\01

Chopie' \l IV\iervotl f 5 .., ,·WI l::f i j 0 V"

\Je_ h~"e lect,\")ed ~1 CJ""v' io estiV11qie__ pqr-qvY/efers

af a. d;sfr,'h~.~f;oV\ trovv-1 a S'ct Y'Vl/}/~, Thesf: J '' I f ll b ~si,·vnC1Td\S were. po,·..,-t es;/mCl-fes ec:c::tus~

3'veV~ +l-le ou-lroVVI€. o{ -1 tH? scr"""f/e; The es-I/1-'Yra.fe-

i.s c:t s/l"lyle poi 1V?i- ct nu~ ber,

No~1 rdi~e(

we wa~t

t hq h

lo

If l t · /..:7\\ SO< J mJ ~ e 5 J I,( e s 5 1 s , ;

s C'JJ // r fJ,,'I"/ k i-is befc-ve@VJ ~~ ctVIcl {'(,\\

-A ra,·"'i es-1,-"""'C'#io, ,·s ~ rc,Vlclovvt Vc:trlotble) e,J, A'. Th e o 4 f co lA'1 e ( e s j /.,.~~ cr i e ) i 'S Cf n~Ynb~r; fC'~J, It,

AV\ in -lervctl es-"1/rn.c,1o .. ' I .S Cl ranc:lovn

a.. I'\ I V1 7 -e r v ct I 1--v' h o s e ! 11 d pa /~s Cl r e

vcn-:c.JJ les ~ e. J· (f=t X) :~ X).

;Y-J-/erva/J

rctndoVV1

pre hQb,'l,'ly l"'a111d0VVl JV1if'(IV<1 ( w,'/1 Cov-t1c,iV1 the

We. co1.4/d tJ,e.., de-l~ rmiV'l e t /,e

thof th~ Pq_'QMP-fe_, a -f l.-'1 -fe f e..st-' CJ. j, /<.. •

'vJ e._ rn ; j 1,-t q Is o m o d i ~~ C) ~ c.. h o o 5 e i " I ~ f' f "~ ( .5 c. 1 ;,., 0( -f ,' +- Vv i II c 0 '(J ~D. i ,

Vv I-I ~ f r c b ~ 6; 1; ) Cf () /o •

1 he rqv1doVV~ 7 1., f' pq rq Meier;

fhe

be O(,(fCoVVIe_ ~f the \qVldov¥1 it--rlervctf wo'-~ ld -ft-,.e,

a_ ctD 'l4 <-ov1 \.ide~ ce ; trd·e fv'ct I io\ 1 hf pq ~ q V"lei e r.

_L-f we re peqi t ~e ex per ln-1 ~ v1 t ma v, j f ,·me 5

5o i~qi -~e have Vlita~~'~) 6u·lro~;qr-:5 fc.x -~L-,e fO;-H::\Ovv/1 !v·1if'\vdl) theVI o{Jffv:X!.VliCJitly rare ~{ ihes-e> ov-lcoVVJe l~l'lec,,als- w;ll c_e;fi~J~,·v1

+ h E' r~ct f Cl ~- VI f=' ·t f' \ ,

ExaM p ( e 1/, I . I Let X\ vll N ~~d EX P(K) I ~··;~l,o

the ll~eii~Vle C·IA vv1ovrfhs)

r-eprese~-1 t ct L/ 0

elt>ciric~ I pctris.

-Tt c_q"' bt" Sh<:J\,./V) thttt. X is an

uVl bi~sed (fo(' ~) ~'-'11-'\C -lro"' c:.-f Cl ..s~.,.~{'-\\c.it..,f

a V1 d c c W1 p I~ -1 f' s I ci -/1 s -/1 c , T h e r e f~ r e J X

is " U.MVU E \c-r A{.

I VI o i I, e r t.-v 0 c d s X ,-s q j 0 0 c:/ e s J,-WIQ for

+()\ .,A..(_.

Recct 1/ H. .. t- f 't" ~ GAM ( "--.J 1"1) ;;.:,

5 i vt c e EX P(~) A.; G A }1\ ( .M. 1 1 ) a vt d ,· f'

t""" iV1depeV1dfo1t. GAfv'l(k ~K~ ~ t~e~

~ Y; 'V' GAJ1.1 ( I-<_ )ZIG~, <AJd Shqpe pq~qme-ft>Lsf

Also fe' c ctll i· h t:Y i J i-t w "' G A M ( eJ K-)) t "' e VI

z;d 'V ~2(2.k:). e Thus ZV\X K

IV\ fC\C""'c.\.((ac/ ~ox "'"'~~(_gc ). M.

rv ~l~Y\)

110

Us;~3 so-f-l"'~.--a\(0 o~ +qlofes we c ctV\ f,,tfl d thl5' pe \C f' V1 1,'/e_s 0 f' [}.__ 'JL17fo) \.v ...

55 = p (J:-:~io) t.. '6~ X< ;;t-~,n(<&o))

- P( 57.15 <- ~ < Joe. t 3)

- f)( _!__ I_ A ( I '\ IOG.t3 <iax 57. 1~ )

- P( ~ ( ~ < 106.6_3

- ) cgox sr.~

-lhvrs f~-Je,e is o 't57o pro bq h;lriy f hC~f the

Ia v1 d oVVl I,VJ i e r "a ( ( <60X '8'0 X )

) -- vv,'/ \ COV\ ~o.r'VI 106. 6 3 s- 1. 15

M.

-1\i+e 1 pe, fo'fwli~ The o \..-{ 1 c__oWt E' o-\' x .::: 'f 3. I

F,)(' p 'fl f 1'yv, 'f' v? t­Movt-fhs,

we obt~i~

Thus ffrtp ow {c o..-vJf o-f il-1~ i11-+ervCl/ ~sf,·Y'Vl~to1 is f~o(.::r3.t) ~o(~3.i)) == (61.'651 130.3'2)

~/06,63' ) f;J,/5

CIVj

Thi5 O!AitCJVV1f' I"S q crs-;~o C:OV1 {;deVI(~ i·dervc:{ I +0\ ).(.

Ill

Def: ..L V\ 5 e V\ e r-q (, ;f

IlL

p[.t<x,, ... ,x.) < e < <..<.(L ... ,x-)}= If

He.o [-Ux,, ... , x,) 1 <..<.(x,, ... ,x.)] If C o vt tide~~'~ c e / vr 1 e r v q I'' -\ c> ' 6.

IS 10() ~Yo

Def: If 1-Je let ~(~) "' - co) Hh 0= P ( ){J)<. el. "'{KJ)"' P( e< "(X)) II II .

C!VIJ V~.(~J 1's c__q ed a OV1P- S1cJed ~pper /00oi6 ' r·d I, '-L '' Covtt-' €VICe 1wr' ,

The_ coV\tide"'cf ,·"' iPrvcrf (c T) is (-(1)1 1A(.c)). Ne'J-fe thr~i ,·i vve k¥Lo~.-v t h. a-+ ft-te )~ct r a rvt e i P r u pos;-f,'vt>_,~ ft,eV'I +he C...L- C-ou {d be 1-v r ; +-1 € ..,

0.~ (o; uf!)J.

l)ef: A 11 OV\e - s ,·Jed I o V\... e r / 0 () '('Yo CC)Vt f,'deYI c e /t'141,·t" 1 s de ti'Yl ed q 1/1 q / q J o u s I y.

Excp'VI pie II, "Z. I f-ro "'1'1 + ke P'e vi0u s -ex o wtp 1~ .~ .. f,·"' d..

f~e ovte -s;ded 100 d'/o Co""t;clf'v1cf'> /;WI,'-/5 qs w-e f/ Q.S C!V)

11e}vC!II-+e~,Jed'1 +""~~ s:cled C.I.

0:: p ( 'iiOX < ~~('i 0~

- P( 130 X x,;(<to)

SOx ~:(coo)

IS

< ~)

~ 1 oo oro

whq-t ijpe of ex wE' h e q de d +'o' ~

o. 1e

oV'te-s;ded lowe\ C.ovt\i·c:Jevrce. }IVI1;t,

I'? fJ/.'J:;t-1"""'1;;;; V./:) fd/f/0:3 '>! ~:'i.L

...so· ~«=> flAb .5h .. =9

~$00'1-; PI h OJ ~/'/\ \..Ad~+ (Olo. ::::1( s (

Sl r? rJ/, J< oo /

J2 = ro-9 ~u a '1 f"VV

( ( og,) :-;(_ / ry

>(Of,) ~1 d c= 9 xo~

• 5 d ? ,I 0 ~ ::> r:l/1 lC Vv4 ~J'Y-)~ ~/VI (I::> fdf,'.5'-0""4+ 10 J O_:J

.J d ddr, (o$).R-~~

.1/ fdft5-dV\0 a'1f-Sl )( 02> os-

2: \ \

ll!.f

A~ + iVVle tkeno is e1 choice>/ OV\e VVtJht ctsk

,'.f -l-i-ter<-"'

----\r'-i e

I 5 0. 1 b e 5 i ' C:: lrr C I C t' , k) t, CJ t Yl'flj h t t h Q i

::; h 0 I iF J t i V1 if \ v 1:1 I ( 0 ' eX p e c -1 f' c1 I Yl 1 e f I) C( l) ~ porClt-V1f'-/e, iS" .)vsi. os- /;ke_j -Ia be qbove

~E" I Oh., ,'VI ~E' f v q I '7.

1 YY1 f'Q r. ,

V5,

.LVI P'ac{cf' 1 e1vC~f -fq,./f'cJ en C>vte-s;c:}ec:f arf'

n fa<" 'J Q I~,-.., ay s v s- e cl.

Excr Wlple 1/,2.2 x, X ;v t:~d I •• I I "'

N(~~a-;t). 1- tVId C/ fwo - s ,·cJ t:>d

e~ fA q/--/c,)ed cr .fcl\ I'<.

t -Xt~ ) 0 =- p l_!.::1 (_ a- (_ ll!l 2.. - :z..

rf;..

r( X+;;.~,_. - ~ ) - /.__At._ )( -r - · r 1+2!' - ~ 7 V'l -'-2.:

I he /oo o 7o c.:r 1'5 (X + ~ • 'lo .!:;.1: ,r;_ Z.. I X - !<:- • i!o !!J: ) ~ -z..•

T~;s l.s -\; V1e. i-F a- IS kvtovvV1) b4-+ -lypicq/ly ,·+ IS not" Jc: V10\vt'l., The<-!5 bctd bec-q r.-•Sf" i "'12 1"'1-f~rvqf -·

I ·'rr..-v15 o-r V"" kV10'v... V> 1?ca cl VJ.-1 ~ 1-e r, 15 t-' Y f r t 5''> ~· a I,;·· 0 v'

!vq·s cctV\ 'be h.~111clled I .o '-!

" I.A.S'1'nJ 0 p,· v; to I ~ '-1 cr V1 -// !J ,

Der \\. 3.1 A -+vtvtci,,o.,.., )

Q== ~(X,, .. ,,XV1, e) 1 of the So.-Hf le

,·tr aVld e ,·.s ca lleJ q p·,~o-lq I

dis-l,;b(.(-lioV\ does not depe111d ofhec uv-~k."'-oL-Vn pctfqWT@-fers-.

[1 f t-tq~? ')

OV\ e ;f 0(' qVI.j

/)5

E;.rc, vY1 p /e; I-f (;) IS k vl 0\..."' v> ,.., <1 d X,_- ""' ~..: t. d ~'(A-1/u-·2) - J

-1 ht' V"'\ X-,.<.( tJ(01 1) ;f ' - """-./ ') S'o IS o--?v,

0: pivofq ( yiAqVJ{-/)·

Tf cr Is t.AVJkno~.-vVJ/ ,· ~ ,·5 VI 0 t-- p~'va-/erl.

tya m p/e: .T -f' X',_' "'"'· l. ~ cl N ( J-11 cr , ) ) -1 0 fc' "'

-j-A-< ,__. t(n-1) s -,rn

4VJd 15 i ~e ~ efo-<e q pivo -/c{( cruq VI -/,')•

OV\Ce' j (jt.A, heJ.vf C1 p1'voiC{ / cy~.~cq¥1~;)) cvCl)e)~ 'j 01..-1.

CCIV\ vv,; te ~ ~ P(~L::l ( 'f,~X ,G) L ~ ~~~) "-vhe re z.

} ... I +~e ., J,

ep~ (' f'VJ--1;/e -ft), i he p1va-lc) cplAcrnl'); Q. ,s ~-

1<. e atr-C( rt j J 'ny -1-() 3 e + e ct lovlf ' v·\ -1 t, (=' CPI1~fr o-f

t-he i~-1 e ~ u ct /;-f) a...! low~ Vt s ~0 ~ I C E 'H? \ ~~"' ,· v? f t·h f' c.r:.

