Chapter 9 - Simple Regression

September 8, 2022 | Author: Anonymous | Category: N/A
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Qtntjstjas bkr Eusjdgss ndc Gakdk`jas Amnptgr :

Qj`phg Ugorgssjkd

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5 >

 

Amnptgr Oknhs Nbtgr ak`phgtjdo tmjs amnptgr, yku smkuhc eg nehg tk6  

Gxphnjd tmg tgst akrrghntjkd pgrbkr` n mypktmgsjs bkr zgrkakgbbjajgdt pkpuhntjkdndc akrrghntjkd



Gxphnjd tmg sj`phg hjdgnr rgorgssjkd `kcgh



Ketnjd ndc jdtgrprgt tmg sj`phg hjdgnr rgorgssjkd



gquntjkd bkr n sgt kb cntn Cgsarjeg U5 ns n `gnsurg kb gxphndntkry pkwgr kb tmg rgorgssjkd `kcgh



Wdcgrstndc tmg nssu`ptjkds egmjdc rgorgssjkd ndnhysjs

Amnp >5-5

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Amnptgr Oknhs (akdtjdugc)

Nbtgr ak`phgtjdo tmjs amnptgr, yku smkuhc eg nehg tk6  







Gxphnjd `gnsurgs kb vnrjntjkd ndc cgtgr`jdg wmgtmgr tmg jdcgpgdcgdt vnrjnehg js sjodjbjandt Anhauhntg ndc jdtgrprgt akdbjcgdag jdtgrvnhs bkr tmg rgorgssjkd akgbbjajgdts Wsg n rgorgssjkd gquntjkd gquntjkd bkr prgcjatjkd Bkr` bkrganst jdtgrvnhs nrkudc nd gstj`ntgc Z vnhug bkr n ojvgd ]



Wsg ornpmjanhndnhysjs ndnhysjs tk rgakodjzg pktgdtjnh prkehg`s jd rgorgssjkd

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-1

 

Akrrghntjkd Ndnhysjs 

Akrrghntjkd ndnhysjs js usgc tk `gnsurg Akrrghntjkd ndnhysjs strgdotm kb tmg nsskajntjkd (hjdgnr rghntjkdsmjp) egtwggd twk vnrjnehgs 

Akrrghntjkd js kdhy akdagrdgc wjtm strgdotm kb tmg rghntjkdsmjp



Dk anusnh gbbgat js j`phjgc wjtm akrrghntjkd



Akrrghntjkd wns bjrst prgsgdtgc jd Amnptgr 1

Amnp >5-3

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Akrrghntjkd Ndnhysjs 



Smg pkpuhntjkd akrrghntjkd akgbbjajgdt js akgbbjajgdt js cgdktgc ώ (tmg Orggl hgttgr rmk) Smg sn`phg akrrghntjkd akgbbjajgdt js akgbbjajgdt js

r   

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wmgrg

(x j  x)(yj  y)  

s xy

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Amnp >5-7

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Mypktmgsjs Sgst bkr Akrrghntjkd 

Sk tgst tmg duhh mypktmgsjs kb dk hjdgnr nsskajntjkd,

M9 6  ώ  9   tmg tgst stntjstja bkhhkws tmg Qtucgdt‚s t cjstrjeutjkd wjtm (d ‑ 5 ) cgorggs kb brggck`6

t

r  (d  5) (> r  ) 5

Amnp >5-2

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Cgajsjkd Uuhgs Mypktmgsjs Mypktmgsjs Sgst bkr Akrrghntjkd Hkwgr-tnjh tgst6

Wppgr-tnjh tgst6

Swk-tnjh tgst6

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Ugfgat M9 jb t 4 td-5, 

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Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-?

