Mathadabi11P1 Book

7/23/2019 Mathadabi11P1 Book

1/92

WOU bL UIMbF 5D `U .dJ b qI bL

uHR*

U?O{U?|d?

uU _ nBKwbMH U WOU uKF

5DK W

wUF rOKF WOd


2/92

5DK W w wUF rOKF WOd d wb UF s b UNb w uU _ nBK UO{Ud U fb

U????????O{Ud UNM* wMu odHU??H UMN jO{ UO UIMbF 5D

d??U UUO WH?OK wK bL qOK wKp???A qz q??I bL bL bL

Ud bz d?

UL UL WUDK Vu; bu rU wU r?OL?B

wb u UNL bO????CMUL UL G rOLB

ZUM* e?dhL(u dL ud d

WOMH dzb

UF d?hL( u rOF ZUM* W?M' fOz

5U ZUM* ed U db

WIM* WO|d WFD

? ZUM* edwUF rOKF WOd u WuH l?D uI lOL

WMb* ed WN s 5LO vK U bUF* UuOB* w ZUM* ed fU uHK 5DK tK

[email protected] uMF www.pcdc.edu.ps WOdJ WHB

WIM* WM d od

bF u qOL U d?

U u s1 UIMdUO bL `U b WU fO wUF qOL `U U


3/92

r?|bI

UuL oOI WOMODKH WOuB) wd UNM l{ d{ wUF rOKF WOd WOMu UO UM U

LN UU bF wMODK UNM UM uFA 5 tUJ cQ v wMODKH VFA

Wb* WdA u* WOLM wU ou WOdu1b rOI aOd UU wMODKH VFAKuK WOL) WD) U UN w

u wu c lL* b oI UN s w rOKFK WOzd WKOu t w UNM* WOL sLJVUD rKF* bO v_ _ rKFK jOu bB* t_ UNM* dUM b wb* UJU WU WUMjzUu s UdO d_ rKF WOK; WUI uU( Xd rKF qzU s dO v WU{

bU*VJ wMODKH ZUMLK UND s WU WKd* oOD UF c u dduD v WU{U wMI wMN* WOU uKF wLKF tdH dA U( uU _ nB

UN UI UF w uU wU nB UNM V UNFO WOU_ WKd* V iF uHBK UF rOKFK WOb* VJ lOL b XKL b wUF rOKF WOd uJWO1uI U qL wdJ rOKF UJA U w WO WOM lOu vK UOU u qLFWOu 5% ud duD WKu* UuL UOI YU* lOL w qd* ZUM* tOKOK%

wMODKH rOKF UU UI Ub v dA b_ uHBK e$ w rKF* W WOb* VJ bFwIDM qN uKQ X{d UuKF UF s tOK qLA U0 rKF rOKF WOKL w WOU eOVOU WDA_ qzUu fb ozdD qB W{ dR sLC WuM d dOu

uc* WOL) WD) U l .uIuuI sc ULKF* 5LKF* 5ud WUA0 UuM Ud UNOIM VJ WFd r w duD qbFK WKU WOd& UF WFd v v_ s UFD u d UNbbI0 e wMODKH wb* UJ WLO UO( UN wuuMJ wLKF bI w dOG lsc WOb* VJ b U w 5BB* s sJ2 b d WUA s uN s tO c UZUM* ed UN WONM0 WFd* s WFu UOKLF s rOKF w Udu dOOG ub

bOu vK qLF c su wd w d n OQ wU wULEM* UR* v dbI dJA qe bI UNF wUF rOKF WOd

ZUM* dA* wU* UNLb UJOK WuJ WU WIbB WOdF b WOb

s wU wMu qLF c U$ w XU w WOMu Wud UHJU dH u ULV yq eOL* ruN rNUA vK rdJA WOb* VJ bS uI w Wud UK qLF Uu 5UA* sd; 5HR* d ZUM*ed Wu ZUM* U' qLA tFuoOD UM bO* s WOb* VJ d w 5UA* 5FUD 5Fd* 5Ud 5LLB*

wUF rOKF WOd

ZUM* e?d

WOLKF YULK WUF

U???O


4/92

WbI

bF 5*UF tK bL(UO{Ud U s _ e' WKD UMzUM_ ULKF* 5LKF* UMze bI UdY* WbF* WCdF uD) o wbM U& WOU uK uU _ nBK

_ wMODKH UNM* WD sL{ UO{UdUL UB K* UOU* w b vK e' c qL bI

WOU* UO{Ud

tu WOU* uNH .bI U/_ UMb K* UOU* v_ bu wHWObMN WOU( K* UOU* vK eOd - U

UOF* d wU( juK WFd UMb UL UB WOU bu wdA* UL wFOD vMM* WUOF* UF vK eOd - tMOu UL

Wze WdEeOd - WOD) UMU* WLE_ WFd UMb WOU* UO{Ud WU bu wbI rNKF& WKD WUI s be WOKL UIOD WUF UFb WOD) Wd vK

UO{UdK WOHOu Vu'WO{Ud rOUH* .bI qON vK UMd bI dI* u; d WId YO s UU/_ WOO{u WK_U UH WIOb WO{Ud 5d nUF v uK bdH YLK dI* hB( b U 5F sc qzU* sUL s VUM* bF

WOKI* rNUUO WKD WdHlOL s ULKF 5LKF UdA 5dAbO* w e qJ qe' dJA bIw bI w qLF W w rNUA s UJ d w uU sc UEU;

UJ u WKD WKB tO U* bF Ud .bI tK d ZUM*edoOD l{u UJ l{ bF Ud UE* s be* vIK UMd vK sbRtK U WOMODKH UMb w

oOu w tK

uHR*


5/92

K* UOU* UFU* UOU*

K* WOU( UOU*

WOU( WKK* uL

WObMN UOU*

WObMN WKK* uL

WU sU9

U???u??;

v?