Tie CI LS thP 5€' t "'f 6 $~c. (, 1 "'C( t

't0- ( ~(~,eJ i. 't!tl. z ~

H-ovv~ v e r J t/,is ~e1 f'Vto.J hoi be Cl¥1 iV!iefval ..

Yo~ c 0t1 fci lhE'V\ (' o /I ' ' C uv1 f; d f'"' c P set ,-r a

0{' co V1 f; de vi ( p reJ io Vl,

Also, Cl... , ~ I PI "'0 ,q, lU~0~;) VV\o j 1\ 0 ·/ f'>t,si,

Ik-t ~

De-\':

A 11 lt>cqi;o'r'1

1.-vhf'\f' +o n. o-{ de pev1d

1 '' J· r. po< o~·\?)r ·1er 5ct11f}lr-s f(x; e);: foCX-~)

a... G. clev,5,'/'1

/

f1.1 ., c flo v. 1) ~ -1-hcr-f

A Scq/e pqraYYte-ler so/;5{; P r f(~j 6) ~ ~ +l~ J · Tl-tc~-1 ,·s1 ,·-r X 'V f(x_; 6) fhet-, ZX,., -f(x:;ze).

})~

do~s

Th eo 'e WI I [, 3 . I Let X:. ""~~cl Note 1 "'cd e rf e is o,

-f(x;e). 5L(pp()s~ e i'l ct.., foiL£ .f:~ ~.

ExG?II"lp le:

Exa vnf le:

is +~~ CJV1i,1 f::":'lftcJvvd"-lpr tt~('" +'-12 cl.is/nb ... ii•~. ./ A

/oct:t'-lfoVl pc:t..~ctVVIE'-Ier, -f~en 6)-e ··s F''vdal.

T f e ,·s e1 5c- q le pa rq vvre-/er,; f·hf>,.. A

-e ' e:- I$' p; VO ~ct ' 1

Le t x~ ~ ~ ~ d IV (A-( I I) . T 1-,e VI X IJ the /11[..£ t"r M

a""d A is ~ /ccq/,o~ potrcrvneler-; - (<"-A.()'

t(xi6)= )rr e -z.. = t0 6<-.M.) ~here

~o (X ) ~ j_ -~ 6rre

Thl-tS X-A. i.s p•'va-lq/.. TV) fCfc1/ ,·f cq..., b~ sho~rvv- -ft1ot X-A-t N 0 1 -;;: • - ........, ( ')

·;J Lei X~ ':__ EXP(e)

X Is thf' 1\AI,..£ -'tl~ e. -XA

tQ<.:b) = 1zx >oJ i e ~

'S6 G is t::t 5c~ le J

- \ \' /1- \ -X -e-ToCB J ~ ke,e forx);- e izx/o}. -X i~ -e

pivc>tct { pa r q vvt e i e -r ct V\ d.

Theo\f'vVI 11.~,2_ It

t~E'V\

a..nd

a0d

where

x~ "-~~d ~(X; f), 1 et) el \.sa.. I(Jcq'lio"'

Gz. I S" C cl 1 F' IS" C!

pq rct me if r pctr q~ei e r

-U . G ) -I r ~- e,) '<..X) .,e._ --re -e eL '2.

"' p·· vo4q l -\o, e, G. -G, -I IS

ez. A

02 Is piv()~Q{ ~O( Gl. ez.

/\. .A

6, I bz. q n' IV\ LE.s to\ G, 1 82.

Exqrnf/e 1/.5. Z. Let Xi,~ t-~d N(~~o-~).

I ht:>VI ,u.

etvH:\ c: ?

IS C\

,· <; Cl

/ o~ q ~ioV1 '5"(' q /~

eq r C( vY/ (! 1 f' I pqroYl'! e1er

X C( V1 J L/X:- -J')z. <ct rf'> -~ 1--Jf' .A11- ~ ..r -ro f" t'\

/4. Cl v1 d < c- •

$,.; X-,<.<_

(xtt~-xY)

CIVId L-((t· -x r-. -" cr-:2.

IS pivo-le~ / for ..c.t

pivo -let/ toe <-15 c-

Tf k.--~ ¥'1~.-tf-l;plj -f~f'se p\v64c,/ c-vuq.,,f,'};e5

C-oV'. ver1lf'v1t con:s+"vds) -/- h f V\ we 3 e -1:

X -A A.. i::(n -f) C(Yld (n-1)$'2. -2

A.-~7-0-1) ,.m crz..

whlc. h 0. re Y' lc.e i 0 1/vO(" k lA/ ; ~ f., Sikce fh.e cl,s-lr: b"i iol1r

I\ 1

b:J

are k~-to~..v Yl.

E.xcrWJp/e: 5\..(ppose :;o'-1 hctve HoiA.. CctVJ !jdt.A. Jet or h!hqf Q ho~-1-L e -A?

q rox ro /o C..L

cr tor­r ~ 7 "to\ e_ '

-r-hi-J +r,'c k v-...()' ks L-vheYI e v'f. r t/-Je t 1.-t ., c -{ ioVJ

rn o 111 o -1 o Vlic.

A,

. IS'

Exerrt?ple //,3.5 Le -L X: /\,- i ~d B£i<{p)

X I 5 the fiLE fo, P; b~f f here 'S

Vl() p,· vo -1 Of I cy v. ~., f,'l_y .

No J.v-e ve 'I hJ fh e CLI

I

-X-p d

fr(~rl N( 01 I)

So ..L J1..t~5S :;o I.A. Cot., f- h /Yl k o-f j h/s I

~s a:s J rvr f to I, ca !!J p ,.v~ fct /.

I-f :Jo\A -f,_} lo use t ~is fo cr-ea~e ct cr , (do t~ i_s YlcJvv J; 'jOl-t see If IS q b:-J u;ly.

' s,h1'1 Fc,, 5i"'p/;,,f) (bui Gl. \.vo<"se C.J_)) Yl o fe fttct-f

-X- f'

J!~) IS aSjMpioiicof0 pivotQI.

II~

(1! 0tJsnJ·I-K) d 7 5hJ'/ +j > Yr' / --

( JA., 2 _11 7 ~o·l) d ::: 0 b . / %-w

-

(t'O)N ~ jJ (110)N ~ 1J rr-Y

fLAb W-X p p

j_;:/ c)'1+ hiA,IS Yl I <AlP(; v

l ·o 1 = ~ 0 2. .:: \J

( )?") -r D d F :'1 ,1 ""' !X ~ j d ~ ~

h/ J o J :.t/ u 0 b ')I ~, v ·' t ~ J J rJ o '"' o F [/' I.::J ~ ' t • 1/ a I d ~ b •<:j

'.r::f:? J~IO~A.A,IXOJddo f;;b 11.1~ V't()h > "' o /" o. ""( o 1 dd b c) I cJ. •• "'s .. L "I ["' • > n "' ~ ~ ""'~ 1-"' H-" ~ "'N

btl

Pivo-Jq f f <--U?VJ f; ·Ir-es ar-e n 1'c e to

;f OV\e is noi Clvc:(l'/qb/r/ -l/,e,-1

a VVlo1-(" Jf?l1f red rv;e thad.

use 1 bv~f IZ-o

you CClVI I.ASf>

R« 1 ~e r i h "" bey i.., >v; ito a. pro her h,' /,') y/cr{en?,-." L i V1 V o / ~"· v) . c:r f / vo Ia/ 'J' r...u:r n ti!J / J '-l <; f: tHe, k (:=' 0!

pro bcrh,f, j :> !"'le rn en t i•H ~lviJ I lte

fC~ICIIIYietf'r o-f' ir1ieres"t: qyrJ the Sai-'Yiple.

TJpic<:t/ly Cfr1 /1/..F

-!he o r o ") t.-t .P .(','(' / t> ,1 { ,s- -lc~ -I; s· I;, c

S' fc~ /p m e 11 t , /t; u e;ed / n

Ex q lN1 p I e II. 'I, l c OvtS1'J e I 0.

n i rc VV\

r~.,cl~m !:'<>w.fl'" .,{' 'i'i~e c..Ls-1-r, hi,.(~/CJVl ,.,.-If, Jev1 s1ly a

f(x) e) = ~ j_ - (x-$)/~Z ez e..

()

X?$ /

X L Q.

~-v;ft, G/o •

t"

f(~; 6): n- icx~; o) : i:; I

_ 1 'C:)(x~ -€)') ~ e~L.: 1-lx,:"' /(QJ e. G2W'\

-VJX e~+tv~

1 t X,:~ 26 f ez"'

5o X I XJ:n <Ire ) oi ~ 1 'J 5\.1 f.f,'c i e~ t for e.

Le1s use X/~VI io C:\eq-}f' C( ci, 1 c '

• cr o = e ( "' (G ) " t,!.. L brb 1 ;

vv hE' I(' q(G) ::: 5~ perc e v71/ If" .£ lo tJ3) : ?2) rJ, r~ f(' !"' n -/,'/e 0 -r ,J, ~ Vl

I

F,->'1 d C( r<&) ',

P ( J,: n f=: x J ~ /- /~ ( J,; y, / X J = / - ( / - Ftx ) ) ~

( -&-G)) h/ h e I e F (X ) ::: ) - e ~ .1_ [ X } 6 f

~ ( -(><-e) )j 5o 1-X',: /") : i f x ) e J l\ - · e e<

- \ - c-It., . -lJ ~ -f ct. (6) '} 1 hE' J ---- F'r r Cf-'t/1 }, )e _; ' VI F "1

. o s- == F ( q(G)) =- i i ct(6) .> e j [) -Xu.,

( 1~)]

7 .~s-= ( ;~JG~)), _ -n(_ c(~-6)

- e_ 6'

) (oct ( '1) J -= - t-~( ~e)-G J _) e"'

-) 8 +- ez lo;(. C()) = q(Q) -h

'5/rn ~ /C( ( /)) t(0) ~ 8-- G -z. f oj(· o:_) h

I::; d 1/-c.-:) d 5

S~"~ 5

<;I I

a""-4 FV''=> s/f"?'JI

0 ( 9 _)OJ \ ~9

~1 ,? 1 h I a s ;;) l11-"" o tt FLAb (~)0 bv,'ff<)/d

i:.h'Z t9'1 e---~------~------~~

. ~--~------------. ------5' 2

(e) a

(e{1

>I

' . . S I z :: V' ~ I X F l/1 t:> OJ~ Vf s j_

( C" o J b o I ~~ -e I u :~ I ( s h )b o I ,;~ -e) c) : 0 b '

I I :;J aVf + Of '1 :Jaa fv\ON

ZZ/

Jz_s

De.f: If <A '""do""' ,-.dn•"/ CoV11<~iH5 fhe P"f""'e/Pr

w. f 0 P' o b"' 'v; I,-) • + I"'"' s -t 0; thPY1

,,

fhfv1 ou-fcovvrf' I/ _/

c-~//ec./ a canservqr/vf' I oo tf;4 c..r.. IS

ExQ l'\1 p le II/-/. Ll XL A/ ~ ~ d i3 E (<: {p} F,Vld o Cf()~ coV'5e r vqf/.,;P oVlt' -);ded upp<::'( c L t~, p .

• 'TO L P( "-(1°) <~X:) c

~,vhe~e C(p) ;;; mo.x~C \ ~.(?) p\1-rr~ ~' ID, c~a:J

Novv to '} !"" -1 a c.:r fc, PJ we ~~ ke /

i p \ CfF) < ,tx; } t' I

T-t h := I 0 Ql'\ z )(~ :: b) t~fYI the

.<313 O'fle · s;J f .J Vff{"\ CL

Th;s i rCJ,.r Yi d

CdV'l OF

if p== .i~ Vv h Cr -/

T-\ p=.cr, wt..cri

,s

b) 4,; ql

iS c.. ( '

15 c~

.£ t' \\Of :

c (.1S) _:: s­c(.9)-=b

\<; be+~eev, • -g C1V7cl • Cf;

C OV\ 5E IV ct!t·v ('

S'o + lrte CL

Th; s Cq VI bE' c o V' ~· \ V1 1.-r f J ic. O\b;~(c;IJ jDff'Cr5iOV\,

A VV<L<C h sivnplf( 5~tq4e_n c( 5;o'Vlf!~ 4t~ o<rk) 10 kcr-/ Vct.fue of

P(~X ~ (,) :;_. J() 7·

-\\ ••') c!f r { -/ ~ f'

p 3; ve S

Ct.. ,· '}

. VIOj~b~!XoJJdv : slllv

i__ "'~'1"'1 : J f Sf f fw.., (/ _l a f lA 11 }-t-J J ~j-yp o '1 ~ ~M S"oJ/j!MoX.;> D-"'"'{ d>l\o&JJ:7 dL-f{ op rto.( //hOJ /V\oH

he. I

(~--z_o~~w)N

( 1 '-o / 'h')(\J

(~ z_p /zw)~ f~., --v A ( -e ~ l,w) (\J r: .1 /V :X

z.ry_ 'rr ..loj .T~ •o pv,~

\ 2)

Ch«p te' \ L - t-\- ~ p<~ t~e 5 is Tc:s-fi "'_)

r,e vfoiA" ( J ' we ho. ve es-1 I .mctfed p"' rO""' -c te r 5 vV ,+Vi P";"' + e sf.irYI Q I e s aVJd ,.., +e rva I e sf,v>tq{ 1'5

(co"~;devtC e iVl+cnl"' Is). Now, we vVCf"'i +o qVlS"'-'"r -these sor·h

.. ~ ~_.es-flo"'s:

:r s if f I a .., s : b I e t h ~ i 0 ::: 5 7

.

