known ondon

Sompl
is e auh have Procedu to
the ando includad
t in birg chnee
of equal on hwa
pup otuit 2ach wich inone S0mpl
is e Tonom
ng: sompli Rgndem A
ono Samele Rondam *
Eoimation Enhmale Elimoton,
terunt@,5atNH Ca 6tahiatiCparamete,
nit, Sanple [amplit [omple, Populaion,
Chuyan Concept boyie
poor Ka V.k. Guptan 5.C.
5iaistis Mothcalemah ’
fndamnta
o 3.
alAsnot
i
(Volue-11)
Stuant,
A: 2
A. 6G.
t. M. ’
and Thonykanda,
Avanee s.
Reetoneed Bok
Enimalion 6ineup-
ttheup-
A:
Intoronee-I statisi
Cal Stal-3o
 Md. Shahtehf Moque 2
Assotate Prate sso
Dept of Statistics
Unverity of Chitiagong

n Binuusion (allad papulo ion

umae 1 in animalo Undott

* somçle: AreMestntetive ond Conhdraolt

pop is Knowh a a Jomple o that pop.

Bechnique af dradi rg a somple

Callad. Jompng
* Sample Uit: Evty Unit of the sample whih we ant
is Called SompleUnit.
unit: £veo unit of the pop?, Which a
Wonto btudst Urut.

OR, The
chatactubic d a pop? is called apanet
Unknon chuaeoubico t he pop otu wbually colle porsmelevs.
Po. mean (), Pe! voTioneelo),oalation Caßiient
), rsge Coient (e), ete. ane the yonples
oR, The erohnate of a Cernfun paanetor
tpnanau
Calad slahnhc
* 5tatibtie
nunTi cal value aebci bi ng a choraa
Sample mean 0, Jompla votçon ea (s, (otu eeef
 sampling Biat of a Statistic u KnDQn

6
i2, s.E() = (o)

*ingenee: Ingenee
Inttntu is the Juokfiad logical Proces
(49,35, ne PropOSii on to anothe
proposihon. In inferane genno moTe than

stotinh cal. intrunee is the fprocas

Slatieh
"f making Conclwion
Conclusion rega
haraeteNinhies (Parontn) f a pop! on t
o obhurved Jomple dratn Jom that
populoi n

uetive intauneeiS a
a Judqzment mAnnaizhm
bassd on oxioms, emisen, ond
L aBumpkon
e t
the fometim then
Pop
fom omple Tandom
Qoimao:
ate w»odto uwhich Statiptie Ebhmator:
T *
Eshmahion )
Conshuete
maon
a inrune induehve Moinly ingnnee
ollbo populat
won the obont Conclude
impoilonte stahoial noAan
vad deri(oncluion
u Conctubion
a
3
R
 mumetü al val ot an lphimat
Colled aimate.
on enhmat

whi h Caed lohmate.

wt Entimai on: Eoimati on the pYous f enhmaing

the pTLUnmw boma techiquen o ant stajahal
devie.
drawn
Vig dt. fe),'Where tue
tom a pop"
The opr. No Weinteustto
patmete ot pop" th
ind on shmate o on the byis ot the
he.
Sample Valnes, then the proe% to be
Caled lpimaim
eohmat on: 4 poai ble to make 4wo
typea of lotimatiny st a pop vo paOnte,thse ane
O point eatimain
 stahahal Intee
Eim'ai
Point enhnain
Inevel entahon
ASingle Value wsg Atung t ValLN stutennt
enimsho a pop
palomel:.

sciininls Tk. 55 000; ANeyAgesnlary

(52000, 55 o0D) ih CnNally

Stendad eno pla% on imnportont ole in Stahshca

to ot the Gtatinic. t impattone in eohaion

ivn bln:

tt enhmate ot the parane.

Prpbable imits within whieh the popn Paramen nay

peut to lie.
 enhna to

nbiasad
Conbi

Tknnt, eri letia or prapnhio.

qual to th paramta.
oR, Te nbisel eotmato u on whOre mean value
9qual to the value of the popr Potameter to be
latmata.

d t be a point etimato of a ponameae.

a

then tn Jaid to be an Unbiased Ahimat o

Exomple:
dyath om a ntal pop Wth mean anl
Vonion oY, ten fomple mean ý an Unbianed
lotimao of pop" maan
Nou
E) E xi)

E)+ E)+ -:
+

wih implis that an ungiased enimate it .