_bu

WOU* UO{Ud sdOG dOG0 v_ Wb s UMU*

WOD) Wd WOKL UIOD

UFb

WOUbu

UL UB WUOF* WF

bF* wFOD vMM*

tMOu UL uNH WFd

dA* UL

Wze WdE

WKI* u(

WU sU9

WUbu

WU WFd UOF* wFOD vMM* X% WU* oK bu UNM q bI UFb WOKI* WLOI oK bu UNM q bI UFb WOU( WLOI oK WOLKF WU( W b oK ld*


6/92

b u b???u

K* UOU*

equen es


7/92

q b_ s UL u_ s dO w UMNu

, ?

wH t WOU WH b VOd bUNO b q wL WFU WOU UL_ c s wL

wU b u bF edU t ed WOUL w _ b u bF k oU UL cJ edU t ed UNO UL w bF VOd U

? UL w WLO Ub s b WUJ dL lOD YO b t VUF UU UDL UI UM k UL

cJ u UL w u UL w L bu WOUL w UA

? UL w WLO UUL UL w WOUL b b bF q

U tuWMOF bU jL o b_ s VOd w WOUL

U

uJ UM Ub WuM vDF vK UM b uM Vd qLFK nu bUF w tKL s v_ X uM w uM nuL V WOU

U Ub wU b WOUL w _ b

SequencesUFU* UOU*


8/92

General TermWOULK UF b(

WdF lOD YO WMOF bU jL o b c uJ UEU WOUL b vu U tu

b wuM b vL Vu `O b YO tOd c b b VOd d WOUL w b

edU t ed WOULK UF

x u lU b x u ld b uJ WOUL w U

u UF b x u wuM b

U

q WdF WKD b bd dUM wU uU dN s wU U w v_ WFL u o U_ WKUIL dNA U VOd VUD U UC WFL u tH dNA s dAF sU u uJ

wK UL WFL

aU ZMU WOUL b s b sJ r bF VUD b sOuO WU WFL o t_ UF u ou UL WFL u ou ucL

WFLv_WOUWUWFdWU

uO aU

UF Ub w WOUL V U

UF b uU w rO uF WOUL b vK uBK x x

x w WOUL uJ

UNO sU b V r WOULK UF b V U

b_ UFd s U UN k WOUL b v dEMU

uJ WOULK UF b uJO


9/92

_ Ub WOU UtI c bU b W ULKJU n

WOUL s v_ WF_ b V

ULKJU uI dU t oU b b q sO WF `{u bUI dU t oU b U W tO UUC bF U WOUL w b

cJ b b rOIU iuFbUI c oOD x UbM

x UbM

x

UbM

WOULK UF b V U

k

WE

WO_ b_ WOU w uNAL WOUL q WdF bUI lC UOUL iF


10/92

qzU? sU9

WOU UOUL s v_ WL b V

?

WOU UOULK UF b V

?

s b UF Ub d WOU UF Ub WOU

sOOUL w UNH WLOI t c b u U ?

wU{ UA

WOULK UF b V

WOU W b V r WOUL w jLM nA


11/92

x x x x WKKL WUJ ed b U

x qJAU WKL WU sJL

w WKKL

UN WIdL WKKL uJ UF Ub w WOUL

uB vK VJ

U Ub sO qBH WMOF bU jL o b_ s VOd w WOUL o ULO UMd

U WOU LWKKvL WOUL S lL US U UMb sJWKKL UNId WOUL WKK vL uB

uLL e vK b ULO dI U ed b sJL dB uB KL WUJ

uB vK UNJ WKKL L

WLO d v w WLO s dOG _ UMKF uLL e U UF b UM UM kd q w Ue w

WKKL uL U U

UF b w rO uF WKKL b WUJ x x x x WKKL uJH

WKKL uL S tOK

SeriesK*


12/92

WOU KL s q uJH b

?

WOU KL s dOFK ed b

XU

?

s oI

WO KL s q uL b

qzU? sU9


13/92

wK ULO UdO s WOU UOUL eO U

UF Ub w WOUL WO_ b_ WOU

XU bI tIU b q sO dH _ WOU WOUL

L UU fO tIU b q sO dH _ WOU XO WOUL

ULMO

L UU fO tIU b q sO dH _ WOU XO WO_ b_ WOU ULMO

ULMO _ WOU XO wN w WOUL

UM dH d s UM b UGK v{UI vKU UNIL d dH UI oH

cJ YU dL s UM wU dL stK d dH tHKJ r d s U dL dH tHKJ r WdF d VU bd

U u tH oU jLM Ub dH d q s UILK UNFbO w d_ d VU V WOUL VJ UM Ub WLEM

w UNK d dH nOUJ UM tHKJ U dL d VU ZMUM

t oU b b q sO dH _ WOU WOU WOUL vL tH VK WOU UOUL d WK c ULz XU bI dU

ndF

UU bI dU t oU b b q sO dH UNO uJ w WOUL w WOU WOUL edU U t ed WOUL U XU bIL vL

Arithmetic Sequences WOU( UOU*


14/92

b WOU WOUL w UWOUL b b u bF q

U_ _ b x

WUJ oI UNU _ Ub w WOU WOUL w U b b UWOUL s v_ W b

x x

w WOUL oI

WO WOM U b

WOU( WOULK UF b(

wU jLM k UNU w WOU WOUL cQM _ b

x

x

wU b x x YU b

x x lU b

U x b W b b q kp s oI _ b vK UC oDM c

YU b wU b S UNU u WOU WOUL _ b U U tu u wuM b UF b uJ cJ dUF b

u UNU _ Ub WOU WOU W_ UF b


15/92

uJO u tLO c b dH x

x

tLO WOUL b s b bu tS Uu UO b XO WLO UL

UNM dA ld b WOU WOU b s lU b U UUNO sdAF b b r WOUL V

UNM u WOU WOULK UF b

uJO dDU sOUFL qsOUFL b w WLO uF

x WOUL

x

Arithmetic MeansWOU( U_

sd sbK UOU UD uJ UNM j_ b S WOU WOU s WOU b W Uc sUL sbFK wU ju u bF k WOU WOUL wH

t sUL sbFK wU ju u bF UL _U tu

b???????????_ q???J???A???

j????u???? c????? u???J???O??? s????b????F???? s???O??? w???U?????? j????? U??????? s??J??L?? s??O?? b??????? b???? U??? ??? U??M??M??J??L?? t???S??? s???b??? U???? W???O???U?????? W??O??U?????? b?????????_ w???L?????? W???O???U?????? W??O??U?????? W??????U???M??? W???O???U?????????L??? q???J???A??? Y???O??????

