Business Statistics

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3 y "Intelligent business decision makers use statistics to plan, control and reduce the uncertainty involved in business decisions."

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10

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11

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39

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41

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mrdih } ixLHd iuQyfha jeu w.h - ixLHd iuQyfha wvqu w.h

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44

y A wdh;kfha je fofkl= wvq jegqma ,nk njy tjeks iajrEmhla ykajkq ,nkafka Ok l=l;djla f,i nj'y B wdh;kfha je fofkl= je jegqma ,nk nj'y tjeks iajrEmhla y+kajkq ,nkafka iDK l=l;djla f,i hy C, D yd E wdh;k" jegqma jHdma;sj, fYaI keUqrejla olakg fkdue;s nj'y tu iajNdjh " iu;sl iajrEmh" jk nj'y iu;sl iajrEmhkays o O;d we;s nj'y jHdma;shl uqka jf m%udKh jlS%uh f,i yekafjk nj'

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jHdmdr wdh;kh ffoksl l=Kq flfrys wjOdkh fhduq lrkak'

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46

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yels nj'

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47

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meye,s lrhs'

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m%ia;dlj o ksrEmKh l< yels nj'

48

bf.kqu i|yd fhda; Wmfoia ( my; i|yka Wmfoia isiqkag ,nd foka isiqka bf.kqfuys fhdojkak

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fhduq lrkak'

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x wCIhg yd y wCIhg .e

49

(1) (2)(3)

(4) (5)

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50

bf.kqu i|yd fhda; Wmfoia ( my; i|yka Wmfoia isiqkag ,nd foka isiqka bf.kqfuys fhdojkak

Tf lKavdhug ,efnk jHdmdl o;a; we;=

51

Ih lreKq meye,s lr .ekSug w;aje,la ( O inkaO;d j. ykd .ekSug is ;s;a igyka fhod .; yels nj'

is ;s;a igyka u.ska ykd .kakd fh inkaO;d" Ok inkaO;d fyda iDK

inkaO;d h yels nj'

mQK fh fjkqjg fndfyda g ykd .; yelafla m%n, fyda n, fh inkaO;d

nj'

iyinkaO;d fkdmj;sk wjia:d o fujeks is ;s;a igyk u.ska ykd .; yels

w;r" tajd Yaf,aIK i\yd fhdod .; fkdyels nj'

52

ksmqK;d ugu 4'3( p,H folla Yaf,aIKhg iyinkaO;d ixl,amh Nd;lrhs'

ld,fo ixLHdj ( 02

bf.kq M,y fokq ,nk p,Hhka m%udKd;aul yd .=Kd;aul f,i j. lrhs'y p,H folla w;r iyinkaO;df Ydj yd tys m%n,;ajh kslaIKfhka fyda is;s;a

igyka Nd;fhka m%ldY lrhs'

y p,H folla w;r mj;sk iyinkaO;dfjys m%n, nj fyda n, nj o< jYfhkaixLHd;aul j m%ldY lrhs'

y o;a; hq., kslaIKh lsfuka ;SrK .kshs'y ldhhka myiq lr .ekSug iyinka; p,Hhka Nd; lrhs'

bf.kq- b.eka ls%hdj,sh (msiquy my; i|yka mqj;am;a isia;, mka;sh bfha m%oYkh lrkak'

mqj;am;a isria ;,

y i,a , by

53

y p,H folla w;r O wdldrfha iyinkaO;d ;sfnk nj'bf.kqu i|yd fhda; Wmfoia (y my; i|yka j.= lKavdh j,g fnod ta ms

