjgjhbdks, sjhl `lh uhb mjstrjcu`jðh mk nrk`ukh`jbs, y sk qujkrk `lhl`kr db fkmjb, sk bpdj`b db nlrfudb sjgujkhtk1

Fkmjb Ylcdb`jlhbd

   −    .   6   

Kh mlhmk1

 

»6fkmjb plcdb`jlhbd H6 `bhtjmbm mk mbtls kh db plcdb`jðh F` jj    6 fbr`b

mk `dbsk mk db `dbsk  j

N jj    6 nrk`ukh`jb

mk `dbsk mk db `dbsk j  

Fkmjb Fk mjb Fukstr bd

Kh mlhmk1

 −6  .  ̋  

 

 

̋

6fkmjb fukstrbd 6fkmjb

h6 `bhtjmbm mk mbtls kh db fukstrb F` jj    6 fbr`b

mk `dbsk mk db `dbsk  j

n   j  6 nrk`ukh`jb

mk `dbsk mk db `dbsk j  

Kakfpdl1 Sktlfbhml Sktlf bhml dls mbtls mk db tbcdb, rknkrkhtk b db kstbtu kstbturb rb mk dls 09 bdufhls, sk `bd`udbrb db fkmjb suplhjkhml quk mj`els mbtls slh plcdb`jlhbdks, l skb, tlmls dls kdkfkhtls mk jhtkrês kh kd kstumjl.

̋   2.;;;3 + 2.>3;23 + 2.>09:;7 + 2.7>;; + 2.50;= ̋

 

6 2.>55 ≅ 2.>: 

Kvjmkhtkfkhtk kd rksudtbml mk db fkmjb pbrb mbtls hl bgrupbmls y pbrb mbtls bgrupbmls ks fuy mjnkrkhtk, kstl mkcjml b db nlrfb kh quk mjstrjcuykh dls mbtls, tkfb quk sk vkrâ kh kd `bpætudl J^.

Fkmj Fk mjb b Ylh Ylhmkrbmb mkrbmb Kstb fkmjb tlfb kh `ukhtb db jfplrtbh`jb mk `bmb uhl mk dls mbtls bhbdjzbmls, mâhmldk fbylr l fkhlr jfplrtbh`jb kh kd `âd`udl mk kstb db fkmjmb, ks mk`jr, db fkmjb plhmkrbmb sk utjdjzb `ubhml hl tlmls dls kdkfkhtls kd kfkhtls `lfplhkhtks mk dls quk sk prktkhmk lctkhkr db fkmjb tjkhkh db fjsfb jfplrtbh`jb.

   −   −6   .  6 

Mlhmk1 W    6 j  6 U jj  6    6

vbdlr mk pksl pbrb

mbtl

j

x j j l    l

plhmkrb`jðh

 

 

Kakfpdl1 Kh uhb fbtkrjb mbmb sk bsjghbh pksls mk jfplrtbh`jb, mk db sjgujkhtk nlrfb1 [hjmb J (39% mkd `ursl), [hjmbm JJ (0;% mkd `ursl), [hjmbm JJJ (39% mkd `ursl), [hjmbm J^ (2;% mk db `bdjnj`b`jðh), [hjmbm ^ (29% mk db `bdjnj`b`jðh). Pj dbs `bdjnj`b`jlhks mk uh bdufhl slh 5 kh db prjfkrb uhjmbm, ; kh db skguhmb, 5 kh db tkr`krb uhjmbm, 29 kh db `ubrtb uhjmbm y 5 kh db ñdtjfb uhjmbm. Pk tjkhkh db sjgujkhtk tbcdb1

[hjmbm J JJ JJJ J^

Ylhmkrb`jðh (Wj) 39% 6 9.39 0;% 6 9.0; 39% 6 9.39 2;% 6 9.2;

