A_ b 1·\.! I _[' ? ·· .. _ _ ... ----f+ -f_(_Q_Qa.J:>~ 5 p a C E' _ ~o_Vl5.is_-LS____s:L.L ..2 . pet L js_ ___ _

- -~ ---------~----~-------------------------

___ _ _ ~ T~e 5cr VV~i I e __ .5f"' c. er <-v h ic h _is___"'- sei ____ _ _ _ Co<'~-/"'in_iJ __ gjl_~ .h le .QJL-( CA>ti~ ····----------------------

=--~~-~=~=~=-=E-) x···· c:-tTe~-JiQn __ Qf __ L_e~~q~ ~,_c£ _____ ~~-:--~---f ( r-.., ________________________________________________ ltvlu.c._h _____ _l~_s___ --B- _ ___ .5_S:i ___ Q_$.S:'- L ___ czt ___ ~~----- ____ _ _ __ ____ _ ____ _

-~· =~-~~~--~=~· . ~~! ~£=~::~"'\:n~.-~=l_~t~-~~~~,~~in;__fLo.1 q 1', ~~~~~-~ =

_____________________ Ih e c o ll e c ·Li_n_Vl_ ____ s::/£ ____ :e__v..el1is.~,---1=-7 ___ 5Q-fis__fies ___ i6__gs:e ____ _____ _

________________ pro peilies_~------------- __________ -- --- ------ ------------ -- -- --- ------------------------------------------~-,------------------------------- -------------------.------- ....... a:>---------------·--····---·····----··---------- -- ---··

_____ --------------------· ________ l.L r+ Lt__c __ E __ ~ ____ the_ij ______ y _&_~------------------------./ '--=1

······-------· ----------·-------··;]_------------------ -- ---···---- --·-····· .. ·····--- ---···-

z If It E F _c ~ F - -----·-- ------------~----------- ---------- ---------7---thf:_!(} ___ il ________________ _

. --~- -- ·- -- ~~--~~- ---~-}]-~_ ):~~-~~~~~-----~-

L ' . __________________ . ___ V} __ ~ ec ____ Vv-a£_d_.s__, _____ E __ )~,5..--~--s- lc;--0 a_o_ ___ gj!J eo r C<.~------

_E;_x?~nfJE_; ____ i_S_b_Q_l-,.L _______ T_fJC(~I ________ L_-L ____ l-tj_f5 ___ ~--E=)- ~:..-~.-:, __ ~~-----_A_(L[l_ E.._E__ ___________ _

-------------- -------~----------~--·-··--------------------·-···-------- - .. ··--- -------- ''' ----------·-------·----·- ··-- ----- --~------·----~

---·-----···------------·--------- ·------- --------------·--------------------------------------- -~- -··- --------·----~----- - -- --~ --- --·-- ----------------

Skip the review on pages 1-7. If a review were desired, a better one could be found elsewhere.

2..

• j

,llhe {uV\ciiov./ /?1 ;. r

5 crt I 5"1 I (! 5 -these prof~'~ t'~s: I I

••

.. \) rr/ S2. \ = 1 , . . 1/-:z~-------} ·- -• t

. ! 2) ~f., ~~= J! 6i~jo•" i) t£,e1-1 r(~;4.) i"• IP(A.) ·

4 ~

s_)o ~ tP(A) L 1. +a, q 11 A6 r=.

'I

Fxcu·1rle: .1 50o0/ -(-Ju:{f fP(¢) :::=O. ~ t

.; A rctv\dovvt 'JO.\ia~\e t'-5 a -c~VIc-lian LA.)~ os-e_

. ; ck~'Vtet_r'n t's tf-t~ 5.ctVV1.f 1 e ::pete e af\ cL [A./?tos e

.i.;coda~~~~ ;'s /R: f;.~~~ . • i

i IV\ cw cl e ' ~-o b e :· [vJ t':SL I ){(w)" t j

fo(" ~~I t E /R.

c:::.t r QV\ clov¥1

Mvt..s-t b~ Cl.v'\

'/ q f ,· c{ b r e-' e (e VVI e t;') t o {' t=

i 'I

·!The ~~Ph<.((q/,ve dis-ir:be<-iioV> L -fu""{,·oV\1 F(t)) .l is dE>-t' Lt~t>a as F(LJ=- !P(X--t) 'i i ' e . C{ s F (t ) - IP ( ~ cvt':_ S'L I )(("-') ;_ /:; n . ••

I t·t-·

. ' ••

-i·'

7 .__,

__ ____ A c. d. -f. f(tL; ' ('~

5cr i 15 -1~_ ----------·-···----~·--·--- -----·-------

~-=~--~i]~~mi-;::;·--_-F(~ )-~-o·----~;d·--1,·~~-F~) ·~··1· :; . -~-~~.-~~-=--t:. ~ -co t; ~ a:>

. :~ ........ ?:LS;i =~=- F&i.~Eft~1;.-~... .... :.-- --·---------------------------

····-~-~~·-~-~~--··-·~·--·· ·~~--~~---~·~~] ___ F(~) -------;~----~-iJ~_f~~~~o ~ __ f __ ;~;- t~-~~~~:__:_~_,_--~~·-~-·-·--~·-~-·---------·--~-~-~--~~-·---______ ..;__ ____________ /_JVvj ______ E(iJ ____ :_ ___ f03:)_ _ ____ fcLc ___ s__(_L _ _3__6 /fl. ____________________________________ _ -- --------- -- - t: ~a_+

----------- ------- ------- --------~---------- ·------------------------ ------------------ --------~----···----·-------

' ~ ~ _ _________ _ ____________________ an_d:___JLVVl. ____ f-(t'} __ __ !2J.5{s ___ qt1o_. ~s_ ___ :b_/1 '_l_·e ______ \L:3:_S:-/C_ __ .