Ts ,1 p/"'"s;h/e fl-,q{ G ~ 0 1.

l~e f'oce~5 o~ q-<sv.,er•'J fhese '/Ltes-liovtS 1<;

c,__lkd hJpc.1k>iJ +est;v'j' a 5+a~e-VVtell t.

Vef l z.( .1 A s+A•~+r.o .. l _ "h•jpC tf..te s•5 '' 's a.ba.,f a cLs+r;b...~;o~. :C~ :he _sfqfer>"t'n"t corvtplefe~ sf~c;\tes -the ,,

d,s+r,h"'-l-'o"', J (S called. c< S1v11p/"' h'jfa-fheri1, o-fhe,vvise ;t i5 ccrlfeJ o. 11coY'1fortle hjpo-fhes•':s.'

fxaWiple ~ 1{ .. : 'X• ~;,;,d N( 0 1 I)

fxa "'f k : "c: Xc"· ti.d N(.u 1 I) vJ ;th ~ !:. 0

(:ir>lp le)

[coMposite J

Def 12.1. (

Novv1 e-ve r'Jeed "- WC<j f-o deierll"1i~ ;+-ft.., ~'Jfofl,.,si5

Is pl~l-1:s;hle.,

Pf'for to the exper ,·~en t J

o...,tco~es ctre ,·YlcoV\SJ·s+evtt

l~ese ot.A+cavvt~s co/lec-{,·,,e!J 1/ J \\ re j e c Tio., reJ, 'o ""·

The I

:;ou need +o

Vvl. tt--, t l-1e

ct '"'e re .f'ered

dec ,·de what

n Lf II "J tof·Jtes/r.

i-o cts tt-t e

t20

re j ec'-i o" r r,J' •~ ,-, "s <-r<> lly s pee ; ii ed ,.., -{,. r ""'.r

of a... n 1 C e s-1-ctf..,·.s i ,·c s~..~ c h ctS a St.r .P·f'i cl ~ 1/1 t_ 5fa--lt's--lt·c

or~""' MLE.

-ExqVI1p/t:_: if<.e'eci t)

1-t: ~ ~ 0 if X > Z..

Det

~ x / x > zJ is fhe reiec 'lio11 "~Jio t1.

It is rass;_h)e -fo cc..-v~e -fc OV\ i Vl co'~' r~c + <o;'\C I <..t.5"'1'dn

II

A tjpe .L ,,

error is 1-v hen the nu 1/ hjfofV!esi 5 IS

+rue b--1t i.s re.Jr:c7ed. TV~ i.s is Y\o-fofed ~--..,~,'fltl 0(.

A 1'+jpe E error''

Is +olse b~t· 1s

wlit., (3.

1s VJhetl\ fkf' rt'-lf/ h'jpo1hesis

n.o+ rejecied. Thi-s ir nofct1ed

.

121

Def 12.1. 3 Fo<" 0- 5I oN! pie h 'J fD -f ~e sis/ 0(" p(l:tpe r err, r I H.) 1 :5 Cq 1/eJ +I-.e 11 Stjllli fie q.-. ce lev~ I~'

F'oc 0 - coVVIfos i -le h /fo fl·wst's J t ~ i :r pro bQ b ,'/ /+ J IS ct .fu'1cit'M'' of fi--le Sft'clf,c d,,sir;bl.,(-/,'6,.,,

fX'o WI p le ')._, ppo s e X',; ~ t. Lcl N( """1 I) o., d I have o

SttW/f/e ,f 5i~e n::l.-f, .I r>~i')~t cle~,'c/, +haf :r ~.'/( reJ~c-1 l-Ie: /(~0 if· X. <-Z.

P( X.(- z I A) " P(i~"' <-~I }<) ::: ~ (~.-p· \. ~ 2 ~ )

5,-~ce k t's c.o1kvtowV?) fh,-s 1'5n 1i ver1.1 vsef"'(. d r 1 ~ ..1

e "~' V1 f' + Vrf"' s ~·~ ~ c f + he 1 to s t--r V'lstec,d we

mo_x,'VI1t.tll"'l r~~s; hi~ r~r-o .bab: N-J_y Uvtdt!>r fo be ft,e

Hfj.

ex:= j 1 -l.t = Y'V\ :x J:i( -z. ~A:..() :::: £ (- 2 ) = ~ (-u\. A - <' Yz_ \; '2. '()

D(:{ ll.[, 'i The qrO"ve/ c{ " -fest ( 1-;3) IS +he pro ba b,);~ hat the n ~I/ h jfo fh~;is ,._,,'/! be rejec1ed. This is a tt-ft~c-lloVl of fhf' flvte clis1rlht.-t-.fiovt, s 0

'N e c cd I i + -1 h ~ /po 1.-v ~ 'I f u v1 c + i 6 Yl; TT (- ) •

I"' fl,, frev iou s e xq mf /e) IT(.«)= P(X ( -l) A)= f(-~1:-«} · No~ t~q t n-(o) == 0<'. I"' je t11.Pr:q I l o(" =- VVL~X Tr(e) •

e~..no

ExqWI p/e:

Qu.es4 i o 111:

N(~o,.k·) N(,t{,, -k)

P..o c ,c.{ I

r ~ Ht.. <l bove f 1 <.H e } j"d;,d1e f t-.e

fo o<, ~,)jl-~(~,J. X~ ,...,;~d N(~, t)

re:yc"" ,,s [;;: I x > c}.

r~jioVl

ov1d

ll~

corr-espoVIdivrJ

-tke c..r-i1ic:..t::tl

::c -t- /-( 0 ) c..} ~,

vv he{i t;vO u /cJ

a \e he /.J CCI.AS-./r..tVJ t bt.tt h fV\ C. rea S'~ S~

hoppe"' fu 0(1 fat.."'-,~, l-(3(x,)?

If Ao and C 0.\t' he (d Covts~q.,., t b~..t t 1--f 1

-le 5 ; ?;f.' Jtr p-o we r 1 i V\ c ,e ~ se s J

\Nhof \~ /At

l.vha1 will hofpe~ dec rea.se s?

Oe~: .People do ik!-l q(~.v<>ys ":Jree OV\ wkai 0(" shoi.A/d be. A /'l1or<" ~~~or»ia4tve "'"'/ .f r<-por-1;'] -f~.,.. re>ult of et hjpo 1 hes-is {e s-i o"' "'"' ex pe rt>"'fe" t 1s -/u f<" pod. +he Sr><.q 1/e s"t <>< af Vv I, ic h OVI e coc; /d re j<• <-f Ho. ~~~ S is CG! /fed the 0 b5erved Si 'l€ or the 11p-vq '" e_ .''

I Z ~

ExQwtp/e 12.2. I X I,,. 'I X 2.~ ,_, ~ c:.d N ( ,t.( I I 6 )

QuestioVJ;

We 'vvq 11t lc +es-t Ho: 1-<. !:: I o vs. 1-!~r: A. > 1o.

If live IA!o~~tf o +~s-t of' S 1 i!:.e c;:.y' :-, 0 s- vv-e c q P1 )

{,-nd fhe qfprofr,·~-1~ r-f'jectiov~ re-y/on.

~os-= fY\qx A(X > c (A) == p(x-~ 1o > c -I o A= 10) '< ~ 16 j_ 't;>

,......

:; 1-I((_-/o) '1;4

~:; ~ :::: 'C. Cfb' '-f/5"""

=:.) c = {{) +- j_. :c.,~ 5

So wE' will rf'_Jec-l u. It X I 10+~·-c.qs-. The IEs'-lft 's x :- /1, Lf 0 avtd llt.R re ~ ~c + Ho.

'IN h) YJ o-f use a re j l!'c -lloV1 re J /o¥\ of fh~

[K(><<.dj~

trV""

E)( 12.2.1 (co~~~ii~ued) If o. persol/1 be/;eved c:<-=.01 was Mor-e

"ffnlpriQ4~ _~ +~e v1 he t-vol.( /d nof ~ CoVIvi~-tced w hC'VJ

we -/elf hiVV'I thaf we r-~jPcted l~c ~.v,'+~ O("==.o~ ~e IA/01..{ {d waVIt. to kV\Olv ; \ we WO~.{ \d rej f' C f 1'f the ~~jecilov-, fe_Jiolll \A./ere. cJto.selll Sttc"' -ft,at 0<'=,01,

J<epo~+ittJ o p-..,.,fue will ~"1,-s-IJ hiwt as

ot~ers 1-\./ho rvtoy hav~ d;.fferet~~f /dects ~;vhq+ ~ Shol.-fld he.

we I I tt s

a hou i

Al'loher ..,_,aj .o-f def.-11/:J or fh;Yik,·"'J abou-f

p-vo/~Ps IS

p-vct f~e - P(+est s-k,1.-d,'c ~.-v.i /I be cd leost as extlf>I"Yle as observe J. H.j

If the pro b b; /,!J d ""f., 11cl s of 1 he s pe(" ,'{',c: d,·5ir,1bl-ff,·olrl iV~ 1-/o J ihe" f~e p-vq/CAe ,.s fhe t'VlQ Y ,·~ u YV?.

I"' 1 ~ r'.s ex-q wr p /e )

p-· vq IIJ( e : rvta x: .. p/x- ~ IJ. '16 1<~10 ., l' A)

== P(~ zu.Yo -16 ~s- ttl'S""

~ ~ ~ IO)

= l - I( I . '-1 0 ) ;::. 'tfs-

O.D'-/ o l

S,) a :Ju ~ ""'"" i ~'"' k >' ""' sl<ou ( d be • 0 I or , 0 3 v,.,o._.l.l

l3a

rto t re j e c1 H () 1 b u f a vr J hod 1 w h o f hi 1-1 k ~ 0(" S ~o v I d be . OL-fOI (a("' yreqier) wol,.( {d reject Ho.

tsl

1 n +' . '-\'~A es 'OV\ •

Does it """aHer 1·5 the bau111d~ o"' o cri~'""l '~J•'OV\ <H" s-fric t 7_

I

AV\svve( :

f:(~fle:

No+ , f He -h-rt s.Jols-hc is <'a"' j,-.,,d:" S',

Xc ~oLd. N ( .<( 1 o- '-)

(\ ::. s-0 o-'" :: \ b ( k """""' i"' "d v"< ~<e) \) e v e I o ~ a. +e s -t o -1' H • ·. 1d / 0 "" . 1-l A : A I' I () •

• p,·,.j a. cr:4ic"l re~l""' $il that 0("'.05',

• F,'t~d +he po\.A.-'1?' +!.-( r c + !'OV\

' j::";,d + ~e p-vq f "'e i.f X. = II.