Mbtls (Wj) 5 ; 5 29

^

29% 6 9.29

5

2 95   9.35 +9.9.30+;;9.;0+; +9.9.335+++9.29.;2+;;929+9. .29   7.3;

 

 

2.=.3 Fkmj Fkmjbhb. bhb. Db Fk Fkmjbhb mjbhb mk Mbtls Mbtls Hl Bgru pbmls Db fkmjbhb (Fk) ks kd mbtl `khtrbd kh uh `lhauhtl lrmkhbml mk lrmkhbml  mk mbtls. Pk eb`k êhnbsjs kh quk kd `lhauhtl mkck skr lrmkhbml mk fkhlr b fbylr bhtks mk dl`bdjzbr db fkmjbhb plrquk mk hl skr bsæ, `lfl db fkmjbhb ks uhb fkmjmb mk dl`bdjzb`jðh, kstb mkpkhmkrâ mkd b`lflml quk tkhgbh dls mbtls kh kd flfkhtl mkd bhâdjsjs. Ylr kakfpdl, sk tjkhk kstb skrjk mk mbtls1

0=, ;>,

32,

2:, ==

 

sj hl sk tjkhk db prk`bu`jðh mk lrmkhbr dls mbtls sk mjrâ quk db fkmjbhb ks 32, dl `ubd ks nbdsl plrquk sj sk lrmkhbh dls mbtls qukmbræb bsæ y db fkmjb skrâ 0=

2:, 32, 0=, ==, ;>

@ubhml db `bhtjmbm mk mbtls kh kd `lhauhtl ks uh hñfkrl jfpbr, sk tkhmrâ uhb fkmjbhb rkbd rkbd   `lfl nuk kd pkqukòl kakfpdl bhtkrjlr?  bhtkrjlr?   `ubhml db `bhtjmbm mk kstjfbmb, quk sk kakfpdjnj`b kh db kdkfkhtls skb pbr sk tkhmrâ uhb fkmjbhb kstjfbmb, sjgujkhtk skrjk mk mbtls.

2:, 32, 0=, ==, ;>, 79

 Bquæ db `bhtjmbm mk mbtls mbtls ks >, quk ks hufkrl pbr, plr dl quk db fkmjbhb sk tjkhk quk kstjfbr mk db sjgujkhtk fbhkrb1 sk dl`bdjzbh dls 3 mbtls `khtrbdks mjvjmjkhml kd hufkrl mk mbtls khtrk mls. >÷3 6 0? kd mbtl quk l`upb db plsj`jðh 0 y kd sjgujkhtk slh dls mbtls `khtrbdks, fjsfls quk mkckh skr prlfkmjbmls pbrb kstjfbr db fkmjbhb. Dls mbtls mk db skrjk bhtkrjlr quk l`upbh dbs plsj`jlhks 3 y 0, rkspk`tjvbfkhtk slh kd 0= y ==, quk bd prlfkmjbrdls mb `lfl Fk 6 0: quk vjkhk sjkhml db fkmjbhb kstjfbmb. Kakfpdl1 [tjdjzbhml dls mbtls mk db tbcdb 2.2 pbrb kh`lhtrbr db fkmjbhb sk tjkhk quk slh 09 mbtls, dl `ubd jfpdj`b quk db fkmjbhb ks kstjfbmb plr skr uh hufkrl pbr kh db `bhtjmbm mk mbtls. Db fkmjbhb ks kd mbtl quk sk kh`ukhtrb khtrk db plsj`jðh 2; y 2>, quk `lrrksplhmkh b dbs bdufhbs Ykrdb y Nbcjldb, rkspk`tjvbfkhtk, y kstls mbtls skrâh prlfkmjbmls.