54

j.= wxl 5j.= wxl 5j.= wxl 5j.= wxl 5j.= wxl 5

wdh;khl ldh uKav,fha fiajd

ld,h iy ,nk udisl jegqm

fiajd ld,h ^wjq& udisl jegqm^re oyia&

10 12

8 20

7 7

8 6

12 8

15 18


cE ksIamdok wdh;k myl cE l=Kq , yd udisl l=Kq m%udK

wdh;k , ^re& udisl l=Kq ^oyia&

A 90 60B 85 80C 92 60D 95 50E 88 48


wdh;khla hka;% w;ajehdjg fhojQ

fiajl ixLHdj iy thg .;jQ ld,h

fiajl ixLHdj .;jQ ld,h

12 3

6 6

9 4

55


wdh;klhl hka;% kv;a;= msjeh iy hdka;s %l fodaI we;s jQ jdr .Kk

kv;a;= msjeh hdka;s%l fodaI

^re' oyia& we;s jQ jdr .Kk

25 2

12 3

20 1

18 2

23 2

28 1

y my; i|yka Wmfoia isiqkag ,nd foka isiqka bf.kqfuys fhdojkaky Tng ,e w;s p,H m%udKd;aul p,Hhka o" ke;fyd;a .=Kd;aul p,Hhka o

hkak meye,s lr tfia i|yka lsug fya;= olajkak'

y o;a; kslaIKh lsfuka fyda is ;s;a igyka wefuka iajdh;a; p,Hh u; mrdh;a;p,Hh fjkiau meye,s lrkak'

y tu iyinkaO;dj ixLHd;aul j o< jYfhka m%ldY lrkak'y jHdmdr ldhlaIu lr .ekSu i|yd p,H w;r iyinkaO;dj Nd; l< yels

wjia:d folla olajkak'

y Tf wkdjrK ksudKYSS,Sj o" iduQyslj o iuia; mka;shg bm;a lsug iQodkjkak'

Ih lreKq meye,s lr .ekSug w;aje,la (y p,Hhka m%udKd;aul fyda .=Kd;aul f,i j. l< yels nj'y iajdh;a; p,Hfha o;a;j,g wkqrEm j mrdh;a; p,Hfha o;a; wkqf,dau$ m%;sf,dau

wdldrfhka fjkiah yels nj iy p,Hhka w;r iyinkaOh ksrEmKhg fLdjla

.e

56

ksmqK;d ugu 4'4 ( .=Ks; >QK iyinkaO;d ix.=Klh m%udKkh lrhsa'ld,fo ixLHdj ( 04

bf.kq M,y we;s o;a;" iyinkaO;d ix.=Klh fiug myiq jk m msqK iyinkaO;d ix.==Klhg ,efnk w.hka u.ska fh inkaO;djfha

m%n,;ajh yd Ydj jrKh lrhs'

y ffoksl ;fha yuqjk O p,Hhka myiqfjka ikaikaokh lsug tallj,skaf;dr kq fhdod .kshs'

y we;e by< iyinkaO;dj, o ksr:l inkaO;d ;sh yels nj'

bf.kq- b.eka ls%hdj,sh (msiquy my; lafjk foni iafjPdfjka isiqka fofofkl= ,jd mka;shg bm;a lrjkak'

lqK ,laIHh ^uOH ,laIHh& f,i ),( YX ,laIHh mok lrf.k .=Ks; >qK

iyinkaO;d ix.=Klh .Kkh lrk nj '

57

bf.kqu i|yd fhda; Wmfoia (y my; i|yka j.= lKavdh j,g fnod tajd flfrys wjOdkh fhduq lrjkak'

j.=j 1

m%pdrK l=Kq

hou ^re' oyia& wdodhu ^re' oyia&

20 30

18 20

20 28

30 35

25 35

j.=j 2

jdykhl jdykfha

jld,h jkdlu

^wjq'& ^re' ,shk&

5 1'0

3 2'5

2 3'2

4 1'5

3 1'8

j.=j 3

fiajd ld,h udisl jegqqm

wjq' ^re' oyia&

3 7

10 12

7 7

8 20

12 8

58

j.=j 4

, ^re'& l=Kq

m%udKh ^ oyia&

5 7

6 10

8 10

4 12

10 7

y iajdh;a; p,Hh X f,io" mrdh;a; p,H Y f,i o k fldg tu j.=j ;jrg;a > lrka ;Sr 22 ,, YXXY imQK lr .kak'y .=Ks; >qK iyinkaO;d ix.=Klh r .Kkh lrkak'y r i|yd ,efnk w.hg tallhla mj;S o@y r .Kkh fldg ,efnk m%;sM, jrKh lrkak'y .=Ks; >QK iyinkaO;d ix.=Klfhys ,eh yels ms