Unbiasedn Unbianed
Rptnato;
An estimatm Sod to b
as the somple ie(n) approches
that . An eoimatm tn padto be antmp
Un biased eohimaton t the potameter o
Lim
E(th) =
Wkne n is the somple d, and bn The tunchon
af the somple valws.
* Bios ephmato; tn be a pant ptiratm 6f
Joid. to bd bion Opinatm

Et)# 8 and th

There a e two bion

bias
poßiive bion
point eaimat t
* poi ivea bjos: t, tn be
b said to b pofitve
va
patame , thn t
bias

bian neans the oVer

Posiive and ngahve

kn
underenimate
 Uhbi hed ebhmator bLUE)
* Bert inear thL Poe! fanam
point eotinaor of
at, tn be a
b BLUE
O, thn tn paid to
funeion t jomnple obs.
) th u a ineat
() E(th) = 0, a, tn
an

wherne
other

folloo: is not
An Unbianed aatimatomynceso to Uniqy.
() An nbi aedainattr my on ma not
onbnt.
i) tn be a hbiased lotmafom ot &, then it

(Dleunt Unbiad. stinatm rt a por meter

Aave di}fnt Verienee.
prof ()
7 dtinihun we have
E(th) = 9 i.,

Von(tr) = Eftn - E]

2tn t

= E(th) - 28 E(tn) +g

E(tN)-29Y+ 9v

tat tn Bnb aned oina

Whih impl:en

poved
201y
t N(,), Show that Y

Ss1" Sine X N ,1), thn teP.d ot

E)= end Vor ( ) = 1 ti:,2,- ,n

Von)=

Wih implin that Ix n Unbi anad nhmar

 Gennguli vaiatex takird the Vale 1 With
9 and the Value o dith Prob
(4-6), Sho) that
nln-)
Sine dran tm bournsdli
X}j-Xn
pop with pataa 8 then
X~B,e)
T= Li ) IX ~ bl n, o)

Now,

neh) nn-)

Md.Shahid
H es
o lq u e
E(T)
os
íitagong
l e v e r s io
tyC
f hi
Asstc

n(n-) [Vant) +no-ne

nel-o) t+nor no
|ne-noY+ n ng

ntn-) no(n-)
Wih inplin tha
Poam Sho thaf th nt Urhiobad 2ohna
5o that
T)=EK)

Tx) Lo odd

Pe) =
Now, T) =k)

EKo)*
 E9]= +[2Eg)+
E) ’+
E(24+,
traf
. ahel Unbi
)=0,it ond =25Vor(xi) . E)
[ine
bedt onewkith entimabr true tke
ohmatm Unbiased blue And
On
Tsthat Bueh n whe
2
der 25. Anee vaTË ond mean akoWn With pop? the
Cahimatr
med. =(- Tt)
K) implin
tha Whith
(K)8 -
21
 1E)+ El) + El

+ El).
(ut ) +
2 t

Vn1 ( +h+19)|

= 833
 t(r) +nl)] t )

q:38
the abDVe We have,

Vor t) ve T)
Sinee Vap ) miimm ahmatr
in the Bryse of. minimm
VaTene

but Y not Unbimed enhmato

Vorbe) =6v
We have

(i-t u-)
)

=Vorli). -n
n
n )

Ey , hh implids that
Unbiaed. 2nhimat somple vnneay
I ki-)

(n-i)

-.
not nbiae hma
 = A .
(nR-) n+
Ii(TN-) A=
n
nn
2(2-1) nV A
11
n A=
(2-1) 2
212-)
Av
22-)
An
e-2)! 2l2-)
A.P) =(4) E
Then Lohinato nbiased the be AKd
Kt,
5inee
eshmate biaded Unthe find 2014
 3 5

ind the bent oimath omng foun phmah.

be a rA. of B2e n fom A pop

+ V)2
n
Hneo
f
Tonom

ofkori0

eohrar.
2

b-a 33

tabt+a

L+abtaV bt)
12
 N í,2,- 7=
P)=
n fm 5ample TUnom
-a)v
12
n
12
vh+-)
of
thin Vapione The krwwn.
(27-) u enhimatvp
b of (nbinsod the Henu,
2 2
12
6-2ab4-V
 MdShsh.
A>
Des
Universay uiciitagong

Sufficitnyd/sufieienl eolinatnt :
saia to be. Sufiintis t wses
An Colimatm is ponameler Contain
aboul th population
Ql the intormalion
in th sAmple
TIanlom sonple. dran fom th
funehon f (x, o) A slaibic s= B(X,,)
5 detine sufiiart Statinica i ond
to be a
the Coni Honal Bioh. o
indepinden st e.
r.A. oadn tom
a