  2 . > 7 + 2. 2 . > 7   3    2.>7

 

Db Fkmjbhb mk Mbtls Bgrupbmls

 

@ubhml sk tkhgb uhb mjstrjcu`jðh mk nrk`ukh`jbs y b pbrtjr mk kddb sk qujkrb dl`bdjzbr db fkmjbhb, sk utjdjzb db sjgujkhtk nðrfudb1

  − ∓    +       

Kh mlhmk1 Fk  6  6

fkmjbhb

D j 6

dæfjtk jhnkrjlr mk db `dbsk fkmjbhb

h  6  6

`bhtjmbm mk mbtls

Πn    f-2  6 n    6 f  6 `  6  6

 

nrk`ukh`jb b`ufudbmb kh db `dbsk bhtkrjlr b db `dbsk fkmjbhb 

nrk`ukh`jb mk db `dbsk fkmjbhb

bh`el mk `dbsk

Kh db mknjhj`jðh mk dbs vbrjbcdks mk db nlrfudb bhtkrjlr, bpbrk`k uh `lh`kptl quk ks mk vjtbd jfplrtbh`jb khtkhmkr, db `dbsk fkmjbhb, kstb ks db `dbsk kh db `ubd sk kh`ukhtrb db fkmjbhb. Kakfpdl1 Ybrb kakfpdjnj`br dl bhtkrjlr, b `lhtjhub`jðh sk rkprlmu`k mk fbhkrb pbr`jbd db tbcdb mk mjstrjcu`jðh mk nrk`ukh`jbs

Hl. mk `dbsk 2 3 0 = ; ]ltbdks

Jhtkrvbdl mk @dbsk _2.;3, 2.;:) _2.;:, 2.>>) _2.>>, 2.70) _2.70, 2.59) _2.59, 2.57)

Nrk`ukh`jb Nrk`ukh`jb  Bcsl  Bc sldu du tb  B`uf  B` uf udbmb ud bmb 3 3 23 2= 7 32 ; 3> = 09 09

@lfl prjfkr pbsl mk mkck jmkhtjnj`br `ubd ks db `dbsk fkmjbhb. Kh kstk kakfpdl sk sbck quk kd tltbd mk mbtls ks 09, bsæ dl fukstrb db sufbtlrjb mk dbs nrk`ukh`jbs.  Buh sjh `lhl`kr dls mbtls lrjgjhbdks, lrjgjhbdks, plr dðgj`b y plr `lh`kptl sk sbck quk db fkmjbhb, uhb vkz lrmkhbmls kstls 09 mbtls, kstâ dl`bdjzbmb khtrk db plsj`jðh 2; y 2>, plr dl quk eby quk kh`lhtrbr kh `ubd mk db ; `dbsks kstâh kstbs plsj`jlhks. Mk b`ukrml b db mjstrjcu`jðh tbfcjêh sk sbck quk db prjfkr `dbsk bgrupb b dls mbtls

 

fâs pkqukòl, ks mk`jr dbs plsj`jlhks 2 y 3, db skguhmb `dbsk bgrupb mk db plsj`jðh 0 b db plsj`jðh 2=, db `dbsk 0 bgrupb mksmk kd mbtl mk db plsj`jðh 2; ebstb db 32, y ks kh kstb `dbsk kh mlhmk quk sk kh`ukhtrb db plsj`jðh cus`bmb (khtrk plsj`jðh 2; y 2>), plr dl quk db `dbsk fkmjbhb ks db 0 pbrb kstk kakfpdl kh pbrtj`udbr. D j 6

2.>>

h  6  6

09

Πn    f-2  6 n    6 f  6 `  6  6

2=

7 

9.97

   2=   2.2.>> +  7   9.97

 

 

2.=.0 Flm Flmb. b.

  2.>7

Db flmb ks kd mbtl quk kh uh `lhauhtl mk mbtls sk rkpjtk `lh fâs nrk`ukh`jb. [h `lhauhtl mk mbtls pukmk tkhkr uhb l fâs flmbs, k jh`dusl jh`d usl pukmk hl tkhkr flmb, kstl `ubhml hjhguhl mk dls mbtls tkhgb fâs nrk`ukh`jb quk dls mkfâs.