59

lsh jH.=Ks; >qK iyinkaO;d ix.=Klh (Product moment correlation cofficient) fifuka

wmg ta p,Hhka w;r mj;sk inkaO;dj, osYdj iy m%udKh uek .kak mqjka

Product moment ^ .=Ks; >QK& f ix.=Klh fydhkafk ta p,Hhka fofla uOHkHhka^ YX , & f. isg we;s wm.uk j, .=Ks;hka f. ftlHh tajd fh iu; wm.ukj,.=Ks;hkag orK wkqmd;hla hghs. t;fldg >QK ,CIH ^uOH ,CIH& ;uhs ^ YX , &LKAvdxlh fuh my; lafjk isrs ;s;a igyfkys oelafjkj

>QK ,CIH u; isg wm.uk .=Ks;hka iQK

iyinkaO;d ix.=Klh f,i ykaj, ;sfnkafk'

f ix.=Kl .Kkh lrkafk fldfyduo@

ta i|yd f.dvk, ;sfnk iQ ;% Nd ;d lr, .Kl hka;% fyda mrs.Kl Nd;fhka b;du

blauKska ms

60

ksmqK;d ugu 4'5( ;rd iyinkaO;d ix.=lKlh m%udKkh lrhs'ld,fo ixLHdj ( 04bf.kq M,y we;s o;a; ksjer j ;rd .; lrhs'y iQ;%h Nd;fhka ;rd iyinkaO;d ix.=Klh .Kkh lrhs'y ;rd iyinkaO;d ix.=Klh Nd;fhka .=Kd;aul$ m%udKd;aul p,Hhka folla w;r mj;sk inkaO;df m%n,;ajh iy Ydj jrKh lrhs'

y fm

61

j.= wxl 2

mrSCIKhloS fmkS is Ihhka i|yd isiqka

fofofkl= ,nd .;a ,l=Kq

Ihhka A f.a ,l=Kq B f.a ,l=Kq.Ks;h 80 58

oHdj 70 60

isxy, 55 85

iudc wOHhkh 65 50

bx.%Sis 70 70

wd.u 85 60

;dCIKh 45 90

fi!kaoh 50 70

j.=j msgm;a lrf.k tla tla p,Hfha we;s ,l=Kq i|yd ;rdjka fhdokak

tla tla w.h i|yd ,e we;s ;rdjka w;r fjki ^d& ,nd .kak ^d2& ,nd .kak my; oelafjk iQ;%h fhd f.k ;rd iyinkaO;d ix.=Klh r .Kkh lrkak

( )161 22

=nn

dr

d = p,H foflys ;rd w.hka w;r fjki

,efnk m%;ssM,h wkqj p,H fol w;r iyinkaO;dj ms

62

ksmqK;d ugu 4'6 ( m%;smdhk fLdj wkqiSykh lsug wkqmldr l%uh Nd; lrhs'ld,fo ixLHdj ( 02

bf.kq M,y is ;s;a igykg .e

63

y tu wjia:d fol wkqudksl inkaO;d jk nj'y by; lS ish inkaO;d oelaug .Ks;uh ilrKhla f.dvke.Su W; nj'y th m%;smdhk ilrKh f,i k lrk nj'y m%;smdhk ilrKh ,nd .ekSug O l%u mj;sk nj'

bf.kqu i|yd fhda; Wmfoia ( my; lafjk is ;s;a igyka isiq lKavdh j,g ,nd tu igyka fol fj; isiq