TRe pn8. funeti m of poion Biat

X 0, 1, 2,

50,
haszA) =
Whre,

31
Poi oson With patane ne.

independent
goint denit funhn: dl K,.,
Size n a n m a Pop ith denatt fneion ta,o)
n

thtn the jont Benhilyt funelion wdsfne

fr4)
n dradn tom the denitt fonekon se) then the

Wsually danaiel
n

= t , 9)
tunetim
ano i u Varuable but in ikolihog funein
h Variable and i fxed,
th paraneta
but fo liklibnk,

L=
F()
Md.ShahidulHoque
Associate Professor
Dept.ofStatistics
UniversityofChittagong
{(x,) The Sinhatho 5,= Al,

facni athon Theoren :; oR, Asker-Nynan Cribin

TRe

and doe not

involve the parome a.
 3

thyough the valn

on
&Tun 5=A4n)
Staioic.

SuiintE Stotaie;

be a r.A tom the denitynein

Con be atbni 2ed a

(5,5- Sr ) h(4,4n)

non
Whre t h(4)
indepondent
Only
tHrngi the f A() b(4a) An(K )
A(H-- a) inde endent oft 8. Then
SS S jönH,
 a rA. orm the dinity
Then

E |20]=0 ony when P[2t)0]1

whi, 2) 15 a shatinhc. Ap te sJatiah T ú

Gmplate.

binonial Bhos?
io4 ith parame e,tun show tha
T= X% : nat a a lomplafe sutfiient Stohnhic
bt T, XtX; Conglate 5wticient Stutinhe.

’E() E -E%) = -9 =0
not CsS Sine E)=D but Ti D

Ex +f)=0
Ex+E)=0+0 24 20O
22ro T 0 WhihtptUy
Complete suficnt sbatiohie.
prphien o suficitn statiohie
Su4hünt slahiohe Uniqud
sttiohie alo a
funein of sutiint
WAny paramt.
staiatht tn th Sam
Siient
the mant efiint
Sutfiient buitic u
xint.

a Conintent e hmator
 K,,.

0,J,2,

1 0,1,2,

Hind the Sufiuent Stohohe fure o.

() btiyen that
a) = (-ay
The oint

(=)

6(-0)
-P

) (-) n
 -(.e) A()

r (n-e)! Which
n indspman
sufint sb~ohe fn 6.
ABttrnaiye 6fi vemthat

=t (-9y',wan,Tst Z
5ine X ~ B , ) ten
8(n, e)

(44 Xn, T=t)

2) st(-8)t
With
2) siven that,
tae)= ()6-)"- ,1 o,.2. m

fune Hm
n

4 Con,)

-()-(m)-e.()ö*
/m

Whre, mn and

) With ndepindnt f.
a
 (3) 6iym 4hat,

The joint

- , ) ={u) 4) (Inb)

skere,

) tivn that,

The oint denity funim

h

t(n)
V
-4-e)v
tE-28i+9)
 ) ; ( n g 20T)
n).ohl4 1n)
wWh.

Whith

(5) ivn Hat,

e)
TReJoint

hlH- n)
where 1 anl
) 2n)2 whih indeprdmt
4 Ku x,x, X bt a rh. from a uni tom pop? on o,01.

, 8) =Bn

au) = n[ Rio)] fk6)

We hav,

(
n[

Hene, Fhe-dynan erilerion tke Slatiati

 b ro tom th d.

0bluin the sutfiient Slatahe tur o.

-Z n0

h-l

) o-e)
n-+
m-1+|
1.
n-)T

0-8)

n e
-Dy
n o - 8)
n
 Xa bo a P.b. tom the normal. popustion
With
funeion
(e.6) i) 1

Fnd the ointy snfhiint stehinhe

The joint Benit
si-)
h

282i+ney

Wire,
 ~wlo,)
t.e,
)
N XSinee
. meon pop the enhnator
o lontintent a
mean
X Sample thin o), N(U, X~ VExomgle
im
im
a be toboid wtn eaimat An
be to aid
pop th ofDimat
si2 mple so
 Thus
evi Buni 9 notmaly 8iotibutod. with meen 22p7

p(2)
No).

MS
e lh
. ahi
H du
olqu

gona
Unive
o rC
sf
ih
tyittagor
DotS
.ftatistics

im

n’

Whith implien that a lonisent ahnao ou.

