\01 Chopie' \l \Je_ af trovv-1 d;sfr,'h~.~f;oV\ J ~si,·vnC1Td\S 3'veV~ i.s ~1 CJ""v' io h~"e lect,\")ed a. No~ 1 rdi~e( we wa~t 1 t hq lo h AV\ -1 We. co1.4/d tJ,e.., th~ thof Pq_'QMP-fe_, bec:c::tus~ scr"""f/e; tH? The es-I/1-'Yra.fe- I,( i 'S ' .S 11 (f=t X) ranc:lovn Cl d pa /~s :~ CJ. Cl r X). t /,e w,'/1 JV1if'(IV<1 ( -fe f e..st-' n~Ynb~r; Cf de-l~ rmiV'l e l.-'1 Vc:trlotble) e,J, rc,Vlclovvt I os e ! l"'a111d0VVl a -f ll Thesf: J If mJ ~l e 5 t J e s 5 1·s /..:7\\ ,; s C'JJ // r fJ,,'I"/ k i-is befc-ve@VJ ~~ ctVIcl 1--v' h vcn-:c.JJ les ~ e. J· pqr-qvY/efers SO< in -lervctl es-"1/rn.c,1o .. I V1 7-e r v ct I j 0 V" nu~ ber, ct lA'1 a.. I'\ i Y'Vl/}/~, I f es;/mCl-fes A ra,·"'i es-1,-"""'C'#io, ,·s ~ Th e o f co e ( e s j /.,.~~ cr i e ) 4 S'ct o{ poi V?i- s/l"lyle c:t a po,·..,-t ou-lroVVI€. 5 .., ,·WI l::f estiV11qie__ '' were. +l-le f IV\iervotl j, e {'(,\\ - A'. fC'~J, It, ;Y-J-/erva/J rctndoVV1 pre hQb,'l,'ly Cov-t1c,iV1 the /<.. • rn ;j 1,-t q Is o m o d i ~~ C) ~ c.. h o o 5 e 1he rqv1doVV~ i "I ~ f' f "~ ( .5 c. 1;,., 0( -f ,' Vv i II c 0 '(J ~D. i , 71., f' pq rq Meier; Vv I-I ~ f r c b~ 6; 1; ) Cf() /o • 'vJe._ fhe be +- O(,(fCoVVIe_ a_ ~f ctD 'l4 the \qVldov¥1 it--rlervctf wo'-~ <-ov1 \.ide~ ce ;trd·e fv'ct I io\ 1hf ld -ft-,.e, pq ~ q V"lei e r. 110 re peqi t ~e ex per ln- ~ 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/ !v·1if'\vdl) theVI o{Jffv:X!.VliCJitly ~{ ihes-e> ov-lcoVVJe l~l'lec,,als- w;ll c_e;fi~J~,·v1 _L-f we 1 rare 1 +h E' r~ct ExaM p(e 1/, I . I ~- VI f=' f Cl ·t f' \ , X\ I ~··;~l,o vll N~~d Let EX P(K) r-eprese~-1 the ll~eii~Vle C·IA vv1ovrfhs) elt>ciric~ I pctris. ct t L/ 0 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 cc pI~ -1 f' s I ci -/1 s -/1 c , T he r e f~ r e J X Tt c_q"' W1 is I VI +()\ o i I, e r 1/ 5 ivt c e t""" ~ Y; J z;d e IV\ X ,-s f 't" ~ s t.-v 0 c d A{. j q 00 c:/ e s J,-WIQ for .,A..(_. Recct Also U.MVU E \c-r " fe' 'V H. .. t- ;;.:, EX P(~) A.; G A}1\ ( .M. 1 GAJ1.1 ( c ctll I-<_ i· h i t:Y ~2(2.k:). a 1) )ZIG~, vt 1"1) d ,· f' i-t Thus <AJd Shqpe pq~qme-ft>Lsf w "' G AM ( eJ K-)) ZV\X K fC\C""'c.\.((ac/ "--.J GAfv'l(k ~K~ ~ t~e~ iV1depeV1dfo1t. 'V' GAM ( ~ox "'"'~~(_gc ). M. rv t "' e VI ~l~Y\) Ill Us;~3 so-f-l"'~.--a\(0 pe \C f' V1 1,'/e_s 0 o~ +qlofes we c f' [}.__ 'JL17fo) \.v ... 55 = p (J:-:~io) - P( <- ~ - f)( _!__ I_ f~-Je,e Ia v1 d oVVl I,VJ is ( 57. 3) '\ 1~ ) ~ < cgox- sr.~ pro bq h;lriy 't57o o i e r "a ( I ( <iax 106.6_3 -lhvrs < Joe. t A IOG.t3 - P( ~ thl5' '6~X< ;;t-~,n(<&o)) t.. 57.15 f,,tfl d ctV\ f --s- 1.X15 ) ( <60X ) 106. 6 3 ) '8'0 hC~f vv,'/ \ the COV\ ~o.r'VI M. -1\i+e 1 pe, fo'fwli~ The o \..-{ 1c__oWt E' o-\' x .: : 'f 3. I Thus is ffrtp ow {c o..-vJf f~o(.::r3.t) O!AitCJVV1f' I"S p'fl f 1'yv, 'f' v? t- we obt~i~ CIVj Movt-fhs, o-f ~o(~3.i)) ~/06,63' ) Thi5 F,)(' il-1~ == i11-+ervCl/ ~sf,·Y'Vl~to1 (61.'651 130.3'2) f;J,/5 q crs-;~o C:OV1 {;deVI(~ i·dervc:{ I +0\ ).(. IlL Def: ..L 5 e V\ e r-q (, V\ [-Ux,, ... , x,) He.o IfC o vt tide~~'~ c e Def: If 1-Je u A 1 Excp'VI pie 11 ,·i vve II, "Z. I . OV1P- ~pper S1cJed is (-(1) t h. a-+ ft-te (c T) k¥Lo~.-v ft,eV'I /00oi6 ' +he C...L- 1 1A(.c)). )~ct r be {d C-ou a rvt e i Pr 1-v r ; +-1 € .., uf!)J. OV\e - s ,·Jed de ti'Yl ed s 0= P(){J)<. el. "'{KJ)"' P( e< "(X)) II a 10() ~Yo IS 6. Hh iPrvcrf ,·"' pos;-f,'vt>_,~ (o; IIed ,x.)] If I,1wr''-L ,'' thr~i 0.~ l)ef: c__q coV\tide"'cf Ne'J-fe <..<.(x,, ... ~(~) "' - co) let r·d €VICe Covtt-' The_ 1 v q I'' -\c> ' / vr 1e r V~.(~J 1's C!VIJ p[.t<x,, ...,x.) <e < <..<.(L ... ,x-)}= ;f q 1/1 +ke f-ro "'1'1 f~e qs q Io V\... e r / 0 () '('Yo / q J o u s I y. ovte P'e Q.S s -ex 100 d'/o -s;ded w-e f/ vi0u C!V) CC)Vt o f,'deYI c e wtp 1~ .~ . f,·"' d.. Co""t;clf'v1cf'> 11e}vC!II-+e~,Jed'1 /t'141,·t" +""~~ /;WI,'-/5 s:cled C.I. 0:: p( 'iiOX < ~~('i 0~ - P( 130 X x,;(<to) SOx ~:(coo) IS whq-t ijpe of ex wE' h e q de d +'o' ~ o. 1e < ~) ~ oo oro 1 oV'te-s;ded lowe\ C.ovt\i·c:Jevrce. }IVI1;t, ? ,I 0 ~$00'1-; I'? fJ/.'J:;t- 1"""'1;;;; ...so· PI h OJ ~/'/\ ~ ::> .J ~u r:l/1 d V./:) ~«=> \..Ad~+ r? Sl J2 = ro-9 •5 d .1/ fd/f/0:3 flAb (Olo. '>! ~/VI xo~ >(Of,) s( ~1 d fdf,'.5'- 0""4+ (I::> ~:'i.L .5h .. =9 ::::1( rJ/,J< oo / ry a '1 f"VV ( ( og,) :-;(_ / ~J'Y-)~ lC Vv4 10 c= J 9 O_:J (o$).R-~~ ddr, a'1f- f d f t 5 - dV\0 Sl )( 02> os- 2: \ \ ll!.f A~ +iVVle tkeno ,'.f -l-i-ter<-"' 1 YY1 f'Q r. , ----\r'-i e 1 0. be 5 i ::; h0 I iF J t ~E" I Oh., ,'VI .LVI P'ac{cf' n fa<" 'J x, 1/,2.2 e~ 0 =- p -- ay s fA a- - 15 !vq·s Der \\. 3.1 (_ -· t-' Y -\; V1e. cctV\ 5''> ~· I Yl -Ia 1e be t hQ f I) C( i l) ~ qbove arf' I,;·· 1- tVId I'<. ~ )(- -r - ~ (X + ~,r;_ • IS 'lo .!:;.1: Z.. bctd I .o '-! " o-r · ,·.s ca lleJ does uv-~k."'-oL-Vn 1+2!' 7 ) -lypicq/ly ,·+ b4-+ bec-q r.-•Sf" p,· v; to I 0 p·,~o-lq I not depe111d pctfqWT@-fers-. i V"" kV10'v... 0 v' I.A.S'1'nJ q r X- ~ !<:- • i!o !!J: ) -z..• I kvtovvV1) ·'rr..-v15 h.~111clled e fwo - s ,·cJ t:>d C/ ) /.__At._ a- I dis-l,;b(.(-lioV\ ofhec ht C>vte-s;c:}ec:f en A -+vtvtci,,o.,.., ) Q== ~(X,, .. ,,XV1, e) aVld c1 c -1 f' /;ke_j .fcl\ The<-!5 a 'be Yl'flj :z.. 1'5 i-F Jc: V10\vt'l., f rt ll!l -'-2.: o not" 0 ' os- cr rf;.. /oo 7o c.:r l.s eX pe -fq,./f'cJ q/--/c,)ed 2.. ( t CJ t:~d N(~~a-;t). ;v "' l_!.::1 (_ k) t, v s- e cl. X I •• I I t' , ctsk I '7. vq f VVtJht OV\e .)vsi. t -Xt~ r( X+;;.~,_. I he IS 1:1 iS" ~E' lrr C I C C:: if \ v I i V1 V'l T~;s ' e1vC~f 1 I~,-.., Q choice>/ e1 porClt-V1f'-/e, V5, Excr Wlple I 5 is f 1 t-tq~? OV\ of [1 ') e 1"'1-f~rvqf "'12 1?ca cl VJ.-1 ~ 1-e r, V> ~ '-1 cr V1 the -// !J , So.-Hf ;f ,·tr 0(' qVI.j le /)5 I-f E;.rc, vY1 p/e; -1 ht' - X,_1) ""' ~'(A-1/u-·2) ~. : t. d ;f S'o ') ' IS t.AVJkno~.-vVJ/ N( J-1 1 cr , ) ) ,· ~ -1 0 fc' - t- ,·5 p~'va-/erl. VI 0 j-A-< "' ,__. t(n-1) -,srn j OV\Ce' vv,; CCIV\ } I ,s ... (jt.A, heJ.vf i~-1 e ~ CI Tie 0{' Also, cvCl)e)~ Vt thP 5€' i ct lovlf ~0 s 'j p1va-lc) ' v·\ -1 t, (=' 6 t "'f "-vhe re he I \ C~ E 'H? ~~"' ,· v? 01..-1. cplAcrnl'); CPI1~fr t·h f' f $~c. (, o-f c.r:. 1 "'C( t ~ t/,is fci lhE'V\ co V1 f; de vi ( Cl... a...! low~ VI - / , ' ) • 'f,~X ,G) L ~ ~~~) -ft), e cruq ~(~,eJ i. 't!tl. H-ovv~ v e r J c 0t1 /;-f) ( (' f'VJ--1;/e 3e + -1-() LS 't0-z ( Yo~ ep~ ~- u ct P(~L::lz. ~ ~ ., J, +~e pivo -/c{( q p1'voiC{ / cy~.