wjOdkh fhduq lrjkak

^1& ^2&

^1& ^2&

tla tla isrs ;s;a igyk fyd|ska ksrSCIKh fldg p,H fol w;r inkaO;dj

meyeos,s lsrSug fhda.Hhehs Tn is;k ir, fLdj ir, rhla Nd;fhka w|skak

w|sk ,o fLdj mok lrf.k tys nEjqu ^wkql%uKh& ,nd .kak th m%;smdhk

ix.=Klh f,i k lrkak

Tn w|sk ,o fLdj Y wCIh fPokh lrk ,CIHfhys Y LKavdxlh ^wka;#LKavh& ,nd .kak

w|sk ,o fLdjg iqiq ilrKh ,shkak

tu.ska iajdh;a; p,Hfha fjkia u u; mrdh;a; p,Hh p,kh f m%udKh

meyeos,s lrkak

64

Tn ,nd .;a ilrKh wod< p,H hq.,h w;r inkaO;dj meyeos,s lssrSug flf;larg

m%udKj;a o@

y Tf wkdjrK ksudKYS,S j yd iduQysl j iuia; mka;shg bm;a lsug iQodkjkak'

Ih lreKq meye,s lr .ekSug w;aje,la (y my; lreKq u;= lrka iudf,dapkhl fhfokak'y p,H fol w;r inkaO;dj o< jYfhka muKla ia;r lsug ir, fLdjla

wNsu; m weu wkqmldr l%uh jk nj'

y p,H fol w;r inkaO;dj meye,s lsug .Ks;uh ilrKhla ksudkh l< yels nj iy ta i|yd += baY fyda 110 xY

+= jeks ilrKhla Nd; l< yels nj'

y ;s;a kslaIKh lsfuka wkq ,nk fLdj wkkH fkdjQ mqoa., ks(Ys%; jQjla nj'y fuh blauka ;SrKj,g u.mdok" ir, fh inkaOhla f.dvke.sh yels myiq

l%uhla nj'

65

ksmqK;d ugu 4'7 ( wvq;u j. l%uhg m%;smdhk ilrKh wkqiSykh lrhs'ld,fo ixLHdj ( 06

bf.kq M,y wkqmldr l%ufha mj;sk fodaI u.yrjd .ekSug iqiq l%uhla fhdackd lrhs'y fok ,o o;a; ksjerj ilia lr .kska wvq;u j. m%;smdhk ilrKh ,nd .kshs'y m%;smdhk ilrKh Nd;fhka iajdh;a; p,H u; mrdh;a; p,H .Kkh lrhs'y fodaI wju jk wdldrhg ldhhka is lrhs'y .Kkhka myiq lr .; yels fjk;a l%ufo fidhd n,hs'

bf.kq- b.eka ls %hdj,sh (msiquy my; olajd we;s rEmigyka y;r m%oYkh lrkak'

ixy 10 += exy i ++= 10

ixy 10 = exy i += 10

y my; i|yka lreKq biau;= jkfia idlPdjl fhfokak'y wkqmldr l%ufha mj;sk j,;d u.yrjd .ekSu i|yd fodaI wju jk l%uhla

Nd; lrk nj'

y fuu l%uh wvq;u j. l%uh nj'y wdkqudksl inkaO;d mj;sk g m%;smdhk wdlD;sh ieY ++= 10 u.ska lafjk

nj'

y ie hkq m%;smdhk ilrKfhka meyeos,s fkdlrk n,mEu we;=,;a jk moh nj'

66

y m%;smdhk ilrKh weia;fka;= lsug ie wju l< hq;= nj'y p,H w;r inkaO;dj meye,s lsug fodaI j. ftlHh wju jk wvq;u j.