 

Jdtrkcuatjkd tk Ugorgssjkd Ndnhysjs 

Ugorgssjkd ndnhysjs js ndnhysjs js usgc tk6 

Yrgcjat tmg vnhug kb n cgpgdcgdt vnrjnehg ensgc kd



tmg vnhug kb nt hgnst kdg jdcgpgdcgdt vnrjnehg Gxphnjd tmg j`pnat kb amndogs jd nd jdcgpgdcgdt vnrjnehg kd tmg cgpgdcgdt vnrjnehg

vnrjnehg6  tmg vnrjnehg wg wjsm tk gxphnjd Cgpgdcgdt vnrjnehg6  vnrjnehg))   (nhsk anhhgc tmg gdckogdkus vnrjnehg Jdcgpgdcgdt vnrjnehg6  vnrjnehg6  tmg vnrjnehg usgc tk gxphnjd tmg cgpgdcgdt vnrjnehg  

(nhsk anhhgc tmg gxkogdkus vnrjnehg) vnrjnehg)

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Hjdgnr Ugorgssjkd @kcgh 





Smg rghntjkdsmjp egtwggd ] ndc Z js cgsarjegc ey n hjdgnr budatjkd Amndogs jd Z nrg nssu`gc tk eg anusgc ey anusgc ey amndogs jd ] Hjdgnr rgorgssjkd pkpuhntjkd gquntjkd `kcgh

Zj 



  ε>x j  η j ε9 

_mgrg 9  ndc >  nrg tmg pkpuhntjkd `kcgh akgbbjajgdts ndc   js n rndck` grrkr tgr`.

Amnp >5-=

Amnp >5-:

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Qj`phg Hjdgnr Ugorgssjkd @kcgh Smg pkpuhntjkd rgorgssjkd `kcgh6 Ykpuhntjkd

Ykpuhntjkd Cgpgdcgdt Rnrjnehg

Z jdtgragpt

j

Z

Jdcgpgdcgdt Rnrjnehg

Qhkpg Akgbbjajgdt



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9

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j

η

j

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Amnp >5->9

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Qj`phg Hjdgnr Ugorgssjkd @kcgh (akdtjdugc)

Z

  ε>]j  η j Zj  ε9 

Kesgrvgc kb Z bkrRnhug ]j

ηj Yrgcjatgc Rnhug

Qhkpg 0 ε> Undck` Grrkr bkr tmjs ]j vnhug

kb Z bkr ]j  Jdtgragpt 0 ε9 

]j

]

Amnp >5->>

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Qj`phg Hjdgnr Ugorgssjkd Gquntjkd Smg sj`phg hjdgnr rgorgssjkd gquntjkd prkvjcgs nd gstj`ntg kb  kb tmg pkpuhntjkd rgorgssjkd hjdg gstj`ntg Gstj`ntgc (kr prgcjatgc) y vnhug bkr kesgrvntjkd j

Gstj`ntg kb tmg rgorgssjkd

Gstj`ntg kb tmg rgorgssjkd shkpg

jdtgragpt

yj ˂

Rnhug kb x bkr kesgrvntjkd j

e9  e>x j

 

Smg jdcjvjcunh rndck` grrkr tgr`s g j mnvg n `gnd kb zgrk

gj  ( y j - y˂ j )   y j - (e9  e>x j )

Amnp >5->5

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Hgnst Qqunrgs Gstj`ntkrs 

e9  ndc e>  nrg ketnjdgc ey bjdcjdo tmg vnhugs kb e9  ndc e>  tmnt `jdj`jzg tmg su` kb tmg squnrgc cjbbgrgdags egtwggd y ndc y˂ 6 `jd  QQG  `jd  gj5 5

 `jd

(y j y˂ j )

  `jd  \y  (e j

5  e x )P 9 > j

Cjbbgrgdtjnh anhauhus js usgc tk ketnjd tmg akgbbjajgdt gstj`ntkrs e9 ndc e> tmnt `jdj`jzg QQG

Amnp >5->1

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Hgnst Qqunrgs Gstj`ntkrs (akdtjdugc) 

Smg shkpg akgbbjajgdt gstj`ntkr js d

 (x  x)(y  y) j

e> 

j> d



j

 

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e9   y  e>x 

Smg rgorgssjkd hjdg nhwnys okgs tmrkuom tmg `gnd x, y

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5->3

 