-~:::.~~-

-- -------------- -~ It_ Cq.-. __ 0_ ___ Oc_G_g___b_e__ _;; hQ~j/}__ - J_b~~ _______ l __ f ___ ~ __ {L!_'2s=.ifq~J--f;--- - -_______ Sg --f is fie 5 J+Z,_ __ <;LYI d ___ 3__ ________ :::0 __ Q_(Jv_~-J-------j_hc<: __ t3_ _____ i~ _t:e _ _ ____________ _

... .... ... . E'Y tsfs g_.p.o b~ b; /,~;·;,-.p~ .(-R,.E;-.1£) _ ..... ___ _ __ __ _ _ ___ ,_::L~-~-----__c~~-d q_~----- v a. , __ i<:i_~l_ f:____ _ X Qfd__ _j ___ h_g-j_ s_pg__-:.__;:~---- ____ ____ __ __

~ td_<:._h _t hq t ______ f __ L~ _____ i-_0_g__ cd-{' <) -t ){_..______________ _ ________ _

-- -----; . . -f' ! --J_ ·! 1 ___ '_Lbe [_§___ ____ ~ \~-~:_ ____ L-~ __ (..':"1 ~-~~ V\_ _____ ~-j-F-f>.s___ _ _~..LI_ d t's tn o v _ · t' OV1_s:-___ ______________________ _

_ ___ _ __ j_l qj ___ ~-~----~;__l_L_.s)j_5 c__u _:;_~~ _____ . _______ _

.~-E· ;:~~·==i s~~.di·~~~j~·········iL. Ix§~~T-~iL~~..siL~.~·~;}J;~l .. _____ _ _ __ --~--~_{j~_de__ ____ QC_ ---~-!":__i_sLb~----iJ1-j i V2_~\i__~ _ _gnJ___.Z_(X:=.x) __ =:)- .. _ _ _______ _

- -

_ J=-25_g~plt=>s:.. ____ Bm o ~-!_a, __ L~---- _rQt~::59 v-L j _ ne-j'-~+~: ~-~---h h o ~ :a 4--~---P ~ei_r_Ls_ .

----4---------------------------------- ------------- -----. ---------------- -------------------------------- ------------------------ ---------------

-------.--,-------------------

~

: . X ~ l3 J N c V) tf) ; f KX=x)~J~ ((l-r)"-x x£io), ... , ¥1J' C>/L.V,.

;~X A/ poLe)) ,· ~ rr(X =x) =- ( e -;\ )( X!

X ~ l o, liZ I ~ • ' 3'

0 o(v.

:: X~ N l3( r, p) (X \ ((X-I) i f If =t<) ::: ) r- t

r x-r f (I ~f0 X E: ~\In l; ... JJ

0 oftv ..

:.X""'0EO(p) r·f J?(X"x)"' [(t~r- 1 p x e: t 1, Z1 ., J, D I~-,_.,; •

'.

.. It c v& t's c.orff in vro lAS if -t-herE ('

·. -1, 5ue- h f-t_ _1/ I

-f hoc-/- F{t) ~ J lkS') dS

. iNa+: fh.cti f > 0 !

ctno

'l

Quesi 'oVI: . 1I:s f ' ? _ t..tnt~f4e ,

? : ;ls F 'dtlt:~'.Le ___ ._

-C()

<:\:)

S +es) ds ==- ~ • -co

No.

Y~s. -

e)C t's-/s q .f 4 V1 c. ~ I o ").

+a, q { I -t E IR.

.LNCl-~~ -1 ~-( if F ,·s d!~e\e~~'.'itq_ble) tf-ten F ft.) ~'s

.. ~ de~~:iy . .f'~..tvt.e--fioVI -t'o, X . • •

c:; ._.i

Exq1111p/e5; No\VV\ct! 1 expoVJ evt-/;et(, (...(vlt.forvvt --------··--·----

- J-: f.t. -J.f-JZ· . - ·---··-----·------ ·--·---·-----------··----~---·-- -- ------- --. --l---------z:cr-\: ----------·-·----------·----·--··--.

-----·-·-·---------·- X IV f\t A., o- 2. ) if ftt) :::,rz-;0".,_, ~--------4--------·--·--·-----··-·----·------·----

~~=: .. ·~=~-£~.~~)~···· .'!~ ... ~=~t~i~~~~ {~-==~~~=~-~~~~~=·· ------------- --------------- -- ---------------------- --t -- --------------------- ----- -- ----------------- -__ _ __ _ __ _ __ _l_~__lil) 1 f ( cc~) _ _i_f __ _fEU__ ~-- _ -k~:Q.-- __________ i ~~4---_ _ _________________ _ - ------- --- ----------------------------------------- ---- -- --- -Q ___ --- ------ -- C/t.,..; • -- - ----- ----------

Q u f=>s_i~QI!J;_ A r'C _±.h~ re_ _____ rq !/'~a VV\ __ _.u_gs_Lg_b_if;"'J _____ Ill'! i ___ a ~s-. ____ ~-f: _ _._:_fJ_e_t__ _ _________ _

----··--·-·· .. ···----- ... _J 1'5 c r-e -fj2 ___ {lo:c_ __ .~i vl u o ~~] ___________ .YE ..S:~- ---··------------··---------------·· . ______ ----------·· _

§~q_~fjf,;___~?!.fp-ClSf. __ fj!Lqr_ __ c__q__'C_ __ _,_~:s:__J~t~.--.s~:, __ C:_LScsh _____ SL!::l_Q.._ __ ~L~Lg-f _ __±~~-------·-----·--------------- _ • dq •• (~-~-_jt:;_ ___ ¥~---~su _________ _,.:; ____ f_~-- ±:::X P(5;mo) ._ ____ ··-- _______________ _ __ _________ _ _____ _I.f_ __ ~nu_c_ __ d ~d,~I:.{J.Je_ __ i_s $LQOfi~----th_e_l'j___ ·f_be____ _ ___ _ _______ _