Db Flmb mk Mb Mbtls tls Hl Bgrup bmls @ubhml sk `ukhtb `lh uh lrmkhbfjkhtl mk mbtls ks nâ`jd dl`bdjzbr db flmb lcskrvbhml db rkpktj`jðh mk `bmb mbtl. Kh kd kakfpdl quk sk eb kstbml vjkhml b dl dbrgl mk db uhjmbm, db flmb ks 2.>; `lh uhb nrk`ukh`jb jgubd b =, buhquk eby ltrls mbtls quk tbfcjêh sk rkpjtkh pkrl `lh fkhlr nrk`ukh`jb. Pj eucjkrb ltrl mbtl `lh nrk`ukh`jb = skræb tbfcjêh flmb y mk mjræb quk kd `lhauhtl mk mbtls ks cjflmbd, ks mk`jr, quk tjkhk 3 flmbs.

Db Flmb mk Mb Mbtls tls Bgrupbmls Bgru pbmls  Bd jgubd quk quk db fkmjb fkmjb y db fkmjbhb, fkmjbhb, db flmb sk pukmk kh`lhtrbr kh`lhtrbr pbrb pbrb mbtls bgrupbmls utjdjzbhml db nlrfudb

 

Kh mlhmk1 Fl 6

   + (Ν  Ν+ Ν) 

 

flmb

D j  

6 dæfjtk jhnkrjlr mk db `dbsk flmbd   Ν2  6 mjnkrkh`jb mk db nrk`ukh`jb mk db `dbsk flmbd y db `dbsk jhfkmjbtb bhtkrjlr  Ν3  6

@

mjnkrkh`jb mk db nrk`ukh`jb mk db `dbsk flmbd y db `dbsk jhfkmjbtb sjgujkhtk

6 bh`el mk `dbsk

Mk hukvb `ukhtb sk prkskhtb uh `lh`kptl quk eby quk jmkhtjnj`br bhtks mk eb`kr usl mk db k`ub`jðh? `dbsk flmbd. Kstb `dbsk ks db `dbsk quk supukstbfkhtk jh`duyk b db flmb, y sk jmkhtjnj`b `lfl db `dbsk mk fbylr nrk`ukh`jb. Pj kh bdgñh flfkhtl sk prkskhtb db sjtub`jðh mk 3 `dbsks `lh jgubd nrk`ukh`jb y kstbs skbh dbs fbs bdtbs, khtlh`ks kd `lhauhtl mk mbtls skrâ cjflmbd y ebcrâ quk `bd`udbr bfcbs flmbs.

Kakfpdl1 [tjdjzbhml mk hukvl kd kakr`j`jl mk db mjstrjcu`jðh mk nrk`ukh`jbs quk eb vkhjml sjrvjkhml pbrb kakfpdjnj`br `bmb `bd`udl mk dbs fkmjmbs, sk tjkhk khtlh`ks quk db flmb skrâ db sjgujkhtk1 Ybsl jhj`jbd eby quk dl`bdjzbr db `dbsk flmbd, quk mk fbhkrb `dbrb ks db skguhmb `dbsk `lh uhb nrk`ukh`jb mk 23.

D j  

6 2.;:

 Ν2  6

(23-3) 6 29

 Ν3  6

(23-7) 6 ;

@

6 9.97

  29   2.2.;: + (29+;) 9.97

 

Fl

6 2.>=

 

 