~cq¥1~;)) C1 te 1<. e atr-C( rt j J'ny t-he i ~e ~ efo-<e 15 4VJd p PI, "'0 ~,q,I J ?v, ~ cl l. d tJ(0 1 """-./ Is X',_' "'"'· ,.., <1 yiAqVJ{-/)· cr .T -f' 0\..."' v> o-- pivofq ( Tf p/e: vl X-,.<.( V"'\ 0: tya m k IS (;) ~e1 (' o reJ io f'Vto.J /I ,-r' ' hoi a be Cl¥1 C uv1 f; df'"' c P iV!iefval .. Vl, lU~0~;) VV\o j 1\ 0 ·/ f'>t,si, set Q. })~ A Ik-t ~ 11 lt>cqi;o'r'1 +o 1.-vhf'\f' A Scq/e Tl-tc~-1 G. a... ,·-r X Let X:. ""~~cl e Note 1 "'cd ExG?II"lp le: /oct:t'-lfoVl f e ,·s 5c- q )rr t(xi6)= ~ j_ 6rre sho~rvv- X Is J pa rq vvre-/er,; f·hf>,.. I) . T 1-,e X VI IJ A 6)-e ··s F''vdal. A -e I$'' p; VO ~ct e:- the /11[..£ t"r M potrcrvneler-; 6<-.M.) p•'va-lq/.. TV) 0 ~here -~ i.s X-A-t - ........, (0 ') N 1 -;;: ,·f cq..., b~ fCfc1/ • ·;J ':__ thf' G -f~en =t -z.. EXP(e) 1\AI,..£ -'tl~ tQ<.:b) = 1zx >oJ i e 'S6 ct.., foiL£ tt~('" +'-12 cl.is/nb ...ii•~. pc:t..~ctVVIE'-Ier, /ccq/,o~ e -ft1ot X~ Lei IV (A-( I le .f:~ ~. i'l f::":'lftcJvvd"-lpr - (<"-A.()' X-A. Thl-tS Exa vnf le: e1 ~ is A (X ) ./ is o, a""d ~o CJV1i,1 e x~ ~ ~ ~ d Le t +~~ e 5L(pp()s~ -f(x;e). is do~s ZX,., -f(x:;ze). fhet-, rf T -1-hcr-f flo v. f(~j 6) ~ ~ +l~ J· so/;5{; P r f(x_; 6) 'V f1.1 ., c / pqraYYte-ler ,·s1 Th eo 'e WI I [, 3. I clev,5,'/'1 ~ f(x; e);: foCX-~) J· r. 5ct11f}lr-s 1 '' ·1er o~·\?)r 1) de pev1d n. o-{ De-\': po< is t::t e. -XA 5c~ le ~ - \ \' /1- \ -e-ToCB J pa r q vvt e i e -r ~ ke,e forx);- e -X izx/o}. ct V\ d. - X -e i~ pivc>tct { ' 1 I\ 1 Theo\f'vVI x~ "-~~d ~(X; It 11.~,2_ el 1 et) pq rct me if r I(Jcq'lio"' \.sa.. Gz. f), I IS" S" C cl C! 1 pctr q~ei e r F' e,) -U -I r ~'<..X). G.,e._) --re -e t~E'V\ eL "'I -G, G. a..nd '2. e, p·· vo4q l -\o, -IS ez. A 02 a0d ez. /\. where I ht:>VI C( c: V1 Gl. .A 6, ,u. etvH:\ X ~O( I bz. q n' to\ IV\ LE.s G, 1 82. Xi,~ t-~d N(~~o-~). Let Exqrnf/e 1/.5. Z. piv()~Q{ Is IS C\ ,· <; Cl ? / o~ '5"(' L/X:- -J')z. J eq ~ioV1 q /~ q C( pqroYl'! -~ 1--Jf' <ct rf'> r vY/ (! 1f' I e1er .A11- ~ ..r -ro f" t'\ /4. Cl v1 < d c- • X-,<.<_ $,.; pivo-le~ / IS (xtt~-xY) r- L-((t· -x - CIVId . - X -A 2 A.. whlc. h 0. c- <- " cr-:2. con:s+"vds) i::(n ..c.t toe pivo -let/ 15 Tf k.--~ ¥'1~.-tf-l;plj -f~f'se C-oV'. ver1lf'v1t for -f) p\v64c,/ -/- h f V\ c-vuq.,,f,'};e5 we 3e b:J -1: (n-1)$'2. A.-~7-0-1) C(Yld crz.. ,.m re Y' lc.e i0 1/vO(" k lA/ ; ~ f., Sikce fh.e cl,s-lr: b"i iol1r are k~-to~..v Yl. II~ E.xcrWJp/e: 5\..(ppose HoiA.. CctVJ h!hqf -r-hi-J :;o'-1 hctve Jet or !jdt.A. -A? ho~-1-L e Q +r,'c k q ks v-...()' ro /o rox r C..L e_ "to\ ~ A, 7' t 1.-t ., c -{ ioVJ t/-Je L-vheYI e v'f. r tor- cr . IS' rn o 111 o -1 o Vlic. Le -L X: /\,- Exerrt?ple //,3.5 X Vl() i ~d B£i<{p) fo, the fiLE I5 p,· vo -1 Of I cy v. ~., f,'l_y . No J.v-e ve ' I hJ fh b~f P; f here 'S CLI I e - X-p d N( 01 I) fr(~rl SoI ..L a:sJ rvr ~s :;o J1..t~5S f to I, ca !!J I-f :Jo\A -f,_} lo (do t~ i_s YlcJvv J; I.A. use t ~is fo 'jOl-t see If cr-ea~e IS q C.J_)) ct b:-J Yl o fe - J!~) IS j h/s p ,.v~ fct /. ' s,h1'1 Fc,, 5i"'p/;,,f) (bui Gl. \.vo<"se X- f' o-f f- h/Yl k Cot., aSjMpioiicof0 pivotQI. cr , u;ly. fttct-f (1!7 5hJ'/ ( (t'O)N ~ p +j > JA., 2 / jJ Yr' / _11 -%-w fLAb rr-Y j_;:/ tJsnJ·I- K) d 0 7 (1 1 0)N c)'1+ J :.t/ J o u 0b I cJ.•• "'s .. c) L"I ["' •>n "' - - ~o·l) d ::: 0 b . ~ p hiA,IS Yl l ·o 1 = ~ 0 2. .:: \J ( )?") -r D d F ,1 ""' !X ')I ~, v ·' t ~ J JrJ o '"' o :'1 h/ J~IO~A.A,IXOJddo '.r::f:? >"' o/" o. ""( o 1 ddb ~j d~ 1J W-X I <AlP(; v ~ ~ ' t •1/ F [/' I.::J 11.1~ f;;b aI d~ b •<:j V't()h ~ ~ ""'~ 1-"' H- " ~ "'N btl Pivo-Jq f ;f f a noi is OV\e f; ·Ir-es <--U?VJ n 1'c e Clvc:(l'/qb/r/ Jf?l1f VVlo1-(" ar-e red to use -l/,e,-1 you IZ-o bv~f 1 I.ASf> CClVI rv;e thad. R« 1~e r iV1 i h"" bey i.., >v; ito a. pro her h,' /,') Vo / ~"· v) . f /vo Ia/ 'J' r...u:r n ti!J / J <; f: tHe, k j pro bcrh,f, '-l c:r :> fC~ICIIIYietf'r o-f' TJpic<:t/ly /t; Ex q lN1 pI e ir1ieres"t: -!he /n c II. 'I, l OvtS1'J e I n qyrJ /1/..F o r Cfr1 u e;ed i•H ~lviJ !"'le rn en t i rc f(x) e) o the .P.(','(' /t> e 11 t , a c..Ls-1-r, hi,.(~/CJVl e..- .,{' 'i'i~e ,.,.-If, (x-$)/~Z X?$ Jev1 s1ly / X L Q. () ~-v;ft, ,s- -lc~ -I; s· I;, c ,1 { r~.,cl~m !:'<>w.fl'" =~ j_ez 0! Sai-'Yiple. 0. VV\ (:=' I lte ") t.-t S' fc~ /p m y/cr{en?,-." L G/o • f(~; 6): n- icx~; o) : _ 1 'C:)(x~ -€)') t" i:; I G2W'\ e. e~L.: ~ /(QJ 1-lx,:"' -VJX ez"' 5o X XJ:n I <Ire ) 'J oi ~ 1 e~+tv~ 5\.1 f.f,'c i 1 t X,:~ 26 e~ t for f e. Le1s X/~VI use vv hE' F,->'1 d C( P(J,: - ~ -f ) J ~ /- /~ (J,; f=: x F (X ( ) ::: ) - 1-X',: /") : i f x ) eJ \ 1 hE' ct. (6) . o s- '} F ( q(G)) == perc e r~ rJ, c ' v71/ f ( ' !"' n I .£ If" -/,'/e -r ,J, ~ 0 Vl r<&) ', he I e 5o -) n ?2) 1 ci, C( brb 1; L q(G) ::: 5~ I(' lo tJ3) : 7 C:\eq-}f' =e( "' (G ) " t,!.. • cr o h/ io Xu., =- y, J = / - ( / - Ftx ) ) ~ X / e -&-G)) ~ .1_ [ X } 6 f ~ l\ - ·e ( c-It., J ---- i i (oct ( '1) J -= _) e< e_ e"' q(Q) -h '5/rn ~ /C( ( /)) t(0) ~ 8-- G -z. f oj(· o:_) h ' -n(_c(~-6) _ - -lJ VI F "1 ( 1~)] - - t-~(~e)-GJ 8 +- ez lo;(. C()) = )j F'r r Cf-'t/1 }, .)e _; ct(6) .> e j [) ( ;~JG~)), .~s-= -(><-e) 6' I::; d 1/- c.-:) d 5 S~"~ 5 a""-4 0 ( 9 FV''=> _)OJ s/f"?'JI \ ~9 <;II ~1 ,? 1h I a s ;;) 1 1- "" o tt FLAb (~) 0 bv,'ff<)/d l t9'1 i:.h'Z e---~------~------~~ . ~--~-- --------- 5' 2 - . - -- --- (e) a (e{1 >I ' .S z :: . ( C" oJ bo I I ~~ - e I :~ I V' ~ IX ( s h )b o I u I F l/1 t:> OJ~ Vf ,;~ - e) c) I :;J aVf + sj_ : 0b ' Of '1 :Jaa fv\ON ZZ/ Jz_s De.f: If <A '""do""' ,-.dn•"/ CoV11<~iH5 fhe w. f 0 P' o b"' 'v; I,-) •+ I"'"' s -t 0; thPY1 fhfv1 ExQ ple l'\1 II/-/. Ll ou-fcovvrf' XL F,Vld A/ c-~//ec./ IS ~~d o i3 E (<: P"f""'e/Pr I/ a _/ canservqr/vf' I oo tf;4 c..r.. c t~, p. {p} Cf()~ coV'5e r vqf/.,;P oVlt' -);ded ,, upp<::'( L P( "-(1°) <~X:) •'TO L c ~,vhe~e mo.x~C \ ~.(?) p\1-rr~ ~' ID, c~a:J C(p) ;;; Novv to '} !"" / c.:r fc, a -1 i p \ CfF) < ,tx; } z )(~ : b) T-t ~~ we PJ ke t' I h := O'fle · I0 s;J f .J CdV'l rCJ,.r Yi d +lrte Th; s VV<L<C h 5;o'Vlf!~ 4t~ Cq bE' o<rk) P(~X ~ 10 ' be+~eev, c ~· kcr-/ (,) :;_. c~ 15 \<; o V' J() \ V1 1.-r f Vct.fue 7· c c.. ( 5~tq4e_n sivnplf( 4,; ql .£ b) iS wt..cri CL VI hCr -/ Vv p=.cr, S'o A OF .<313 ,s CL Vff{"\ p== .i~ if T-\ ct!t·v (' Ql'\ i Th;s C OV\ 5E IV the t~fYI J t' \\Of : (.1S) _:: s- c(.9)-=b • -g C1V7cl • Cf; ic. c( of O\b;~(c;IJ -\\ • ') c!f p r { jDff'Cr5iOV\, -/ ~ f' 3; ve S Ct.. ,· '} i__ "'~'1"'1 :Jf Sff S"oJ/j!MoX.;> D-"'"'{ fw.., (/ af d>l\o&JJ:7 _l dL-f{ lA 11 . VIOj~b~!XoJJdv }- t- J J op ~j-yp o rto.