Bjdcjdo tmg Hgnst Qqunrgs Gquntjkd 

Smg akgbbjajgdts e9  ndc e> , ndc ktmgr rgorgssjkd jd tmjs amnptgr, wjhh eg bkudc usjdorgsuhts n ak`putgr  





Mndc anhauhntjkds nrg tgcjkus Qtntjstjanh rkutjdgs nrg eujht jdtk Gxagh Ktmgr stntjstjanh ndnhysjs skbtwnrg and eg usgc

Amnp >5->7

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Hjdgnr Ugorgssjkd @kcgh  Nssu`ptjkds 





Smg trug rghntjkdsmjp bkr` js hjdgnr (Z js n hjdgnr budatjkd kb ], phus rndck` grrkr) Smg grrkr tgr`s, ηj nrg jdcgpgdcgdt kb tmg x vnhugs Smg grrkr tgr`s nrg rndck` vnrjnehgs wjtm `gnd 9 ndc akdstndt vnrjndag, ρ5  (tmg akdstndt vnrjndag prkpgrty js anhhgc mk`ksagcnstjajty mk`ksagcnstjajty))  5

G\η j P  9 ndc G\η j  P  ρ 5 

bkr (j  >, , d)

Smg rndck` grrkr tgr`s, ηj, nrg dkt akrrghntgc wjtm kdg ndktmgr, sk tmnt G\η jη f P  9

 

bkr nhh  j   f

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5->2

 

Jdtgrprgtntjkd kb tmg Qhkpg ndc tmg Jdtgragpt 

e9  js tmg gstj`ntgc nvgrnog vnhug kb y wmgd tmg vnhug kb x js zgrk (jb x 0 9 js

jd tmg rndog kb kesgrvgc x vnhugs) 

e>  js tmg gstj`ntgc amndog jd tmg nvgrnog vnhug kb y ns n rgsuht kb n kdg-udjt amndog jd x

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5->?

 

Qj`phg Hjdgnr Ugorgssjkd Gxn`phg 

 N rgnh gstntg nogdt wjsmgs tk gxn`jdg tmg rghntjkdsmjp egtwggd tmg sghhjdo prjag kb n mk`g ndc jts sjzg (`gnsurgc jd squnrg bggt)



 N rndck` sn`phg kb >9 mkusgs js sghgatgc Cgpgdcgdt vnrjnehg (Z) 0 mkusg prjag jd $>999s Jdcgpgdcgdt vnrjnehg (]) 0 squnrg bggt 



Amnp >5->=

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Qn`phg Cntn bkr Mkusg Yrjag @kcgh Mkusg Yrjag jd $>999s (Z)

Qqunrg Bggt (])

537

>399

1>5 5?:

>299 >?99

19=

>=?7

>::

>>99

5>: 397

>779 5179

153

5379

1>:

>357

577

>?99

Amnp >5->:

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Ornpmjanh Yrgsgdtntjkd 

Mkusg prjag `kcgh6 santtgr phkt

379    )   s 399    9 179    9    9 199    >    $    ( 579   g   a    j   r 599    Y   g >79   s   u >99   k    M 79

9 9

799

>999

>799

5999

Qqunrg Bggt

5799

1999

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Ugorgssjkd Wsjdo Gxagh 

Skkhs / Cntn Ndnhysjs / Ugorgssjkd

Amnp >5-59

Amnp >5-5>

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxagh Kutput Ugorgssjkd Qtntjstjas

@uhtjphg U

9 .? 2 5 > >

U Qqunrg

9 .7 = 9 = 5

Ncfustgc U Qqunrg

9 .7 5 = 3 5

Qtndcnrc Grrkr Kesgrvntjkds

Smg rgorgssjkd gquntjkd js6 mkusg prjag  :=.53=11    9.>9:?? (squnrg bggt)

3 > .1 1 9 1 5 >9

NDKRN  

cb

QQ

@Q

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Ugorgssjkd

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>=:13.:13=

>=:13.:13=

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>?9=.>:7?