', Ques-lio": Whal wou /d w-e c~o•Je iM f[,t> ex:qwoflr: "hove

I ExctMpJe:

I Ex a VVIP le:

~~~ Et o..-v"f le: I

i ~ l-J, ·. t1< '!. I 0 a " J ~ A ·, .<-<. <.. I 0 1

R.e p"a -t (,,{--) b..ri- Vv; t h a- '"" LH1 /c VIO"'-'"'·

Th« rowe<' fu~~-!ioon f~ ,of e«s;'J ex-p;.-.-ted, so j,s-t

~iV>cl rr ( 10·5). ( V\5e 5<>-l'vv-.<e J

Rep""1 (If) 6-A wifh 1-t: u:f(J vs. H~= x~1o. . IIJ I ~ 1

lh;s )s Laffe~J <> -ro-vo--taiii"J r-est

o' <> +wo- 1-<>: / e>J q l f€ n-• a -I• ve,

Repea.f (:;1' J 4-t.vo r -\--q; \ ed

b J-f- Vvi 1 h ~ 2 tA ~ J< tAOWVI

a\ {t2 r V\ a+ ,· ve ,

c:t ill J 0

X~ h ~~d N(~t-t,cr~) l'Sl

ExaWiple; T e5t H o •• o-z. ~ 1 vs. (-tA: cr?.. / I

'Exqlfvtp le. : Repeat +t)' l~ A : a- 2. < \

ExCiVV!fle: R~ rectf f'O, q +IA./o- .fc,'/ecl ct!ieriAafl·ve..

awd ~ SIWtple n....,(f h;pot/.tes•'S

troW1 X.:.. "- C.:~ J 1\) (~., tTi 2)

y ~ "'- i .:d J.J ( 1-1 l. ' oz.~) f--x a ""'f/e : Te.--t H.: o; ~ = o-~ L

rv1'i ~ a_ t~.vo - s; ded o / ter ..,a i 1've,

Exq l-'l1 pIe ~ PQ; red +-test tor 1 n P<7JsQL.

fro w, base /;,., e chavtje

bx,. WI p f e : A ppcox;.,.. <11e f e sf; o-f' ' - J_ [.((). p- 2- vv,'fh t~BfER(p)

Exo. WI p /e: Coflservcxflve fes-t of' fh" saw.e h~pa-l-hc:-s·is­[p-ve4l\.te c:1ppu1o.et-.. ,·s s,·vVJp(est]

l::)(plqiVJ: lwhj !'.s ih~ qbo"e -l-est Covtservatt've Q~~td why (_s ;t ..SoVVI~-1-/mes called exctci ~

No+e! Approx,'V"'atf' OVId 11~X:Qc t'' {e.sis CqV1 be oh}aiilrf:'J

~or o Po\S56t'\ p()p~{c,+lovt QS Lvef{.

l33

u .. 1rl rtol.v ~~ve ~crve Sitmf/J deviv.ed He t.VV1 at +he re j-ec'fioVl reJ,·OVt at..._i o~ -fl11VI o,·r.

ko'-' do we kl'\oV +~e best ch.br·c~ 1,

We w ovd t k e. r-tos t po e-ve> t'u ( b-t possib /e .

For now .I a !:Su me t.v-e

~jf(){~~sis l~o: 8 =G'c.

v._qnf -fo fest a. s,·Y¥1f/e v~. H,A: G = <8, .

TheoreWI 12.6.1 (NeJ""'cr"- PearsoVl LernWl.x)

The. c i \ + l c q I r- e J I 'o VI, 0 f 1 h.~ -f'c.,. liV\

J ~ / A(~ i Gl, e ,) "' f(~ i ed) ( I< ) f~;e,)

pf"ovide>5 ff,e Wtost po~--ve ..-+4 ( iest cf Si'~e ~.

No+~ + hq-t K . Is se l~c -led +a 3e_ i ike des,'recl t:>(.

De f: (I '2. . 6. I) Le { 1l (e) be i he po Vve (" C)-\' 0\ -J-e s tl "'--'; t "' c,; i1 c C( I re-3 iOV\ C .

A k-St w;-fh c.ri1icotl re.J,- 0 "' C ~ ,·s t h~ VVLost

poYv-e r f l-1 ( i est o-f si-t_ e e><- ~~

I) nc*(eD) =<X

.2-) TTc¥(9,) ~ 1l_(EJ,) -\:..- q,.,j .t~e, s~"' <t-j~et'o" ~J"•"''·

\31-{

proot: Let C1C~ be_ '' 11 'ca ( r e; i oVls o t s- ; -t:e o( ~.-v~· t.h (.. * of

t ~~ -fun-'1 c~ £ ~ \ ~e0._ ~ k(<><)}. -t~; e,)

Ld A~C". T~e~ P(X~A/e.) ·

~ \<("") P ( ~ ~ A I e, ) . ( t)

t bj fhf -fu\WJ oT C *

Lei 8c ec. TheVl PU E B l e") l kC<><) P( fE Bl G1). (tt)

c*=- (c~()c)u(c~nc~) c = (cnc'*)u(cnc~<).

)C)

Tlu ~,) ~ P ( ~ E c-11c \ e.) + P ( ~ E en c' / 0, )

rr,:.(E:>,) ~ f:J{Jfc ne l G)+ P (l (, c !)c"c / e,) OV1d

-

rrc*'(e.)-nc~')= P( (_~nee I e\) - P( ( (\ c ~· c ( 61)

·~ ~ p(c•nc/e,) -~1 P(cnec{Qo)

"' L ( rrc,(e. l - lTJe.)) ~ ~o<l ( =-"') ;: o 0

Exo~ylt (2,6./ Den've The WlosT powe(\v \ +~si of Si-l~ o< of l~o:e.::eo vs. HA: $.:-(8,

~\OW\ ~ ral'ldoW\ 5q Mf Je ~+ EX P(~ ).

A S'v.VVIe e, ) eo . -x·

r\ (~)eo/ e,) =- f(~ ~ 6cJ _ f~; e,}

Tf J.. e ~0 t eo n- l e-x~/e.

J_ e~'~

0

-~ ') )(' eoC- ..

e_

l :.k~ Qf\ e $ 1 LX~e I

\Je h,; I ( reject- ~o

) f- LX: is )a 'j e. •

l_X~ ""' GA"'1 (e J n l

2. 'L x~ ~ ~2.(ZV1 J 8

So) \...-.... e \~_)ect Ho

1<.. G,

(L.x; )(_l_ - 1 )

~ (~ r ~ e, e. J

j f A i .s s~a II ; i, e. 1

(Tio>s "vDu /d do"Je if $1 <Elo ,')

if z_ l~~ > l-,:[2n). e()

\35

1\J of I a we M v Q r ad" ct {{J ca.lc 1..( I afed !<&<). Ab~ ~trf-lce -fkoi the VVtosi po~Pr\ul +est of si~e ~

------·--

of Hv: eo:::- l V5 H,t: e,; z.. ,'s th~ Soto~~e el.S fc~

li~~: Go= I vs, HA: e,; l T. T~is St.t:f,('Sts tkGt ~ u.ni-\'ot-v-'11~ rvtvst fo..-er-\1.1( te.s"t <:alii be cr-eeded, ~

l3t

Oef lZ. 1. I A test w1 t~ Co.-! posif" 11 ~'J P.o the ~n . ( 5 ca /1 f> d

11 lA VI j +or>"' J'J WI 0 5 f po vvectl.( I {o,

rc)l,ve r t 4 I I f I + I .s 1'7-1 OS 't ct ll e €: ..52., ( Q I\ ross,'b/e o. HernQ-h've.s)

Ex q ~ p I e ( { 2 . 7. I ) F,·VId C( unito\VVl!; W/ost pot-ver.f.., I f.es"t:·

iv'l t-he prev,·o...,_r exaY"''{Jie {or H 0 : e = 3 "5, Hit : e / J ;

Ex CtVV\ pie: Repe-crf- .-J; ih Wo: e f:: 3 VS', HA: e / 3.

As "'-"' k.,.v,.. 51"ev> iv> tt.e exa,..,f/er crhave) q_ UMP

fest coY\ Some-l-iMes- be C>hjqinF"cf b~ us;'~J +tte N e J""" " - p.,"', ,- oYI Le .t~(Yl q -l-o .f1., d i he /1A cJ s i f • "'-""r .f:, f te~t fo, s;W!f)e hyp~-fl,ese1' <2111d -ft1f',., -/G ar-ywe iA~"f· -the Yl1ost po~.-ve,fl({ ff>S't \<5 not ct f-4vtc1io"' C)f tt.e r~4)c..,lr.tr ol~tvv~1•ve v·crlt.~te.

Exoh1p/e : T,j -fo rt>peo-f· a I 1f'rvlo·hve,

e/C•·s-i (

t (,,., a ba,., e >< a VI/I p /f> h..-i1 1- c. -f-wo-5; d eel

h!Ay does a_ U .M f fe s"t 1'10 i

De-fQ'Z.t.z) It dens~f) +~.~~'~c1io~ ho~ a 1rf'l.10tAo4·Df\e likelih();d raf,·o''

lh 0- s~-fp-/,·c: -t{K) it +C!; eo)/f(~;e,) = ~(1}eo;6) depe~d,- oto !! o".'J thro~J~ ~) a~d i-t' .,\ IS' a. novt- Jecreqs i:J ~UVIc4t'oV\ ~f ~) ltvheV\ Go/ e, a.n d bot~ 9" Qv'ld e, -o.re foss;.b/(" val~es of' e .bctsed o"' ~.

L3t

Theorem (\1.1.1) I\ X,.._; ~)e) hcts ct /Yloa.~ofo"'e 1;/ce/,"~o/Jc:l \q1;,. iVl t(~) J the~ q UMP +es-t of r;~e tX o-f /-1,: ~~eo

V5, H~r: (f) ::>eo ; s fn r-e j eci Ito i t t(~) ?. k(O<')

~~E"C"€ p( ifl) ~ l<(ot') l6o) = o<.

Th~ pr-evitJl.lS def,·..,;J;()., a.-td +heore-,_.., s;V'I"'f~ s-fq+e.s wAq-f we

a.fr-eqdj /eQrt'led abcu.+ e)(teYrdiJ <A Mos+ po\,./~f;.ful +-~st af Simpk

~pofl-1-eses -fo ct UM..P f-est. Well, ;-t exte11ds g simfli{;es ;.,, StJW~i!?' cases.,

ExaMple (t Z. 7. :S) tt ---ci.d E'X p( I ,?L) " · n=?L6

l e. t ll.. )Jl, •

Ho •

HA: ~ /ll.

- X.+7l,

)(~; Jl, ,Jlz) ::: rr e .. 1-ix~?ll,J

Xu., \-\o

IT -xl( 'f'Jl, e 1-zx~ '7l,.f

:::: \e n (1..,-it.-.) if X',;,_ > ~, -' 0 it Jlt < X,;" .5 ll,, DN£ ,·f X,~,. £: ?Z~ •

c::t s C\ V'l-1 o 111 oto n e li ~e I i ~ oo d ratio I

it x,:n is /a 'j t:.

"71.., - LX;

- e 1i x,,~ )7l,j -

"llz - ~X'w. 1-t x,:" > .?£~) e

ar'ld we ~~v\1\ -reje-c.1

rreeder

Typewritten Text

There is important content missing here, e.g., generalized likelihood ratio tests.

ExqiM p IF

Chapfe, IS GoodVIess of F/ t

Uf Lp'd;/ Ylo~..J~ we q t;,s(..(vned f~q-f we

hqve q/Wlos-t qf,_ctJs

d ,· s -f r ; b <A i ,· o Vt s-- j e . J. hove fcc (..,~ s Pd 0 V\

kt1o"'-" the f""" i/J of nor-mq f, CfVtd we e s -1; Y'VI er -I/J i h e 1~ q rcu:-n e -fc r ( s ).

T~ -CV/cf +~--Je /VfLE ot c=t pQ v CJf m f' fer 1 ~r f' XCti-14/) /~ Jou mus i kYiow tt, e

{(...(nc 1t'ovr,

.fo f'W1 of f he dPtrt ~~·~;

Iv'l -!his chqpfPr, ~-ve_ ~iII /eC~rVt htJ~.vt +c +L"st wh€'-lher a ce,-Jqin mode> ( -(/5 -fhe

d~io..

Yr,"'/ l~ ~ i~d 11.E"£(p)

Wo: p = P' uA~ p ~ pa

We h~ve <;ee~"'~

he~ve SeeVl

fo re . f' c-+ J

/;::'t willq/e.,f ~)