W???O???U?????????L??? w????? t????? s????U????????L???? s???b???F???K??? U???O???U?????? U?????D????? u????J???? U???N???M??? ?????? _ s????b????F???? s???O??? W???O???U?????? U??????U??????


16/92

sbF sO WOU U q U

uM vK WOUL `B sO WOU U U bM

uJ x

w U_ uJ WOU WOUL `B

WOU WOU UOUL ?

WOU WOUL w dUF b b b WOUL w tLO c b W WOUL b sL{ lI bF q ?

s q WLO b WOU WOU b XU

sbF sO WOU U q

bQ WOU WOU b_ Xu WOUL c b b WLO

WuM v{UI c UM b uM Vd UdA b w tKL nu b UM nuLK uM Vd B WM r bF UM Ub WU

qzU? sU9


17/92

v s WOB b_ lOL uFL cO s rKF VK U w

UD Q YU nB w c U U vb cO b QU v WKOK ozU u iL ru vK XKB nO UAbM rKFL tQ u `OB u

wK UL q U VcJ uJF qJA tH uLL V r ?

? b b ? lLU

x ? x ?

w UO{Ud w uNA ULU bF ULO ` c U UN V w WIdD b UMMJL cJ ? edU t ed WOU WKK b s uL b w bUI UI

UNO dO_ b ? ?

d ? lLU x ?

? YO d WIdD bUI WU sJL

? uJ tOK

?

Sum of Arith.SeriesWOU( WKK* uL

u UNU _ Ub WOU WKK b s b uL bU ? ?


18/92

WKKL b s b uL b U

_ Ub WOU WKK w WKKL

? x x ?

x

v_ WOU w Ub WU lDI d UuI b UH s r jI UcJ WU WOU w Ub WOU WOU w Ub

WU WOU w r UNFDI w WUL bv_ fL wu w WuDIL UUL uL

_ v r q v

UN _ WOU WOU c w WuDIL UUL WOU XU U_

WU WOU w r UNFD w WUL Ub x

? v_ fL wu w WuDIL UUL uL ? x x ?

b x

se c w WuDIL UUL uL uJ WOU bF _ q r dH

?

x x

tuI su bF _ v r q


19/92

U UNd{ qU U WOU WOU qJA b W uL U Ub_ c b

VOd vK w W b_ dH UNFL qU

ZM vK WLIU UNd{ qU

iuFU sOUFL q

uJ s WUFL `B ZM vK WLIU

UNM

S UbM

w WuKDL b_ uJ w WuKDL b_ uJ S UbMqK d WId

j_ b Ud dB WIdD UL q sJL YU b _ b uJO

uJO

uJ vK WLIU

v_ WIdD w UL q qLJ


20/92

WOU WOU KL uL b

?

UNM b uL UNU w WOU WKKL w _ b b

UNuL uJO UNU _ Ub WOU WKK s c V b r WQLK r

vK tLI qI w sO uBL WOB b_ uL b

b_ c b UNd{ qU U WOU WOU qJA b W uL U

uM Vd wU UM b XU uDI uM Vd _ Wd w qLF UHu b WOU WM q w UM b XU bIL ev_ WM w UM mKL b

UuM sOHuL U U WM r bF wU nuL U{UI U uLL UU _ nuL U{UI U uL uJ WM r bF

nB w U S UH w Wd wdJ s b UNO bL b w UUL WU cJ UOd YU nB w UOd wU nB w wd _WUI uH s sU nB w uuL wdJ b U

WUI w wdJ uL U uHB sOLKFL U fK UC_ v_ W uHB w wdJ XBB ?

WKD b UL WuGA WUI bUI lOL XU WKDK d_

qzU? sU9


21/92

WOU WU w v_ WU w UL s ZdNB w l{ ULU ZdN WF d

WOL wHFC WU WU W w ZdNB w l{u w UL WOL uJ YO cJ WU WU wUN WIU WU w l{u w UL uM vK w WU q w ZdNB w u{uL UL UOL kUU fO sOOU sb q sO dH _ WOU WOU XO UN b WOUL c b UMKQ

k sJWObM WOU UOUL s uM c wL WU W w dU t oU b b q sO WM

WObM WOU UOUL s q uJ V fHM

ndF

WU W dU t oU b b q sO WM UNO uJ w WOUL w WObMN WOUL

edU t ed WObMN WOUL U vL WU WMWObMN WOULK UF b(

wU UL b UM WObMN WOULK UF b bU v quK WU W wN _ WObM WOU w WOUL

_ b x wU b

x x YU b cJ x x ld b

jLM fHM b VOd s bIL qI v Uud U_ x_ b b q k x

fU b uJU tu

u WObMN WOULK UF b S UNU WObM WOU s _ b U

Geometric SequencesWObMN UOU*


22/92

UNO fU b b UNU _ Ub WObM WOU U

x x

WObMN WOUL b s tLO c b VOd U U

YO u tLO c b dH

vK WLIU x

UL

U b u tLO c b

bFL WM q hIM UO WLO XU S UM mKL UO q d UUNb s WU WM WUN w UO WLO b UN WIU WM w UNLO s

UM , x x v_ WM WUN w UO sL

UM ,

x

WOU tM WUN w UO sLcJ UM , x WU tM WUN w UO sLWObM WOU w w WM q WUN w UO UF WOU

_ Ub , UNU

WU WM WUN w UO WLO

UM , , x


23/92

wK ULO WObMN UOUL eO

, , ,

UF Ub w WOUL ?