Akgbbjajgdts

Jdtgragpt Qqunrg Bggt

Qtndcnrc Grrkr

t Qtnt

Y-vnhug

Qjodjbjandag B

 

9 .9 > 9 1 :

Hkwgr :7%

Wppgr :7%

: = .5 3 = 1 1

7=.9113=

>.2:5:2

9.>5=:5

-17.7??59

515.9?1=2

9 .> 9 : ? ?

9.915:?

1.15:1=

9.9>91:

9 .9 1 1 ? 3

9.>=7=9

Amnp >5-55

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Ornpmjanh Yrgsgdtntjkd 

Mkusg prjag `kcgh6 santtgr phkt ndc rgorgssjkd hjdg 379 399

Jdtgragpt 0 :=.53=

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Qhkpg 0 9.>9:??

9 9

79 9

>999

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59 9 9

57 9 9

1 9 99

Qqunrg Bggt    9.>9:?? (squnrg bggt) mkusg prjag  :=.53=11

Amnp >5-51

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdtgrprgtntjkd kb tmg Jdtgragpt, e9 mkusg prjag  :=.53=11   



9.>9:?? (squnrg bggt)

e9 js tmg gstj`ntgc nvgrnog vnhug kb Z wmgd tmg vnhug kb ] js zgrk (jb ] 0 9 js jd tmg rndog kb kesgrvgc ] vnhugs) 

Mgrg, dk mkusgs mnc 9 squnrg bggt, sk e9 0 :=.53=11  fust jdcjantgs jdcjantgs tmnt, bkr mkusgs mkusgs wjtmjd tmg rndog kb sjzgs kesgrvgc, $:=,53=.11 $:=,53=.11 js tmg pkrtjkd kb tmg mkusg prjag dkt gxphnjdgc ey squnrg bggt

Amnp >5-53

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdtgrprgtntjkd kb tmg Qhkpg Akgbbjajgdt, e> mkusg prjag  :=.53=11   



9.>9:?? (squnrg bggt)

e> `gnsurgs tmg gstj`ntgc amndog jd tmg nvgrnog vnhug kb Z ns n rgsuht kb n kdgudjt amndog jd ] 

Mgrg, e> 0 .>9:?? tghhs us tmnt tmg nvgrnog vnhug kb n mkusg jdargnsgs ey .>9:??($>999) 0 $>9:.??, kd nvgrnog, bkr gnam nccjtjkdnh kdg squnrg bkkt kb sjzg

Amnp >5-57

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

@gnsurgs kb Rnrjntjkd 

Sktnh vnrjntjkd js `ncg up kb twk pnrts6

QQS    QQU      QQG

Sktnh Qu` kb Qqunrgs

QQS 



Ugorgssjkd Qu` kb Qqunrgs

Grrkr Qu` kb Qqunrgs

QQU    (y˂  j  y)5





(y  j  y)5

QQG 

 (y  j  y˂ j )5

wmgrg6   y

0 Nvgrnog vnhug kb tmg cgpgdcgdt vnrjnehg

yj 0 Kesgrvgc vnhugs kb tmg cgpgdcgdt vnrjnehg y˂

  j 

Yrgcjatgc vnhug kb y bkr tmg ojvgd xj vnhug

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-52

 

@gnsurgs kb Rnrjntjkd (akdtjdugc) 

QQS 0 tktnh su` kb squnrgs 



`gnd, y QQU 0 rgorgssjkd su` kb squnrgs 



@gnsurgs tmg vnrjntjkd kb tmg yj vnhugs nrkudc tmgjr

Gxphnjdgc vnrjntjkd nttrjeutnehg tk tmg hjdgnr

rghntjkdsmjp egtwggd x ndc y QQG 0 grrkr su` kb squnrgs 

Rnrjntjkd nttrjeutnehg tk bnatkrs ktmgr tmnd tmg hjdgnr rghntjkdsmjp egtwggd x ndc y

Amnp >5-5?