 Coryintnt.ehmato
prprtin of Consintntoimar

ma
detndfmmi Jonple.
(@ onsiotent eotmato is not Vnigue.
A Conioent eohimatn is not necesainiyy Unbi nhed
iv) Varian ee Coninte eaimat tendnto

l
Corbiatnt tolda invatúanee
propnyt
a Consinteat enhmati
diatr. then to
 K6
hae, We
inaguali
Sira then
po}oihve ont
Vor(t)
D
en im
onthat Shb ORy
y the thConliion
Conin
fore for cent Aufi the »What OR
Prafiv)
 on vnhin .tnimat af o, ie,
thim
E(ta)=o
We o ,
Ky n

Vonlt)

Vontn)

Now, lim
no Var(tr) =p thun We hre m
lim
n’
be
nogaive
Whieh implien that ta ú. a Coniatent-lohmar of
im

to 2e n>
 (V) We hve.to sho that to Coindlat

W n00 thal
lim

im

im

im

im
n
wshre,
im
et2te
im

im

inplien that
WiLh imoien to ý a ConJintent eha
ne nbianed mat
 y X BCn, ) ,then Show that n a Con inent enhimatm
Ournouli 8int.
> Sinee xn Bn,)
tun E(u)-n,Vor(i) nol) Vor) oll-o)

Vor )= ven h Ix)

=n. n - n . oli-e) e )
no

E() = 0 k

Vor a)
hY
Yor ) n

ny

&l-e)
Ver )

k
81-0)

im
Hw

Xn be a r . tom a poi%on iN
Witth paname o. sho that Somple. meen

meen an apienee

KV

nev

im
ba aat ton one
. Wich implin ttat

Vor ()
 An eatmaton badto b men squa tn

oriotent mean

an n tenl to inti 0n inatn

sito be mean

im
/
E(th-e)o
 X~ e, , thn ShDw that n Siple lonislt
MsE Content eotimatoy Sf 8.
as
5in x~N(0.o)

N(o,1)
NOW,

6
 6

im
n>

na Simple Coninat ehmatr ot &.

MSA E(-e)=.Voe (7)

Cm o = o
£(-)2im
MSE( ) : m gR-)Y:
 Hathe of Momento(MH) *
The methad otof mom&nte into Aueo A karl peoitJe

ra mDment
a Kinon
fumeim

tuether lat m be the rth Somple raw morentn

gual to Ex then th metho& ot mongnts Eont
Coniat in

oblinad
methos f momens. so that tue
Kporameters denitt
funeion we areL [olveLte
=m
M m

Thws
On eepoental
dintr ith dut to,8) =
 not butmormal
OTe MH The
enhnaten Coniioy,
the Aenena quite Unden )
enimaty.
pnvitn mH
ie,moment populahon pong Covreh
ot
me lohmato Convinfent aremomen% Sample (
dave, We MM Hen
Here
ah n a
2-1
We
2
 +sa -2
u+62)
2% -
E()
No),
have We ’
moment mtho
imate an
fom r.Ar.A onide
a
Bintr.
momnt the
mlhod Thin
iklihooA
meihbd.
f ne tho tom bthine o tho than
 meth of momnt

2-2u

-o
) -

26Y
Va
6n
J

6 Sine e) &

No). MM,
 Am.

X be

Eohmate « and tu mtd ot mo mintr

’ We ave,
and
Now

Ta

Kl)

the MH
MM

m
 ’m ) m
m

- M m,'

Fnn
nz
Xi-)
ny
Zoi-)

<
ni
z-)
*Eahnate aB n he Ca ot paarorn T
diotr methd of moment.
tahmate the MH,
We have
m x
Now,
t d

7e t -1

the MH

m
m -m
m'
 44Eh-)v

<

Iu-)

a, ) pla)
Eoinate MH.

ond m = Iu

pla) (-)

pla-b)

aty latp
 lath atp

1atb lat
1a atbt (at)lot at

Tat at)a a
a
atbti) atb) ats
acati)
th
MH atb) at bti)

m
alat)

bolvi
(-) (En)

+
 b

Fom )

1-X

+) +bri)
76(b+-z)
(I- )

(I-)
76(ab-7+1)
b(b- +)
(

bn

(R-1)Y-01
(7-) -n+)
Ti-)
1-X
 Aorple
bvori ate opulaim or en

DE
=0 2,

Hene
bhal the erhna
usieh Cala leat squa lohmato.
éxomçle

a Conbtant ond is Kngdn as the tgrehon Coskgiuen

 The laant squane Qpinale providen the Valw
in Sweh hat th NO \e) sum

are

3-2 Z-8,*) =0
- 2 Zx(-Rx) = o
From

Fom ()