( //hOJ : slllv ~ ~M '1 /V\oH (~--z_o~~w)N ( 1 (~ '-o / 'h')(\J z_p /zw)~ ( -e ~ z.ry_ 'rr l,w) (\J ..loj .T~ f~., r: .1 --v /V A :X pv,~ •o he. I \ 2) Ch«pte' r,e,+Vi J' P";"' + esf.irYI vV t-\- we ho. ve vfoiA" ( (co"~;devtC e Q ~ p<~ t~e t hi ,1 p/"'"s;h/e l~e f'oce~5 o~ A Vef l z.( .1 0 ::: 5 G fl-,q{ hJpc.1k>iJ a sor·h 7 . ~ 0 1. q-<sv.,er•'J fhese +est;v'j' s+A•~+r.o.. l_ "h•jpC tf..te s•5'' a.ba.,f -these qVlS"'-'"r ~ :r s if f Ia .., s : bIe c,__lkd Tc:s-fi "'_) is iVl+cnl"' ~_.es-flo"'s: Ts 5 es-1 I .mctfed p"' rO""' -c te r 5 I e s aVJd ,.., +e rva I e sf,v>tq{ 1'5 Is). we vVCf"'i +o Now, .. ~ \L - cLs+r;b...~;o~. '/Ltes-liovtS 5+a~e-VVtell t. a 's :C~ :he _sfqfer>"t'n"t corvtplefe~ sf~c;\tes d,s+r,h"'-l-'o"', o-fhe,vvise ;t J (S called. i5 ccrlfeJ 1<; c< -the ,, S1v11p/"' h'jfa-fheri1, o. 11 coY'1fortle hjpo-fhes•':s.' (:ir>lp le) fxaWiple ~ fxa "'f k : 1{.. : 'X• ~;,;,d "c: Xc"· ti.d N(.u1 I) N( 0 1 I) vJ ;th ~ !:. 0 [coMposite J t20 Novv 1 e-ve r'Jeed "- deierll"1i~ ;+-ft.., ~'Jfofl,.,si5 f-o WC<j Is pl~l-1:s;hle., Def 12.1. ( Pf'for to o...,tco~es l~ese 1/ r r,J' •~ ,-, l-1e n Lf "J tof·Jtes/r. II i-o cts s pee ; ii ed ,.., re .f'ered ct '"'e "s <-r<> lly s~..~ c h s-1-ctf..,·.s i ,·c n 1C e tt-t e ctS a > Z.. St.r .P·f'i cl ~ 1/1 -{,. r ""'.r 5fa--lt's--lt·c t_ MLE. if<.e'eci t) zJ 1-t: ~ ~0 is fhe - if X reiec 'lio11 "~Jio t1. -fc It is rass;_h)e -fo cc..-v~e II A tjpe .L b--1t +rue A '+jpe E 1 Is tt--, t \\ ~x/ x> Det dec ,·de what need +o reJ, 'o ""· I or~""' Vvl. co/lec-{,·,,e!J re j ec'-i o" a... :;ou J ,·YlcoV\SJ·s+evtt ot.A+cavvt~s J of ExqVI1p/t:_: ctre re j e c Tio., The exper ,·~en t the +olse wlit., (3. ,, is error i.s 1-v hen re.Jr:c7ed. error'' b~t· 1s 1s n.o+ the TV~ i.s is VJhetl\ i Vl OV\ nu co'~' r~c 1/ + <o;'\C I <..t.5"'1'dn hjfofV!esi 5 Y\o-fofed ~--..,~,'fltl fkf' rejecied. rt'-lf/ Thi-s IS 0(. h'jpo1hesis ir nofct1ed . 121 Def Fo<" 12.1. 3 1 :5 Cq F'oc 1/eJ pie h'J fD -f ~e sis/ +I-.e 11 Stjllli of .fu'1cit'M'' ct IH.) p(l:tpe r err, r lev~ I~' 0(" fie q.-. ce h/fo fl·wst's J t ~ i :r pro bQ b,'/ /+J coVVIfos i -le 0 - IS fX'o WI ple 5I oN! 0- fi--le Sft'clf,c d,,sir;bl.,(-/,'6,.,, X',; ~ t. Lcl N(""" 1 I) o., d I have 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. ')._, ppo s e P( X.(- z 5,-~ce fo IA) " P(i~"' <-~I }<) : : ~ (~.-p· \. ~ t's k r V'lstec,d be o c.o1kvtowV?) d e r"~' we ft,e V1 ~ 2 fh,-s +Vrf"' f' 1 1'5n 1i ~ s ~·~ ~ r~~s; hi~ mo_x,'VI1t.tll"'l vsef"'(. ver1.1 cf ..1 +he r~r-o .bab: ) 1 to s t-- N-J_y Uvtdt!>r Hfj. ex:= D(:{ ll.[, 'i j 1-l.t The = Y'V\ :x J:i( -z.Yz_~A:..() : : A - <' qrO"ve/ 'N I"' e is a c cd I fl,, No~ t~q t ) =~ (-u\. '() -fest ( 1-;3) IS +he pro ba b,);~ hjfo fh~;is ,._,,'/! be rejec1ed. tt-ft~c-lloVl i+ -1 h~ frev iou s n-(o) \; '2. c{ " hat the n ~I/ This £ (- 2 of /po fhf' 1.-v ~ 'I e xq mf /e) == 0<'. flvte clis1rlht.-t-.fiovt, f u +i6Yl; v1 c IT(.«)= I"' je t11.Pr:q Il s0 TT (- ) • P(X ( -l) A)= f(-~1:-«} · o(" =- VVL~X Tr(e) • e~..no ll~ N(,t{,, -k) N(~o,.k·) c P..o ExqWI p/e: r ~ Ht.. <l bove f 1<.H e } j"d;,d1e f t-.e fo o<, ~,)jl-~(~,J. X~ ,...,;~d N(~, t) re:yc"" If i V\ c a c..} ~, and 0.\t' C /At do ik!-l q(~.v<>ys ":Jree A /'l1or<" ~~~or»ia4tve "'"'/ .f hjpo1hes-is +he ~~~ S {e Sr><.q 1/e s"t is CG! c..r-i1ic:..t::tl 0(1 t fat.."'-,~, bt.tt rea S'~ S~ l-(3(x,)? Covts~q.,., t b~..t t -le 5 ; ?;f.' hofpe~ fV\ C. h 1--f 1 Jtr p-o we r 1 dec rea.se s? .People et CCI.AS-./r..tVJ he (d l.vha1 will ,e ~ se s J \~ he /.J \e hoppe"' fu /cJ t;vO u Ao \Nhof -tke ov1d I c}. 0 ) vv he{i corr-espoVIdivrJ r~jioVl [;;: x > ,,s ::c -t- /-( Qu.es4 i o111: Oe~: ,c.{ I s-i <>< /fed the wkai 0(" shoi.A/d be. r<-por-1;'] -f~.,.. re>ult of o"' "'"' ex pe rt>"'fe" t 1s -/u f<" pod. af Vv I, ic h OVI e coc; /d re j<• <-f Ho. 0 OV\ b5erved Si 'l€ or the 11 p-vq '" e_ .'' I Z~ XI,,. 'I X2.~ ,_, ~ c:.d N(,t.( I I 6) ExQwtp/e 12.2. I We If 'vvq 11 t lc +es-t live IA!o~~tf o +~s-t of' {,-nd ~os- = qfprofr,·~-1~ fhe A(X >c (A) fY\qx '< ~ 16 Ho: Io vs. 1-!~r: S 1 i!:.e c;:.y':-, 0 s- vv-e r-f'jectiov~ re-y/on. 1-<. !:: p(x-~ 1o > c -I o == 't;> j_ ,...... ) A= A. c >1o. q P1 10) :; 1-I((_-/o) '1;4 ~:; ~ :::: 'C. Cfb' '-f/5""" c =:.) So u. It will rf'_Jec-l The IEs'-lft 's x:- /1, Lf 0 QuestioVJ; 'IN h) wE' YJ o-f use avtd llt.R re j l!'c -lloV1 a ={{) X +- j_. :c.,~ 5 I 10+~·-c.qs-. re ~ ~c + Ho. reJ /o¥\ of fh~ trV"" [K(><<.dj~ E)( 12.2.1 (co~~~ii~ued) If persol/1 o. "ffnlpriQ4~ _~ +~e v1 we -/elf ~e IA/01..{ 1'f the 0<'=,01, hiVV'I {d he thaf waVIt. ~~jecilov-, be/;eved c:<-=.01 t-vol.( was Mor-e /d nof ~ CoVIvi~-tced w hC'VJ we r-~jPcted l~c ~.v,'+~ O("==.o~ to kV\Olv ; \ we WO~.{ \d rej f' Cf fe_Jiolll \A./ere. cJto.selll Sttc"' -ft,at l3a J<epo~+ittJ p-..,.,fue will ~"1,-s-IJ hiwt as ot~ers 1-\./ho rvtoy hav~ d;.fferet~~f /dects ~;vhq+ ~ Shol.-fld he. o Al'loher ..,_,aj .o-f def.-11/:J or p-vo/~Ps IS p-vct f~e - P(+est fh;Yik,·"'J ~.-v.i /I s-k,1.-d,'c extlf>I"Yle we I I tt s as be cd leost as observe J. a hou i abou-f H.j the pro b b; /,!J d""f., 11cl s of 1he spe(" ,'{',c: d,·5ir, bl-ff,·olrl iV~ 1-/o J ihe" f~e p-vq/CAe ,.s If 1 fhe I"' ,·~ u YV?. t'VlQ Y ex-q wr p /e 1~ r'.s p-· vq IIJ( e : x: . 1<~10 ., rvta == ) p/x~ l' P(~ zu.Yo -16 ~s- A) IJ. '16 ttl'S"" = l - I( I . '-1 0 ) ;::. ~ ~ ~ IO) O.D'-/ o l 'tfs- S,) rto t ~ ""'"" i ~'"' k >' e c1 H() buf a :Ju rej be . OL-fOI 1 (a("' yreqier) ""' sl<ou (d be ho f hi 1-1 k ~ wol,.( {d reject Ho. a vrJ hod 1 w v,.,o._.l.l •0 I or , 0 3 0(" S ~o v Id tsl '~A es +' . '~J•'OV\ 'OV\ • t 7_ , f He -h-rt bau111d~ the 1· 5 o"' o cri~'""l <H" s-fric No+ AV\svve( """aHer it Does s.Jols-hc <'a"' j,-.,,d:" S', is : Xc ~oLd. N ( f:(~fle: .<( 1 o- '-) (\ : . s-0 "d v"< ~<e) H•·. 1d / 0 "" . 1-l A : A I' I () • re~l""' $il that 0("'.05', o-'" :: \ b ( k """""' i"' \) e v e Io a. s -t o -1' ~ +e cr:4ic"l • p,·,.j • F,'t~d +he j::";,d +~e ' , Ques-lio": a. Whal i ~ wou l-J, ·. +!.-( r c po\.A.-'1?' p-vq f"'e /d t1< '!. w-e I0 +!'OV\ i.f c~o•Je a" J X. ~ A·, iM = II. .<-<. (,,{--) b..ri- Vv; t h a- '"" ExctMpJe: Th« rowe<' fu~~-!ioon f~ ,of e«s;'J ~iV>cl rr ( 10·5). ( V\5e 5<>-l'vv-.