m%;smdhk ilrKh jvd;a fhda.H nj'

f.kqu i|yd fhda; Wmfoia (y isiq lKavdh j,g o;a; j.= ,nd my; i|yka ls%hdldrlfuys fhdojkak'

j.= wxl 1 j.= wxl 2

jdykfha jhi kv;a;= msrsjeh re.oyia wuq jH 1kg l , re b,a m%udK2 5 2 25

4 6 4 20

6 7 4 25

3 5 3 24

5 8 5 28

8 7 8 10

Tng imhd we;s o;a; j.=j msgm;a lrf.k mrdh;a; p,H Yi f,io iajdh;a;p,H Xi f,io k lr .kak'

tu j.=j ;j rg;a m%idrKh lrka X2, yd XY ;Sre .Kkh lrkak iEu ;Srejlu ftlHh o ,nd .kak tajd XYXYX ,,, 2 f,i k lrkak tu w.hka my; oelafjk iu.d ilrKhg wdfoaY lr th i|Sfuka 0 iy

1 fidhkak th ilrKhg wdfoaY lrkak'

iy ykajkak'

by; ldh ish,a, lrk ,oafoa l=ula i|yd oehs meyeos,s lrkak

0 iy 1 fiug my; olajd we;s m%u; ilrK Nd;d lrkak'

y Tf wkdjrK ksudKYS,Sj yd iduQysl j iuia; mka;shg bm;a lsug iqodk jkak'

je r wOHhkh i|yd(-

wvq;u j. m%;smdhk ilrKh .Kkh lsrSug Nd;d l< yels fjk;a iQ;% ms

67

Ih lreKq meye,s lr .ekSug w;aje,la ( my ; lreKq u ;= lrka iudf,dapkhl fhfokak'

/ia lr.;a o;a; ldKav j, iajdh;a; p,Hh u; mrdh;a; p,Hfhys yeisrSu

.Ks;uh ilrKhla u.ska meyeos,s l< yels nj'

fodaI j. ftlHh wju jk wdldrhg m%;smdhk ilrKh ksudkh lrk nj'

faIhka wju jk neka th o;a; i|yd jvd;a .e,fmk ir, fLdj nj'

1 m%;smdhk ix.=Klh f,i ye|skafjk nj' .Kkh lrk ,o ilrKfhys 1 ys w.h fhd f.k iajdh;a; p,H u;

mrdh;a; p,Hh fjkiajk m%udKh iy osYdj jrKh l

68

lsh jH m%;smdhk wdlD;sh fhdod f.k m%;smdhk ilrKh ksudkh lsu m%;smdhk wdlD;sh fhdod f.k m%;smdhk ilrKh ksudkh lsu m%;smdhk wdlD;sh fhdod f.k m%;smdhk ilrKh ksudkh lsu m%;smdhk wdlD;sh fhdod f.k m%;smdhk ilrKh ksudkh lsu m%;smdhk wdlD;sh fhdod f.k m%;smdhk ilrKh ksudkh lsu

m%;smdhk wdlD;sh (-

ieY ++= 10

ei hkq fodaI mohhs tys ksudklh f f

ei

ta wkqj mrdh;a; p,Hfhys ishu w.hka i|yd jQ m%;smdhk wdlD;shg by; fldkafoais

fol wdfoaY lsfuka fuys jk neka

^1&

m%;smdhka wdlD;sh Xi j,ska .=K lsfuka fuys jk neka

^2&

fuu ^1& iy ^2& ilrK hq.,h m%u; ilrK f,i i

69

ksmqK;d ugu 4'8 ( wkqiSykh lrk ,o wvq;u j. m%;smdhk fLdjl fyd|lu mlaId lrhs'

ld,fo ixLHdj ( 06

bf.kq M,y wkqiSykh lrk ,o m%;smdhk ilrKh Nd;fhka mrdh;a; p,Hfha w.hka weia;fka;= lrhs'