-8
Zny -
 n indeponnt vóriel):
me

tom
be
knowh moy be Unknodn. Aume that 2ach
[amle peint Unil rovi des intmalion on VoruableA
Wheu the Voruable Y is asud.
Bepenant on ohon varuabl en X,, X
X!

planaty Ariables. Ha (ku) Vótüoblas an

studind om esch sample point ond hene 1e iatn
VoriableA is in the incipine of mulhvatiate
Sinee the Variables one ob a tom eath

t ws Coryiden thal -hete exiA a ineah

rulaionship of the vorua blea, were.y
dsprnn on
TRe ineot molel fe the
varuable is asu mad
+ PXK +e -
Wre, A'a(i- o, 1,2, -
- )ae te turadoon
 fortioi

Vouo Ble) which 0t not ineudod in th

In

Whe
Y= |1

hx1
1 yn n

TL pnblam is to Qnhnate the

Tüe eoimaton u dorg in Ah aa atanth Veett p.
Wa that th sum of
15 minmum,
SUm
 We havt
A

=YY- 2 y+' .: YÇ=pxY

belomes

miri mum. Thi poible. y

sofvig tin squation, we shall least

Whe

EU) =0,

ate Cottelaod

(0) TRe x ane meauud Witht noH

iv The Xi (i=2,. , k) Votçables are non-tondom
(v)
otUinlependent.
Unbet
Sho@n ih (i) iVa.
= -2Y + 2 *= o
2xY

('

Here u th
 Squaen
one depsnhent Varuoble onb one

’ zo- -a- bx)- Sum

14 e NO.6), then
20 C-a- bx)

The lkeli hood funetim

20Y
276Y

ACLovint to 'ML methoh of lahraion uWL neot

ffn t Value t a anl b in Zueh a wo hat
L becomA Tin
Tu valu of a on b ü obtai nos bt
thl metthd ot leant sautes on minimsia
Thun tL lotmayn ¢ the potameloaa
in ineat modl ate squvalent to HL eahmaora,
 Aistri buted.
an 2e0 anl Common Narù cnee 40m ee both
method 5quates ond th methol o
maumwm ukeihoo ce idenical.
 Lert 6,
Conatent ate fotyRotimasquaA Least S.
X= Z(x-)
BB=
Here,
= A
tuneiomn ino a
into unetion non-ineon the f
Un
out tound albo eahnar
Con squaeN leot the
, near 1on-i
fnetn
of ineon latinan
is S4uaten The
(LUE). Qslehantnat
Qshimob squates leaat 2.
Votionee.is
Sguates
\ean The
6
The line ot mo&sl aumd
Wkn, E(u)=0, tluu) =oT,

-( ' (xo+u)

6,ne

uu x )
Pronf- 2 The asUm moll

TRe leanl
Squas eninar

5ine(x*)x a mattux t fixd

the

a ineo ebtmatore.
other ineon asd,ins
onbi
Gyy
eatimator .hat Wh an

ahosant DY 5uch that

- +DY [Dw a natrix oft Consaat
=(*'*y +DY

bËnee
have
E()

Sine E)D
(-) )
 + 6D'

v(o) + 6
m)
poihve deii}o matrix, soD 0
=

v (@)

The least squaes eof ma tot P mi nimu m

Varüone. Hnee, 1hat
h Method of Hin m m hi

The meihod of
minimwm ci square 1s
fa fittg the theoruticoad Siat". Frr th mthol we
nul a stutintie KnoWn chi- 59 uate denotod.
The prDebute aeplaine. belo
bo
popr With
of iz n drawn tom
Papame,2nittfunein ) where oin the
Supposehee obapvakor Conthwows
meaumnt gpd. in

in
thin clsen With hypothai cal prba bilHe l)P,e)-pO
ni = obsanad equncien in the ith cla%
fio= yPohical pnbatiitis of the ith clas

Then hi-sqATeis dined

(0i-E).
 The Value t is to be
be enhnatedin Suehh a
that the vale ot beomes miimm. TRe ii y
Value
Porsi ble,
n P;le)
vative font the nied WeVauemnm t
t mimi
method
wLwe thia parameerr th eohmale
s the
iwheh
w eatimate atemt
to WL
mth tti'n 4n
 n-o]nll-8)
intp Avide
The
alue havi x ond
15 chWhi
is ofValw the elenent
, of no thehe
be t
Thun,
1-8) P)=
|-X
thal, Ven 6ti
e. tonpoyane With pophBernoulli
the fn.
n f Exorpla:
bsr.n
(41)4