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

@gnsurgs kb Rnrjntjkd (akdtjdugc)

 Z yj



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(yj - yj )5 

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y

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xj

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]

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-5=

 

Akgbbjajgdt kb Cgtgr`jdntjkd, U5 



Smg akgbbjajgdt kb cgtgr`jdntjkd js cgtgr`jdntjkd js tmg pkrtjkd kb tmg tktnh vnrjntjkd jd tmg cgpgdcgdt vnrjnehg tmnt js gxphnjdgc ey vnrjntjkd jd tmg jdcgpgdcgdt vnrjnehg Smg akgbbjajgdt kb cgtgr`jdntjkd js nhsk anhhgc U-squnrgc ndc  ndc js cgdktgc ns U5 U-squnrgc QQU rgorgssjkd su` kb squnrgs  U   QQS tktnh su` kb squnrgs 5

dktg6

9  U5  >

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxn`phgs kb Npprkxj`ntg r 5  Rnhugs  Z r 5 0 >

r 5 0 >

]

Ygrbgat hjdgnr rghntjkdsmjp egtwggd ] ndc Z6 >99% kb tmg gxphnjdgc eyvnrjntjkd vnrjntjkdjdjdZ]js

 Z

r 5 0 >

]

Amnp >5-5:

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxn`phgs kb Npprkxj`ntg r 5  Rnhugs  Z 9 8 r 5 8 >

]  Z

_gnlgr hjdgnr rghntjkdsmjps egtwggd ] ndc Z6 Qk`g eutjd dkt nhhgxphnjdgc kb tmg vnrjntjkd Z js ey vnrjntjkd jd ]

]

Amnp >5-19

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxn`phgs kb Npprkxj`ntg r 5  Rnhugs r 5 0 9  Z

Dk hjdgnr rghntjkdsmjp egtwggd ] ndc Z6

r   0 9 5

]

Smg vnhug ckgs dkt cgpgdc kdkb ].Z (Dkdg kb tmg vnrjntjkd jd Z js gxphnjdgc ey vnrjntjkd jd ])

Amnp >5-1>

Amnp >5-15

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxagh Kutput U  5

Ugorgssjkd Qtntjstjas

QQU QQS



>=:13.:13= 15299.7999

 9.7=9=5

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9 .? 2 5 > >

U Qqunrg

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Ncfustgc U Qqunrg

9 .7 5 = 3 5

7=.9=% kb tmg vnrjntjkd jd

3 > .1 1 9 1 5 >9

mkusg prjags js gxphnjdgc ey vnrjntjkd jd squnrg bggt

Qtndcnrc Grrkr Kesgrvntjkds

NDKRN  

cb

QQ

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:

15299.7999

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Qtndcnrc Grrkr

t Qtnt

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9 .9 > 9 1 :

Hkwgr :7%

Wppgr :7%

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7=.9113=

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515.9?1=2

9 .> 9 : ? ?

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1.15:1=

9.9>91:

9 .9 1 1 ? 3

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Amnp >5-11

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Akrrghntjkd ndc U5 

Smg akgbbjajgdt kb cgtgr`jdntjkd, U5, bkr n sj`phg rgorgssjkd js gqunh tk tmg sj`phg akrrghntjkd squnrgc

U    r  5

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Amnp >5-13

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gstj`ntjkd kb @kcgh Grrkr Rnrjndag 

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s    s5 js anhhgc tmg stndcnrc grrkr kb tmg gstj`ntg g

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Amnp >5-17

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxagh Kutput Ugorgssjkd Qtntjstjas

@uhtjphg U

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15299.7999

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515.9?1=2

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9 .9 1 1 ? 3

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Amnp >5-12

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Ak`pnrjdo Qtndcnrc Grrkrs sg js n `gnsurg kb tmg vnrjntjkd kb kesgrvgc y vnhugs brk` tmg rgorgssjkd hjdg  Z

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tmg $599 - $199L rndog Amnp >5-1?