y ksKk ix.=Klh .Kkh lrhs'y ksKk ix.=Klh fhdodf.k m%;smdhk fLdf .e

70

y tu p,khka mok lrf.k ksKk ix.=Klh ^ 2R &.Kkh lsfuka m%;smdhk ilrKfhys wkqiSykfha fyd|lu jrKh lrk nj'

bf.kqu i|yd fhda; Wmfoia (y isiq lKavdh j,g my; we;s j.= ,nd ta flfrys wjOdkh fhduq lrjkak'

j.= wxl 1

jdykfha jhi kv;a;= msjeh

wjq( re' oyia

x y2 5

4 6

6 7

3 5

5 8

8 7

j.= wxl 2

wuqo%jH 11111kg , b,a m%udKre'

x y2 25

4 20

4 25

3 24

5 28

8 10

y my; Wmfoia isiqkag ,nd isiqka bf.kqfuys fhdojkaky Tf lKavdhug ,efnk o;a; j.= iy .Kkh lrk ,o m%;smdhk ilrKh mok

lr .kak'

y m%;smdhk ilrKh weiqfrka tla tla w.hkag wod< j .Kkh lrkak'

y fidhkak'

71

y fidhkak

y fidhkak

y Nd;fhka

ksKk ix.=Klh fidhkak'

y m%;smdhkfhka meye,s jk p,kh mrdh;a; p,Hfhys uq p,kfhka lsk m%;sY;hla o hkak meye,s lrkak'

y m%;smdhkfhka meye,s fkdjk p,kh i|yd woyia olajkak'y .Kkh lrk ,o m%;smdhk ilrKfha wkqiSykfha fyd|lu ms

72

ksmqK;d ugu 4'9 ( p,H w;r mj;sk inkaO;d olska mqfrdal:k lrhs'ld,fo ixLHdj ( 02

bf.kq M,y wka; ksfYk iy nys ksfYk mqfrdal:k w: olajhs'y fok ,o m%;smdhk ilrKh fhdodf.k iajdh;a; p,Hfha w.hg wkqrEm mrdh;a; p,Hfha w.h .Kkh lrhs'

y iajdh;a; p,Hfhys w.h ;sfnk uug jvd by< ouka mrdh;a; p,Hfha w.h weia;fka;= lrhs'

y jHdmdr lfIa;%fha m%Yia; ;SrK .ekSug mqfrdal:k Nd; l< yels wjia:d i|yd WodyrK imhhs'

bf.kq- b.eka ls %hdj,sh (msiquy my; oelafjk foni N+ld rx.khla f,i mka;shg bm;a lrkak'

N+ld rx.khN+ld rx.khN+ld rx.khN+ld rx.khN+ld rx.kh

miq.sh mej;s Y%shd isrs wdh;kfha l

73

f isrs ;s;a igyk u; we|sh yels fyd|u m%;smdhk fLdj

ixLHdk Ys,amSh l%u wkqj f.dv k., ;sfhkafk. ta

ilrKh ;uhs f ;sfhkafk.^m%;smdhk ilrKh ,shQ

ldm; m%oYkh lrka& f fLdj remsh,a ,shk j,ska

f.dvk., ;sfhkafk. ^ jeks m%;smdhk

ilrKhla ldm;l ,shd .kak.&

l

74

bf.kqu i|yd fhda; Wmfoia (my; lafjk j.= isiq lKavdh w;r fnod tajd flfrys wjOdkh fhduq lrjkak'

j.= wxl 1

jdykfha jhi kv;a;= msjeh

wjq( re' oyia

x y2 5

4 6

6 7

3 5

5 8

8 7

j.= wxl 2

wuqo%jH 1kg1kg1kg1kg1kg , b,a m%udK

re'

x y2 25

4 20

4 25

3 24

5 28

8 10

y my; lafjk Wmfoia isiqka g ,nd foka isiqka bf.kqfuys fhdojkaky x=7 g Yys w.h weia;fka;= lrkak'y x=10 g Yys w.h weia;ka;= lrkak'y fok ,o w.h mrdih ;=< j.=fjys fkdue;s kEu x w.hla i|yd Yys w.h ,nd .kak'y fok ,o w.h mrdifhka msg; j.=fjys fkdue;s kEu x w.hla i|yd Yys w.h ,nd .kak'

y jHdmdr lafIa;%fha ;SrK .ekSu i|yd mqfrdal:k fhdod.; yels wdldrh ksoiqkla iys; j ia;r lrkak'