. _ .... .... . _iu5JL~:Q_ VIc 1'0_ c;-a •nrq ~j- _ ~;_iL . .p7 _ .Y=: .. m."x (9.1- ':(=.too.i)_.. .. . . ...... . - -······~_.. "-- ------.. - ~-------------~-------------··---------------- - ---------- ----~------ "--------- --- -----------------····--- .. ----------- ----------------------------·-- --- -- - -- -"-- - ~- -- ·-·

Y , -/ 1 , : _;_r t /"'\ · 1 J . ,' I ; 1¥1(-' -4- S::<"'' '(? -----··-·--·-------------------~~--- no d I 5C f P]e ---~~~sd..2'_E; _________ . __ 5e _Q_L __ 1QQ ': _ _f)_~---------------

_ ________ _ ou -!comes ______ l's ___ (d_jj_[ o l.d Y7 ice b 14o_,-D::L)_ ________ ---- ------

-~~1- i~~~-- f\o -1 ___ ~. -j I =~1~~~~~~~- jJ~-~~~-~~s-~_jJ~~-r_e ________ LS _____ .t1 o. _-!~.A ~~~~~~r';'-- __

~- s: ~_t:..b _____ &cLf: . EC±) __ = _}S:2Ld s._ _))_ -t <SJR .. ____ ,____________________ _______ _ _ -CO

-··----------~·--------------............... --.. -·-·-·· -----" ----· ·------ --

.. If X I , + '~ r : I '

' d lsc n"'·; E·, heV1 t e ex-r.)ec. 1ed I

~s /ct lt.tf' IS"

.. j,·VeVt bj ~(X) :: L xJ?«=x) pro v ;Jecl -Tlv.t+ f4e l_x llRX'=x)?oj

.. 5C!Wt COV! VE' r-Jes G...h5o/'-1-;.c/;l,; i,f:,,; z lxl nfJ=x) ( cf)•

.If -f0e 5~W? ciot:.=- Y1o-/ Covtv~~e abso!(.{fe(J, ihevt

.. the expec-{ed VC? ~~~ does 0.o-j -ex"t".st •

. . 1

CvV! I ivl '-( ClU:..)

/ b

.. Sivvr;/ct(l.J, i l-te expec-fecl vctfue of a._

.. r-. v. i 5 E{X) "'~R. x ti'x) dx prov;d.,cJ Ht"'t StxJ-Rx) d }( < <:r> • (K

ExqWfp/e :.. 5"ff"'e 1 i.-..c,1 r(f = L(_-1 /) ='- #t)'l-c -\o<"" L :.o [1 2, ...

~ ~6 :::. Cb L. 6 -TheV\ f,ikl)(fri.)L t:~ \) W - lo3Cz)) ht.{-f

the expecft:!iic>VI does no-f eXi5i.

:.T~e vctr\ctV\c_~ d-\ ~ '· v," XJ Is . .1 ~ .-/x .\L .. c e V", , \q ( m a m e n 1" J t:: (l - f{X; .) ,

.i tt-tq-f . ·{ ~e C'rpcc·-le(( vC?f<..<e excsls •

I. +s SecOV:Q

prav,'c:lecl

. .

.,fvla( kovs .1 "eJ"""UJ 5 i"'t~ s ; nai . 1-f K 7 0 J <1. 7 0

.,<J~d fiX) extds) -the VI !f(X > q) L ~ • Prove -!ktsl 01... ..

. . ~heb:f..s.0c v's irteyu a. I ;i) s-fq{ es +&uoti it r:(X) ert ,t,, ;i va..'(/) ~to; co~ _, aVL d k ? 0., then .. . JP{{Y~tt.l-~J~~-J- L- -~~,[~). Prav~ i·h;sL __

k

((;>-Q:?~

---·----~-v-\ i-'xJJ·--;t--~Tx)~(l ___ F_W/ ________ ..... --s--p·rrf:i:L5Jj--~f-s--===-rx~~-----------------·: -~Tli 'J----SY'IQnWI:~rr:r·---lj-p,.,-!oc~:-->-o}·-· sa·;;-rs\.ya·c:r -~~r--r-;~i --· £o··---'>\ '\) ol·-:a]~w~:=-1 --·

i ' I .~.1

~~X' -(~·XJJ--~!~----;·--r~-~---------·-------------

-· -------• c1oT ~r\q ~J}~s:~r·-·rl'·t:;~ £ 1 Pw--·cn:yT-·-s-1:)--~ p a•u_jJ. u _} --·s7 -·----f! ___ 1Adlf-t ____ ----------------·---------·-----·----------+-.7\--.J·---.nvrouo '1 f .'?Vj ----v;~rnrr;TJ-T!fT--liA~f-;rr---w9!f----pn-.D-;:w,--·-··-----·----·-----------

-·--------------·yr & lo/ ~'J457r----wA~p;/Th-s-·--u---y '-"O r~n ~ ~ sp;! p · d-~f--\1\dlfM·------------·

···-. ----------·-··---------·------------------·-····-------------------·-------------·-. 0?-0::>-

--------------· -··------····-· .... -·-···-----------·····-······-----···········-· -------~-=f_-f-{~19t~j-·f _ \~ lx)~ ···n····-···--·-·····

( ~ ---·-··--·--··---··-·--·-· ---··----~ ~ 11 ----·-·r;3f--w,t------~1-·--· -----_s;..-JI"T ______ SI'Io-;:;·F ~1=--v.-o",------.JCTT.'.j -------------·----------