<e J LH1 Rep""1 (If) 6-A I Ex a VVIP le: . lh;s o' )s <> IIJ I /c VIO"'-'"'· ex-p;.-.-ted, u:f(J ~ so j,s-t H~= x~1o. vs. 1 -ro-vo--taiii"J r-est +wo- 1-<>: / e>J q l f€ n-• a-I• ve, <> bJ-f- Vvi 1h ~ 4-t.vo r -\--q; \ed a\ {t2 r V\ a+ ,· ve , Repea.f ~ Et o..-v"f le: I Laffe~J 1-t: wifh 1 <.. I 0 R.e p"a-t I ex:qwoflr: "hove f[,t> (:;1' J 2 tA ~ J< tAOWVI c:t ill J 0 l'Sl X~ ~~d N(~t-t,cr~) Te5t Ho •• o-z. ~ 1 vs. h ExaWiple; 'Exqlfvtp le. : Repeat ExCiVV!fle: R~ : Te.--t H.: o; ~ n l::)(plqiVJ: =o-~ tor (~., tTi 2 ) i .:d J.J ( 1-1 l. ' oz.~) 1\) o / ter ..,a i 1've, chavtje fro w, base /;,., e P<7JsQL. lwhj f e sf; ih~ !'.s o-f' [.(().' p-- J_ 2- qbo"e -l-est ;t ..SoVVI~-1-/mes Approx,'V"'atf' ~or ct!ieriAafl·ve.. X.:.. "- C.:~ J troW1 L Coflservcxflve fes-t of' fh" [p-ve4l\.te c:1ppu1o.et-.. ,·s s,·vVJp(est] (_s No+e! +IA./o- .fc,'/ecl q t~.vo - s; ded a_ bx,. WI pfe : Appcox;.,.. <11e Exo. WI p/e: < \ y~ "'- PQ; red +-test 1 a- 2. SIWtple n....,(f h;pot/.tes•'S rv1'i ~ Exq l-'l1 pIe ~ l~ A : rectf f'O, awd ~ f--x a ""'f/e +t)' (-tA: cr?.. / I o OVId Po\S56t'\ called 11 ~X:Qc saw.e t~BfER(p) h~pa-l-hc:-s·is­ Covtservatt've Q~~td why exctci ~ t'' {e.sis p()p~{c,+lovt QS vv,'fh CqV1 Lvef{. be oh}aiilrf:'J l33 u. 1rl +he at we w ovd For now.I r-tos a !:Su me deviv.ed best c i \ l~o: 8 =G'c. +l c q I r- eJ I 'o VI, J~ /A(~ i Gl, e ,) "' ff,e pf"ovide>5 No+~ +hq-t ch.br·c~ 0 1, 1 h.~ f f(~ i ed) s,·Y¥1f/e a. LernWl.x) -f'c.,. liV\ ( I< ) f~;e,) Wtost po~--ve ..-+4 ( iest K. o,·r. -fl11VI v._qnf -fo fest v~. H,A: G =<8, . t.v-e t.VV1 He po e-ve> t'u ( b-t possib /e . t TheoreWI 12.6.1 (NeJ""'cr"- PearsoVl The. o~ at..._i +~e kl'\oV t k e. ~jf(){~~sis Sitmf/J reJ,·OVt re j-ec'fioVl ko'-' do We ~~ve ~crve rtol.v Is +a 3e_ i se l~c -led cf Si'~e ~. ike des,'recl t:>(. De f: (I '2. . 6. I) Le { 1l(e) be i he po Vve (" C)-\' 0\ -J-e stl "'--'; t "' c,; i1 c I re-3 iOV\ C . C( A k-St w;-fh c.ri1icotl poYv-e r f l-1 ( i est o-f re.J,- 0 "' si-t_ e e><- C ~ ,·s t h~ VVLost ~~ I) nc*(eD) =<X .2-) TTc¥(9,) ~ 1l_(EJ,) -\:..- q,.,j .t~e, s~"' <t-j~et'o" ~J"•"''· \31-{ Let C1C~ be_ proot: '' 11 'ca ( r e; ioVls o t s- ; -t:e o( ~.-v~· t.h (.. * of c~ £~ \ ~e0._ ~ k(<><)}. t ~~ -fun-'1 -t~; e,) Ld A~C". T~e~ · P(X~A/e.) ~ \<("") P(~ ~ A Ie, ) . (t) Lei 8c ec. c*=- c TheVl PU E Ble") l t bj fhf -fu\WJ oT C * (tt) kC<><) P(fE Bl G1). (c~()c)u(c~nc~) = (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 Ie\) - P( ( (\ c ~· 61) c ( ·~ ~ p(c•nc/e,) -~ 1 "' L ( rrc,(e. l - P(cnec{Qo) lTJe.)) ~ ~o<l ( =- "') ;: o 0 \35 Exo~ylt ~\OW\ ~ The WlosT powe(\v \ +~si of Si-l~ l~o:e.::eo vs. HA: $.:-(8, ral'ldoW\ 5q Mf Je ~+ EX P(~ ). A e, ) (2,6./ Den've of o< S'v.VVIe eo . -x· r\ (~)eo/ e,) =- f(~ ~ 6cJ _ Tf J..eo e ~0 f~; e,} l e-x~/e. t n1<.. -~ ' ) )(' r eoC- .. J_ e~'~ e_ G, (L.x; )(_l_ ~ (~ ~ 0 l Qf\ e :.k~ $ LX~e e, - 1 ) e. J 1 I ~o \Je h,; I ( reject- ) f- LX: l_X~ f )a 'j e. • is A i .s s~a II ; i, e. 1 (Tio>s "vDu /d do"Je if $1 <Elo ,') ""' GA"'1 (e n l J 2. 'L x~ ~ ~2.(ZV1 8 \...-.... e So) 1\J of I a Ab~ j \~_)ect we ~trf-lce of Hv: eo:::- l li~~: Go= I Ho M vQ -fkoi V5 J z_ l~~ if e() > l-,:[2n). ca.lc 1..( Iafed !<&<). po~Pr\ul +est of si~e r ad" ct {{J the H,t: e,; e,; VVtosi z.. ,'s th~ Soto~~e el.S fc~ vs, HA: 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 lZ. 1. I Oef 11 A test lA VI j +or>"' J'J po vvectl.( I Ex q ~ pIe ( {2 .7. I ) F,·VId w1 t~ Co.-! unito\VVl!; prev,·o...,_r H e=3 "5, 0: CtVV\ pie: Repe-crf- As "'-"' k.,.v,.. I e {o, C( ~'J P.o the ~n 11 rc)l,ve r t 4 I ct ll €: ..52., WI 0 5 f t-he iv'l Ex posif" I + I ( Q I\ ross,'b/e pot-ver.f.., exaY"''{Jie {or ca /1 f>d 1'7-1 OS 't .s W/ost HernQ-h've.s) o. I f.es"t:· Hit : e / J ; Wo: e f:: 3 .-J; ih f .( 5 HA: e VS', / 3. tt.e exa,..,f/er crhave) q_ UMP fest coY\ Some-l-iMes- be C>hjqinF"cf b~ us;'~J +tte Ne 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 51"ev> iv> r~4)c..,lr.tr ol~tvv~1•ve v·crlt.~te. Exoh1p/e : T,j -fo rt>peo-f· t (,,., a h!Ay a I 1f'rvlo·hve, ba,., e >< a VI/I does a_ p /f> h..-i1 1U .M f c. fe s"t -f-wo- 5; deel 1'10 i e/C•·s-i ( De-fQ'Z.t.z) It dens~f) +~.~~'~c1io~ lh 0- s~-fp-/,·c: depe~d,IS' a. oto !! ho~ a rf'l.10tAo4·Df\e likelih();d raf,·o'' -t{K) it +C!; eo)/f(~;e,) =~(1}eo;6) o".'J novt- Jecreqs a.n d bot~ 9" Qv'ld 1 e, i:J thro~J~ ~) a~d i-t' .,\ ~UVIc4t'oV\ ~f ~) ltvheV\ Go/ e, -o.re foss;.b/(" val~es of' e .bctsed o"' ~. L3t Theorem (\1.1.1) t(~) J iVl H~r: V5, X,.._; ~)e) I\ (f) the~ q UMP ::>eo ;s fn ~~E"C"€ p(ifl) ~ l<(ot') Th~ hcts ct +es-t of r-e j eci Ito l6o) = o<. /Yloa.~ofo"'e 1;/ce/,"~o/Jc:l \q1;,. r;~e i tX o-f t(~) ?. t /-1,: ~~eo k(O<') 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., pr-evitJl.lS tt ---ci.d ExaMple (t Z. 7. :S) E'X p( I,?L) " · n=?L6 Ho • HA: ~ /ll. l e. t ll.. )Jl, • rr e )(~; Jl, ,Jlz) ::: : : \e n IT \-\o c::t s it C\ V'l-1 x,:n e 1-zx~ if X',;,_ Jlt ,·f o 111 oto n e is 1-ix~?ll,J -xl( 'f'Jl, it DN£ Xu., . (1..,-it.-.) 0 "71.., - LX; 1i x,,~ )7l,j "llz - ~X'w. 1-t x,:" >.?£~) - X.+7l, < X,;" '7l,.f e >~, -' .5 X,~,. £: ?Z~ ll,, • li ~e I i ~ oo d ratio /a 'j t:. -- e I ar'ld we ~~v\1\ -reje-c.1 Chapfe, IS Uf Lp'd;/ Ylo~..J~ we q/Wlos-t qf,_ctJs we kt1o"'-" the f""" i/J of e . J. nor-mq f, CfVtd we e s -1; er -I/J i h e 1~ q rcu:-n e -fc r (s ). f~q-f q t;,s(..(vned d ,· s -f r ;b<A i ,· o s-- j hove fcc (. ,~ s Pd Vt T~ +~--Je -CV/cf Jou mus i F/ t of GoodVIess hqve Y'VI 0 V\ /VfLE kYiow ot tt, e pQ v c=t CJf m f' ~r fer 1 f' XCti-14/) of f he dPtrt ~~·~; .fo f'W1 {(...(nc 1t'ovr, Iv'l -!his chqpfPr, ~-ve_ ~iII /eC~rVt +L"st wh€'-lher a ce,-Jqin mode> ( htJ~.vt +c -(/ 5 -fhe befc),e, otvrd LvE' wht'c. t, 15" d~io.. ExqiM pIF Yr,"'/ l~ ~ i~d 11.E"£(p) Wo: p =P' uA~ p ~ pa We h~ve he~ve fo <;ee~"'~ SeeVl re . f' c-+ J excrc-l ~V1 ClV! /{a 0 ,-f fr I '(O )I --/Fri /vt't C1 X- p, Jr·('~P·) /;::'t willq/e.,f ~) -ir· ·/eri I >t:,_~ z. /~ < (~~-~;--~~~-o((\) ....? x- - e(.) (- = ;;\n X- - Pc )z. e" )"t' X - V1 lJ'~P·)) - (nx -npe)z (hi-n~) 2 - r1 y - V1 e~ )M,~ -n·~·( 1- {'cJ) + po r't (n el1 - f V1 pG X) - (n - VI p< il2. n ()- po) - (c.bser-Ve d5 - C' xpec-1 t>d,} 2- + (obsecvPdi= e.x p'Pc .., e ds -I E· L ihCI-1 o fh~ bseu v P d "/he F'y fPC -f()c-f rh ~ -let m i f'qs; eyp'!?c .f (',.f~ (oz_- ez )<- . fh€' ~ e"'f~"< -1~"' \ 1..::::0 No-h: n p.