~V1

ClV!

/{a

excrc-l --/Fri befc),e, otvrd

0 f r '(O )I /vt't C1 -ir· ·/eri wht'c. t,

,-f I X- p, I > t:,_~ J r·('~P·) z.

LvE'

15"

(~~-~;--~~~-o((\) <

- (nx -npe)z

r't po

- (hi-n~) 2

r1 pG

? - .... x - e(.) ( - )"t ;; - )z.

X - e" ' = \ n X - V1 Pc

+

f

lJ'~P·)) n p.(l-po)

( n y - V1 e~ )M,~ -n·~·( 1- {'cJ)

el1 -V1 X ) - (n - VI p < il2. n ()- po)

- (c.bser-Ve d5 - C' xpec-1 t>d,}2-

e.x p'Pc .. , e ds + (obsecvPdi= ~ e"'f~"< -1~"'

eyp'!?c .f (',.f~

\ - I (oz_- ez )<-. E· 1..::::0 L

No-h: ihCI-1 fh~ fh€' o bseu v P d "/he F'y fPC -f()c-f

rh i ~ -let m

f'qs; /j· o-f

-fe>-t si-L· -/ t.-,_ I' II he I q ry I? whf'vl ~ .· . '~i'(' 'IC

n u ,,.,.., hP r is r L-1, -ff> di f{e"ren i t ~ 0{~ VI w ~-~1 b e r .

-f ~? p ~ f' r i ) k ··1/r I~ c J e VI E r o, /,· :c f s

Exqrv?p/e l3,'1,1 4 6-s;JPd di(O' ,s roll~d 60 II'Yr?t!',S.

We V\_. Q VI -1 to -1 e ") t l -r I _) } ') t q I r, I H 0 ~ p t = I? 2 ~ p J -~ p If ::- p s ::: p 6 -- 6

H A : a + /e c, rt -/tv c ps qre c:J;.f'ferent.

U" d", H o : i (0, -E, J< ~ ~ "- ( s ) L-"1 Et

No-ff? i hcrT tl1e fe Kf (;(Se) +-h C' Sc:;>vY'l F VIO ·fCI'ftth /

Ot... j e i fc.."" bo-1 h r qt·?clovv1

ov-/c omP S • f h P cJ ( t.fe r 'f> s-u· P

l.Jqr;ctj;)f's OVtd Jhf,'r .,f,cv !c/ b~ c lf'Cl r {' rc vn

c o V1 +ex t , A , q /1:: e r VIC! -1 / v e l,.. ~c L(<;c' Q. c. -.') . •;: " / L L 'o' -\:h~ r.v.s.

No-le fJ,qt (Oi -Et-) q {"·(" 6

1 V) -fqc t I_(oi- E ~) ::: 0 VI o -1 in cl ("' f e •" J e Yl t .

becqL(Se [_':: I

joi ::cf£; ==6(). L.: ! -

t::l

B~c_C<'-<5P c;{ 7h,-s coVIstrct/n+, Li-1ere ore ov1) ) de) rePs of -f re Pdo'll'\ lo 'f h P r -} £1 cr VI b.

Here$ ~ +qb t~ of observed Qytd eKfr? ( --I!:> cl

2 3' ,~ Observed 8' I\ 5 12 IS' 1

Exp~c -lee:( I{) ID 10 16 16 16 {t.~~<•c:ler Ho)

5c +h~ c~..t/cornf? o-r' o...,r -/ r:~ ") -t s~ f/sf/c

(¥-Jo)z T (IJ-16)"2.. +- (5--/0)2. ~ {12.-10)~

/6 IO

+- (ls--tu)'t 'o

/6

t

/()

(1- I 0) z -----/0

results:

IS

- {; < J!:;o (s) = '!, Z 'i 5o '-'-P do rd '",;'"'< "f /{ qf a.. 5j-uiffcqnce /eve( o-f 0('=-0./0.

ExqiN!f/e I 3, 3. f

A cer- 'fed,, c ho- r o c~qn·s--1 ic..

2 0 'Ya o-P R ~ ~ e / i Z 0 Yo

S'o J (-/ c : p 1 = pz. = f3 = • 2

We Je-f ~ sow, f/f!' of

q v1c/

'$ f::R lir:: tdd '-lo bt:·· prfC'5"f-'#?t ,,.,

o{ Rc,cp Z 1 q.,J 20% of Rcrce 3,

S (J -froM 50 f f0W1

/00 \roW!

Rctc e I

J<q c e j

Ret ( e 2

He ,e IS 0 -Jq b/e o-f a bse rvecl ( E>x pf>c -ied J cD\...1 Vl -f $,

Prese111i A b se Vl t ___ ., RGc e I

I 20 ( \0 ) 30 ( L-1 0 )

l<qc E' Z 25 ( lO ) 7c; ( ~O)

R.cue ~

' ts- ( to ) 35"' ( L( 0 )

SoJ -1 l-ie oufcovne c-r OlA r -lf!'"si 5-lc,·l:r·lt·c 1 s

3 2. I I (o•J - C:;_j Y j=l '-=' e~J

IO

@ o - 4 o) 2 +- (z r; .. z o) 2 + ( 7 r; - sc~ )'? + (I s- - 1 o) z- + (2:.;jl z

lo "2- G lS 6 I 0 4 (;

_ (zo- Ia )~ t

::: 11,/~

If X~;;t~(3) / -~~PV) r(X .2:11.1~):: o.oa{)b)

So/ f!?e p- vet!(.,( e i-s o. uoa6 5., (Okaj) -+h~ crprrax. p- value)

ThP VlLt W1 be-\ of de5rees of {re e do VVl CaV\ be(?

de ie c VY1 in ed j Vl a COt-t p fe o.-\' Jvvq is.

One

+tJ\ are

Sv w1

odd

IS fu Ylo-f.<? +h~t i (Gs -E,J)' - /V ~2(1) t::-1 E,.:j

eqch , J . Be"'cqusr.? fhe 3 So~/;?/t?s

s-/q-/;5ic /s '/t,p ,·"'~ clepe, c:levti/ it,P ie~-t

o -r t' 11 d .P pe 11 cl f> vt -L ;J?z_ r.v.s 1 56 1..,..,;(?

-) h f> df-3 re p s d -r {ree doy¥1,

/hF o-lher

b~-1 no -JF'

o -p --l 1'v"' 1's

-f lr1et-f -)he

fc 5VYI'?

S~ctri ,_, ,· t h 6 el-f .~ 0 .... .,.. ( 0 i. - Et I) c::t c. c. lei 5' J

J t)

each ro L.V / s ~e r o • 1h~se- 3 conS Yf'erlri'ls

red(.( c P 1 he d f hy "3.

E'XQmp/~ /5, 5 • 2

I~

~'

p.

I+ ~0\

-;~prose '"' -!i.e pre v .·o.,. s ey"'""'j/1~ .. fhc,-f ;+ 1/vc:{S be/;e,.fd 1he~-l ihe /?roferi1on

ot peopiP v...,;fh C( cer-la/YJ clu:~rc,c-/ar/.rflc..

~ Q s +he ~ C( me i 11 r o c h. r ce c <? b '-1 f +hf> C:xoc-f profor~ioV! k...os r1o+

pt;sJukfed, flo: f' = P.z ~ P3 = p

t~;') cqse /

cle1~r v~·?if/ .pcJ f> ~p~c led C 0 I.-I"? t S C ~ YlWO -1

Vr, i ·I J-, ow~/ f,- r s- t t:.o-r 1/y~.r ff~ "7 tt,f' 1'1LE (u~clf'r t-1~). \.Je u.s~

C. ct.,., b~ ~ h(} 'vV"' -f hct-f- -ft,e 1'1 L £ ( U "t<..ifr 1-1~) p \S p =- +o~ { pro por -1 io"' ohSe>rved io h01ve

-thE> c~ooroc-f~, ;S"f;c ,

rV1 7t,;-s excrl'vtr 1~ p ~ .30.

~€t("' I'S + ~E' -Jq}Jft:= o-f observed avtd (eKft'c -led) Cou n+s :

Present Ahsev? t

RQc ~ I I zo ( \ S" ) 3o (ss- )

Roce Z I 2 5'(5o) t5 ( 1-(J )

RQcf' 31 Is- ( Is- ) ~s- ( 3 s- )

T~e Ot-{"cC)mP o"'f -fhf" -iesi. fYetit"<J:'/;·c (s 3, Sr T h e c1 i r f r ,~ b v lt'(tV1 of -f h e -1~ s i s ·It, -1 /r 'I c IS

X z.( z). r<o t.vS c -f 0- r-

red'-fcif1J Thf' d-f .froWI 6 -1-o S .

l3~cq '-1 5e Vv- e es -lt'YV'laled OnF pq r ct VV1 e fer-/ p 1

ihe caluVJ?s af 0-.£ cr/so 5CAvv1 'Ia '2:Pro.

apfroYim4..fe J .Su~ fo ~€ 10/

I -1 15 if'w.f"I,.,J -lo '5u b-frc,d S cJf, b,d 7 hrxt

il ~r-aVtJ· Yo(.{ Cctl/f fhir1k o..bo1.1-/ ,·-; -!his f/vqJ; Hail\...- rneu1y en-lr;es do you. nef:'d ~o know bE=>-for~ yo~..~ C'C? 1<'1 {;I I 1 .. , ·I t, ~ r ~1 )-1. a1 -/ hr -/qf:; k, lA f"

e·Nt > vv p , is 2 . 1 f ~ o .,.. k ~-~ o VI,; Z e V1 I r 1' P s , 7 ~ f' n

lj C• t.• (' q Yl lA -<;; €' -j ~ ~ .f Qo e -f -/ h Q -/- C 0 /14 n1 "' ~ ~ r o "'-- _r

~14 ¥"' ·/u -?E \" Cl --ft:. J~··l i "'~ {e·~t. No+e fhotf by IJ -f ,, evr ries IN~ r11e"q'1 o-e.

No""" \\/€' CQ~ tivJJ +~~ p-vctfue: P(:;ke·•tz) ~~.51)-= 0.161"8

Exco-vr p /e 13. t. I

l<ep 4 b lie~,., DeMo c ret+ r Vtdee@,df VJ t

A 5urve_; d e i F r v¥1 ,· V1 ~ assoc,.o.-IPd

s '"" epo r t-.

IV\~\eC<s~

15 ( ~ )

lo ( \'L )

12( ' ) ~0

~o: No ct~.soc.,'q-/io¥"1,

15 ji,Vf'YI In or cler -/G I.-\ po /,),'t:: t:1 ( q f {// /Cl-/I,GVI IS

v._,· -1 t., Sfq t e p r~Jr~::~ Yl-'1

Sot VVt e Dec If' C( s e

12 (IO,t)) IO ( to.c; ; 30

36 \

,1 (/'1) Is ( 1lf )

6 ( 10. C) ) I L ( lo. 5 )

YC

3s- "35"'

If i~ec-e r's Ylo QSS'oc:J'ct-/,·oVI (1Vlde p-en cJ f'YfC <"') 1 f he-...,

P (An B ) ~ pO\ ) efF~ ) ,

5a, P( 12-.ep~blic .,.., (\ .T.," re6 s ~) ~ p( ~ep4 h/;;:,., ) P(I:, c rPctse)

= (.3) (. 3) Les-lrma-lioVJ -fro""' cla1a.

t = I ('?..:_,d e

:: '1. 5t.{

p-vqlue = P( ~~ ( '-1 ) ~'-{,§'f) :::o • 3TH

Lei XL he ft,e

oir·plqnP perrt.

~ t'Y¥?P5' for qr, Eyq~pl~ 1~.7.,1 rr pct ,.,

~~= l ,_'~d A. Poi(3)

The repf1lir Yt'YnPS o-f 'lo CCJj-?-t(;Wf';-tfs atr:> i~C,;;r-c/ed,

QU f' S i I 0 VJ : lio~ Cq n l--v<:"' -1-e st /lo <; AV15tve\; T {' ~,..._.,e c rt?Of -1 f' b,·....--1 5 J k-(0 C<::rr7 Cq/c!.llc:tff,

·1 ~~F ob~fr veci ot/ld

1es-l s-/oi;s~,-c Co'-1VI-fs -lo cree~-le

J::: z. c/ /s -lr, b u}; 6 r. C(

e)( fe c --led

tr.- / It-.. cr

( (,m d P r /{ ) 1

QL{r?s-1/ov,: No~.,....,- ~ hac.-t ld 'lhP b;'V?s be c ttose~? Ar1s~rve r : So'MPf/i"V!«'S -~-~f'fP qre n~·-lurct-1 plcu:e"5 +() cle·f'/,.1((

-J~f' boi.Ancfc,rr'Ps, 64-f /f nt;{ Choose bJVI5