UF Ub w WOUL s qI s be w bF UHUC ?

wK U b WObMN WOUL w U b

WObMN WOUL w rO

WOUL c V U Ub YU Ub WObM WOU

? s q WLO bQ WObM WOU ? XU

qzU? sU9

wU{ UA

? s q WLO bQ WObM WOU ? XU


24/92

WKKL UNU WObM WOU w WOUL UMd o

sJL UMS WKKL b uL WLO U U w UN WIdLwK UL uI

? lC ZM bF w WUFL wd dC

? ? ? ?ZM sOUFL dD

? XU U tu ? SZM sOUFL dD

? ? ?

YO ?

bU

u UNU _ Ub WObM WKK b s uL YO ?

`B WObMN WKKL sJ ucL bUI oOD lOD UMS XU t kUL

WObMN WKKL b s v_ W b uL b U

?

x

?

Sum of Geo. SeriesWObMN WKK* uL


25/92

b U

w WKKL

WObM WKK WKKL UU uLL U sJL UN x x

? u UNuL uJO UNU Ub

s r w UdOJ b U U WU nB q sOM bu `B UdOJ dUJ U

UU WFU WU w Ub `B r UU WU WU w UdOJ VOK

UU WU WU w UdOJ b UU nBM WU WU w UdOJ b

UU WU WU w UdOJ b

UNU _ Ub WObM WOU c WM_ dUM Ub b U b b

u lU b uJO x

x

WFU WU w UdOJ b

uL fO WObMN WOUL w lU b WLO u WFU WU w UdOJ b k

WIdL WObMN WKKL s b


26/92

WObMN WKKL b s b uL b

UNO v_ WL b uL UNU WObM WKK _ Ub b

v_ WL Ub uL b fU Ub YU Ub WObM WKK WQLK r

w e _ dF U S WM UM dO WMb w r sL U WM UNO _ r sL UL UuM bIL WMbL pK

bF UJ b UL UuM bFL e bF c U WL n WMb UJ b uM

rJ UuM bIL hUM UO WLO XU n dF UO bO d uM d bF UO WLO `B

b_ pb

WObM WOU qJA WIU b_ X sbFK wU ju u U rO bQ sbFK wU ju u U

WLO U WIU WK_ s V WObM WOU qJA d b d

qzU? sU9

wU{ UA


27/92

ed bU x x x x uLL V

WOU KL uL b

Ub uL V r WLO b Ub WOU WOU

Uu Ud b uL bL V Uu UO b uL dOL V p `{ sOuLL sO dH bu

U S UNU _ Ub WOU WOU rO b ?

WUM tOUL X YU b v nO{ UNuL WOU WOU uJ b W W b_ b WObM

WU sUL

WKKL pb

p dO WObM WOU WKKL q Ub uL b

wU{ UA


28/92

b u b???u

UL UB

2

3

4

6

5

8

7

9

2

3

4

5

6

7

8

9

2


29/92

Standard Score (z-score) WUOF* WF

dHLK UOFL d wU ju b U

wU ju

wU b vAM UOFL d U

UOF* d wU( ju WFd bON9

UC X wU ju UNL s WedL WeMK WHK fOUI WOU_ WKdL w XUOFL d UNL s XAK WHK fOUI

S rOI s WuL XU S

rOI uL

rOI b rOIK wU ju

wU UND s rOI Ud UFd uL juL wFOd c rOIK UOFL d

dHL

uLL


30/92

VU qB L sOHK sOOzUB sOuL s sOuQ sOLO sO WUI d vdO UWOdF WGK U w WF vK qB WeOK WGK U w WF vK uHB b w

qC VUD qOB u U sOL wHc sJ d UNO t _ qC WOdF WGK w VUD qOB u dUL WU uJ bsOucL sOF s q UNO wLM w UF WuL u u UuKF W v bM WU

vK UL WeOK WGK w D UF UOFL d wU ju UM{d SUMS VOd vK UL WOdF WGK w D UF UOFL d wU ju U VOdnB UF wU ju s be w WeOK WGK w VUD W b WU c w wUU XAK UOI u UOFL d dOG dO Ue c q U bIL

sOUOF sOd wMF U Ue S UOIL c UMbc U bIL nB UF wU ju s be w WOdF WGK w VUD W ULMO

WMU VUD lu VUD u wMF c UOF d nB qL UevK WeOK WGK w ju u tFu _ WOdF WGK w tM qC WeOK WGK w tH wU

WOdF WGK w ju u tFu sUdOH q s UdHL vMF XO U U WOK_ UF d U_ cvKWdF s b UNO wLM w WOzUB WuLL w UCHM UOU u qL UN YO s UNOK rJ

WUOFL WF WLOIdJ QA UM s WuLLK UOFL d wU ju

ndF

UNd wU UND WOzUB WuL w U WFK WKUIL WUOFL WFvDF WuLLK wU ju s U WF UbF w WUOFL Ud b w UOFL

bUIU

WUOF* WF uNH

dHLK UOFL d wU ju b

Vb

WMOF j u WMOF lL u

WMOFK UOFL d wU juU UNb rO tS dF dO lLLK UOFL d wU ju UL


31/92

WUOFL WF U d bu wU ju U WF sO dH qL WUOFL WF YO

WUuJ wU ju s d U WF XU Wu uJ WUOFL WF S UOFL

wU ju U U WF XU dH uJ wU ju s dG U WF XUWUOFL UFK UOFL d wU ju

U sJL tS WKUI WUOF U v U lu w U UF lOL Xu dH WUOFL UF lOL uL UC wMF c dH WUOFL UF lOL wU ju