(6)9

(8-9
atimotrn. fiient
OE
idenical
HMe and MLE Sonpla arge. hp
biatibutal.
otu
not eahmatory The
necsAi
 MeHo
Qaination
on byian
on
nati nethodo
of -ciCal Non
thenatho& cloical non- In
MLS ()
MLE
spaU paramee the in Volue
take ean Which Unknon an
ixed
etor mhoßs
the claica To
methol;
I, Clossical *
*prim da
be

Witth
funehon
prior

funeim a ronom Variable.

dnily tonehm
 Tien l) lahmate
d onder the be t and
e oVer bete t ?
-t 4= -
Lt) La)
Sam,
tha
negative poihve
on the
Called funehinis lo% :A
Ayolute
n loX LINEX
tonehm Lo%
)70
defined
)
nogai
ve non- vals rual (he funeion
is lox
indetibion
1hio eror means LoX Junelin; LOss
 the
the
D=
otnation
-
uo in aahinah
O, ten LINEX Lo fonetion
Lo) =
Tis
but for a bs
K-0-k,c0,VemianK>0t
in 195.
tunei
wken D-0, i.e wen
on We mut
have Lo) =0
-A
We ha,
at the point o
D=d.
ke
-=0|D=o
Fom )
LO) = k[-e -41 i cD, k>D

Presantel
llner in (986. wherL too
Paramtets
 (9»86) Zallner shon
on
The ()
adit nenti oex Gnd
al ith
acombines tuail tenei
on ThL
D-070 when
alment ond bLD D=8- wisnineang
mont ass uncion los LINEX he
C70,
D=-0>0. When
ineon tur
amA &-020
on D=when
ponentially
ineetim either
in
quile funeion
is LoM ihea Ct0, Eat )
LINEX
 Squane error lox fonen (+:8) =(t-8)
()
0-4 ty

= 0, 4 lt-&|< k
Quadraht funeion:

* Risk tuneion : Risk

meens oVmge the losas
Indieiion theon The 2pRetid valuu of o% tunetin
With reapett i t y tneion ú
Called unk
estimate

The Apeted valu of the

Tìnk funeien with eapee to prion
IS lalle prion rik o
 tuuetiom

the
pOs\.eni o dinitt unefon then poseruo Pinm
is defnad

* Opteior ByeA Lohmat: The eyeputad volw the

funeion is Colled pon-eior bayes

Ond
ehmattgý
dafi ned

* Beyes enhmator An eotiator which minimize the

poateruot Tunk 2Nhimatt.

baráb fmet mthon penkeoin

pink

Lainator Wh mi núm2e. potenln rimK

(aed that
the

Then
Qotmoto

bayes eohmaton nt nbia entimat

ayen ertinaBt ote a
BAN etimaton
Bogen eainao orethe neim ot mi nimal
sufiint 6tohinh'e .
enhmato appDah MLE
prion intr
Show 1hat
6hou tkat Bayen eahma is that enhrnator wkich

be

prim Jta unetion of 8

anl et9)be

the tuN

Rotr8) = E

Jf4ks)
måmmite the 2A pretiom Wihin braeket. But the
Apreion Wihin bracket is the pON-erç ot tunk.
hant that
that Raimab
phmaot
 rolem-o: a . dradn tom
prion

tni of is a beta ints with parantern s

8l-0)
0 sho that aatihr is not nbiasod but
oti aly unbiaseb ond eonintent. AHbo shod
thad po Nerýitt boy eohmar is the inear Combination
og the somple meon ond the mneon of the suggstd
prion

WhvL,
(-,8) =e-)l
and

n-Z4

n
 (-e)

E (8)

ptn-I%-)
(-B)

p(a+ik) ntp-T) put Zitl, ptn - I)

|a+ I1+)

(K+ptn) |otptn
K+tn
ukích is the poatetuoh
iasea not ehmaot
(s Bates Porteriot Hne,
nE)+
E(n)=E
(atptna)(atptn
oslenjon the Wiisth
(1-)
 lim
E(G in n(0+ )

-im
|+

im

Hene, PeE

ntatp n(t +P/n)

im tim

im
Ope)7, 40, PoE conhtent.
Foo &I-8)
K+ni) hp- nz)

E(®n)(Kn)(na-n9)
30,

im
n E(ps) n n(0t%) n-o +/n)
n(Itht P:n(6th+)
(etn)8)+)
im

n( t«) n (it /n - )
(

(Pr) = X -)

n to
PB

A\)+-A)
Tree, pOsterion
and eon
 with papamn . et the prir a
gomma Binl with aranet

Wt hove
) X 0 , 1, - -

and

pooteri toneti en
f(0t) =

ng

len)a
(tn) 9 T
 12

{ot) = -en)e
entinaton
funetim.