2.=.= Skdb`jðh khtrk fkmjb, fkmjbhb y flmb   Ybrb plmkr kstbcdk`kr uhb rkdb`jðh kfpærj`b khtrk fkmjb, fkmjbhb y flmb eby quk sbckr mjnkrkh`jbr dbs `urvbs mk mjstrjcu`jðh mk nrk`ukh`jb mk hukstrls mbtls kstbmæstj`ls mk db sjgujkhtk nlrfb nlrfb11  Kh kd `bsl mk mjstrjcu`jlhks uhjflmbdks, db fkmjbhb kstâ `lh nrk`ukh`jb `lfprkhmjmb khtrk db fkmjb y db flmb (jh`dusl fâs `kr`b mk db fkmjb). Kh mjstrjcu`jlhks quk prkskhtbh `jkrtb jh`djhb`jðh, ks fâs b`lhskabcdk kd usl mk db fkmjbhb. Pjh kfcbrgl kh kstumjls rkdb`jlhbmls `lh prlpðsjtls kstbmæstj`ls y mk jhnkrkh`jb sukdk skr fâs bptb db fkmjb

@ubhml uhb mjstrjcu`jðh mk nrk`ukh`jb ks sjfêtrj`b, db fkmjb, fkmjbhb y flmb `ljh`jmkh kh su vbdlr (»   6 Fk 6 Fl). Kh kd `bsl mk uhb mjstrjcu`jðh cjhlfjbd sjfêtrj`b, ks hk`ksbrjl `bd`udbr kd prlfkmjl mk dbs flmbs.   Kh uhb mjstrjcu`jðh sksgbmb b db jzqujkrmb, db flmb ks fkhlr b db fkmjbhb, y kstb b su vkz fkhlr quk db fkmjb ( » 4 Fl 4 Fk)   Kh uhb mjstrjcu`jðh sksgbmb b db mkrk`eb db rkdb`jðh sk jhvjkrtk, db flmb ks fbylr b db fkmjbhb, y kstb b su vkz fbylr quk db fkmjb ( Fl 8 Fk 8).

2.; Fkmjmbs mk mjspkrsjðh pbrb uh `lhauhtl mk mbtls y mbtls bgrupbmls. Kh sk``jlhks bhtkrjlrks sk eb mjs`utjml slcrk trks fkmjmbs mk tkhmkh`jb `khtrbd. Pjh kfcbrgl, kstbs fkmjmbs hl slh sunj`jkhtks pbrb khtkhmkr uh `lhauhtl mk mbtls y slcrk êd tlfbr mk`jsjlhks, pukstl quk ltrl bspk`tl quk mkck skr tlfbr kh `ukhtb ks db vbrjbcjdjmbm l db mjspkrsjðh mk dbs lcskrvb`jlhks.

 

@lh kd prlpðsjtl mk fkmjr db mjspkrsjðh mjspkrsjð h l vbrjbcjdjmbm, sk mjs`utjrâh kh kstk bpbrt bpbrtbml bml dbs fkmjmbs mk1 Sbhgl (ddbfbml tbfcjêh bfpdjtum l rk`lrrjml), ^brjbhzb y Mksvjb`jðh Kstâhmbr (tbfcjêh ddbfbmb mksvjb`jðh tæpj`b).

2.;.2 Sbhgl Sbhgl.. Kd rbhgl (S) ks db fkmjmb fâs sjfpdk y `lfl yb sk vjl bhtkrjlrfkhtk, ks db mjnkrkh`jb khtrk kd mbtl fbylr y kd mbtl fkhlr kh uh `lhauhtl mk mbtls. Kstb fkmjmb hls mb db jmkb `ubh kxtkhsl ks kd `lhauhtl mk mbtls, y hl kh `ubhtl b db `bhtjmbm mk mbtls, sjhl kh `ubhtl mk mlhmk b mlhmk vbh mj`els mbtls. Zb sk kakfpdjnj`l kstb fkmjmb `ubhml sk tuvl quk `bd`udbr kd bh`el mk `dbsk kh db sk``jðh 2.0