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdbgrgdags Nekut tmg Ugorgssjkd @kcgh 

Smg vnrjndag kb tmg rgorgssjkd shkpg akgbbjajgdt (e>) js gstj`ntgc ey se  5

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Amnp >5-1=

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxagh Kutput Ugorgssjkd Qtntjstjas

@uhtjphg U

9 .? 2 5 > >

U Qqunrg

9 .7 = 9 = 5

Ncfustgc U Qqunrg

9 .7 5 = 3 5

Qtndcnrc Grrkr Kesgrvntjkds

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9 .9 1 1 ? 3

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Amnp >5-1:

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Ak`pnrjdo Qtndcnrc Grrkrs kb tmg Qhkpg  Qe> js n `gnsurg kb tmg vnrjntjkd jd tmg shkpg kb rgorgssjkd hjdgs brk` cjbbgrgdt pkssjehg sn`phgs  Z

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Amnp >5-39

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdbgrgdag nekut tmg Qhkpg6 t Sgst 

t tgst bkr n pkpuhntjkd shkpg 



Js tmgrg n hjdgnr rghntjkdsmjp egtwggd ] ndc Z;

Duhh ndc nhtgrdntjvg mypktmgsgs   M96 ε> 0 9   M>6 ε>  9



(dk hjdgnr rghntjkdsmjp) (hjdgnr rghntjkdsmjp ckgs gxjst)

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t

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grrkr kb tmg shkpg Amnp >5-3>

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdbgrgdag nekut tmg Qhkpg6 t Sgst (akdtjdugc) Mkusg Yrjag jd $>999s (y)

Qqunrg Bggt (x)

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5?:

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153

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Smg shkpg kb tmjs `kcgh js 9.>9:= Ckgs nbbgat squnrg jts snhgsbkktnog prjag; kb tmg mkusg

Amnp >5-35

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdbgrgdags nekut tmg Qhkpg6 t Sgst Gxn`phg e>

se>

M96 ε> 0 9

Brk` Gxagh kutput6

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Akgbbjajgdts : =. 53 =1 1

Qtndcnrc Grrkr 7 =. 91 13 =

t Qtnt >.2:5:2

Y-vnhug 9 . >5= :5

9. > 9: ??

9.915:?

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Jdtgragpt Qqunrg Bggt

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Amnp >5-31

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdbgrgdags nekut tmg Qhkpg6 t Sgst Gxn`phg (akdtjdugc)

Sgst Qtntjstja6 t 0 1.15: M96 ε> 0 9

Brk` Gxagh kutput6

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Qqunrg Bggt

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e>

se>

t

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Qtndcnrc Grrkr 7 =. 91 13 =

t Qtnt >.2:5:2

Y-vnhug 9 . >5= :5

9. > 9: ??

9.915:?

1.15:1=

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tmnt squnrg bkktnog nbbgats

5.1929

5.1929 1.15:

mkusg prjag Amnp >5-33

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Jdbgrgdags nekut tmg Qhkpg6 t Sgst Gxn`phg (akdtjdugc)

 Y-vnhug 0 9.9>91: M96 ε> 0 9 M>6 ε>  9

Brk` Gxagh kutput6   Jdtgragpt

Qqunrg Bggt

Smjs js n twk-tnjh tgst, sk tmg p-vnhug js Y(t 4 1.15:)+Y(t 8 -1.15:) 0 9.9>91: (bkr = c.b.)

Y-vnhug

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Qtndcnrc Grrkr 7 =. 91 13 =

t Qtnt >.2:5:2

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9. > 9: ??

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tmnt squnrg bkktnog nbbgats

mkusg prjag Amnp >5-37

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Akdbjcgdag Jdtgrvnh Gstj`ntg bkr tmg Qhkpg Akdbjcgdag Jdtgrvnh Gstj`ntg kb tmg Qhkpg6

e>  t d5,ξ/5se>    ε>  e>  t d5,ξ/5se> c.b. 0 d - 5

Gxagh Yrjdtkut bkr Mkusg Yrjags6  

Jdtgragpt Qqunrg Bggt

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t Qtnt

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Amnp >5-32

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Akdbjcgdag Jdtgrvnh Gstj`ntg bkr tmg Qhkpg (akdtjdugc)  

Akgbbjajgdts

Jdtgragpt Qqunrg Bggt

Qtndcnrc Grrkr

t Qtnt

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:=.53=11

7=.9113=

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-17.7??59

515.9?1=2

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9.915:?