WUOFL UF lOL UOFL d sOIOI sOU W `{u wU UL

UL UFK UOFL d wU ju UUL dUML WUOFL UFK UOFL d wU ju `{ VOd vK

VOd vK

tUOFL WF YO WUOF U v U UF u

U UOFL d U UU UF wU ju U U sOF s qJ dUML WUOFL WF U

WUOFL WF,

WFK WKUIL WUOFL WF

UOF d , bIL nB UF wU ju s be WF

WFK WKUIL WUOFL WF

b UOF d bIL nB UF wU ju s qI WF

WUOF* UF hzUB


32/92

WUOFL WF ,,,,,,

,,,,,,

uLL

wU b TAM WUOFL UFK UOFL d U

wU UND s UF Ud UFd uL juL wFOd c UOFL d

WUOF* UF WuLUb WUOFL UFK wU ju

dH dH , , , ,

wU WOM XUJ WUOF rO U v rO s WuJ WuL w dH* X=u U ? WLO U ? [ , ,

dH lu w WUOF* UF uLdH ? [ , ,

dH ? ,, ?

p s oI WKUIL WUOFL WF U WF sO w b

,,,,


33/92

bb U VUD W WU{ fHM dQ ju _ D UF bb wU ju

WU{U dQ t_ , tH u D UF bb UOFL d

, , bb WUOFL VUD W

WU{U dQ r bb WUOFL WF

x bb U VUD W dC qU fHM dQ ju _ x U UFK bb wU ju

qU fHM dQ UOFL d _ x, U UFK bb UOFL dVuL dC

bb WUOFL VUD WVu bF dCU dQ r WUOFL WF

U UF dOG WUOFL UF dQ

WOK_ U UF s W qJ q XU bI nO{ WUOFL UF dQ dOG Vu b YO q XU bIL U W q Xd{ WUOFL UF dQ

VU b YO XU bIL U W q Xd{ jI WUOFL WF U

U UOFL d U UFK wU ju UJ tD U rKF d U U WKUJ WF ,

U w t XU c bL VUDK WUOFL WF U sOO sOU s q w bL_ WUOFL WF `B r

nB w VU qJ sO rKFL U{ w VU q W rKFL d{

WUOFL WF

, ,


34/92

UFK wU ju UJ uHB b w tD U UO{Ud rKF d U U w t XU nB b dOL UOFL d

WUOF* dOL W U WUOF* t `B r dOL W `B r nB w VU qJ U rKF* U{

,

WUOFL dOL W

WU{ bF dOL W `B

WUOF* dOL W tKUI* WUOF* WF s dOG U W qJ XU bI WU{ YO, tU{ q UNH w WU{ bF

VOd vK UUOF* UF ULNKUI UF XU UWOK_ UF lu UOF* d wU( ju u U

wU dCU

wU dCUc(U 5UF* q

dDU

ZM 5UF* b w iuFU

UOF* d wU( ju


35/92

U UOFL td U wU tD dJ lu WO U UF rOI s

qJ dUML WUOFL UF rOI w U

r U UN UOF* d r U u_ s WuL* wU( ju

[ WUOF* WLOI tKUI c uD u U

U UFK UOF* d U UFK wU( ju U wzUNM UB U w

VOd vK 5F vK B 5UD UUOF* UF UL U VOd vK, , 5UOF* 5F vK B 5UD U) UF UL U

w b' w w UL YU W w VU U XU

YLwU juUOFL dVUD WWOdF WGKUO{UdUB

W YUL w WUOFL VUD U w U vK_ u tH D tMU VUD qOB u U W YUL w

wU XUJ WUOF U v WOzUB WuL w dH* Xu WLO U , ,

, UdUM* UUOF* ULU XU UOuuMJ U w 5U U XU U w UFK UOF* d wU( ju u U VOd vK

qzU? sU9

VOd vK UL rOI s WuLL UOFL d wU ju WLOIK WKUIL WUOFL WF w U

WF qu q U WF YO WF V U rOI Xu

qu c bF WLOIK WUOFL WF `B r qu bFU


36/92

vMML dJ lKCL dJ bL UNM WdJ b qOL tHK WOUO Ud UIU XLKF

dJWO UJ_ UNL s WHK UJ dJ vMML c

w dOJ rOI u vMML s sL_ dD w rOI rEF UNO ed w WU qJ qL

luw dOGB rOI u vMML s d_ dD w rOI rEF UNO ed w WU qJ qLlu

ju s bOF tdDL rOI U lu j w rOI rEF UNO ed w WU qLO qJ UU WKOK uJ dO vMLO sON s

UOMM UNM dI w WOuLM uB ubFL wFOD vMML vL qJ w vMML dHL b e UbM p Uc V UF _ u_ q WOKLF UO w UFu s dO

b dO U WU q w UNb uI w

qJqUL dO dJ vMM

u Vu sOLO u uK

qJqUL vMM

bF wFO

qJqUL dO dJ vMM

u VU UO u uK

dJ vM*U UOUO b' q wU b' w Wu h n q w b' q1 U

UH

dJrG d_ h q lu dJ b

Normal Curve bF* wFOD vMM*


37/92

wFOD vMM* hzUB

duL q ULKF s dO U vMML c WOL_tb dA lU dA sU sOdI w U U wUL_ rUF v eF tu s dOJ uHAd vMMLK WO{Ud WUFL v uu vMML UNOK ud WOUL WKL vK U rUF uB w

tUF wFOD

oU b qL wU dJ vMML

,,,,,,,, ,,

dJ

UH ed

vMM* u

wU ju qL wI_ uL l qUL u UI WDI jI b WL t qUL vMM luK

dHL c qI r ju w dHL s dO b vK u uN d qJ tA qJ vMMLK qUL qJA sOU vK UOb


38/92

s WUN U v UdE bL vMML wd UFDI wI_ uL s vMML Ud dI sOU

L luK UOFL d wU ju vK p nu tHK UJ wFOD vMML c

sOO sOKJA k

YO wFOD lu qJ c w dHL u

b UOF d sL{ lI a UF qJ u_ s U s wu b UOF d bIL luK wU ju s qI w WLOI sO lI ju wU vK

b UOF d bIL luK wU ju s be w WLOI sOs qI w WLOI sO lI ju wU vK sOUOF sOd sL{ lI U s wu bIL luK wU ju s be w WLOI sO sOUOF sOd bIL luK wU ju

sOUOF sOd

wU ju w UHK tH UOFL d ULN UOFO UOMM

UOFL d w UHK tH wU ju ULN UOFO UOMM

wFOD luK wU ju vK WbK Xb


39/92

w WLOI sO lI ju wU vK WUOF Ud W sL{ lI U s wu ? bIL luK wU ju s be w WLOI sO WUOF Ud bIL jus qI