. , I , ......{

-·-·--··---···-·-····-----·-----------------------------iozt?-;;X-f-s =X/Jr f1.AE> -4--C;7~-+ x) <; l (+'s)t-------------------·-·-·-··------------··

--------------------··············-······-----------···-···-···-············--·--·---·--~-____ i~ct-;f!~;fJ/~ 5-tf:i 'x)j ------

• m --------------------·-.th-=iJ·-···'-X 7 t)a -~E Sb r~TA[J'df _______ _ ---------rr ( r:; 'YJ ~-::1. ')\ p 111)? -X 1 s d 1 q b " nil___ [/1-1 op//1 ru ~ + :; o -------

·trrT!f-!'cr:-r;:~rtJ----rv-:rf)-+:---'df1T--·--s-;;-q-~-s'5r.;;rp· --' ( fltx)_-j-'~u:> T VI )0 r: { J I ~ '•' ,, I ,_)J -r

.. Twa '~ if

rctv1e:i.cw1 vd.~tlct.1ft's_,~ X avtd_ // ctre ;nde;;eV?dent rP(f~x, Y~ J) ;=_ !F(J ~>() IP(Y ~y) V x, :J e_f:.

'EcriAI'v~/eY<tft!c.{ f (x,~.~) ::: r (x) fy1\Jo) . . ~~ ....... ~ ..... _)) .. x,r J . 1x lJ V x,J 6/R

E-)(Q fh1 pf e! ; 5 ho ~;v ± ht:~i 1 Jt e 5 c c c. "' cl. d c tl'1 :' 7 ;'~ • ., I. I ,., ' i.-,.1pie..s ln~· -;,·rst.

''

.. SG(ffose .[ .. ht~s .d~rz5;!J furzcii~-1 / -r·, .af'l d Y == h{X). (i;t,ff'.Sftor~: . , ~Ot-v C~ 1""\ VV e f; Vl d i· ~~ E' O'f v1 S ,· ;__; 9 f r <:

.. T~E'. ~'C5t .. _t:V!_ef~<:)d vve '{{ disc'-< s s /s +he cd t

.. +ec.(,._nL~.Y.t.f. . . 91(ld .. Is hes1 i/lf.LS-Irc:r-/ed· ~;jl,. QvJ

" exqrttf/e ~

••

EJrctY'1~ff :~ )uff as e (t.Z.IJ

Le [ .• FrL'1) =

'I

• I

•l

L" r -2..X

ex~) oc (I -~ X ) 0 _,

o/v.

(.-::ex. Then ;{ ~e h./qv?i 1(&): !P(Y f: j) ~ (f (eX:= J) 7:: rrr (X~ lo~;.~)) j / 0 '

( o :; =o.

~- f fx (./o_.~_ .. (::J)} ~t. /0 .J

( 9 . O/t.v ..

. We cctVl 1 he v- q [.so ;-; •' o .... . ...... . ( .. , ~ l.

I • : I ;' II

~

. i fy03) ~ F('j\ ~-t K ~,c~) I~ j I OJ ~~e·,r~:

~ ( ~··. -~ 7 l ..J I

O(t--J' ~/v. •j ~--"•"'-' ..... ,.. ---~~~«-----.----------.- ··-- -~~--·-~~~ ·' --~--~------- .. ·--~~.----~-~-~.....,_"""" .... -,.~---'V·•~---~"-"""" --"'"' -~"'"

Start here.

• n n --n nn -u nn n -?ft;J r: A -~ ~6'±JI;li1-=-tifr-·• ••••---·-·•••---•• ••-

------------------;;q-----or---kf;--:)-v.,:;p-------~11-r=-----p-:;;s-:-----u-s;y_;·----Cl~-dM ---------------------------

l...., --------------------·

0-z·-p"~--------rc,-F··-··--····:;rr:;-Tr--

'-1 '

---------------.. --------~)-\A"b-.--tr---z----=------------~---·---------~---------------~--------------~-------·------------· ------\"')\_ T T ~ l -J.)..<. .JJ.£ ::r:C ll-

---------X~ -·---; --------~~f~1JT ~ ( ~) XJ " lhfJ~-1' "" -------------------------------

~~==~=~t~Yii_=1J~=~~~7~ ~dT~-Tfi~7; Jr~=r:9Jrl'-\Ta ~ -_L ____ ~~~~-~~~~~~~~--/ -;_· z·9)

------------------------------------------------------------;r-;J.·---1-dl------J;~wo7~-

--------------~----------------

'I , I

Exavvtpl~: , I 't

'

Let Le-t

X,._ UNI F(O;Z-)

y ~ ) \ '*- X 7 1 .,

(o o/~ )',e. y-= h(X) ~ ~ e \e hex\ ;: 1 [X ) I J.

. Let W ~ mqx(o, X-1)

: W.e s:ee + ~ ct-f Y -v f3 F Rli .J avtd ''V\1 ts .l hE>i+h..e r dtsc...reie rtor co..., -1/ v1 u Cl L.{_S) i + is l_e ro v..--r'-1-t,

;j prababt'!d_'j i blAt Cov'\di1t'oVl ~ r OV\ w /0' if hqs , , a_ ()NI F ( 0 1 1) d ts-fr; b~.~...f,'oV1.

h'Vtd 1 ha.-f :JOt.<

10

. I , I a..c>

-f-'-'e de~ivcd,ve o~ Fftv'(vv) qV\d hu-1-"E>

N.oT Jef q dens-ify f'-'"'~c -fie.-,- i-f /0 -f-eJr-q-f E'S ia

fvi_.>tead o-f j_. J_ II ;Z_

'+

; 1 Tht q hovE> ex a. lhff l'f..S 5 ho~v f~ ct--l vve- Y/1L( sf

.. be .5o VIlle ~ h.ctf c ct LC--J i o 4. 5.