1.15:1=

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Qjdag tmg udjts kb tmg mkusg prjag vnrjnehg js $>999s, wg nrg :7% akdbjcgdt tmnt tmg nvgrnog j`pnat kd snhgs prjag js egtwggd $11.?9 ndc $>=7.=9 pgr squnrg bkkt kb mkusg sjzg Smjs :7% akdbjcgdag jdtgrvnh ckgs dkt jdahucg 9. 9. Akdahusjkd6 Akdahusjkd6 Smgrg  Smgrg js n sjodjbjandt rghntjkdsmjp egtwggd mkusg prjag ndc squnrg bggt nt tmg .97 hgvgh kb sjodjbjandag

Amnp >5-3?

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

B-Sgst bkr Qjodjbjandag 

B Sgst stntjstja6 wmgrg

 

@QU B  @QG @QU 

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QQG @QG  d  l  > >)  wmgrg B bkhhkws nd B cjstrjeutjkd c jstrjeutjkd wjtm l  du`grntkr   ndc (d ‑ l - >)  cgdk`jdntkr cgorggs kb brggck` brggck`   (l 0 tmg du`egr kb k b jdcgpgdcgdt vnrjnehgs jd tmg rgorgssjkd `kcgh)

Amnp >5-3=

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gxagh Kutput Ugorgssjkd Qtntjstjas

@uhtjphg U

9 .? 2 5 > >

U Qqunrg

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Ncfustgc U Qqunrg

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Qtndcnrc Grrkr Kesgrvntjkds

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15299.7999

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Amnp >5-3:

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

B-Sgst bkr Qjodjbjandag (akdtjdugc)

M96 ε> 0 9

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@QU

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B.97 0 7.15 Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Yrgcjatjkd 



Smg rgorgssjkd gquntjkd and eg usgc tk prgcjat n vnhug bkr y, ojvgd n pnrtjauhnr x Bkr n spgajbjgc vnhug, xd+> , tmg prgcjatgc vnhug js

y d>  e9   e>x d> ˂

Amnp >5-79

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Yrgcjatjkds Wsjdo Ugorgssjkd Ndnhysjs Yrgcjat tmg prjag bkr n mkusg wjtm 5999 squnrg bggt6

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1>?.=7

Smg prgcjatgc prjag bkr n mkusg wjtm 5999 squnrg bggt js 1>?.=7($>,999s) 0 $1>?,=79

Amnp >5-7>

Amnp >5-75

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Ughgvndt Cntn Undog 

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379    )   s    9

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Qqunrg Bggt Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-71

 

Gstj`ntjdo @gnd Rnhugs ndc Yrgcjatjdo Jdcjvjcunh Rnhugs Oknh6 Bkr` jdtgrvnhs jdtgrvnhs nrkudc y tk gxprgss udagrtnjdty nekut tmg vnhug kb y bkr n ojvgd x j Akdbjcgdag Jdtgrvnh bkr tmg gxpgatgc vnhug kb vnhug  kb y, ojvgd xj 

y 0 e9+e>xj

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y

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  xj

]

Amnp >5-73

 

Akdbjcgdag Jdtgrvnh bkr tmg Nvgrnog Z, Ojvgd ] Akdbjcgdag jdtgrvnh gstj`ntg bkr tmg gxpgatgc vnhug kb y  ojvgd n pnrtjauhnr xj Akdbjcgdag jdtgrvnh bkr G(Zd> | ]d> ) 6

 



 d >

t

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Dktjag tmnt tmg bkr`uhn jdvkhvgs tmg tgr` (x d >  x)

5

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xd+> js brk` tmg `gnd,  x Amnp >5-77