PO

(nt)

nt

n t
 ollo beta
Swppose. he paromejen if a int
ind th
iolr Wth mean
privo
m Bn
With poramtet
The p. 8. beta diat
,n-1

(-e)
elmyn)
(-&)
plm)

Imty n -

3
|mtnt)

m m

Tm(m-tn) mtn
m
(

2) 3m
 Sh
9
+

(0-1) (-u

(u't+u)d
1,

(+u

S)
 Eley=
t20
Again,
r 2-B
-B8
8iat gommna .& The
prion the find Varionee
20 and mNeen With
gomna fpllown bint apopamotna T
)
Pl34)
4(0i3.6)
nonfno
 3
|D

10
2

The
prim Ainhbt
S
e:s.n)
C .
an om qon
-netial &int th poraneten &.onh the prioo 8inily
gamma &int With potameten x,p. find bay

Gn

-nT
Jao
 (pt7)nta
(pn +n+)0
|nta

the

Ele) = (Bn) ptn+)8 n+d414

In+a

n-fat)

nrta
T) Rtn3)
 Con fi denee interva Wig Lage sample meto :
4, , n r.a obb^.s from

is
pop" havi denit fonetionf o ) oes.14
asumed hat . The ike bi hooduneten
of the Sompla
obervoions
L =T i )
1t w ned that the rgu eriy Conitioy are
Vaid r the intn and henet ML oinator
obtai nab solvia the sauation

gin, wao Kno@n that thu ML Qahmatt

iateioute an normad, ond
E (3l -ED1
v(-E
TErafore, the arple ize is

N(o.1)
outined nm the tlloi epuatfn:
 or,

ond p[9 4h247=1-o

Pwbobilit Tabli.
n obseVoim
pop". With

Congidenee inteev ssumi

fiven That

L=

li-e)
2n%
Q3
 VorDloge
2n)

2n9
93
2n

Nd. WL Can Weite

+-9)o)
TRArefre, t00
fm th

Vñ(i-)
ri-)
n(+-o)
 A4
Vn

y Fn the 1o0 ( -a) %(ontdon einteoval of Pin oo)

Woine

We have,
p =() ) 0,J, 2,- , n

|-P
R(r)-W-)P

PC--)
p p (-P)
nel) Nn-n)
 n

n NnlI-)

+
J
Nn I-P +p
P()
PLI-P)
) -- EP) PlI-)
We Con Write
(R-Np) ~No4)
P)

fom the
2=
Vn
Vn(-Np)
VNPlI)

NPI-D)
n (-Np)Y= NPU-D)
 2%) (n+N 2
Pa
have aguation
We thin
Puatin
im quadraie
Dq i
aa isqwaton thu Her
NR-P(2nnNA
* ivotal Hothod to Find Canlidene Tnteval:
a r.s. of n obbepvahonh
5eleeted om a

ObsieVotHory Gnd
Colled pivotad qanit For

the sompla abble Vaions are seletadom

thn

N(o,1) n

qyoot Evn, bomple Jize is Sinal, t

5inee We need to

po bsi laSamples
 the inaqualily 4<ac
inagualiy
, thern

wwlo.1)
piotul
5ueh fhat

Hnee-,R+ ,nais to0 (-o) % onfdsne intenad

 -961
i-)ñ
Weite Con (We No,
-E Vorlog
'=
:-E)
nEl)
nX
-) dloge
-ntnx
J
have We
method. somgle lange U
int: the inooyod
of Confdom
to 9/% the Find
 7-)-(t14:)

22 + 3842 0

2Y- (27+ 3.8y2) +7= o

2
No
Conide the ónal
T, then the Conidenee intals 1s
T=0.45
wher,
 bo n 0689'n

ume
( when 6
When
When
Whon UnknoWn

Cabe-i We hoVL

&=

nHt
tee,
We have two umbA Sueh that,
 5uehthat Thaefre
pilal
Here aiatributed Wich
f t that, Know We
Knoln Whon
that Swh thre
pivaa
NNO,1)
(ase thia In
Iki-) Caimatol Here
6
when
6
Case-:
 Hene|n ,nAYis a1or- ci. for 6.
15 a