Th~ Jllexf VVte-/0od t's f~~ -f-rcu-75 f-oryV!~/,OV1 t11e-fho cl •

. Let Y = ~{X} _ '1!1.~-- G..S5<.JfYt.e. ____ h _____ _,~s_ ______ Q"J(; ~ +o- ov1e •

. . ·tNdJt:' . thctl i he vtas·~

< , ,-. •

. ; vt5 1 J i r (411\.S ·\C)t ~ .,, q ·i i ovts

e~o ·~l'f k rt bove .. ·' ~ .• c <.. < ~-,,cc /V.:-t• .. yc)l --

c. avvt e "\-~c>W] 1-+o- l

•o• . .,,•v•»•_,._,._ ••• "'• "' ~.~

• j

I ·t ~

ll

. : 1./e. ~,'/( L{se itt~ no lct 'I (j n -~{x) C?c.v.d htx) it~fefchq~qbj ,. LJheV\ }} = h(x). Also) X(j) will be used i.rJ

. p fqc e. of h-{:J) ~bert.:e v:er _ .coYJ.-v.evtient- (aVId h_, exis1s).

Tneo,eV"'' .. Tf X Is d/scre-le 1

{ 6. 3, I) .. the" fy(~) = fxCx(:~)) • Y~ ~{X) J avtd )1 ,·s I -+o- I

.. EcruivalenljJ fy6) = +x (h-('j)}.

Proof: f y(~l ~ IP(Y ~ y) :: /P( h(X) = J) = fP (X: h -(:;)) = !P( J= x(:;))

Exq~le.:. Le{ X""'C:SF(](p) (6. 3.1)

So fhe~-1

=-fx(x(j))

flx) 1'-p~/p

0

x: 1;'2,,.'

0/"-'-'.

L e i. '( = X- I. (\) 0 _, e f h C( 1· y Is Cl fie V1 a Is Ci c q I (ed

.. jeaMefr,·c. If X JS" the rl(/ntber o-f' f,J·qls t..~Vlf:l

.. Q S c.t c c. e SS"', t h f' VJ Y J·s -f he rJ v r"l be r o-F' .fc:t; /I.-{ U? s betc;;~0' ct 5u: c ce 5).

J3j the q bovP fi1ec(' e vvr / ~e. hove fy(j) .::- tX( 'j'~-IJ

=(;(1- \~ ~ p ) p :J ::: 01 1/ •••

( () 0/vv 0

'

""70--------------------· -------------------------------·------·-----------------------------~-----------· _______ ,

-------------------------------·------·----------------------·-------------

--------·---------·--------·---------------------------~ _ {h)J};i~ (A)~---f9 -u a A.• £I i ---------------s-~~--V'CY.'T~-u,-s-----r-s-15!;1A _____ 1}1:rr· -·~, ':J ~f CJl ) s.~ r _H _____ '0 :> Q_j ; ,

-----.---~-------·--·---···---···· -·-·······--------------------------------·-·-------------------------------------------~------·---------·-

·····---------------r-r-,r~-0 -------

-----------------·--v~---fi• (J-1)-T} -(t~q;5J "'-wAr snu ________ Et_ __________ ---------------------------------. --------------·--

---------------~-----·-·------------------

----------------·----------------------------~--------------~ --------------------------~-

·----....---;----------------------------- _..._,7o ____ -------------9

··-·----------------·----------------------d---------------·-----------------·------------------------·-·

--r t x ---=---r;:=__==rh,a;::::_,~? ==;::::::;' ;::::::+f_= ----------______________________________ \._I_>!] ______________________________ -·-·-··-··-------·--·

I :::-1-

···------------------cJ f:d-=ij·;;--~\x)>£ -<df"'dt;<"A<J:> of pa.::Ju--~K--------I-.:f ..-I _ I I

-------------·-·------------------·-·---------------_______ fx_ . -·---------------------------------------------------------------------------------------------------

~(1+93 ~-r+h ~!)d/:; (h 7 1-l)JJ: (f:;JJ(}) _wy------------- ----·---------------------· ----------------.. '-----·~·----·-· --

Zl

L3

3 p ( /- p) !j = 0/ L/ 2 ;'·· .·.·. ·

'1 0 C?/S/ ..

As we h C( v e s e E: t1 _, o Vl e Y'V1 e f h o J VV~ ay he

, eQSie' fhaVI fVIe o 'fh e r dt-peV~ cf,-""J o~ fAe J,·sir;"bv~fioot1 of X) fhf !rctr's{~f~"''"'I/()Vl h) aYld

vvhaf /s q/rear!.J kvtowV\,

TheoreVV\.i Tf X /s c.aVJ41vz(,-lous, Y.:::h{X)) q.,.cl h t:S 1-fo-1) (6.3.2_) ., -/hrcYJ fr(j)= fx(X(:J)) /1:; X(j)/, «s lo"J as ij XLJ) '5

•! Con-L'vtt..tCIVtS q.,-.d heve\ ~ero C)V) (3::: l.~ I fy6) >a]. f3 CaV1 a lsc be v-.--r~-iif'V\ qs g=i_h{x) /fx.0) >oj ..

Proof: ,, 8ecause dd x(';1) \5. C.oYI~'\'to.{c,w.s Q'VId YloV\-:Z.t"'co ClV\ £3, j

,, x.('j) is rvzovt6-lo¥tic oV1 E . • j

. : We hJ ,'/ r p r d V("' f h.e res L-r If irder vet l.,

wltel'l h . ' 15 Jrlcrea5-,Vl.J·

.: C\nd B •s ct.V\

';

'I We u 5e 1 h. f' c el-f VVl e -u," d:

:: FytJ) ~ !P{ Y ~ J) : 4~(i..(X) ,_ 'J) "' /!'(! !=. h "(~)) = f( 1 ec x(:Jl)

. : = Fx ( X(j}J -f'o r j ~{3. (I-f .J ~ S J the~ F;(.J) e l_o, Is a.,J ty('1)=0)

. : Thv.5) f yC:JF fx (x(j)) . d~ x( J) = fxC X(.:t)) f:J X(j) /.ib EB].