WUOF Ud

WIU VM `{u w WO UJ_ k

WH UIU d d c h q lu dJ b U v uFU wIOD UL

S rG U UOFL d rG U luK wU ju QULK sO uB rNK uJ UB wu U_ s UdI rG 5u 5 5

sO uB rNK uJ UB wu U_ s wu rG 5KJ 5 x x 5

sO uB rNK uJ UB wu U_ s wu rG 5KJ 5 x x 5

uc* b' w u UODF* l b_ VM c ou b s oI sJ1

Vb

UNO dBM w rOI b UOFL d bUALK wU ju b wFO wzUB lu wbUAL s wu bUAL s wu bUAL s

wU UDu s WF UbF w Ud b U

+

wU( ju

) +

wU( ju

)

+

wU( ju

)


40/92

UOF* wFOD vMM*

s c luK UOFL d wU juK UF bF UJ c wFOD vMML UMLKcoOI UMMJL tO ud d u WOFOD UFu lOL bu vMM vK uB p bB

w WUOFL UFK wU ju wFOD lu w U rOI s b WUOFL rOI UF bU bNUOFL wFOD vMML ZUM wFOD vMML vL b U UOFL d dH U wFO lu

w qJA k

UOF* wFOD vMM* X% WU* b

WUL W sO WU b ULKF l{ b U UOFL wFOD vMML X WOKJ WULWUOFL UF s WMOF rOI bL vMML X

bLFM ldL VJ lOL w b WGO c b cju sO uBL WUL wDF c UJ WUN w oKL b

w WKKEL WUL q Vu b YO WUOFL WF wUvMML w qUL WH bM WU rOI UUL qJA

WUI WK_ w `CO UL

WUOFL UF UOFL wFOD vMML

rOI UOFL wFOD vMML X WUL sO ucL b s e WOU WHB vK lDIL, , , ,


41/92

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Wu* dH 5 UOF* wFOD vMM* X% WU*

, sO uBL UOFL wFOD vMML X WUL b U

uLF , nB lUI bM lu bF v dEMM b s dU WUL dI, bF bM,

, w WuKDL WUL

, X WFu UOFL wFOD vMML X WUL b U

X WUL , sO WUL WuKDL WUL, ,

, ,


42/92

, X WFu UOFL wFOD vMML X WUL b U

, u WUL U , X WUL

UL qJA k,= sO WUL , , u WUL, ,

, , w , X WUL

,,

, w UN WFu UOFL wFOD vMML WU uJ w WUOFL WF w U U

sO WUL uJ Wu , s d , WUL UL , , , w WuL

, bF s Y b w UUL v dEM

, , sbF bM tO b d bM UFUI sc uLF nB v dEM u U UMS , bF v d , bF UL, , , WuKDL uJ uLF , nB bM

UdI , w , UN WUL uJ w WLO

b UOF d wU ju wFOD lu s Wd U U ZzU XU U

s qI U vK uKB sc WKD W s qI U vK uKB sc WKD W s be U vK uKB sc WKDK WuL WM ?

ju

w s qI U vK uKB sc WKD W U qJA w WKKEL WUL

WU WU c w b ULF d{


43/92

s qI U vK uKB sc WKD W WUL c U qJA w WKKEL WUL U

WUOF W v U WF u

, qULU UN WUL WUL b UMS WU YO

qUIL qJA s, u WUL , X WUL

, ju sO WUL ,

, , , , , w s q vK uKB sc WKD W

s be U vK uKB sc WKD W ? qJA w WKKEL WUL W U

WUOF W v U WF u

,

, ju sO WUL , WKKEL WUL, ,

, , w s be U vK uKB sc WKD W

, x, WuL WM

,

,,

,


44/92

U qJK UOFL d rG U h qJ wU ju UsO rNK lI sc U_ W w UL wFOD lu l qJ XU rG

U_ R b U rG rG

qJA w WKKEL WUL W rG rG sO lI rNK sc U_ WWUOF W v U WLOI u

UJ WUN w b s , WUL W, , U_ W

WM xwKJ bF U_ b,x

, UdI UB

UOF dU WU u lUBL b UNM w WOzUdNJ OUBL UL_ wU ju U UOzuA OUBL b dOwFOD lu l UL_ c XU S WU bI

WU s u bL UU vI UL UL

qJA w WKKEL WUL W WU s be bL UU UBL vI ULWUOF W v bF u

,

, ju sO WUL , , u WUL, ,

, , uKDL UL

,


45/92

UJ WUN w UOFL wFOD vMML X WUL b b

WO U s q w UOFL wFOD vMML X WUL b , X WFu

, u WFu , sO WFu ?

WO U s q w WUOFL WF w U

,

w X WUL , w u WUL , w sO WUL ?