2.;.3 ^brj ^brj bhzb. Pj sk qujsjkrb fkmjr kd prlfkmjl mk dbs mksvjb`jlhks mk dls mbtls `lh rkspk`tl b db fkmjb, sjkfprk mbræb `krl, plrquk db fkmjb ks kd vbdlr quk baustb b tlmls dls mbtls. Ybrb kvjtbr kstl, `bmb mjnkrkh`jb sk kdkvb bd `ubmrbml y db sufb mk kstbs mjnkrkh`jbs mjvjmjmb khtrk kd tltbd mk mbtls mb `lfl rksudtbml db vbrjbhzb Db vbrjbhzb ks puks, uhb fkmjmb kstbmæstj`b quk fjmk db mjspkrsjðh mk dls vbdlrks rkspk`tl b db fkmjb, ks mk`jr, ks kd `ubmrbml mk dbs mksvjb`jlhks.

^brjbhzb mk Mbtls Hl Bgrupbmls Ks jfplrtbhtk sbckr sj dls mbtls b lpkrbr slh mbtls plcdb`jlhbdks l mbtls fukstrbdks, plrquk bmkfâs mk db hlfkh`dbturb, eby tbfcjêh mjnkrkh`jb kh kd `âd`udl mk db vbrjbhzb.

^brjbhzb Ylcdb`jlhbd

Kh mlhmk1

    −         6   

 6 Db ^brjbhzb mk db plcdb`jðh

H  6  6



`bhtjmbm mk mbtls kh db plcdb`jðh

 

 

U jj  6    6

mbtl j  

» 6 fkmjb plcdb`jlhbd plcdb`jlhbd

^brjbhzb ^b rjbhzb Fukstr bd

  ̋    −662 2

Kh mlhmk1 P 3  6

 

Db ^brjbhzb mk db fukstrb

h 6 `bhtjmbm mk mbtls kh db fukstrb U jj  6    6

mbtl

j

 6 fkmjb mk db fukstrb

̋

Db mjnkrkh`jb kh kd `âd`udl m db vbrjbhzb plcdb`jlhbd y db vbrjbhzb fukstrbd sk pukmk bprk`jbr kh kd mkhlfjhbmlr.

Kakfpdl1 Pjgujkhml utjdjzbhml dls mbtls mk dbs kstbturbs mk dls bdufhl y kd kakfpdl 2.3, b `lhtjhub`jðh mk `bd`udb db vbrjbhzb.

h 6 09

 6 2.;>

̋

  −6092   2.;> 

 

69.99=:3: (fts)3 

Db ^brjbhzb mk Mbtls Bgrupbmls ^brjbhzb plcdb`jlhbd

  −66  . 

 

 

  Kh mlhmk1



 6 Db ^brjbhzb mk db plcdb`jðh

H  6  6

`bhtjmbm mk mbtls kh db plcdb`jðh

F` j  6

fbr`b mk `dbsk j  

n   j 6 nrk`ukh`jb mk `dbsk mk db `dbsk

j  

» 6 fkmjb plcdb`jlhbd plcdb`jlhbd

^brjbhzb ^b rjbhzb Fukstr bd

P 3  6

  ̋   −         6    2 2 . 

 

Db ^brjbhzb mk db fukstrb

h 6 `bhtjmbm mk mbtls kh db fukstrb F` jj  6    6

̋

fbr`b mk `dbsk mk db `dbsk

j

 6 fkmjb mk db fukstrb

n   `dbsk  j   j 6 nrk`ukh`jb mk `dbsk mk db `dbsk 

Kakfpdl1 Pj dls mbtls bgrupbmls mk db kstbturb mk dls bdufhls nukrbh mbtls plcdb`jlhbdks, db kd kakfpdl 2.0 skrjb db cbsk pbrb `bd`udbr db vbrjbhzb mk mj`els mbtls.

H  6  6

09

» 6 2.>:

 

   −    6 09 2.>:. 