'. i We.ve t-t "

.. ~L~~~· .sl'_)h+G c.heaitd o.v-. The praof. T -f B \5 Y'l()-f ~VI

.. ~L~J V.f.~tJJJ +heY\ )<(j) .. _!Y}3:;} ... JJQJ h(; Y\'l.Q~O .. Jort i c_.

i11 fervet {

-·--··-···---·-····---------------------·-------~-------·-·····---··-·--------······

Q

1:rc-fi~--f---~~ dq :_ T>r31i1Tt/0~bo//f = (h)lj-s-0'11 T' ······· ·-· ····················--( -w·::·-\-h·}~t) ~~--------······-·····---------······ ········ .... -----·····. ---------------------·--------~-----

"'Z. --·····-·------·-· --------------· ... c-c:o+vT-=~----·q-0 --"d/! :~ro" ,-:)-i"d_f___ lJ"ld"~--=-~ArriA··-···--·'sn~:::r

-----~1-T~----~,t -t;;~j"JJ]Tp·---s.,---"'1-TT"~-----(-~Tfio 1-= -rh"J:~-q·----+10'1~[--dt (57\T----------

---________ ... ( '1..--D-t"'H ]\''r~sC:J1·-;;_,·-;----J·o--·-frTTi/r"dp--·a-rt1---piAfJ-~-----··------------

~~=--=~= ==-: --__Z~!=:!~-• -rvo-J,~+ni_!f\1_-:__x__ =t~~= :~-=~~~-... -----···-·· . ·············------·······------------.-------{----. ---··-··· -···--···------·-------·-·····----·-· ----------------·-···-···-·····----·-~---.. --· -· . ·---···--· -..... .

I:;-h o

------------------·---------· -----·---------· -·-------------------------· -··-------------.-f~dW·-·-vrrT!FoT:v.~o}-sT11ETf_, ---------·------------·--·-

--------------····-------:itt+·------:ds-n·7v-rm;r·---Of'1-------·e-r-· z 7 --~-wr-1::>)(3) pol-Jf Jw ----·-----------

--------J-F=' -drfr -t:~--J7sn "''""'J d--f:rv)y F"' '"T-;,!"\·~----------

----------------------····-----------------------------------------------------------------·-· -------------------------------d :::-_A ---··--cc-:t--=-;) l------· ---------·-···----------------

-----------------------------------------------------------------\Xrr;pm-3 c:t s lA bJ +--a-Tr+·----1~

$ ·' (h )X ~'P. fb'1 + dt-ON

hi

Is-

E~q""ple; ·: Lei X~ PAJZ(I1 1) So -f h_OI_ic . to<) = ( cdx y~ )(/()

Theo\e VV1

(6.~.3)

Roof:

0 0/~

. Le -l Y = l D_J /X)

.. \hen h -(j) ~ e ~ / \..-../hie"' IS J,·-r-rf'reV?~,-q.b I~ ~,v,'f~

.. noVJ- ~e fO C OVl 4-,oVl !..10U.5 · de rr r'va -le On [S = /R •

.. Thvs . . -fy6) = fx(e ~J e:; ~ ej

Q-;_ e~y· .

. ' r-t X is Q Cor'ttlvt UOI-<.5 r. v,) fhen F(X) 'V UNir:-(0,1)

.. We prove? tl-ie rest.< I t o,.., ly , ·t'\ inver-1,'JJ/e. •

fhe S"fec,"q ( CCL se .. llv'~ere. F i.s

• i

· Let U ~ r:(X).

. ' TheV1 . '

0

FJ'")"' !P(FfX)~ -.,) ::.jlf( X =p-·(~) \

4 L6J

4. ~(01 t),

u.. z. I •

,;Sivtce F curd he_v.tce_. __ F-' a.r_e Vl10v1o+oV!ice~l&- /Vlcr~g57':J·

·i u 4 _£-0/

. ::: ~f( F -[4)) = u. u.. ~ {o,') 1 .l 1 j . ____ L{ Z. l •

·A--

-----------------------------------------'"'"? / o------------------------v-~(-----------------------------~-------

---... ---·-·-·· ... '02 -~ -· ·· ······ · c~/o h-)~ _ · (~/,h) ~I -~=-~=~-=-~-= ~ ... ~~--------~-/·;;~--~ r~;<~;;,~-id-~-v~~ .~.r2~~~~=> ... -··· ..... --------3--c;-:s;-p-.;;-----vr----f5-o·qr·;;;7;;;;~--·---c;.-r::ri·;-v~::t:rr

5--~-,"A-"Pr-r. -----~~lrr·--·--as-rr--~r-----?"Vfvq--------------------

---~ATBC>l----ll17VT___ -d~---J-~f-b~r-------·t::>oqt.;; -Vr/1----]-p--.,-----------:;;)o/f-___ asn--7TT;:;~ ------------------------------:;; -.., --·-o:s ---------~-[h.,.r=-;-·-·-rxc·-·---,;-=ry·-:::r--+uu----s~r---11 x ~XJT{___ -----------------------------

---A Jcs·-----~~_-rs-t1-~-r----dlff ___ P~8---------------------------------------------------------------------------------------------------------------------p-h·x-~~r-----7J1 --_ ----------------------------