UOF d r U wU ju wFO qJA UdI u e tM c e nO WK rU

r s UNK qI lMBL UNM w WH_ W U

r s UNK qI lMBL UNM w WH_ W U

U wU ju wFOD lu UdI l UO{Ud U w uHB b WK U XU UOF d

sOUM WKDK WuL WM UL U w UM W XU s d t uJ UL UL UOzuA WKD b dO

ezU UMO VU UNOK qB W q UL WdbI ezu WKD s qC UD dI ?

r UOF d r U wU ju wFOD lu l WKD s WuL u XU r r sO rNu dBM c WKD W U

VU u tKDK wKJ uLL U WKD R b U t WKD s , lI c uD u U ?

qzU? sU9


46/92

tMOu UL uNH WFd

WOzuAF U vL U UU UL WdE UM

WuL bb lOD UMMJ UNzd q UNO bb lOD w Wd w WOzuAF WdWOze WuL q vL WdK wMOF dH WuLL c vL WdK WMJLL ZzUM lOL

UUwMOF dH ssOOF U ULvL U c u Wd WOUJ s dF b U qJ jdU u d b sO WM UK wM dJvL U v UOKL uK sJL UL cdJ tM dI c bF vL UK wd ULvb U c Wd d d b

U ULvL b dO U Wd d d U bM UK wd UL wMdH wL UMS uu w UNH WdH wMOF dH dUM lOL UNO uJ w WU WU w

U dUM bdUM b

UdE UL czbM UK sOF ULEM UOMO Ud wMOF

UL 5u

qOL U UL

bRL U UL sO dBM U UL

wU UL _ UL sOKBHM sOU U UL S U t

sOU lUI UL wU UL _ UL sOU U UL

U p WLL UL U UL


47/92

uLL

dJ

sOuKF sONu vK sbF uL WE sOLEM d d UI s Wd uJ UWO ZzUM q d Wd dJ VU U

uL vK uB U wd UL U uL vK uB U dEM UL U

uLL vK uB d bWd d d b UK wM dJ wd UL

, dUM bdUM b dEM UL

{ } { }

,

dL s b dO b Wd Xdu dEM UL wd UL wLO sO dH k dEM UL s wd UL d

v s U_ qL WUD t bM s UOzuA b WUD X UWuL WUD vK bF uJ UL U

UO b

vK WLI qI b

vK WLI qI UO b

{ }w b vK uB U sJO

vK WLI qI b vK uB U

dUMF b{ }


48/92

dUMF b { }

dUM bdUM b

dUM bdUM b

{ } sJ

qzU? sU9

dUE bF ku d_ bu Ud X WLd WKUL d t fO U l V U WdK wMOF dH u U

WOUO dAU sF U V U dUE W t_ XEu WLEM uI lD XOI

WdK wMOF dH V UNM q UL V r WO u V

q_ vK b u vK uB U UU u vK uB U

d_ vK sOU vK uB U

[ , , YO sOU U b


49/92

b d UU Q ULK dA sOF U u UL u dAL UL UIU XdFcJ ]dF edU UL cN ed l

U dUE bF WE b d rEM d d UI Wd w U

s qI b uN U w b uN U bQ

sJ

dUM b

dUM b

x

s uJL UMLK b b Wd UM YO WuN q W WE pMJL

WU d sO s b Wd UM oI b dUM W

x

x xx

xx

ndF

Conditional Probability dA* UL


50/92

ULMO

b oU UL wH sO d UM uu UMLK S wFO d c WLO s dO l b Q WOU{ UuKFL

fO u bb hKI wMO UC v UuM u UL qFdAL UL uUI d WGO oOD UMMJL oU UL w UL rEML UL U w wUU w

dUM b dUM b

dUM b

dUM b oU UL wH

rEML UL U w UNULF lA WDO WGO w

bF uJ UL u U sdUE sbF WE b d sOLEM d d w Wd w U sdUE sbF uL Q ULK sONu b vK dUE

sbF uL U sONu b vK bF uN U sJO rEM UL UC dUM b

dUM b

sJsONu b vK vK uB U

sbF uL

oI UU UM q W WE pMJLsO s Ud UMN oI b UMdF bF sOOM vK uB bM

WU d

x

x x x x

x

x

x

x

_ dM d

wUd

Md

U?E

x

x


51/92

dA* UL dC bU

rKF

uJ WE sOUFL w wU dC WOKL dS

x x

_ u dA d u ULx

ULb u UL U UF sOU u UL

U UOzuA wu vK Ud X UCO d u d UN WOI Usu UdJ uJ UL U

u v_ dJ U sJOu WOU dJ U su UdJ uJ UL

x

x

UCO

u

u

UCO

uUCO

V WOv_ dJ

V WOWOU dJ


52/92

b , , , YO w UU U

x ,, x,

, , ,

,

d b u dUE bF dA UO b dUE bF uJ UL U b d rEM d d wI

b , , , YO UU

b b d sOLEM d d UI Wd w U _ d vK dUE bF Q ULK U wU d vK dUE bF uJ UL U _ d vK dUE bF Q ULK UO b sdUE sbF uL uJ UL

wu vK Ud bMB s X dL d UCO d UNM WKUL d t bM b U

sUCO UdJ uJ UL UCO d_ dL sOdJ b uJ UL

ub sOb s WeOK WGK ub sOb s tO UGK rOKF wUI ed UODFL c `{u UUM J r UF sOGK ub sOb s WOdH WGK

b UOzuA edL w sOb b dOq_ vK sOGK b U hA c uJ UL

WOdH WGK U U WeOK WGK U hA c uJ UL

qzU? sU9


53/92

sOOzuA sOd s WKK d s nQ UNK w UL vK bLF w WOKLF JAL iF

WUML dD b sOd sOUN jd sOF U u UL U UNK WKJAL VKD d WO WK_ w UL dAL UL d{ bU WOUO dA b l{u c nu

bMB w WOF ÒUB WU ÒUB _ bMB w UbM UUB tM V r UOzuA sObMB b dO WOF WU ÒUB wU

UU UBL c uJ UL U UOzuA b

w jDL w UNzU Ò{u sJL sOOzuA sOd s WKK c

Ù

VOF

,

Ù

VOF

,

,

,

,

,b

s Ù

s Ù

w UNH WdH ULN sOOM wNM sObMB b UO w v_ Wd `{u jDL, bMB UO UL bMB UO UL uu UL VOF U sOOM wNM sObMB b s b UB UO w WOU Wd UL UL , s U UOF uJ UL , u s U UU UBLuJ

, u s U UOF uJ UL , u s U UU uJ bMB UO U UM{d tOK

bMB UO U UU uJ UB UO U

s UU uJ s UU uJ U sOKBHM sOIdD r sJL u S s UU uJ UL s UU uJ UL UU UBL uJ UL

x x

, x, , x,

,

,

,

Partition Theory Wze WdE


54/92

WO

WOC

?