75

Ih lreKq meye,s lr .ekSug w;aje,la (y we;s o;a; ldKav foflys iajdh;a; p,H m%;smdhk ilrKhg wdfoaY lsfuka mrdh;a; p,Hhl w.h weia;fka;= l< yels nj'

y mqfrdal:kh wka; ksfYk mqfrdal:kh yd nys ksfYk mqfrdal:kh jYfhka j. l< yels nj'

y jHdmdr lafIa;%fha b jIh i|yd b,lal weia;fka;= lsug nys ksfYk mqfrdal:kh fhdod.; yels nj'

y m%;smdhk Yaf,aIKfha fodaI .Kkh lsug wka; ksfYk mqfrdal:kh fhdod.; yels nj'

76

ksmqK;dj 50 ((((( jHdmdrsl wjodkug uqyqKoSug we;s iQodku m%oYkh lrhsksmqK;d ugu51 ( ( ( ( ( jHdmdrsl wksYa; ;d weiqka iNd;d ixl,amh .%y lrhsld,fPo ixLHdj ( 03bf.kq M, ( h yelshdj wkqj is j.SlrKh lrhs

tla tla j.hg wod< is fhdackd lrka tajdfha hyelshdj ;SrKh lrk idOl

f.dkq lrhs

tu is i|yd iNd;d w.hla wkqudk lrhs

hyelshdj mslaiSfuka wjodk mej;sh yels w.h mrdih olajhss wksYa;;dj uek lah hq;= isoa ;SrKh lrhss

bf.kq-b.eka l% shdj,sh ( msiqu my; kafjk isoa mka;shg fhduqlr isiq woyia uikak

bfnla mshdir lsu fyg jeis we;su fyg;a kef.kyska ysre Wod u

u;= lafjk lreKq biau;= jk fia idlPdjla fufyh jkak

f;wdldrhl isoa by; ia;r lr ;sfnk nj

fjd ksYa; is, wksYa; is iy lsisfia;a h fkdyels is f,i ykajk nj

tfkod ;fha yuqjk fujeks is j,ska fndfyduhla wksYa; is nj

bf.kqu i|yd fhdacs; Wmfoia ( my; i|yka ud;Dld isiq lKavdhj,g ,nd we;s Wmfoia wkqj isiqka bf.kqfuys fhdojkak ksYa; is wksYa; is lsisfia;a h fkdyels is Tng ,e we;s ud;Dldj meye,s lsug WodyrK 5 la ,shkak jHdmdr is ta w;r ;sfk tajd ykd .kak jHdmdr is .Kk my olajd jelsug fjk;a is ,ehsia;=jg tl;= lrkak ffoksl ;fha tlS jHdmdr is j, iq,N;dj uYkh lrkak ^Analysis of Frequency& Tn bm;a fldg we;s is j, hyelshdj uek laug iqiq w.hla fyda w.h mrdihla fhdackd lrkak Tf wkdjrK ksudKYS,Sj iy iduQyslj iuia; mka;shg bsm;a lsug iQodk

jkak

Ih lreKq meye,s lr.ekSug w;aje,la ( jHdmdr yd nef|k wksYa; is iq,N nj ,dN$w,dN, wf, u $fkdu, weKjq ,eu$fkd,eu jHdmdr yd ne is lsysmhla nj jHdmdr wjodku wju lr .ekSf Ys,am l%uhla f,i wksYa; isoa ixLHdkuh jYfhka Kqug ,la flfrk nj fuu Ys,am l%uh iNd;dj f,i k flfrk nj ksYa; isoahl iNd;dj 1 jk w;r lsisfia;a is fkdjk isul iNd;dj 0 jk nj wksYa; isoahl iNd;dj 0 ;a 1 ;a w;r w.hla .kakd nj