---------------------------------------------------------------------------------;-c h ' ,_) ~-rrrn-~x----1-·a·=r-=~;3-~ ~;(~----

-------------------------------------------.-;;1 d wr-o-x.d-~~ j o vtr--~-lilo ______ ----------

----------------u-p··---~Fefr··'si.Xffj r--~ ~1-A:IDj-~iU-f-+-\Xf<R---{)t-VP 0 ,3/\01/!/f-~-d~JOJ~-------------

----------------------------------------r-uv'l-rf<rr:rrs-yp·---(r,.o}:rrr,rn ____ ---1)---'JF ___ b v _, p J (J :J :; o -----------------------

------~--------------------------pFor)·---fu_~f~-s~!Ad-15--~~JTj---kq--v-orrnrr:-Jrs?fY-kl-1 o-----q_----------------------------------hrp-"JOY5o----i5+'or-·d+~TcrtrdG---of :s n Sl '10/J b 1/ilfdTO"dlf+-~ ~------------------------

----... ?J(Cf-.']-sa/1-u~t --sT ________ -;j__ -J:i------;~-.=1 :::-9-1-b '1~---dt 9f\T~---------------------

----------------------------------------------------------------------·-s---dS_T7--sax-a··----s15 ___ +Jdr--J-cuj~r---_ --------------------------

---------------------------------------------"=i rv ( n) 9 ;;: r UJ '1_j_ .. ------[h-~~-9)

--------f:_J"o)3J.Nn -v n P\AB tcx).:f-:; nTx~ U!\AA = (n)~ +dl lMJ.)?Ja~l

9l

' ----------------------------------------~-------------'"-. -----~---·------------

---------------d-------------\:AO!)PwdJ314Prr--ddq·r--j-0 ab'V1b:J ~r :3~1\0 ---------------

------------------------------------------------------------------~--e---------------------------------------------------r-+-------------

------------v·y~--=~-ov·--y.s\,.11;:5 --··-s-Y\-on-\11-,iF\Ao-")----->' ~1~----------->--o----f,-vvr--> v ----~-------------------

-~· -~x ~ __ "Xe, ________ _

- -------!A-o~~--w-5'JS7TfjTr--a11r--tr;5____ ------------------------------------=-----=-------hf _____________ ----------,.. -----------------------. ... ?;-flAO.JHd'jf-_.,t-~ ~ "'!::-; ;-; n ~!t -'~ ~-(1\)J<\: d > ay "'\ ---· -

'-. '-"

--------------------------7"1/ 0 -----------------------·-·---------·-------------------v---. ------------------------------··-----------------------------------------------r·-----c -~---------------------------------------------------------------/-.7 ------------------------------------------------------------------------------------

~-. h "!l 3 h/£-h T · ~ -------------· --------c··----r·-----h -------··---------------.:s---------------------1=;--·~------------------------------.-------· -----

1 0 J ~ /-,~ _ !=. • ~ • 1 + it;. _ h 1. t = -··--------------------------I_ ________________ . -_____ £. ___ -

------------·--------~-------···------··--------------------·-------------------------------------------

/o 9

t1

Theo\e WI ·I If A ~~ disc,ei e 1 Y = J..(6) 1 ct~ d ~ t' s -~ ~fa-:- .1. ,

(6. 3. 5) ·; t~ eY\ fy (t'):: tx(!(~)J - -

P\1lo-\: : .tyi'i) ~ lP(Y = ':l) = tf(h(K} = ~) ~ JP(1"' nY)) = IP(K =_ !!:_(t)) .

~ ~

• + = tx c~c;t) J. Q

·!

..

18

Proof of ~~3. 6 = Le +- '8 h~ . i ~e 1a Vl) e of ihe {rqnsforVI-1q{o..,

. . . i • j

• !

i .. 4 ~

••

•·i

• i

~ ~

. .l

' . i

.. • i

h 61'\ J. le i DC. [3 • Le+ C· be. the

pre lwt ctJ e of D LJI(ld~r ~ : <=-= f! t h(!) E; D J.

T~e VI 1 rp((~ D)"' j ... S fyc'!.) d~, dj. ,,, d:J,. --~ .. bt

= f(!. 6: c J ::_ L. s~ {i) b. dk .. ~d.t., c -

== t . .s fx(:(~ )) c1:tl dd~ ~ 1)) 1 dy .... dJ~ D

by a. cku::tv:Je- o1'"' V~f ~· q1/es ~;:: h~). D ...

:I+ eql'l be hel~tl.l[ +a .do t~~ p\"oo-\ in iitis wa.J .. +o\ Cl t..<x!ivc;:cr ;~1i2 . +tq6''\.$"' \o~·vvl c{bon .f,,,s:t ..

.. ... ~.

•rv,;o

"0( "h _:_z:h 1 ()~ 'h ::LI

"'" _,..__,. ~--~-... --...... -~'"'

r-

0

bl

·-. zF.e -zx f'

-"Zh e -...__

'xe

'he 'l:xe

'r, e ~e ---(. h7x f( 'xe -

'h-"Zh =C~;-zx

'. j •

. .

'' I

..

. . • i

, .. t.

i ! ' i ••

'h =l~)'x I· ;

'.

'.

ta1 t J/ :a,JJMa:>X:1

oz

9:;

0(

0

.. t ~ .. j

!·•

1'

'.

lll ,;

_ ·hr <·~ "~J ,l~s) = ( ·htA ~f' r -~

'h 0 '.

'h d - 'h--

. ;

'Jl

:::: -..~:? ('h ,.~,, '/11 j = NJ'A_r

••

. _,....