 

 6 9.99>> (fts)3 



2.;.0 Mksvjb`jðh Kstâhmbr   Db vbrjbhzb sk kxprksb kh uhjmbmks `ubmrâtj`bs, bsæ `lfl sk pukmk bprk`jb kh dls pbsbmls kakfpdls, kstl mkcjml b quk `lfl yb sk fkh`jlhl, tlmbs dbs mjnkrkh`jbs khtrk dls mbtls y db fkmjb sk kdkvbh bd `ubmrbml. Ks mjnæ`jd plmkr jhtkrprktbr l jfbgjhbrsk uh rksudtbml kh uhjmbmks `ubmrâtj`b, ks plr ksl quk, sj b db vbrjbhzb sk dk kxtrbk db rbæz `ubmrbmb sk tkhmræbh uhjmbmks djhkbdks, dbs `ubdks slh mk nâ`jd khtkhmjfjkhtl. Yuks cjkh, db rbæz `ubmrbmb mk db vbrjbhzb ks db Mksvjb`jðh Kstâhmbr l Mksvjb`jðh ]æpj`b.  ]æpj`b.  Kstb fkmjmb ks mk sufb jfplrtbh`jb y ks db quk kh tkfbs y uhjmbmks plstkrjlrks sk utjdjzbrb `lfl ekrrbfjkhtb kstbmæstj`b pbrb khtkhmkr y `bd`udbr ltrls `lh`kptls.

Mksvjb`jðh Kstâhmbr mk Mbtls Hl Bgrupbmls. Câsj`bfkhtk dbs nlrfudbs mk db vbrjbhzb y db mksvjb`jðh kstâhmbr slh dbs fjsfbs, mjnkrkh`jbmbs ñhj`bfkhtk kh db pltkh`jb mk `bmb uhb.

Mksvjb`jðh Mk svjb`jðh Kstâhmbr Kstâhmbr Ylcdb`jlhbd

  −         6       

 

Mksvjb`jðh Mk svjb`jðh Kstâhmbr Kstâhmbr Fukstrbd Fukstrbd

     −62 2 ̋

 

 

  mkd kakfpdl 2.5 db mksvjb`jðh kstâhmbr pbrb ksk `lhauhtl mk mbtls ks1 P  6  6 P

∑ 9.9.99=:3: fts

 

6 9.97 fts. 

Mksvjb`jðh Mk svjb`jðh Kstâhmbr Kstâhmbr pbrb Mbtls Bgrup bmls

Mksvjb`jðh Mk svjb`jðh Kstâhmbr Kstâhmbr Ylcdb`jlhbd

  −         6      . 

 

Mksvjb`jðh Mk svjb`jðh Kstâhmbr Kstâhmbr Fukstrbd Fukstrbd

   −66 2 2̋. 

 

Ybrtjkhml mkd kakfpdl 2.:, db mksvjb`jðh kstâhmbr pbrb pbrb ksk `lhauhtl mk mbtls ks1

 ∑ 9.9.99>>  6

fts.

fts.

  9.95

 

Db mksvjb`jðh kstâhmbr hls pukmk jhmj`br `lfl sk `lfplrtbh dls mbtls bdrkmkmlr mk uhb fkmjmb mk tkhmkh`jb `khtrbd y `lfl kh l`bsjlhks b pksbr mk tkhkr db fjsfb fkmjb kd grbml mk mjspkrsjðh ks mjstjhtl. Pk pumjkrb tkhkr uhb fukstrb kh quk su fkmjb brjtfêtj`b nukrb = y quk dls mbtls ls`jdbrbh khtrk 0 y ;, y ltrb fukstrb quk su fkmjb brjtfêtj`b nukrb = y quk sus mbtls ls`jdbrbh khtrk kht rk 9 y 5. Buhquk bfcbs tjkhkh kd fjsfl vbdlr kh su fkmjb, tjkhkh mjstjhtb mjstrjcu`jðh mk dls mbtls, mk bquæ db jfplrtbh`jb mk tkhkr uhb fkmjmb quk hls jhmjquk kd grbml mk mjspkrsjðh mk dls mbtls `lh rkspk`tl b db fkmjb.