77

ksmqK;d ugu 52 ( O mCIK w;ka iiNd mCIK fjka lr olajhsld,fPo ixLHdj ( 03bf.kq M, ( iiNd iy ksKdhk mCIK i|yd ksjpk bm;a lrhs

tla tla mCIK j.h i|yd jHdmdr fCIa;%fha WodyrK u;= lr olajhs

mCIK foj.fha fjki ikd: lsug WodyrK fhdod .kshs

jk wNsfhda. j,g fkdie,S uqyqK u i|yd iiNd is ykd .ekSf iQodku

m%oYkh lrhss

WodyrK u;= lrka lKavdh idlPd fmdaIKh lrhs

bf.kq-b.eka l% shdj,sh ( msiqu my; i|yka m%Yak mka;shg fhduqlrkak

ue.akSish mhla oykh lsf ,efnk m%;sM,h l=ulao @

remshf,a ldishla by, f ,efnk m%;sM,h l=ulao @

my; lreKq u;=lrka idlPdjla fufyhjkak

ue.akSish mhla oykfha ,efnk m%;sM,h ksYa;j lsj yels nj

ldishla by, f ,efnk m%;sM,h ksYa;j lsj fkdyels nj

f wkqj ksKhk mCIK iy iiNd mCIK hkqfjka mCIK

foj.hla we;s nj

bf.kqu i|yd fhdacs; Wmfoia ( my; i|yka mCIK wjia:d fol isiq lKavdh j,g ,nd my; we;s isoawOHhkh weiqfrka isiqka bf.kqfuys fhdojkak

ksKdhk mCIK

iiNd mCIK

tu mCIK wjia:dj isf;ys r|jdf.k my; i|yka isoah wOHhkh lrkak

kl ;udg ,efnk meKs,s Yaf,aIKh lrk ksIamdok wdh;khl l

78

Ih lreKq meye,s lr.ekSug w;aje,la ( m%;sM, fmr lsj fkdyels mCIK iiNd mCIK k jk nj

m%;sM, fmr lsj yels mCIK ksKdhk mCIK jk nj

jHdmdr fCIa;%h ;=< fuu mCIK ish,a,u is lrk w;r jvd;a iq,Nj we;af;a

iiNd mCIK nj

iNd;d ixl,amh iiNd mCIK j, jeo.;a jk nj

79

ksmqK;d ugu 53 ( kshe wjldYhg iy isoa wjldYhg wh;a wjhj l=,l wdY%s;j ksrEmKh lrka isoa ixhqla; lr jHdmdl wjodkug

uqyqK ug iQodk fjhs

ld,fPo ixLHdj ( 10bf.kq M, ( iiNd mCIKhl oS ,eh yels ish m%;sM, ksrEmKh l< yels O l%u k lrhs

kshe wjldYh O l%u u.ska ksrEmKh lrhs

kshe wjldYh ;=, tla tla isoag wh;a m%foaYh fjka lr olajhs

O wjia:d ixhqla; lrka ixhqla; isoa m%oYkh lrhs

ixhqla; isoaka f.a Wm fldgia j, wjodku flfrys ie

80

A is fkdjk g B isu B is fkdjk g A isu A fyda B fyda hk foflka tllaj;a isfkdu A g yd B g wh;a fkdjk isoa isu Tf wkdjrK ksudKYS,Sj iy iduQyslj iuia; mka;shg bsm;a lsug iQodk

jkak

Ih lreKq meye,s lr.ekSug w;aje,la ( iiNd mCIKhlska ,eh yels ish m%;sM, O l%u u.ska ksrEmKh l

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