(l-po) /j· o-f -fe>-t ~ si-L· -/ 'IC .· n u ,,.,.., hP r VI w ~-~1 -f . '~i'(' is be r . ~? p t.-,_ I' r ~ f' r i L-1, II he I q ry I? -ff> di f{e"ren i ) k ··1/r I~ c Je whf'vl t ~ 0{~ VI E r o, /,· :c f s Exqrv?p/e l3,'1,1 4 6-s;JPd We V\_. H~ p 0 HA : Ot... j i hcrT ov-/c omPS • co V1 +ex t , 'o' No-le fe Kf (;(Se) -fqc t I _) } ') t q I r, I -- 6 c:J;.f'ferent. q /1:: e r VIC! +- hC' Sc:;>vY'l F l.Jqr;ctj;)f's -1 / v e l,.. -.') ~c VIO ·fCI'ftth / Jhf,'r c lf'Cl r {' rc vn OVtd b~ .,f,cv !c/ L(<;c' . •;: Q." / c.L L r.v.s. q {"·(" VI 6 1 V) qre II'Yr?t!',S. Et (Oi -Et-) I_(oi- E~) ::: fJ,qt ps -r p6 p s ::: ::- bo-1 h r qt·?clovv1 f h P cJ ( t.fe r'f> s-u· P A, -\:h~ l 60 (0, - E, J< ~ ~ "- ( s ) tl1e fc.."" ei -/tv c t -1 e ") =I? 2 ~ pJ -~ p If L-"1 No-ff? to -1 a + /e c, rt i U" d", Ho : t VI Q roll~d ,s di(O' 0 o -1 in cl("' f e •" J e Yl t . becqL(Se [_':: I joi ::cf£; L.: ! ==6(). - t::l B~c_C<'-<5P de) rePs 7h,-s coVIstrct/n+, of -fre Pdo'll'\ lo 'f h P r c;{ Li-1ere -} £1 cr VI ore b. ov1) ) Here$ of +qb t~ ~ 2 Observed Exp~c {t.~~<•c:ler 5c -lee:( 8' I\ I{) ID Ho) +h~ 16 16 16 o-r' c~..t/cornf? -/r:~ ") o...,r (¥-Jo)z T (IJ-16)"2.. +- (5--/0)2. /6 IO 'o ExqiN!f/e a.. A cer- 'fed,, c I 3, 3. {12.-10)~ ~ (1- I 0) z ----- t /0 5j-uiffcqnce /eve( do rd 5o '-'-P o-f 0('=-0./0. '",;'"'< "f f ho- r o c~qn·s--1 ic.. 2 0 'Ya o-P R~ ~ e S'o J IS /() - {; < J!:;o (s) ='!, Z 'i qf s~ f/sf/c -t /6 +- (ls--tu)'t /{ results: --I!:> cl IS' 12 10 eKfr? ( ,~ 1 3' 5 Qytd observed (-/ c : / p 1=pz. = f3 We Je-f ~ i Z 0 Yo '$ f::R lir:: tdd o{ Rc,cp Z bt:·· '-lo 1 of -froM 50 f S (J f0W1 q v1c/ /00 \roW! ,,., q.,J 20% of Rcrce 3, = •2 sow, f/f!' prfC'5"f-'#?t Rctc e J<q c e I j Ret ( e 2 He ,e IS -Jq b/e o-f 0 ___ Prese111i RGc e I l<qc E' R.cue ~ ' I I to ) 35"' _ (zo- Ia )~ Vl -f $, 30 ( L-1 0 ) ts- ( 2. cD\...1 Vl 7c; ( ~O) c-r OlA ( L( 0 5-lc,·l:r·lt·c -lf!'"si r ) 1s (o•J - C:;_j Y e~J '-=' j=l pf>c -ied J lO ) oufcovne 3 E>x 25 ( -1 l-ie SoJ ( Ab se t ., 20 ( \0 ) I Z a bse rvecl t @o - 4 o) 2 +- (z r; .. z o) 2 lo IO "2- + (7 r; - sc~ )'? + (I s- - 1o) z- + lS 6 G (2:.;jlz 4 (; I0 11,/~ ::: If X~;;t~(3) So/ f!?e / -~~PV) p- vet!(.,( e ThP VlLt W1 be-\ of de ie cVY1 in ed j Vl i-s r(X o. uoa6 5., (Okaj) -+h~ crprrax. p- value) of {re e do VVl o.-\' COt-t p fe Jvvq is. de5rees a .2:11.1~):: o.oa{)b) CaV\ be(? One fu IS +tJ\ eqch are ,·"'~ odd -) h f> df-3 re p s o-lher o -p --l 1'v"' b~-1 no -JF' -f lr1et-f each ro L.V / s red(.( c P -)he '"' ;+ t~;') p. \.Je .,..0 C( ~C( me +he '/t,p /s 1..,..,;(? 56 h (0 i.J - Et I) 3 conS pre v .·o.,. s be/;e,.fd 1he~-l v...,;fh i 11 flo: f' 6 el-f c::t c. c. lei 5'J Yf'erlri'ls ey"'""'j/1~ .. ihe /?roferi1on cer-la/YJ clu:~rc,c-/ar/.rflc.. profor~ioV! +hf> C:xoc-f pt;sJukfed, I~ .~ .... -!i.e 1/vc:{S ot peopiP Q 1 ,_, ,· t 1h~se- ro• -;~prose ~ s r.v.s S~ctri 5VYI'? hy "3. fhc,-f s-/q-/;5ic t) ~e ~2(1) {ree doy¥1, fc 1's /V 3 So~/;?/t?s ie~-t ;J?z_ -r df 1 he E'XQmp/~ /5, 5 • 2 d E,.:j fhe it,P t' 11 d .P pe 11 cl f> vt -L o (Gs -E,J)' t::-1 Be"'cqusr.? clepe, c:levti/ Sv w1 /hF , J . -r i +h~t Ylo-f.<? r o c h. r ce c <? b '-1 f r1o+ k...os =P.z ~ P3 = p led C 0 I.-I"? t S C ~ YlWO -1 ~' cle1~r v~·?if/ .pcJ Vr, i ·I J-, ow~/ f,- r s- t t:.o-r 1/y~.r ff~"7 I+ ~0\ C. ct.,., p cqse u.s~ b~ \S f>~p~c / tt,f' ~ h(} 'vV"' p =- -thE> c~ooroc-f~, ;S"f;c , +o~ 1'1LE (u~clf'r t-1~). -f hct-f- -ft,e { pro por -1 io"' 1'1 L £ ( U"t<..ifr 1-1~) ohSe>rved io h01ve rV1 7t,;-s ~€t("' Cou excrl'vtr 1~ p~ .30. +~E' o-f I'S n+s : -Jq}Jft:= zo ( \ S" I ) (ss- ) 3o Roce Z I 2 RQcf' 31 Is- ( Is- ) T~e The (eKft'c -led) avtd Ahsev? t Present RQc ~ I observed t5 ( 1-(J ) 5'(5o) ~s- ( 3 s- ) fYetit"<J:'/;·c (s 3, Sr c1 i r f r ,~ bv lt'(tV1 of -f h e -1~ s i s ·It, -1 /r 'Ic IS X z.( z). r<o t.vS c -f 0- rapfroYim4..fe Ot-{"cC)mP J .Su~ ~€ 10/ fo l3~cq '-1 5e I -1 red'-fcif1J af if'w.f"I,.,J 15 ~r-aVtJ· il Hail\...- rneu1y yo~..~ C'C? 1<'1 e·Nt > vv p lj C• t.• (' -f q evr ries \\/€' Yo(.{ Yl ·/u ,, .froWI pq r ct VV1 e OnF cr/so 5CAvv1 '5u b-frc,d -lo fhir1k Cctl/f do you. en-lr;es 'Ia S nef:'d a1 r ~1 )-1. 2. 1f ~ o .,.. k ~-~ o -j ~ ~ .fQo e -f -/ h -<;; €' -?E \" Cl IN~ --ft:. r11e"q'1 J~··l '2:Pro. cJf, ~o -/ hr ·I t, ~ lA 6 -1-o S. fer-/ p1 b,d 7hrxt o..bo1.1-/ ,·-; -!his f/vqJ; {;I I 1.. , is , ~14 ¥"' No""" 0-.£ d-f Thf' es -lt'YV'laled Vv- e ihe caluVJ?s IJ -fhf" -iesi. o"'f VI,; Q -/- i "'~ {e·~t. -/qf:; k, Z C0 bE=>-for~ know lA f" e V1 I r 1' P s , 7 ~ f' n /14 n1 " ' ~ ~ r o "'-- _r No+e fhotf by o-e. CQ~ tivJJ +~~ p-vctfue: P(:;ke·•tz) ~~.51)-= 0.161"8 Exco-vr p/e 13. t. I A 5urve_; 15 In ji,Vf'YI -/G or cler po /,),'t:: t:1 ( q f {// /Cl-/I,GVI v._,· -1 t., Sfq t e p r~Jr~::~ I.-\ d e i F r v¥1 ,· V1 ~ assoc,.o.-IPd IS Yl-'1 s '"" epo r t-. ~ IO ( , 1 (/'1) Is ( 1lf ) YC I L ( lo. 5 ) 36 15 DeMo c ret+ r Vtdee@,df VJ t lo ( \'L ) 12( ' ) \ VVt 6 ( 10. C) ) If' C( to.c; ; 30 "35"' 3s- ~0 se ) l<ep 4 blie~,., ( Dec e 12 (IO,t)) Sot IV\~\eC<s~ ~o: No ct~.soc.,'q-/io¥"1, 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 p-vqlue = :: '1. 5t.{ P( ~~ ( '-1 ) ~'-{,§'f) :::o • 3TH Eyq~pl~ Lei 1~.7.,1 he XL I 0 VJ : AV15tve\; lio~ Cq n l--v<:"' T {' ~,..._.,e c rt?Of ·1 ~~F dP r ( (,m QL{r?s-1/ov,: Ar1s~rve r : /{ ) P>-~-5-l Q j. • , ( u ~.~ tcl qvtd Yes J s t h c t, "o "+ b Mq d ,'-~F r €- • -t iMeoctcmi), \11..-; II It-.. a+ be k-(0 Cq/c!.llc:tff, C<::rr7 cree~-le -lo Co'-1VI-fs c//s -lr, bu};6 r. J::: z. cr be c ttose~? n~·-lurct-1 plcu:e"5 /f 64-f P;rppc QVI ;se I ,I -' $ kP ,._ J //be I0 d~+iV1F l-he d~tel, /Vlva/;d VI -/he Cc~,u--rt c;. ! I vl J I\ r~~ cle·f'/,.1(( ;.,,) bJVI5 pos-s, b /<" s of' ''? y'VI lf' r t-o Is ~ 2 cc0 ,, j" 5 -l!'f} -Jc J e "l"' €> r.! 0 b,·Vls ~-v hj fi,E V1 13t::PoRE rn 0 ~ t c.! yov. Ot~E>I~.~~.-is~, 'jo""' p-vc:, L. . . e a v1C\ VVI~ct v1 i~ l e ~"":~, I or· k-• .i bl'.,-1 :r~ 5ur€ Q-5 ff·d +() Choose nt;{ h,/7s Y11cr "';/ r;;C( r ,, s I look 5 J fe c --led /,') b [o~ 1) VLz 1 3) [ I, 2) \) L5; '-! ) I 'S e,Jc A-1'\.s v-vP r ', tr.- / qre Q v..,,· :r e)( b;'V?s boi.Ancfc,rr'Ps, 1h<r f 5o r tAfS · '0 V'1 -~-~f'fP So'MPf/i"V!«'S i~C,;;r-c/ed, atr:> CCJj-?-t(;Wf';-tfs 1 No~.,....,- ~ hac.-t ld 'lhP -J~f' b,·....