Case-V: Unkeno n
We Kno E(-) (0-)AV
-)AY Jih is o,buto
6Y
pivotad

qporh
that,
fr (a) t two Values 2 ond2 5uh

6Y

(h)

|00
 ndomy seleetd ne born babi

D,2-4,323.2.9,272:8, 2:y, 23, 2-?,28

Ssf
We Kno),
-304. = 248

E0Y*% 08 -

unknn 45/%

) 2-78- |:96
11

2:St, 299
goin,
-)Y

4ere,
(-1)x009 S6
4b-)x 0.0956 3-247
20.5
 Interval
).
nterva enimalion is a 4ehnique in whih
attempt to hn oul On int00Val of on enhmato
5uh 1hat the pon dno nonol Will ontun the
Unknon papame}e With bome
nqecifod. probabii
be a 6f bite n
noem poputahon with
ih mean u andarisne
Lntmate
Buh that tn Knon s on the ain sfsnpl abn
the probabiit that
in the intepval

The nethod
inepvad eoimaHon.

gl Metho& of intevad eoimation:

Method ot inteevol enimation atethee tpe
Sueh
Contden inteval
) Aducial intenval
Botoion inkoval
 NUessily of intteyal alimaion :
In poinlt eoimai on We nimate por poromd for
a sirgle valw (poit) Fom thi's e (onnat fina
onlg
that h point eotirat is
pnbabiil 5ons0
We nse intov

LRimotin 5ine in intrva lofimaiem we to

find on ineova of the parametep on ban

O} the S0nple obAraho With Jome speifad

Probuoit Sueh that thL inteval ContunA theu
Valne papameer. we also toy to miumie
the paA of the in tenval

#Confdtnee inteovad ond confdenee otfiet

t si2e n dran om
the popr. havi funehin

T7 A(4, , - , L) be two 6ttiotiCA

T2 T Sueh that

W ? does not depnd on g ond

o. le) (s som
funei n f&, then the ronom ineval T, T) 15
 Here., =(|-x) is colled. tho onfene0. lo- fheientt
For the Cantne inteovah (T,T). T, ond T arL
Called lo@er on& upp Confiden. imto

reapekvetPLT -alo)T =8, then Ti lower

Confden inteo va fon

A Valu (th,t) ot the ron&om inteeval T,T) ISs
ao eoled a sd4fs Confilenea inteoval toe g The
beteen
the Con8inee ength or ge.

For frad (-«) the ait ot the value t on

t fom &fent sonplan deterine a fed within
0), the fa& is Knowh o the
lontdenee belt
k
Centod Conimto . intoval,
A Confidenee inteeval is sail to be. cantral
iKthe tail areas of the inteovel ae. agual. fir yomple,
Co fiint t a Coni&une. intevod is (-)
3 1h
and t tail
aneCanta.
aqual then the inteva

Non- Cantal, Conilenee intOVai

A Con idene interval is saüd to bl non-Conal
u the til aran

Then theinteyal aTe caled non -centra

Coniden intervad

There may o4
bome Coni denee lo-fent. Amorng these contidlnte
intepvad With he shertent length o colled
Shontent Conflene inteoval.

Whre,
 be. choosen
(n on infnihe
numhe sahinfga Conitions)
Heneo Hhere aeniniBe
'nteevalh OnVoulo be With the sam Conioen
Co-iiet («) onstht Confdenes nierval

too alValues &thun T,) is baid to be Shortert

Congdeneinterval for
to ale) with Contidene cofüut

Confi ene intenal

A 95/ Contidne inteova tor a popr. povta
-mL 9 is intenval sueh that the probebili
that an inteeval ill Contain he value ot t
paramet s 0:95.
A 9Sy: conidenee inteVal is an intkoa
(ortruahvey jom the somgle sueh that the Sonols
Were tepaetadby rawn ond. en inteval is ConAue
fop eachh somele hen a qs% of thin interval
parpamete
On t h
ovg
 e
=8
E() i.e, 8. enhi6fmato Then
Unbiased an
'e sample it and
a Is that Connde
Es f*o), funeHm
foam
obsen n r.s
sualit chiyokav'a the ig Consitn
in
d problem buionak Sintrü where
inteovadh,
tontiden
eeconstuetig af
methad qenenad
A
a onienee Consructirg of
Methad Genepal A*
metho& quantitg WPivotal
methol sample Lange
metho 6enepa
method simple O
methada fun pllodi
(onbirvets interval (onidene A
Intenyal(onfidones Corvrueti
of on
Hen ee he

Wwre, LI) ond vt) ane the funtion

Tuio nethod lan be ned

a pop ih Pd.

And 100(to)
meth

6tiyen that,

2 2
J
 -[E0)]

(=2£
E ) 2 x =0
Henee T 2X

Vtt) = y Ve) = 4x 12 =a3

Tknte,
-a

2 2X

4en
is the 10 (-a)

2n
2