WO

WOC

WO

WOC

wU w bu WOC WO UU _ w bu WKUL oUM W UWOC WU WO UU ? YU w bu WOC WO UU

WO WU c uJ UL U b WU tM X r UOzuA bM dO

bMB UO U d vK

bMB UO U ? bMB UO U WO WUUO U

x x x S

x

x

x

s WO

s WO

? s WO


55/92


56/92

d U u b uu dQ UU b b x YO s qI U uI WU c w

x S

, u bN UL VOB UL , u Ub bL VOB UL U Ub bN u b d UL bL s q ]u

UF bN UL bL VOB UL bN UB UL

bN bL VOB UK edU Ue bN UL VOB U edU

S tOK I UU S UF bN UL bL VOB UL

, x, x ,

bN UL bL VOB UL bN UB UL

, , ,,

Independent Events WKI* u(

ndF

U I UU

x

S sOKI sOU U WO


57/92

sOO sOU s q w bQ , , U UsOKI U

sbU sOKBHM U

S sOKI U

x x, , ,

, , ,, ,

,

,

, S sOKBHM U

, ,

,

UF ULNu b wMF UBHU ULNI sOU UBH sO d UM x d uu ULb dQ b wMF I ULMO

d b uN U w b uN U U b d rEM d d UI bM Lb d w WO uJ sJL BHM UU S

_ Xu fH w Ud UOBHM ULN `{u s x q I q

dH

x

sOKI dO x qF UL

BHM

U?E


58/92

UC WKI WO u s vK tMd sJLO sOKI sOU U

, , , YO UU I q

b

b

YO I UU

sO sOU s q w bQ

U

BHM UU

I UU

U l d_ bu Ud X UCO d dL d t bM UsdL UdJ uJ UL b

dL v_ dJ U sJOdL WOU dJ U

sdL UdJ uJ ULI _ x

x

dL w V WO dQ v_ WU wH U l V U V sO dH kdOQ c q uJ U l V WU w WOU WU w ULMO v_ dL w V WOM WOU

v_ dL w U UL bMB uFO bMB v dJ U V

qzU? sU9


59/92

lDI s q vK u vK uB UL U b d WLEM bI lD XOI

b WOU d bN vK hA c u , u Ub h VOB UL dL w bN hA VOB UL

WU dL w tOB sOd w bN hA VOB UL

dL Ud UCO d tO bM UCO d dL d tO bM tH uK s UdJ uJ UL U bM q s UOzuA b d X

U l d_ bu d X UCO d u d dL d t bM

bdL dJ uJ UL su WU WOU dL v_ dJ uJ UL


60/92

U UOF d U wU ju UOFO UFu Wu U U w VU ZzU XU

U w t VUD WUOFL WF U WUOFL t VUD U WF U ju s WF UbF w WUOFL Ud b U ? UOF d , bIL ju dM w WF U

sOUM WKD b U U w UM W UL U U w UM W XU ? UL dbI uJO VUD UNOK qB W q UL UL dbI D s s wD

U sObIL s qC sL{ uJ UL UL UOzuA VU dO

WOKL U qU rb mU lUBL b UL tU c seK wU ju U bQ wFOD lu UdI l WOKLF U s lu U ozU u UOFL d WIO u WMOF

WIO s q w WOKLF eM sc ULF W WIO v s d X w WOKLF eM sc ULF b

w b w UL sOu UOKJ b w uKUF uHuL U

b UOzuA WOKJ w sOHuL b dOv nuL uJ UL

wLU v nuL uJ UL uc s tQ ULK U nuL uJ UL ?

YO UU

sOKI UO sOKBHM UO UU sO

uUwLU

qU

WU sUL


61/92

b u b???u

WOU* UO{Ud

100

100

100

100

100


62/92

sdOG0 dOG0 v_ Wb s UMU*WFd XLKF UL sdOGL b dOGL v_ Wb s WMUL q WMUL uNH oU n w XLKF

sdOGL v_ Wb s UMUL s UEM q WuL b nOWO WOU_ hzUB vK bLF UMUL q kwIOI b ? ? ? S XU

Vu wIOI b ? ?x ?x S XU VU wIOI b ? ?x ?x S XU ?

q d rOUHL cN WFd wQ ULO

b dOG0 WMU q uJO sOdD vK lL WMUL c q v_ Wb s WMU w WMUL

qB vK WLI WdFL WMUL c q b dOGL v_ Wb s WMU w WMUL

WO uDU uI WO U WMUL qF w dOGL rOZMO WMUL wd s qJ lL

ZMO WMUL wdD lL

ZMO

bF w WMUL wd dC

x x

U s UNM q qI w WOIOI b_ lOL WuL w WMULK q WuL wMF c

qJA w UL b_ j vK WuLL c qOL sJL


63/92

UNKL WMUL q WuL sOb tM bF

j U v WFu qJA w WKKEL WIDMLuO lC vK uO lC WMUL kS c wIOI b uJ sJL vK

UIM qLA q WuL a

sdOG0 WMU q UOU

WuL w WMUL q WuL sdOGL v_ Wb s WMU vL WMULUOUO WMUL c q UOIOI b YO WMUL oI w WdL _ lOL

WO uD lrOIL j vK sODI sOOF p wUJb uL w rOIL j rd

b w UL oIK WU WDI WUFL UII

v uL rI rOIL j bb WMULK q WuL qL ULb sOIDM q_ WDI q U WDI b WIDML c

WMUL q WuL wUJb uL w UOUO q U

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64/92

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67/92

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71/92

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73/92

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75/92

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76/92

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85/92

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Lexington .MA.1989

3Ehrhardt M.,Brigham E . Corporate Finance A focused Approach Southwestern.2003

4www.dhurley. com/normal.html5

www.willamette.edu/N mjaneba/help/normal curve.html.


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