--1 ot/ld 'lo /lo <; -1 f' s-/oi;s~,-c 1es-l C( o-f Yt'YnPS -1-e st ob~fr veci qr, Poi(3) A. repf1lir The QU f' S i ,_'~d l for ~ t'Y¥?P5' perrt. oir·plqnP ~~= rr pct ,., ft,e /I ore, txc. ~pie r~~ 7., CC)Vl'-( 1'vlC-( f<efc:t ;r-"Time ( Oe~.J s) ed : C) (f I Obs€rved Pro bq h; l,"tJ .os- Fxpf>c{ ed 2 3 .1!./9 IV1 s· of 'vi-/// Comb;..., -E> CoW7 ~ b1~<? {?,·.., 3 5 (5,', 4 L.f 81,11 /"V1 /s .5 I ' e9 c llr? '::1 t, /n ~c q b/VI I ?:s- s- /_ z. t ~ ) ~ ~- e ~ 2 L e. = r. 2 2 .. ~=I "' p(;:;'( L.j ) >t- 0 • o 3 '-/ ---··-----~--. 6 . os2 h fJ_S ......~----- I, 3(" eyfto c f~d h1'y,, s~--e.r-s (~.Yfec --led) q+ S + J !k· ( IM~c Cf b,>-, qnd t.ve ~,·;; 2 s- p- VQ ( !<P IS b ,~ 0 6/ .Jlc t:-r o/ I ~~ ihof J<e ep /3,·VJ I (S,•VI z . 6 1 • Io I l.f. 0'1 , ' rY1 ofe 0 I +o or de r ~e s-- , 5,C({, The part o-f i he ~b/e t;.t:Jt..fVIIS , \ ;> "(, 22) : (), ()~ _s- ~ ExC(~M pIe /),=f.Z-! il-1€' F"r-GlW1 f.lo VP So th~f h ot.v ;-t-J .1- pr~spec:.fj Opii<.lY"I '"1 .,u- h,n s b/11s ~o/ If t=x pe c·} f'd Lf 4 •~ 4 VV!~ selt>c-f "'-'hcr-f Opt;,, 2 - e- ~. vse -lo be SE't"rv> 1'¥1 r:::,dvc11~, c re<=>S'cn<>tl~ 1 he: n ovr ..J. co llo-;s e neP.c/fifd, CI'S 2 3 1 I 6 6. rz 7 c: ho:s e P1 , blvzs a.-re qi~",;-1~M forA J&{',r?tr~ b·~sj a0 +~~ o..lyord~m Observed e Yc:r vvt f If' 1 1'11o d,fy no-1 q s.51..f 1-rlFd /e,.; Poi (P. ) I.J /1 ho vt -1 k n c; ~.v 1' / AV\5'-'-e(': violAS knowVJ, }-tc: Y~ ,J,VA QlA. e 5 i,'u VI ; pre I '-1 >s- 10 '3 -r, o<? /2,12 I 1 g--, ~--; 1 1 i:= /,62_ p-vq/~f T~P = P(;tY3) ~ /.6z) de;re@s ol freedoM ~ ()Jstt1 wo~ we Co VV1f/• fe ~ Sfec; Drc!/ b..,f es~,·mqfe ~:: 5., 65 J redLAci~ \Je lese i €5-J imq-fed, deJrer_o ot -\reeclcwV'l /d b@ if c a i hP {c:.~\ ,, se /f '-f we J-f ecrc l--. bj /Jo h01d fG OVIe ~or~. ~rQ vvtf''/~r C ~~ p+er Ylo~.-v_, UVI-1,'/ ~1)fP -lhf' m_qde- w~ v~ l;d Qre d ;5 ·lr:b..,~l,-oV! T 1-, ;s Is k 1'1 0,..., ~e ~~ctve yenerct/j o-·r d/s- ir~bv~on ,·,..,fer f'n<' PS I VI rei! ar t--vq._r abaw i /n fhe'f\ he~ ve Met r row e r V1.oV1- pqr~1-11etflc Als", VI'! ,· -floe Yl ce 0 1 ho-vve ve fj +~, + cl 1 "' e do ·/ f ,' ih~ no-1 /1 (., i1 J;)c<:"', pqrcrme-.ff,·c -Jttu~ -lt,P"' .JCI, crdvctv,c@ 1 OV7d !'f' r f' r~ c lerss cr VI d qy,d pet rq YY~~ fers . use+:! bf'cc,use ~.t~e o.P~e-, c:::r J vq lr'7 c e ~- ~ 01 -/ 1t1 r- d / ! fb ihQI-f crs.rwi/Y;t"d f.t.s -fh~ d,~s-Jt; b\A~~CJ~ fe V/d VVle-ft-?ods Kridt-...,J-7 mot k;YI_y a-t the d le$s hP CJVt..J5' }e;dk Vv;lf lv1 pq ro, me ir,·c. proc e ~ht res for- So¥'11 e If Y1ov1 ILl ore be i ir: r /~-:J,; no1~o~l) j')CJ ,_,_, P e Y11t:ihods havf:" po..._er ID-s"t iYJ mc.'!"'-/1--,ods -}e., i {-for q ca., fevver~A...-;cler w1/f /Y?J) Ho ~.-v e v e r in i e r va / s . co 111 .f; de n c vql;d r 1'~ kYI~"'--1-1 1 ctss'-'rnff/ovts se+ of cqs<:=>.;-, fJ? '7uik 5JI11q//, F ><9.., ple tv1ed 1""" 1v. coWl e £ . - j "'~ ;), <5 Some cfa,.W7 fhq-1-k!ler people eC!rn -1~1CtY\ Shorter }?eopl€. A Sctrnplf? 11.-1. Z, I is $2'f,>da VVIclf' o-r n-- 2() tcr/ler'\ l')eoflf> (b +i +) WG1s oblo/VJed qvtd "ih~,r anVJuc:~f t;cp/qr/es recotded(iYl f~ou-sa11cls) / 1 I() <0 2 '1. 6 23.!) 3o. 1 '10.3 55,5 so.t Is. 9 l <g I lcr,Lf 2ii, 6 25,() 2 5, t.f 21.1 I "3 3' 5' 52.3 k)e zt. 5 1 12 I 5L.f '1 1 WctVI _, I D ss-, '3 /-/t~ ; m :: 2 '+. 5 I< 1e s t f-/A: P(X; >2Lf,5 k / m" 2'1, :;) (Y) := / Z"f,!) k i_ zo s;OJ Li(l; / z'f.)) ~ BLN(zo, {:) Ho lAnder !=t UYidf r Wov f hf> 1-J A) lei fo b? /ct rJ e r? -fe,,._j p _v"' I<.( e ou-f c UYVJP of- := p(f3I IJ (zo/ 7hf' -/ esi 5 ~) > I Z. ) S ~ 1/s '/ 'c Cl .:: • 2 5 F{- Fx9~p/e /V1edict~ 1'-/,Z,/ i / 1 s he; f -1 e' tcr/ler'\ l')eoflf> 12 2..,. 6 2.~, I r I 1 Exa ""'~ ple I t..f. 3 . Z A s l <g I tcr. 4 25,() 2 5, t-f 21.1 I = L i,=).. 0 -r io I....., ·~ i1 -- 2 () '1h~,r 23.!) 3o,t I wo.-1-f X.~~ CCJVVfftJ rrr -JJ;,€>' Je VIOl.{$ q C.-I e..t-Q~f?le, P(F Uz:zo) ~ F(x.2,)) fi "'- ..:~d F Uz: 5ec ovtd of" de r 20 3o. 1 ..$2~)(}() VVIcif> pI~ zt. 5 \ - P(Ume ~ he\e C( h-1 1'5 oblo/VJed qvtd recotded(IYJ f/tiou-sa~'~cls) ~ (J'z: zo =. X,zs) L-Jhe~e eC!rn 9 We -fo\ people WG1s Is. 6 - , .. Y""ll''io'f 5 (C +i +) t;cp/qr/es I() <0 -k!ler I? f" 0 p IF, t1er Yl anVJuo f p fhq-1- cla/W7 Some £, ,·V!come 20 has (~ok2s)c (Js) lc- • 2 5" ) J /5 --/r,' bulan of of ~ Uf\JI F ( o, i) . fhe s}o ~is J,c • ~ i.. :: .1151- --lh~ $JJ~. = I - [IJSb' -/]- /Ji>b' 1- tSib• - u~th' -1] -I - [C"'":f; >z•KjJ -[~'~/Z:J!]cJ _/ < n ~e1'7 oz':Y)]d -( - [J 01 "t?'''X) ~ oz:o~ 7l>'X) lJ(~'X 7 o"'X jc1 :: ( o.:o')' 7 ( ;) o-z ? ·o-z ( \ sz•x) j 17 5 ,. (sZ'),(st_~ · ; 7 7 ; ( sz. ~ oz:'X ) d coS oz;·,n)d = o"Z v·::,r)J )d = \ <t•x < oz:o•x )d 'hi Jol nuls: Excu-.,ple 1'-L"i.l ~~ p/e (?f'o fo dC) C" Q f-' r s-urvey. [ere ·Ia /.,.., e5/,'J;c q ~ 5pPec II 5co~ed_ Ll Sc_o 3 :Sco,ed Of v'l f~E' r-ed him t...-v f' c.q I-f +-hp QV1 e 5 f!'o An~wt\ VI No vv : 1~ Plsj p - v I~ e C( t.~ 5 do uq :c D pe t-c p--vaf'-'tf' )~ 0 fhe-y, / h t..-v ~.V/t~ fes-t q s wf' ~,-v,'/1 Co VIc e f'-ff> r: -.J,·v e. ha11dle J = 5co-<e f,·e_s (X'~2=;~,)? 1-~f'V'Yl, 2_ 11Di >oJ Sf/~C t, befor-e). D,: ::>o -- I\ j_: I Su (s<o'('e qf'1ec-- p ( rsr N(I ~ l) 1 t I~ f t, P s-~ rv1e, 51)111 ct vvf? r.d ctftf>l~ =o ff'jec-/ ~A,...e fhof ·I /,f' do of HA: yVIechq"' 5"/'f'Pc h . . lo~ ~ r. /Aer I) do J;V7 }o.ve ·/ t1 F h,yhf'r p(Jf,·J,·c,·qy, n r;~ r 0 (1.')' '1 before sccH·e ~ ,J ct /.r'l d crf per:ScY1 t, Dt =X{z- JL, rnediqV) c.~ ll.): /C(Y? h _,·VV1 /her I e-t. J Tes-t) SF)rcft:d we'e C?. a.. F, ·~ s i Na w s.,VI ( Pa,'red o_ os9z J j ~here /1-fd'<?. N de : _:[f -1 i e s +hfVI ~,· s j r e p o H; b le ( e q 0 P(l); ) 0 jl~,) ~ tk J, e Y>ta) no·f Covt -1 lnttat.~ s f) ,·s Aa r ro 4 VI d;J e f'(o (' 5 lz . Of V1 ) ~.-v~ r-{' 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~) C~VI the S1jV1 +e.s"t be +f'vte(y "'-SeJ <:<s a_ ~-ert c.l l~ .. :~ed;.,"'• '<" rd P{D.: / o / H,) '*' j -ties:, be + d.,~-\=o tn'lot~ Cfcc~nrl~ : 1 . _c:. 1 • '/ _ 0 V1e w 1i~' -I- hi:' r~ired· SiJIII ie5'i ctsQ T e>!:t (i\ >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'( -1~1; 0 Vl ,·V1 'IJ1f" /?r~cec:/,7 .I-f ·Les ? ,e 'L/ pc'S,Pte h\..(+ d ls-y'-(crr J d eq) _, + C'XctMplc::- (a.sSI.(fYilVI_Jfie_qre t ~ (17 impo~sihle) t ~1 f' p<l~; ~,\~e. <:.-·r n_p P'f' rejec.-f/:; d : VI ;I:+ ,·s -r · / /-, -f e f .f /r1J VV} ~ Q f' () T~e~f ~ f<;'l,.