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Yrgcjatjkd Jdtgrvnh bkr nd Jdcjvjcunh Z, Ojvgd ] Akdbjcgdag jdtgrvnh gstj`ntg bkr nd natunh kesgrvgc vnhug kb y  ojvgd n pnrtjauhnr xj Akdbjcgdag jdtgrvnh bkr  y˂ d> 6

 

y˂ d>  t d5,ξ/5sg

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Smjs gxtrn tgr` nccs tk tmg jdtgrvnh wjctm tk rgbhgat tmg nccgc udagrtnjdty bkr nd jdcjvjcunh ansg

Amnp >5-72

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gstj`ntjkd kb @gnd Rnhugs6 Gxn`phg Akdbjcgdag Jdtgrvnh Gstj`ntg bkr G(Zd+>|]d+>) Bjdc tmg :7% akdbjcgdag jdtgrvnh bkr tmg `gnd prjag kb 5,999 squnrg-bkkt mkusgs Yrgcjatgc Yrjag yj 0 1>?.=7 ($>,999s) 5

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Amnp >5-7?

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Gstj`ntjkd kb Jdcjvjcunh Rnhugs6 Gxn`phg 

Akdbjcgdag Jdtgrvnh Gstj`ntg bkr yd+> Bjdc tmg :7% akdbjcgdag jdtgrvnh bkr nd jdcjvjcunh mkusg wjtm 5,999 squnrg bggt Yrgcjatgc Yrjag yj 0 1>?.=7 ($>,999s) 5 ˂y d>  t d->,ξ/5 s g > >   (]j  ])  1>?.=7  >95.5= 5 d (]j  ])



Smg akdbjcgdag jdtgrvnh gdcpkjdts nrg 5>7.79 ndc 359.9?, kr brk` $5>7,799 tk $359,9?9

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-7=

 

Bjdcjdo Akdbjcgdag ndc Yrgcjatjkd Jdtgrvnhs jd Gxagh 

Jd Gxagh, usg YMQtnt | rgorgssjkd | sj`phg hjdgnr rgorgssjkd … 

Amgal tmg ‘akdbjcgdag ndc prgcjatjkd jdtgrvnh bkr x0‐  x0‐  ekx ndc gdtgr tmg x-vnhug ndc akdbjcgdag hgvgh cgsjrgc

Amnp >5-7:

Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.  

Bjdcjdo Akdbjcgdag ndc Yrgcjatjkd Jdtgrvnhs jd Gxagh (akdtjdugc)

Jdput vnhugs

 y

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bkr jdcjvjcunh yd+> Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-29

 

Ornpmjanh Ndnhysjs 







Smg hjdgnr rgorgssjkd `kcgh js ensgc kd `jdj`jzjdo tmg su` kb squnrgc grrkrs Jb kuthjgrs gxjst, tmgjr pktgdtjnhhy hnrog squnrgc grrkrs `ny mnvg n strkdo jdbhugdag kd tmg bjttgc rgorgssjkd hjdg Eg surg tk gxn`jdg ykur cntn ornpmjanhhy bkr kuthjgrs ndc gxtrg`g pkjdts Cgajcg, ensgc kd ykur `kcgh ndc hkoja, wmgtmgr tmg gxtrg`g pkjdts smkuhc rg`njd kr eg rg`kvgc

tmg gxtrg`g pkjdts smkuhc rg`njd kr eg rg`kvgc Qtntjstjas bkr Eusjdgss ndc Gakdk`jas, 2g © 599? Ygnrskd Gcuantjkd, Jda.

Amnp >5-2>

 

Amnptgr Qu``nry 

Jdtrkcuagc tmg hjdgnr rgorgssjkd `kcgh



Ugvjgwgc akrrghntjkd ndc tmg nssu`ptjkds kb



hjdgnr rgorgssjkd Cjsaussgc gstj`ntjdo tmg sj`phg hjdgnr rgorgssjkd akgbbjajgdts







Cgsarjegc `gnsurgs kb vnrjntjkd Cgsarjegc jdbgrgdag nekut tmg shkpg  Nccrgssgc gstj`ntjkd gstj`ntjkd kb `gnd vnhugs ndc

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