5o 1 h<r f Q s Y11cr "';/ h,/7s Q-5 pos-s, b /<" k-• /I P>-~-5-l v..,,· t h QVI P;rppc ff·d Cc~,u--rt of' s or· y'VI ore,

;.,,) ''? lf' r t-o Is ~ Q j. • ( u ~.~ tcl :r c t, "o ;se

I ,._ J , I c;. r ~~ r tAfS · '0 V'1 , $ -' !

e,Jc [o~ 1) V Lz 1 3) /,') b I vl .i qvtd [ I, 2) \) L 5; '-! ) I 'S bl'.,-1 2

A-1'\.s v-vP r ', Yes J b "+ Mq kP 5ur€ :r~ cc0 ,, j" 5 -l!'f} ~-v hj fi,E d ,'-~ F r €- • I -t r;;C( r ,, s //be I 0 VI J I\ -Jc J e "l"' €> r.! 0 V1 c.! rn 0 ~ t

iMeoctcmi), d~+iV1F -/he b,·Vls 13t::PoRE yov. look a+ l-he d~tel, Ot~E>I~.~~.-is~, 'jo""' p-vc:, L ..... e \11..-; II be /Vlva/;d VVI~ct v1 i~ l e ~"":~,

I a v1C\

txc. ~pie r~~ 7., CC)Vl'-( 1'vlC-( ed : C)

\

I 3

f<efc:t ;r-"Time ( Oe~.J s) Obs€rved

Pro bq h; l,"tJ .os- .1!./9

Fxpf>c{ ed 2 5,C({,

The part o-f i he ~b/e ihof t;.t:Jt..fVIIS of s· 0 I rY1 ofe

IV1 or de r +o J<e e p b , ~ ~e 'vi-/// CoW7 b1~<? 0 !k· ( Comb;..., -E> ~ 6/ .Jlc t:-r /n ~c

/3,·VJ I o/ I (S,•VI z 2 {?,·.., 3 5 (5,', 4 L.f

~~

81,11 s- ?:s-

s- /_ z.

t ~ ) ~ ~- e ~ 2 = r. 2 2 L e. ~=I ..

(f , ,

, . '

IS

' llr?

s--1

• I o I l.f. 0'1

6 >t----··-----~--.. - ...... ~-----

6 0 . os- • o 3 '-/

2 I, 3("

/"V1 /s .5 I '::1 h fJ_S

e9 c t, h1'y,,

eyfto c f~d

s~--e.r-s (~.Yfec --led) q+ S + J

IM~c Cf b,>-, qnd t.ve ~,·;;

q b/VI I

p- VQ ( !<P "' p (;:;'( L.j ) ;> "(, 22) :: (), ()~ _s-~

ExC(~M pIe /),=f.Z-! F"r-GlW1

f.lo So

il-1€' pre violAS I

th~f .,u- '"1 e Yc:r vvt f If' 1 1'11o d,fy

no-1 q s.51..f 1-rlFd /e,.;

VP knowVJ,

}-tc: Y~ ,J,VA Poi ( P. )

Q lA. e 5 i, 'u VI ; I.J /1 h o vt -1 k n c; ~.v 1' / ;-t-J h ot.v a.-re blvzs 7

c: ho:s e P1 ,

AV\5'-'-e(': Opii<.lY"I .1- pr~spec:.fj a0 qi~",;-1~M forA J&{',r?tr~ b·~sj +~~ o..lyord~m VV!~ vse ~.

Opt;,, 2 - selt>c-f "'-'hcr-f SE't"rv> -lo be re<=>S'cn<>tl~ h,n s 1'¥1 r:::,dvc11~, c e- ovr ..J. 1 he: n co llo-;s e

~o/ If

Observed Lf

t= x pe c ·} f'd 4 • ~ 4

i:= /,62_

b/11s CI'S neP.c/fifd,

2 3 '-1 1 I 6 I 10

6 .. rz 1 g--, ~--;1 1 -r, o<?

>s-

'3 /2,12

p-vq/~f = P(;tY3) ~ /.6z) ~ ()Jstt1

T~P de;re@s ol freedoM wo~ /d b@ we Co VV1f/• fe ~ Sfec; Drc!/ b .. ,f if c a ,, s e

es~,·mqfe ~:: 5., 65 J redLAci~ i hP J-f

'-f / f /Jo we h01d fG

bj OVIe ~or~.

\Je lese i deJrer_o ot -\reeclcwV'l {c:.~\ ecrc l--. ~rQ vvtf''/~r €5-J imq-fed,

C ~~ p +er ILl Y1ov1 pq ro, me ir,·c. VVle-ft-?ods

UVI-1,'/ -lhf' w~

Ylo~.-v_, ~e ~~ctve yenerct/j crs.rwi/Y;t"d ihQI-f ~1)fP o-·r d/s- ir~bv~on t--vq._r Kridt-...,J-7 qy,d -lt,P"'

m_qde- ,·,..,fer f'n<' PS abaw i f.t.s pet rq YY~~ fers ..

So¥'11 e p roc e ~ht res for­Qre v~ l;d rei! a r d le$s

d ;5 ·lr:b..,~l,-oV! hP CJVt .. J5' .JCI,

mot k;YI_y a-t the

/n !'f' r f' Yl c e 1 h o-vve ve fj

r ~ VI'! ,· 0 + ~, cl + 1 "' e

T 1-, ;s Is use+:! bf'cc,use ~.t~e o.P~e-, do no-1

k 1'1 0,..., I VI c:::r J vq lr'7 c e ~- ~ 01 -/ 1 t1 r- d / ! ·/ f , ' ih~ /1 (., i1

Vv;lf }e;dk J;)c<:"',

If -fh~ d,~s-Jt; b\A~~CJ~ c lerss /~-:J,; no1~o~l) 1'~ kYI~"'--1-1 lv1 crdvctv,c@ 1 fhe'f\ pqrcrme-.ff,·c mc.'!"'-/1--,ods w1/f

fe V/d fb he~ ve be i ir: r j')CJ ,_,_, P r {-for -}e., i /Y?J) cr VI d Met r row e r co 111 .f; de n c e in i e r v a / s . Ho ~.-v e v e r 1

V1.oV1- pqr~1-11etflc Y11t:ihods havf:" fevver- ctss'-'rnff/ovts

OV7d -Jttu~ ore vql;d iYJ q ~A...-;cler se+ of cqs<:=>.;-,

Als", -floe po..._er ID-s"t ca., fJ? '7uik 5JI11q//,

F ><9.., p le 11.-1. Z, I tv1ed 1 """ 1 v. coWl e £ ..- j "'~ ;), < 5 is $2'f,>da

Some cfa,.W7 fhq-1- -k!ler people eC!rn VVIclf'

-1~1CtY\ Shorter }?eopl€. A Sctrnplf? o-r n-- 2()

/1 tcr/ler'\ l')eoflf> (b +i +) WG1s oblo/VJed qvtd "ih~,r anVJuc:~f t;cp/qr/es recotded(iYl f~ou-sa11cls)

12 I 1 Is. 9 l <g I I

2ii, 6 25,()

lcr,Lf zt. 5 23.!)

so.t I() I <0

2 '1. 6

52.3 "3 3' 5' 5L.f '1

2 5, t.f

ss-, '3

21.1 3o. 1

'10.3 55,5

k)e WctVI _, 1 D 1 e s t /-/t~ ; m :: 2 '+. 5 I<

f-/A: (Y) / Z"f,!) k

P(X; > 2Lf,5 k / m" 2'1, :;) := i_ zo

Li(l; / z'f.)) s;OJ !=t

~ BLN(zo, {:) lAnder Ho

UYidf r 1-J A) f hf> ou-f c UYVJP of- 7 hf' -/ esi S ~ 1/s '/ 'c

Wov lei -fe,,._j fo b? /ct rJ e r? 5 Cl

p _ v"' I<.( e := p ( f3I IJ (zo/ ~) > I Z. ) .:: • 2 5 F{-

Fx9~p/e 1'-/,Z,/ /V1edict~ ,·V!come £, Y""ll''io'f 5 1'5 ..$2~)(}() Some cla/W7 fhq-1- -k!ler people eC!rn VVIcif>

i t1er Yl s he; f -1 e' I? f" 0 p IF, A s C( h-1 p I~ 0 -r i1 -- 2 ()

/1 tcr/ler'\ l')eoflf> (C +i +) WG1s oblo/VJed qvtd '1h~,r

anVJuo f t;cp/qr/es recotded(IYJ f/tiou-sa~'~cls)

I() I <0

2..,. 6

12 I 1 Is. 9 2.~, 6 25,() r - , .. ~

l <g I I

2 5, t-f \

tcr. 4 zt. 5

21.1 3o. 1 I

23.!)

3o,t

E xa ""'~ p le I t..f. 3 . Z We wo.-1-f io CCJVVfftJ Je -JJ;,€>' rrr VIOl.{$

q C.-I

e..t-Q~f?le, -fo\ X.~~ I.....,

p (J'z: zo =. X,zs) ·~ P ( F Uz:zo) ~ F(x.2,))

L-Jhe~e fi "'- ..:~d F

- P ( U me ~ • 2 5" )

~ h e\e Uz: 20 has 5ec ovtd of" de r s}o ~is J,c

fhe

of ~

J /5 --/r,' bulan of U f\JI F ( o, i) .

20 • lc- i.. = L (~ok2s)c (Js) :: .1151-

i,=)..

--lh~

$JJ~. = I -/Ji>b' 1-tSib• -

[IJSb' -/]-u~th' -1] -I -

[C"'":f; >z•K jJ -[ ~'~/Z:J!]cJ _ / <

[J 01"t?'''X) n ~e1'7 oz':Y)]d -( -

~ oz:o~ 7 l>'X) lJ(~'X 7 o"'X jc1 :: ( o.:o')' 7 5,.1 7oz:'X ) d coS

( ; ) (sZ'),(st_~ 77

; ( sz. ~ oz;·,n)d = o-z ? ·o-z · o"Z

( \ sz•x) j ; v·::,r)J )d = \ <t•x < oz:o•x )d 'hi Jol nuls:

Excu-.,ple 1'-L"i.l ( Pa,'red s.,VI Tes-t)

~~ (?f'o p /e fo dC) C?.

Q C" f-' r ·Ia /.,..,

we'e SF)rcft:d crf (1.')' '1 do J;V7

s-urvey.

e5/,'J;c /C(Y?

[ere t, per:ScY1 }o.ve

q sccH·e before a.. 5pPec ~ Of v'l d ~ ,J ct /.r'l 0 r;~ r ·/ t1 F 5"/'f'Pc h ...

II 5co~ed_

Ll Sc_o r-ed

3 :Sco,ed

f~E' p(Jf,·J,·c,·qy, h,yhf'r ctftf>l~ f t, P Sf/~C t,

him /Aer lo~ ~ r.

h _,·VV1 /her ·I /,f' s-~ rv1e,

F, · ~ s i I e -t. Dt = X{z- JL, (s<o'('e qf'1ec-- 5co-<e befor-e). N a w t...-v f' c.q n

J do ct 51)111 fes-t ~.V/t~ D,:

ll.): rnediqV) c.~ I) =o HA: yVIechq"' of D ::>o

I-f ~A,...e ff'jec-/ )~ 0 / fhe-y, wf' ~,-v,'/1 Co VIc /1-fd'<?.

fhof +-hp 5 pe t-c h t..-v q s e f'-ff> r: -.J,·v e.

Q V1 e 5 f!'o VI : No vv d o vvf? ha11dle f,·e_s (X'~2=;~,)? An~wt\ 1~ Plsj uq r.d 1-~f'V'Yl,

p - v C( I~ e :c p ( rsr N (I ~ J l) 1 t J j ~here

I~

t.~ 2_ 11Di >oJ - I \ -j_: I

Su p--vaf'-'tf' = o_ os9z

N de : _:[ f -1 i e s q r e p o H; b le ( e

+ hfVI P(l); ) 0 jl~,) Y>ta) J, e ~

f) ,·s no·f Covt -1 lnttat.~ s ) 0 r A a 5 ro 4 VI d;J e f'(o ('

~,· s j '<" r d -ties:, P{D.: / o / H,) '*' j be h ,., yer or s Y"'cdle r H."" ~. The,e-\ore, only,·{ D ,·s ca"d;111uo~r (or- Q-1 ~~~s-t -fiE'S ar~ ii'YI/Jos-r,'bl~)

tk Of V1 lz . r-{' ~.-v~

C~VI the S1jV1 +e.s"t be +f'vte(y "'-SeJ <:<s a_ ~-ert c.l l~ .. :~ed;.,"'• 'L/ h d J d _, + d.,~-\=o .I-f ·Les ? ,e pc'S,Pte \..(+ ls-y'-(crr eq) ,· is tn'lot~ Cfcc~nrl~ + .

: 1 . _c:. 11 • '/ \ _ 0 V1e w -I- hi:' r~ired· SiJIII ie5'i cts- Q T e>!:t (i\ i~' n_p >o )- P(O~o). Ho~ever 1-t -then~ ct\~ ~E'W iies, ;i 'IYIC(kf:>s /l't+ie <:Lrter\"Hce, I-\ 0\-1 tl.-!~ o-lh-e~