{ f t's i t'l ~- () .' ~ p r. lof:J er ~lui j s~c on~ ( ~ Exa...,pl~ r ~ f'~ t i ilctV1 z. 3 5E"COiff LJ 2_ ~ 3 5, 6 $ 1 fo pos_,.)· h}_y -Jhf' n1etJ,·~"~ · hF WJe rhetrl -} I~ ....... VV1E'cu1S .J 15o?0 'rov ps h.c::tve.. I'VIf'cliq.., of ~E>ro bu1 Vl-te d;Q..., of d;fter(>ntto_s 0 z the Vlu// d ; ~rif' r (' VI C f' rv1 CoVIcJwclr:' one ExCfrnple..: + . is i51nq+ I'VlcDtrec+':J c-t ,· d no+ ~ero B() t" ~tov (.)s. h.ct ... ~ diff Y'vifd,crV1 of J;.ff hv.1 15 med;ql'\ ..:e.e re>. Se~VI/1 Paired- Lei S 1'5 t1 f pie d- +es i: vtl<. rq dc::::.x~-j:. RaVIk. \cL\ ~ .1_ to<" Srrtq//€st' Z -\Ci, Ce~ll fh;s roVlk(ldd)~ ) 0? i~c:l - 13£ R. ( J..2 \)' '·" 5 ;,-e, "~j se~ ... e~,e .. .; + "'"'<L - ii D.. As5l.fltl1€· Lei T== Z f~p r<>n ks o-1' .If f here ct , 1-fo C{_J cue ~e.q. F (X,'j)::: F ('JJX) ) •. . J 0y ~y . e't..-.,l~y /,)c.e'J· 1[d, >d] '"'"k«dt/) o-f' 3 per irs I 5" -3 ~t: Te +esi dA+~re.,c<' rned;q.,. l-IA: rnedlc:,VJ p-vq{l.{e - r( T r;..._::. I f'.CI'IIJ 5r jVl S ... \ 2 3 t + -t- + + - + ---+ . ,,.,-·- ---- + + - - ?. I 3 - i-t + 5"- t- - I ' ~1 >o l iiD>:- )oJ j_;d ~ '-1 dd. -)- I r'o. n k ' I +- ~ - 6 z .3- o- - si~"' 2. -3 s'J"""'e-I'J} 8~1<.) -- :::,. - /t_, "! af l,·je.~/j (t.t~t~c/er Ho) ~,·efcl. ~ I o.f T b - (<>~d d,·f\E>cevr<:(" Sl'"'~t.f> +- - "0 -f h e s '-~ ..,_, d/f[e rew c~.S:. )< ·~ ; ; . e be pos,'-/,t,~ fhP SVJ;f~l/&si> efc. 111e-x7 ft,(f/' g" etwr,/~ e;y-/tCJVl-?f"S' I ()\ »~ore. I~ Y\ lq '1 e_ , . How-ever, 111 i-f' vzs, ve b¥? <:qV1 /?- vc,/.., e the \s /a ry f' - be wi // s ~~~··V?f /e C o.-npv-/q-/,g,.,,J Cfff roy, ~q /,o"" 5 dOVlf"~ T- f(T) d J N(o/ 1) 4 Vor(T) E (T) Ho ( =E [ ~ i-" <J i. u~ j n "' ~ ~ ~i ~T) 1-Jo V <:> r == ( h ( tt +I )(2 ., . ;. J._ , '1 ::: ~ ~L t ke C L. T VI <- ) J z. 2. f, i.. U ,;"\. l3Er<.(i_ n(V~t/) _L4 - ... .::, v Cj r U~-· where v o.. < ( U. ~) = ~ ~i2. i) 6 Q~~t P5~(CJv1~ If ~.~ve fest) t1 ee d '-'t,J nu-l J·~Ast e-( ~ , ct(Jf r ox q pctt'red -fu j LtSe +-i-e si ~ , Ah5w~\ • A/fhoJ~ "'-'~ ~J;/1 +e si less /s ~"'-es-/iovt: W~af a.~ f~Epcti red AVI5~A..-e(: _j_t 0 pros sijnecl rem k is ye1 fo 5c:>ns/ft've Ji:-. coV\S +e 5t 1'S of 'Jlf I o )'/wo ·h t1 (J i h,·5 >'I, o{,fflre-rs. i~f' r~ ai fPd Stjn +est vs-, ~ SjVVI~Mei{!c. Qboui \'<l--vf:'f'~vf) {)c;v.lt1-5 1d~ CIV! i-l-_s VVIedra"'~ 1 5i<J..,_ecf rctvtl< tJ.tctf- p-~q{wes d~pevtd OVl tl.,i..s 1'~ 1'!4cte extrct QS_sq_,l'f,OvJ 5(\ dtte j \q If r- SWZt!l/( vq/14e Co vt!c/ s~'YI"'p/'j be avr ctsymetr/c d,·sfr,'b!.(l,-~111 a-P D t h 'f v1 ~ / c c q 1/o V? S h; { i , £t +o th-e\ U.Si\1\j \'C(VIk..s expec.f +1-tctf LAsinJ rY\\'J~t 3''ve ctn T: Z IJ()I/vfJI' c::rcl~ctl +ttf' 'eveY? shoi..(IJ 01;1P cl/-fl'~rence..r pv~;vPr·{v~( VVZcre test. iid.~)oJdt IV\de-ed t~;5 11: ,·V)1f(ov~ Cqv1 r'Y\E'dr.,.., ct~n does -9''ve dif{eie~ce =(; Q~esoli'VIe +e.si o{ (o.d s~ ""'"""~r1) ~A: med;ct"? d,·{-ferevzce < () It ,·~ qfso a/ f.er VIQ~,ye ExctW~ple~ ~xce!IP"?i o, If 1 V1.,..- o ct 0 bserved --2-o 1 -to./ The VI There q,e +~~ d;1~~t"evt ces. Some Q L'ke I)_ -for ~I s,'Jed q - 1- I 5 I OV?e--s,Jf:'J I ferna1tve. cL"ifer e-1c er z<~=zs{ (),.J~~ oih-er fhe are -lj I poss; b/e. s ij"' 1-1 Q) Ovt ~ 21 ~S5iJii!VV1'E'"'~.J .Se~ ue.,.,c("> is ftJ e$ vq/ly •-€ T -Le; -lo -2-0 -(0 -2-o -10 -~ -g -~ -7-~ -t- -~ -s- _, -')... -l - Lf +2. -\ -t.t -2... +I -~ 0 z. I \ a.7 is v' u~j We /eqst .:: \ z p-vqf~Ae -t' 4 +o \A {VI The \J;/c_oxoV1 Q let VI This tesi IS e~lA_q/;) o-r /c,_ o Cl : b~+ <1V1d \~(,: X j L<..- h 'c ~ 1 (T::.f). v VI- Vl +~ fe s k- 't ,__,.It, S 1.1 n-1 +e s t: q/iy,, technic Jf ,· c /':)q ( q Y"?f q q VI lo J ,·nJFfe""c:l~n -t- Mq•n-\J hi~••j (o I U.) -/~f' of ff'st ct J /s1r,b\A-J Jov1s F. = F; i-t 1.5 5e is lly at L1 5 (.,{ ct e~~qli0 ExqWlple a_ _ 0. OLf 1- -2-5'& q J,l 5 IJ VI Q5 f ~ 6- Sct vYip/e Sq VV?f le s: 0 ~ )}J 'r'1 WI e 11 YJ -tC \ ex-lr-ei/Y?e qs observe~ 0~ pn~po\~J'o"' P"vaiiAe::: Y1 s,· /,·v c. CJ vts io Io c --e._ ,·de r-e J C( ~ ,· o V) +est .fo\ Cl sh;{-ls fhe me cLans. 1~,1-.l Fx;Fy ~l)+e Sct~-'V!fles are 1- I '2 3 't l6 55" ~Zo -a ~ ,, 1 l 20 I I '1 'l 6z I ~~~ 2ZO._lt5~ t..j1_ 1.3 Q!) ~ 6s ,·r-1df?pe11devtt- no/ fa/r€d l.f l z 2. 2'--/( 111t ' @J) 5@)~ f- I II 1182 IS' \ (16. I 2 iJd .~ 5 3s- 2.1 J Poe..! Q II 6 let \Jx = of fl-w SL1 W1 ~A: mx < 111 y 1 wx I~" ,· .s o{ IS l--lo '"") +~,e \C?11 i hcri io I+ e~ Si~e o{ +a I +6 t he the 0 hse r v q-f,·o VI t'\ n5 X. ov1 t "'p -/o '} e-1 be $ h0\-V VI C(J-7 -i h T 0! ThE' of l\ 0 this Is +est MCI~f\- The i5 a i h s~crLd 1c bu-1 s..Jq-/;s-/,·c .S o 'F IN h;-f.., f' _j 0 5 X) obsNved {lo"" p- va f~ e vo!ttf>S I 1 f)oss:b!e Vct I ~.-,e o bse f vfd +cr ~ t~ s ~CJ' c,: ~i(" ct I So5t-IVvCI re o{ L¥1;: nx~~J pro po ,-} i ~E' (V\uVI1ber of b~-1 l 2 L-1 . ·~ ·' I ' f' p~ v~ I~vt e . e'KocT I - n)<.(n'l-+1) ----.: efuivCJifvrl U.-5ioJ,s-l,·c.. ')"~"' f)~c/}J 1Ca 10 V\ O' v'VIct be -1-!-e r / II := \ \.A. X V\J X 3 1 v~s Cl•" 5 Wx is 2 d;1~erewL s- ,·-r ll f't'j-ec-f f' -\roO\ th(!j+ SWJ.cr/f~\ CCIV\ 'vi.,....- 1_1,2, ~~ .•• , v(O/ a. "f' \..,{ 5-e" of ks 12/ S~Abse-f /,·ke0 Cl. \~ There the wx;: k..s e~ v~ll'j +~f\E"+o((" lo VI 5wtqf/. 1his example Uvtde\ ~J r~ a X. f,om I-\' X t ., bse cv 0 i,oV? .~ f. . .Jf'si. C\ Ux Naif i hqf n.x I L U, :: ~;I F,o n!f i h,· s /)., 1k.,t o. be a: s e,{pcessE'd j_iX; >~j j~ I M w;i h CoV' -far rv1 ; I +c q qs'jYVIf-/,-1/cq/ly is 5'j Wlpfc i,'c VV1Eq V1 n)( n 6e S nor""al ho ~ n 1-l,, lAMder Yl_j 2.. Q nd G sj ·h1 p+o -1/ c V Cl f i q rt c e Y\c v'J (VIx + V1J + I) )2. This 51~es 3'ws qre ern appra>i~q-lf' lary-e. +e<;t cvhe.., Sq'""fl~ SLfppo5e iv1dfpt'11 I-1 deVJ t fqJio n is are (C\ff'(q1f>c/ L{se fke •) -Jq -1/s- 1/c. if' 5 t ov r X ovtd Y f05;Jt'veiJ -1-u r·easoV?q}/e, as ~t: fest 1-+A: X V5. S e' e Vl--1 s Co ''e J(., WQY?-/ "'-'e Vvi~Vt Sc;.mpl~ \~~ Jvq r(x) (t Vqr{Y) x, ~t\ L-1 /\ r ::- - nXj ) ~~xJ ~t~z)-hjz] We (.. .( se CQV1 fl-le> ctS rqndd w;/~q l/oV? a -fest 1/Ld 'f h s -/-c, l/s It' c.. fe:s-t There o..-e n] foss;ble pcrir>";s ~-(' H., v1 )f'S -}hf'' n 'jts, Yov CqV1 think crbo'-1f 1h,·s- Cfs- ;,·n/~rJ "'- ".s u e ;'"' (\ Chokes o t'ir''alct' -h,, x, , 0' d p r : X, I X2 I Xt I (" lc • I he V1 I 1- h f \ f' ~f;)\ t-v~lc ~ j iD eai\ ~;-it-? X,. NC?v!YI) (o, )(, 1 i~frf' are (ri-1) chotces .fo, Q e-1 c.. r""' n; s '" Tiw be,,;, pri"c ; plr ,1-\ eoc{"'-/;,J. 1--v ,' -1 ~ fhe q I e chosf'y; i"'v1o-fc~ r. F'o por -lio The of V1 A. ll'\ r a.V\ Q.5 ""\ ohs~rve d }a -fhese 'J E' lhis I5 ql/l exac-f Th;s is q -fesi ,._, r-eet lly J's ct/-1-erVtq-l,v-e X ctv~d if of V f. indeftY~clerlce; power-fu I ovd j Ylovt-~ero o-f -vq/IA the -fest-. 