. l¥tVld iloiPre wer~ 106 -J.Ies q_o~~.:::\ +~e ofV!er- 'l..O ~-ve'e neJq-five; t~e itJ1-ecpwfq-f;() 111 is

rl"t)p_o<"fctd. T~e co,\e~ + ,·VI ~-f'( P'f' -1~1; 0 Vl ,·V1 'IJ1f" /?r~cec:/,7 C'XctMplc::- (a.sSI.(fYilVI_Jfie_qre impo~sihle) i51nq+ rejec.-f/:; the Vlu// VV1E'cu1S

t ~ (17 t ~1 f' VV} ~ d : Q VI () -r ··/ /-, f' d ; ~r if' r (' VI C f' $ 1 .J

p<l~; ~,\~e. ;I:+ ,·s -f e rv1 f .f /r1J fo pos_,.)· h }_y

I'VlcDtrec+':J CoVIcJwclr:' T~e~f -Jhf' n1etJ,·~"~ ·

c-t one ~ f<;'l,.{ f t's lof:J er i ilctV1 -} h F WJe rhetrl

<:.-·r i t'l ~- () . ' ~ p r.

ExCfrnple..: ~lui j s~c on~ 15o?0 'rov ps h.c::tve..

0 I'VIf'cliq.., of ~E>ro

z (

z. bu1 Vl-te d;Q..., of d;fter(>ntto_s ~ 3 I~ no+ ~ero

Exa...,pl~ r ~ f'~ t 5E"COiff d .......

B() t" ~tov (.)s. h.ct ... ~ diff

LJ 2_ Y'vifd,crV1 hv.1 med;ql'\

~ 3 of J;.ff 15 ..:e.e re>.

5, 6

Paired- Se~VI/1 pie S 1'5 t1 f d - r q vtl<. +es i:

Lei dc::::.x~-j:.

RaVIk. \cL\ ~ .1_ to<" Srrtq//€st' Z -\Ci, 111e-x7 SVJ;f~l/&si> efc. Ce~ll fh;s roVlk(ldd)~

i~c:l J.. \ ~ ) As5l.fltl1€· ii D .. ) 0? - 13£ R. ( 2 )' C{_J 1-fo e.q. F (X,'j)::: F ('JJX)

'·" 5 • . . J 0 y ~y . ;,-e, "~j se~ ... e~,e .. .; + "'"'<L - cue e't..-.,l~y /,)c.e'J·

Lei T== Z 1[d, >d] '"'"k«dt/) be o-1' f~p r<>n ks o-f' fhP pos,'-/,t,~

-f h e s '-~ ..,_,

d/f[e rew c~ .S:.

.If f here ct , e ·~ ;;;;.. 3 per irs )< ~1 d d. I r'o. n k I ~ -)-5" '-1 ' 3 6 -3 -

Te +esi ~t: rned;q.,. dA+~re.,c<' "0 (<>~d s'J"""'e-I'J} l-IA: rnedlc:,VJ d,·f\E>cevr<:(" >o

p-vq{l.{e - r( T ?. I l iiD>:- )oJ j_;d 8~1<.) - :::,. /t_,

2.

I

~

si~"' -+----

f'.CI'IIJ r;..._::. I

T Sl'"'~t.f> "! af ft,(f/' g" etwr,/~

l,·je.~/j (t.t~t~c/er Ho) e;y-/tCJVl-?f"S' \ 2 3

5r jVl S t + -t- b ~,·efcl. ~ I o.f I ()\ »~ore.

+ + - 3 -... + - +- i-t + - - I -- + 5"-- + .. ,,.,-·- '

- + - z - .3-- t--- o-- - - -

I~ Y\ \s

111 i-f' vzs, ve . lq '1 e_ ,

How-ever,

dOVlf"~

the /?- vc,/.., e wi // be C o.-npv-/q-/,g,.,,J /a ry f' - s ~~~··V? f / e Cfff roy, ~q /,o"" 5

<:qV1 b¥?

T- f(T)

J Vor(T)

d 4 N(o/ 1)

Ho ( n ~ E (T) = E [ i. u~ j

"' ~ ~ ~ i - _L4 2.

... .::,

i-" <J

where U~-· ,;"\. l3Er<.(i_ J n(V~t/)

z.

1-Jo ( VI

v Cj r ~T) == V <:> r f, i.. U <- ) ::: ~ ~ L v o.. < ( U. ~) = ~ ~i2.

J._ , h ( tt + I )(2 ., ..;. i) '1 6

Q~~t P5~(CJv1~ If ~.~ve t1 ee d t k e C L. T -fu

fest) '-'t,J nu-l J·~Ast LtSe

j e -( ~ , ct(Jf r ox

+-i-e si ~ q pctt'red

Ah5w~\ ,

A/fhoJ~ "'-'~ ~J;/1 ye1 CIV! 'Jlf I o )'/ wo t1 ·h (J >'I, i h,·5 • +e si /s less 5c:>ns/ft've fo o{,fflre-rs.

~"'-es-/iovt: W~af a.~ f~E- pros Ji:-. coV\S of i~f' r~ ai fPd Stjn +est vs-, pcti red sijnecl rem k +e 5t ~

AVI5~A..-e(: _j_t 0 is SjVVI~Mei{!c. Qboui i-l-_s VVIedra"'~1 5i<J..,_ecf rctvtl< 1'~ 1'!4cte

\'<l--vf:'f'~vf) {)c;v.lt1-5 1d~ 1'S tJ.tctf- p-~q{wes d~pevtd OVl tl.,i..s extrct QS_sq_,l'f,OvJ

5(\ £t j

dtte +o \q th-e\

SWZt!l/( r- vq/14e Co vt!c/ s~'YI"'p/'j avr ctsymetr/c d,·sfr,'b!.(l,-~111 a-P t h 'f v1 ~ / c c q 1 /o V? S h; { i ,

be D

If U.Si\1\j \'C(VIk..s Cqv1 ,·V)1f(ov~ IJ()I/vfJI' 01;1P shoi..(IJ expec.f +1-tctf LAsinJ +ttf' c::rcl~ctl cl/-fl'~rence..r rY\\'J~t 3''ve ctn 'eveY? VVZcre pv~;vPr·{v~( test.

T: Z iid.~)oJdt

IV\de-ed t~;5 does -9''ve ct~n Q~esoli'VIe +e.si o{

11: r'Y\E'dr.,.., dif{eie~ce = (; (o.d s~ ""'"""~r1) ~A: med;ct"? d,·{-ferevzce < ()

It ,·~ qfso

a/ f.er VIQ~,ye

ExctW~ple~ If

~xce!IP"?i -for fhe oih-er OV?e--s,Jf:'J

o, ct 1 V1.,..- o s,'Jed q I ferna1tve.

0 bserved cL"ifer e-1c er are

--2-o 1 -to./ - ~ - 1-I I 5 I -lj I 21

The VI There q,e z<~=zs{ poss; b /e. s ij"' ~S5iJii!VV1'E'"'~.J ftJ +~~ d;1~~t"evt ces. (),.J~~ 1-1 Q) Ovt ~ .Se~ ue.,.,c("> is e$ vq/ly L'ke I)_ Some Q •-€

T -Le; -lo -~ -7- -~ -~ -')... -l 0 -2-0 -(0 -g -~ -s- - Lf +2. - \ z. -2-o -10 -~ -t-

_, -t.t -2... +I I

P"vaiiAe::: pn~po\~J'o"' \ is a.7 /eqst

0 ~ 5 IJ VI a_ ~ )}J 'r'1 WI e 11 Y J -t C \ L<..- h 1 'c ~ Q5 ex-lr-ei/Y?e qs observe~ (T::.f).

p-vqf~Ae .:: \ z _ -2-5'& 0. OLf 1-

We v' u~j 4 \A {VI +o Q Vl v VI- /':)q ( q Y"?f J f ,· c q VI ct lo J + ~ fe s 't ,__,.It, ,·nJFfe""c:l~n -t-0 -t' q f ~ 6- Sct vYip/e

Sq VV?f le s:

The \J;/c_oxoV1 let VI k - S 1.1 n-1 + e s t: (o I Mq•n-\J hi~••j U.)

This tesi

e~lA_q/;)

J,l Cl : F. = F;

b~+ i-t

IS technic q/iy,, q ff'st o-r /c,_ o J /s1r,b\A-J Jov1s

of -/~f'

i s 5 e Y1 s ,· /,·v --e._ io I o c C( ~ ,· o V) sh;{-ls

<1V1d 1.5 L1 5 (.,{ ct lly c. CJ vts ,·de r-e J Cl +est .fo\ fhe

e~~qli0 at me cLans.

ExqWlple 1~,1-.l

\~(,: Fx;Fy ~l)+e Sct~-'V!fles are ,·r-1df?pe11devtt- no/ fa/r€d

X '2 3 l6 1- l 2 0 I '1 6 z l.f l z 2. 5@) f- I I 2 iJd 2.1 't I I 'l I ' ~ II .~ J

j 55" ~Zo ,, ~~~ 2ZO._lt5~ t..j1_ 2'--/( 111t 1182 -a ~ 1 1.3 Q!) ~ 6s @J) IS' \ (16. 5 3s-

Poe..! Q II 6 bse cv 0i,oV? .~ t .. , X a ~J r~

let \Jx = SL1 W1 of fl-w lo VI ks of the 0 hse r v q-f,·o VI s-

f,om X.

I-\' ~A: mx < 111y 1 the 'vi.,....- f' f't'j-ec-f ll ,·-r

wx ,· .s 5wtqf/.

I~" 1 his example wx;: 12/

Uvtde\ l--lo '"") S~Abse-f o{ Si~e t'\ (V\uVI1ber o{ obsNved {lo"" X) o{ + ~,e \C?11 k..s 1_1,2, ~~ .•• , n5 L¥1;: nx~~J

IS e~ v~ll'j /,·ke0 +a he -\roO\ X. ThE' p- va f ~ e 1 I 5

+~f\E"+o((" e ~ v(O/ I +6 t ~E' pro po ,-} i ov1 of 0 l \ f)oss:b!e Wx

i hcri Cl. \~ SWJ.cr/f~\ th(!j+ t "'p o bse f vfd Vct I ~.-,e of l 2 L-1 ..

There a. "f' +cr ~ t~ s ~CJ' c,: ~i(" ct I vo!ttf>S b~-1 ·~ ·' I ' f' be -1-!-e r

/

io \..,{ 5-e" So5t-IVvCI re -/o '} e-1 C(J-7 e'KocT p ~ v~ I ~vt e .

I+ CCIV\ be $ h 0\-V VI -i h 0! T II := \ I - n)<.(n'l-+1) \.A. X V\J X ----.: is C\

2

d;1~erewL +est s..Jq-/;s-/,·c bu-1 3 1 v~s Cl•" efuivCJifvrl ... Jf'si.

this Is The MCI~f\- IN h;-f.., f' _j U.-5ioJ,s-l,·c..

i5 a i h s~crLd 1 c .S o 'F 0 5 ')"~"' f)~c/}J 1Ca 10 V\ O' v'VIct f.

Naif i hqf Ux CoV' be e,{pcessE'd a: s n.x n!f

U, :: I L j_iX; >~j ~;I j~ I

F,o M i h ,· s -far rv1 ; + c q n 6 e S h o ~ n I

1k.,t /)., is qs'jYVIf-/,-1/cq/ly nor""al lAMder 1-l,,

w;i h o. 5'j Wlpfc i,'c VV1Eq V1 n)( Yl_j

2..

Q n d G s j ·h1 p +o -1/ c V Cl f i q rt c e Y\c v'J (VIx + V1J + I) )2.

This 3'ws ern appra>i~q-lf' +e<;t cvhe.., Sq'""fl~ 51~es qre lary-e.

SLfppo5e "'-'e

iv1dfpt'11 deVJ t WQY?-/ J(., fest ~t: X ovtd Y are

V5. 1-+A: X is f05;Jt'veiJ (C\ff'(q1f>c/ Vvi~Vt r. I-1 S e' e Vl--1 s

Co ''e fqJio n

r·easoV?q}/e, -1-u L{se fke Sc;.mpl~ a s o v r if' 5 t •) -Jq -1/s- 1 /c.

/\

r ::::-

We

ctS

\~~ Jvq r(x) Vqr{Y)

(t x, ~t\ - nXj L-1 )

~~xJ ~t~z)-hjz] CQV1 (....( se a rqndd w;/~q l/oV? -fest

fl-le> fe:s-t s -/-c, l/s It' c..

1/Ld 'f h

There o..-e n] foss;ble pcrir>";s ~-(' H., v1 )f'S

""' r

1--v ,' -1 ~ fhe -}hf'' n 'jts, Yov CqV1 think crbo'-1f 1h,·s- Cfs- ;,·n/~rJ

"'- ".s u e ; '"' 0' d p r : X, I X 2 I X t I (" lc • I he V1 I 1- h f \ f' q I e