1e5"t v.-ol( /cl Tl11s p rest.( It fhaV) lo 'Je r o\ 1he IS 1A(J+ raj\iYLJ S were have bvf 4J q/J--?S t QV] Co('re/d--f,·oVJ. -}f' r r; b /~ depev?deH r per-far ynQHc ~ VLt+ wn corr-e/a-fecJ, If- (~ YJ ,.s Di 'v a. r; q }e Ylo~ Ma I) -fhF Y7 ~~ A/ A'-~ lhi3 afsc evev1 AI'\ t(VI-Z) pro\.ides wi·f ~ modt: ret-If' "lvl q/o__5dV s ~;; ~·,) f" f'o ~· .-:: o CtVI h., t )peq rmqn~ V1- '5 , eX c .sae-~-r/P co v1 cppro)(/VV~<'I f f' r)!Pt1-t bf c. c f r elet/ ov1 .5 i -e'e c;- rr·\ J/'' r 1 f', "! ('., \ J· f., st I 5- Dd VVl!;.". l ra f~fr ··t haVI • 5pearmqvt5 r-rrlaci.-.J a"~ d AVI~~-ve r rctv? w; t h t~e f-ly a has Q 1~e {rom I-f p 01' X, fS q ,·s VJ d q fHOYlo he .1.. Ce; rre (of/cv? fW?ap1 -lonicq/ly fhci1 fest /sa Inc rt?Q sin/ (c:lt?c re"s'"J) rqn k s ttr e h"" caVI \ \ ~f the Cetv1 +esi: J./a: I-lA; ( q x~/ 2) j bovQ ,10 t Is OV) /lee -led I,S Y~ donE' as ov r·Qn 0 1 e to efc; ~ r- IPs t 1/VV?e tloiiV1 5"o M~~~ CoYls-,·Jer -the ~ • +esis bQseJ +r-e1-1d, i he f"-(? co -fhe0 I 1/o ~;v s q ,·-r ,·-j po pu lalioVl. oc co/lee-fed ~C·~st A'VI.j V5, be--lt.... . e t=>,r' 'Peq r· so vt -:s co r-re /Qf/a v1, Some (x \ I 5pea So set of dct-Ja +;e ys. P Co rff' /qfed A shy ~t qJa plrr/lD"' if nJ k.s~ C/PfroXJ'.mcde "'1>'1 -} h corre lo~ioVI -1he \tvhe~-1 ~ is tLA~ttc-f-t'oVJ of X per+'ec tor ~ QWl e t hf=' ~ Vvh€1'7 ~,v/// (~, -l). : 1\Jde; yy c. c VV1 p 1.1 -le -fhe QrAtos-lioVJ fhe b'j dojVl_Y Ti-?e¥~ compufed b'j w/ih fhf',·r rqnks ~/, ..., Is Corre{q-l;olfl /P 0... OV1 a Cc'\re(qfi'6V1 C.oet.f,·ciet1t x~ ""'~.:,.J +r-end (LA p~.va \ d 0' d-1 ~a r cl) dC)lA.. Arwi her neo t test vJq/J-Wolto~~vit-e- ruV1 o-f 1 preseYJied w,·j"" C<. 5eJKence a VI cl 1/\/CJ w i -lc o Ll f c o Vl1 e of a S e ~ I.A.. -en c e Perhops 1.-v'C?ni fc Y'V1-C/Kf' 1 +os5es. ~7ou. 1 friewd COlVJ 5vr~ pet ,-i ,·c VI Ia 1 The -}f'$--i II of r= o ~ -\ r ; yov- ,·-:; ~ Xct tNJ f le J )QfJ'E' J q no-f ~~ed s -1-c, f,-> t/c ,, of rL1V15 e v1 d yo(.,( c h~ctf,-7 et -f q nu VV7 j'Hr7e. Q :; ld~-E' - IS e le rn "Pv? H-HTH-1-JHH If 1hE' nc..-tr'Y] be, o' Cit f-es L, you ctre ~ cJs . g J;,is f f r'__s t h E' IrnaJ''vte of k VlO w ca./ led ,- S WL~ ou b o v_-/ D-r c r GfYIS need heq f, ~j, 15 -1-o the z' . be r ---. ) hos thre~ ~ '(;1:_'/c /ot-t COIIl -frcVl t s r )t you.r f..,(n_s. 5 VVlcr II C h pfe~ \5 - ~e J re s Sian Q This chapferbe+~~v~eV\ Th~ I-f )c Ie vfh 5 the relq-/-/oV15h,p Mcdef,·vtj Vqfiqb/es +e r C 0\1\1 q6oll1f \c:, L ;.,., e ct r M o cl e f ~ o Ill d II M r eS t Fr· Cl V1 c j s G I he 'J his \E-5re s sio111\\ ro VVl t h E:' 0 + Q +Of 7he r.s q,., lAse . o 1 rf'YJ l ~ !!J_ I~ 0 V"1 I d r _/ ' 1s /rw.l c_ f' H Vl Lf f rl- '4 J; e d h6 cl . s 0 f J -t~e s, • SOV\S he~~t • -- _3---i -~~-~---..-~·--·~· 0:. fctthe("5 ht>,-Jh-t +~at +q f{e r ""~ '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 no f, c e d. Frq VIc \ s +eVldtlllCJ fV\ this fu CCtse ~.< s e d a. s M {.de li vt_s • a. \e+!..-lffl1 -\-o ct tIt C( vr o prevlo'-l5 fhf' VV\eaYI, R.ejre~SiOVl J e ~ e 'i q I +e r VV1 fo ' -t h; s I. .S skfeJ YIOIA./ -f ~ pe o-r A SiWip/e mode ( /iv1eC{\ -+ ~P cf IS fur Vl-1 \L::: (30 t- 0• x.~ t- £.r Xt Lo\.1/J Y~ <:cu £~ is the d: 1f'preSf>V7{s ld Co ~ ld d OVl a_ { s c he,jl-tt . dev,·o1J'oV1 iroVVl d,'-f-f'~re.-tce tJ.te fr"" e -l??e c o VI s , .d P I V1 ;: : rJo tf3, Xt -t t, ,·s 1 f- /Jz.Xi I/ heiJht VVICld€ rvtoi~er5 -t- 2 (->3 Xi3 t.Je T h;_s ,, s s '2. +; II the cce-\-\tc leVI +- VV1 Wleqn) cb..serv-ed li...,e, /.,vl'i h c de ( VV'lo re + £t ~1~er5 have jrav1cl{c(/hfr:S /""c /4de -the. he,jht1 he!Jh-t. s'/;ut:f\e c-f +he" pre d/~-1or : £. L 0. ts. a. (" +~e be.\;'-'~Vl r-eJ"@SSJ'ovt rn'J 'l--ri Ot1d 4hoiher- Wlocle/ YVIijhi ~ct1hf\~ h-ejh-L Cf s q Y: ;; (3. t ft. X;. CIV?d vctrfo.ble.s)~ pr-e dtc+c,5 (,·r? df"pe11dent Y~ he,yht S'oVI'S rq111dom po,·v? f.s \J e +~e be f.c,f~er~ +he 1epresev1t )iV]e'Q\ I(V'l~d€ \; +hE> /r't?e~d ~ t'5 ,·., L-e ~ 5 '~ f-u {' i ~ :: ~tj T hi 5 \M -t v> (3, X~ p/i 1:' :5 s ,· VVl f(:) pfe ' I VI e a e:; <~e s .s ,·v"' E.L ~ L.~d N(O; + £~ YL t he/~ c.q·e iv1de pe111deV1 NC)+e t~ctf E(f~)-=-0. !~Is ,·s hC~-ve {?;c. i [j, X~ ~c.) des<-f;be i c D u tc::i Cl 1fo v-v E (z.~) 1o ~e CVI (J "e t-ff>c -f- 1 hQ f t.vo v / d \,Je l r , cr- 2· ) t. \A..)~ becquse h~ bf"' {Vlf'q VIO V'/- hove ,·r VJ. ~-f'f() / bv-f· ~ sh;{+ f3~. As5u.., in) +(o.,-j t h~ Vlex-1 s-fep (?;0 Cll1d (j,. i I s a he b o-ve -e -s-f/,.,.~-~ o -If> io I he rvto de i •·..s ra co-r-rec.-1,~ r C(l/}1 ~ -fp r 5~ VV1 e -} h" d o-f Jeq s i Srfi" r e <;, 0¥\P vY>e-/ hc.cl ,s -1 1-, We ckocs~ f'o, f, ,·V\ de' ~u VV']tVli.Vl'?l~f' +h.t> 5uWJ i, = :1~- j; o j~ -(~. +~,x,). o-t He S(; ua. <eel r-e .,,J" q (> A, "" C)\ q ( 5) j (-;o t p, )( A. X "' ~'1+ 0 == 'Z C) -= :xz 'Z 'gJ = '2. 5\.4 - _h_!VI - -z ~X? ~r: ~X? (? ~ :h:x? (_::;. 'U- 'F)~! v. + 5X \A- :'h ~X/ ~x 7 D " - 'y- (1' 'y -F) 3u - s.. = 'x [~x·sJ- •g)- 7~] ~·~ z- ~ 17)_ 8 =- 0~ {_~ () ~ ~ " V '<(} - of} U v - ~ 1..4 .n-t ~ IA.A 0 t Y'\ QfJ e ""2 Z- :: $=t h ~3 ? lN1 9 ~'~5 Jd~tQ\AV V\O!tda OIV) .J V' }0 d~+ Jf"lo:;~b p\ cH1_ ( =:. v 0 ::: [ ~x•y- q~- ~h] • rar,;~l\ j-9 d 2; / l--tA The 5 (A of VY) deVloit>d 5V'v_ared Noie ,.._ 'l... a- - SSE VI ~ l'S \,ftll su oi'-1-eVI 'S 5~ <Y.f IN1 -fo\ b,'C(sed res) (.,(Q o-Z.. -2. (Le-t ~ 1 ('1~ be the lheo"f' VV1 15. 3. l E(~,) ::~I ') ~ lfof" sst as Is s-; d t-1 o , f ~I - - \ cl ~ ) £-~X'~ x ) C{ I"' - -/ " ) - t: V~, prou-F; e s-f,-,.,. {,,5 /es-l,'rnaie l..-5 = I((~ -J }((30 -tr,x~) Z..(x~ -x) 1. L~~ -xrz- - z~.(x,-l)+l~.(x.:'-Xx~) 5) =_9 -rLf3.(r!•X'----Zx,x) II "' + 0 = f-., L{x,·-x-) z) E{f3:)= (1 -; )z_ IX!~ - /i, 2 rro.1': 6 F{~.) =- E ( y -~X) ;:_ E- ( ~ Z@A. X; + ~;) :: f3o s) Y) E(c,r, i c,~ C 1 "' {5o ~0 \ ) =c,~, "'\ -l C 0 z (J, c, ~() i c 2. IS r, c. t thp t-{?1 X A ,, gcyf._ .5 t'Y\q 11~5 t 0 -{3,,7 .:: (3, i J /Jo r:1 c,(Jo -+Ct. p, Vfvt b1'qsed for c,f3o "" 1'\ "5(',:) uVI 1/t?t'qc be>t /f t - i~t~ the 5t'~-J.s~ Vcn 1q f1C €, I j tc~ (3,. bl.:~5eJ e~J-,>~~{r (BLLI£) f haf if ha_s -fltlf:'