(\ Chokes ~f;)\ t-v~lc ~ j iD eai\ ~;-it-? X,. NC?v!YI) chosf'y;

o t'ir''alct' (o, )(, 1 i~frf' are (ri-1) chotces .fo, Q i"'v1o-fc~

-h,, x, , e-1 c.. n; s '" Tiw be,,;, p ri"c ; p lr ,1-\ eoc{"' -/;,J.

The F'o por -lio V1 of -fhese raj\iYLJ S 1A(J+ rest.( It A.

} a 'J E' lo 'Je r fhaV) the ll'\ a.V\ r Q.5 o\

ohs~rve d "" 1he p -vq/IA f. \ IS

lhis I 5 ql/l exac-f -fest-.

Th;s is q -fesi of indeftY~clerlce; bvf ,._, J's r-eet lly ovd j power-f u I 4J q/J--?S t QV]

ct/-1-erVtq-l,v-e o-f Ylovt-~ero Co('re/d--f,·oVJ.

Tl11s 1e5"t v.-ol( /cl have -}f' r r; b /~ per-far ynQHc ~ if X ctv~d V were depev?deH r VLt+ wn corr-e/a-fecJ,

If- (~ Y J ,.s Di 'v a. r; q }e Ylo~ Ma I) -fhF Y7

~~

A'-~ A/ t(VI-Z)

lhi3 afsc pro\.ides CtVI

e vev1 wi·f ~ modt: ret-If'

AI'\ "lvl q/o__5dV s h., t ~;; ~·,) )peq rmqn~ f" f'o ~· .-:: o V1- '5 ,

eX c r)!Pt1-t

.sae-~-r/P

co v1 bf c. c f r e let/ o v1

cppro)(/VV~<'I f f' f., st

.5 i -e'e c;- 1 f', J· I 5- •

- rr· r Dd J/'' \ "! ('., \ VVl!;.". l

ra f~fr ··t haVI

5pearmqvt5 Corre{q-l;olfl Is compufed b'j r-rrlaci.-.J fhe yy w/ih fhf',·r rqnks ~/, ... , nJ a"~ d b'j dojVl_Y t hf=' ~ QWl e tor -} h P ys.

Ti-?e¥~

-fhe c. c VV1 p 1.1 -le

rctv? k.s~

-1he corre lo~ioVI be--lt... ... e t=>,r'

QrAtos-lioVJ ~ Vvh€1'7 ~,v/// t~e 5pea fW?ap1 Ce; rre (of/cv? he .1.. (~, -l).

AVI~~-ve r : \tvhe~-1 ~ is q fHOYlo -lonicq/ly tLA~ttc-f-t'oVJ of X So fhci1

per+'ec f-ly Co rff' /qfed

Inc rt?Q sin/ (c:lt?c re"s'"J) rqn k s ttr e

1\Jde; "'1>'1 C/PfroXJ'.mcde fest caVI h"" donE' as w; t h 'Peq r· so vt -:s co r-re /Qf/a v1,

A shy ~t qJa plrr/lD"' if a set of h as Q + ;e VJ d o c

{rom 1~e Some

q /sa dct-Ja

q 1/o ~;v s OV) e to IPs t co /lee -led o v ~ r- 1 /VV?e

,·-r ,·-j I,S 0 r·Qn tloiiV1 5"o M~~~

po pu lalioVl.

I-f X, ,·s co/lee-fed ~C·~st -fhe0 Y~ 1 efc; CoYls-,·Jer -the p 01' f S (x \I \ \ ( x~/ 2) j t I • ~

A'VI.j ~f the q bovQ +esis bQseJ OV1 a Cc'\re(qfi'6V1 C.oet.f,·ciet1t Cetv1 +esi: J./a: ,10 +r-e1-1d, /P x~ ""'~.:,.J

V5, I-lA; i he f"-(? Is 0... +r-end (LA p~.va \ d 0' dC)lA .. d-1 ~a r cl)

Arwi her neo t test vJq/J-Wolto~~vit-e- ruV1

,- S ca./ led f-es L,

Cit

IrnaJ''vte you ctre preseYJied w,·j"" C<. 5eJKence

of ~ cJs . g J;,i s a VI cl yo(.,( 1/\/CJ w i -lc k VlO w 1 f 1 f r'__s t h E' o Ll f c o Vl1 e of a S e ~ I.A.. -en c e

o-f COlVJ +os5es. Perhops yov- 1.-v'C?ni fc Y'V1-C/Kf'

5vr~ ~7ou. 1 friewd ,·-:; no-f c h~ctf,-7 et -f q

pet ,-i ,·c VI I a 1 j'Hr7e.

The -}f'$--i s -1-c, f,-> t/c ~~ed - the n u VV7 be r IS ,, of ld~-E' e le rn "Pv? z' ---.

of II

) . rL1V15 Q

r= o ~ ~ Xct tNJ f le J H-HTH-1-JHH

If 1hE' nc..-tr'Y] be, D-r r GfYIS

o' )QfJ'E' J :; ou WL~ need -\ r ; e v1 d q b o v_-/ c h eq f, ~ j,

15

-1-o

hos thre~ r f..,(n_s.

~ '(; 1:_'/c /ot-t s )t 5 VVlcr II COIIl -frcVl t you.r

C h Q p fe~ \5 - ~e J re s Sian o Ill d L ;.,., e ct r M o cl e f ~

This chapfer- \c:, q6oll1f Mcdef,·vtj the relq-/-/oV15h,p be+~~v~eV\ Vqfiqb/es

Th~ I-f

+ II \\ ' r _/ d e r M \E-5re s sio111 1 s o 1 rf'YJ lAse .

C 0\1\1 e S r ro VVl t h E:' l ~ !!J_ c_ f' H f Lf f J

SOV\S

he~~t

) c I e vfh 5 t I F r· Cl V1 c j s G Q I~ 0 V"1 I /rw.l h 6 rl- '4 J; e d he 'J his 0 + +Of 7 he r.s q,., cl . s 0 Vl s,

•

• _3---i

---~~-~----..-~·--·~·

0:.

fctthe("5 ht>,-Jh-t

-t~e

Frq VIc \ s no f, c e d. + ~at +q f { e r tIt C( vr o ""~ 'o. J e +q-t hecs t· f Vld-e::/ fc have +q 1/er +"' ctVI ave rqy ~ 5' c VIs b \.A f -t-~ha -f· the i' s oYl5 -1 e"" de d + o be

c._ \o Sf(" +c q ve \ C<. j e, R.e.J re S5io.r~ i $ -the +eVldtlllCJ fu \e+!..-lffl1 -\-o ct prevlo'-l5 skfeJ fV\ this CCtse fhf' VV\eaYI, R.ejre~SiOVl I . .S YIOIA./

~.< s e d a. s a. J e ~ e 'i q I + e r VV1 fo ' -t h ; s -f ~ p e o-r M {.de li vt_s •

A SiWip/e /iv1eC{\ mode ( IS cf -+ ~P fur Vl-1

\L::: (30 t- 0• x.~ t- £.r

Xt Lo\.1/J 1epresev1t +he f.c,f~er~ he,yht CIV?d Y~ <:cu ld be +~e S'oVI'S he,jl-tt .. £~ is the rq111dom dev,·o1J'oV1 iroVVl +~e Wleqn)

d: 1f'preSf>V7{s tJ.te d,'-f-f'~re.-tce be.\;'-'~Vl cb..serv-ed

po,·v? f.s OVl d -l??e fr"" e r-eJ"@SSJ'ovt li...,e,

\J e Co ~ l d a_ { s c c o VI s , .d P (" a. VV1 c de ( /.,vl'i h VV'lo re

pr-e dtc+c,5 (,·r? df"pe11dent vctrfo.ble.s)~

Y~ ;::: rJo tf3, Xt 1 f- /Jz.Xi2 -t- (->3 Xi3 + £t

I V1 -t t, ,·s rvtoi~er5

VVICld€ I/ heiJht

t.Je rn'J 'l--ri have ~1~er5 he,jht1

Ot1d jrav1cl{c(/hfr:S he!Jh-t.

4hoiher- Wlocle/ ~ct1hf\~ h-ejh-L

YVIijhi /""c /4de -the. s'/;ut:f\e c-f +he"

Cf s q pre d/~-1or :

Y: ;; (3. t ft. X;. '2. +- £. L

T h ;_s ,, s s +; II 0. )iV]e'Q\ I(V'l~d€ \; +hE> /r't?e~d ~ t'5 ,·.,

the cce-\-\tc leVI ts.

L -e ~ 5 '~ f-u {' v> f(:) s ,· VVl p f e ' I VI e a , r e:; <~e s .s ,·v"'

i ~ :: ~tj -t (3, X~ + £ ~ E.L ~ L.~d N(O; cr- 2·)

T hi 5 \ M p / i 1:' :5 t he/~ YL c.q·e iv1de pe111deV1 t.

NC)+e t~ctf E(f~)-=-0. !~Is ,·s becquse \A..)~ hC~-ve {?;c. i [j, X~ ~c.) des<-f;be i h~ {Vlf'q VJ.

\,Je l ~e f3~.

c D u tc::i Cl 1 f o v-v E ( z.~) 1 o

CVI (J "e t-ff>c -f- 1 hQ f t.vo v / d

bf"' VIO V'/- ~-f'f() / bv-f·

hove ,·r ~ sh;{+

As5u.., in) + (o.,-j

t h~ Vlex-1 s-fep (?;0 Cll1d (j,.

I s

i he io

a b o-ve rvto de I -e -s-f/,.,.~-~ o -If> i he

•· . .s co-r-rec.-1,~ ra r C(l/}1 ~ -fp r 5~

0¥\P We o-t

j

vY>e-/ h c.cl , s -1 1-, (> VV1 e -} h" d o-f Jeq s i Srfi" q r e <;,

A, "" ckocs~ f'o, f, ,·V\ C)\ de' ~u VV']tVli.Vl'?l~f' + h.t> 5uWJ

He S(; ua. <eel r-e .,,J" q ( 5) i, = :1~- j; o j~ -(~. +~,x,).

A. "' (-;o t p, )(

X

'2. 5\.4 --z ~X?

'gJ = _h_!VI -~r: ~X? (?

0 == 'Z :xz 'U-'F)~! v. + 5 X \A-:'h ~X/ (_::;. " -s..

C) -= 'Z ~x 7 'y-( 1' 'y -F) 3u -: h :x? ~

D = 'x [~x·sJ-•g)-7~] z-~·~

~ 17)_ 8 =-0~ {_~ "

() ~ ~ V '<(} -of} U -~ 1..4 ( =:. v v

QfJ e 0 ::: [ ~x•y-q~-~h] ""2 Z-:: $=t

~'1+ j-9 rar,;~l\ Jf"lo:;~b

d 2; / l--tA .n-t ~ IA.A 0 t cH1_ p \ Y'\ OIV)

• ~3 ? .J 9 h V'

d~+ }0 lN1 ~'~5

V\O!tda Jd~tQ\AV

The 5 (A VY) of deVloit>d as

Noie ,.._ 'l... -a- -

lheo"f' VV1 15. 3. l

') E(~,) ::~I

5V'v_ared , f s-; d t-1 o Is 'S oi'-1-eVI

sst ~ lfof" s u IN1 <Y.f 5~ (.,(Q res) SSE ~

\,ftll b,'C(sed -fo\ o-Z.. l'S

VI -2.

(Le-t ~ 1 ('1~ be the l..-5 e s-f,-,.,. {,,5 /es-l,'rnaie 5) -/ " ) - ~I - - \ cl ~ ) prou-F; t: V~, - £-~X'~ x ) C{ I"'

L~~ -xrz-= I((~ -J }((30 -tr,x~)

Z..(x~ -x) 1.

- z~.(x,-l)+l~.(x.:'-Xx~) = _9 -rLf3.(r!•X'----Zx,x) II

"'

= 0 + f-., IX!~ -; )z_ - /i, 0

z) E{f3:)= (16 rro.1':

F{~.) =- E ( y -~X) ;:_ E- ( ~ Z@A. X; + ~;) - (3, i J

L{x,·-x-) 2

:: f3o t-{?1 X t 0 -{3,,7 .:: /Jo r:1

s) E(c,r, i c,~ ) = c,~, t c. A "" 1'\

"5(',:) c,(Jo -+Ct. p, I j

Vfvt b1'qsed for c,f3o tc~ (3,.

Y) "' "'\

C 1 {5o -l C z (J, IS thp be>t 1/t?t'qc uVI bl.:~5eJ e~J-,>~~{r (BLLI£)

~0 \ c, ~() i c 2. r, /f ,,

gcyf._ i~t~ the 5t'~-J.s~ f haf if ha_s -fltlf:'

.5 t'Y\q 11~5 t Vcn 1q f1C €,