\Z

0/o o

= ( ·h "r.) A 'A J-

[: "2

~h + 'h = (~')(

/(~:e)~dp/ (1F,yx '(~/'x)A''3-:: (•!?·t;);{') ··

, (~J.t'h) ;..;'/;.{l J fv1!j

(1)dX3 p:!--v~/~ ---·-------.... 7Y.; 1x :: -z ;\

~-=:f:::;{

L -------. -------<; V ~-< o 'f J'' -;J,-i;;>:;; ;.--it ~ ,-(h)d <u -~ ('A --. -·---------·--·-····--

------------------·--···--------·--. --------------··-------------· ------------------------------------·····-·· ·-. --. ----------------------------------------------------------------------------------.. --------------------

------------------· ----.. ·--....... ----------------------------·--·-------~----t.J710 <d---,x0---·-------------·-· ··

.. -· ... -------· .. --------,.-=-o-r=l-----sr ~ -t-0'1+ -"tf"Jn-s; ------...ty---n-r-L'11-'-t-v-----s--r~scrn-~--.. ----------.... ------------. --------fvt:«Jf'3rrs-------·oF~---v-·--o-o:rr.rJ..,::rcl· --o:r----;jrr:Ts sod -----v ;;ryo--->1 + J_ ------..... -----------

·fo ( Cx<j=J xr-=v-· -------·-·--.. -

--------------• afu b--1 ~ !1-f. f o /4 %>ptA~(;) q aq f of dow ----------------

-· --------------. 7?'17'f.ij'_(~ b 1M :1, :.S Ubj f "S"I"tD h U/"f HO:; ----'t) J Q/) PI 1'7 /' {1 '/?VI 1

·--------·-----/hFbty;;-rro-----:.:J'Dit------z.;;;fJfJ:/;.;_~Jr----o..l ... __ TI DO/. bWfJO · SlilbJ.f . ~+ To ·-----------·· q ~ -r -v · 1-I · ' · ,.. J ~'/-v . , . ,... r'

-------------· ~b~bJ O>Cf+ OSI'i!w.J<3fop o+ A S7Vi h'Sb;; M#o +10·\ft::--~for:J-----------------

-zz

Z3 ,--·

Exq wrp/e.; 5'-'~fpos~ )( 1J dis c..re -1<- t....-J,·-;~, f~l = j ~ x~[-21 -l;s ;/ zJ cJ/v.

let Y-= X 2_

The" f r('l) = -f/ x(•)( 'i) J + f~ (xi'\('i))

VvJ1ece A,= lo/ I, zJ A~ _:: j -z I - I J

ih><s fy('t)=~(z) rfj-z) ' . !, $

- ?- ~- -?-- J . )

::: ;y5

" \ .,1 . /T S'Y¥11 q \ _; ,• '. -_. . -if

CXil?t-rf,a •'1 CXl.5 t.5 vv ~e V)

.. r ~ ,.... I ~c (~) ) I

.Ty&F f~U'b:Jl)· Jet~~ "M

A' ' I • 15 Covt'IIV!WI1U.S:

.fo, j C /3

. ,Tn ,·5 Cctf'l q /so erfend ·lo J o~·v, .. :.

.. t'n ihe ohv/ouy J...vQ..~I../•

_1 ,...., • •

trCft75":drYncr lto;ns·

- ~/

~x~~ple ., Lef X--- UNLF(-1 1 /)

( 6, 3 0 8) . y = 12-, A~ =-(--II a)

A l :: [01 t) x('b):: -6

.xz.)(j) ==Jj

•· frr1)~ ~(-Jj )-/-tJ-~/ + ~(oGl '/-b-y1 = ~· ~be(o,1)3

{o, ~ f (o1 1)

/

~ \ '?~h t~i~--T-r--;------------·--·-·· ----·--·-----·-·---·

--·ioz-tt:~-r-ro·z-cr~--~T ~ot. ·C::~1"-r _J ~-------------------------------------------------·---·----------· -·-·-·-·---·--·----·--·-···--·-··----------··· .. ~L ··--··--·-·-·---··--------------~~----· ···--···-----· -·----

---l<f;:.•h :__ (rf(-;1---(-"£ •h) lt~f9-~ t-"'}.J_ ~3::! ::: (fk•.f:f.·)'A'~''J..-----·· ----.--· ' I J -z J I I \ ~ ' "j -··r= -..,;:;-----···-----·------·------• ----)<) ... -· -. ->0.. -. --------·-------·-·----... --·· ·--·------·--··---·----· ·-··----------------.

-·-----·--·-·--·-------·------·---r:t~-=v-~~-~--------=------·------·----·--------··-----·---·----·---·----·----·--·-----··-·-·----·------------------· ·-------···-----···--r--· ··-·-·-·· ·--·------r---------------------· -tfi~,:;>~ ··--·-·--··---·----·--·--

l.r-~h 0 ' ---------------------------------------·--· ~~----------Q-----------f[ ~---~~l-x ~-----·--------------·--·----·---·-·-

---rf,-.. ~-::: 7H ~ z h -~-----------~~--:::--r.r-----------·-l"FTF--::_-T--------·---··-·---·--------·------X _/\ rn_x

----------c~-------------·---, --·------·-·-----------·-------·-·-------·--·····

-••. ==-= . ·-~···· •.. _. Jo~-_'_?]I----~ x =~~-~;_'X -:I:::=_~,~~~~ _1-~~-f----d-~o~-= ~·-=~=-~-

---------------. --·-·----·-·· -----··---·----·-··------··--·-------·------------··-----------rTAAhj------~fr·-----

19----·----------·----·-----

kl v! f j -.....,;

vt 0 j ------,J-717:------~~ 11A J.CJ~:S L.J 6Tf ____ J "j--~il.r ---v-------------------

· ----------"Js-oo~--::_;-----:~~-----37. ------· -~~r;-,-r-----··:s-~-~--t~-0------J-9 -------'Ct7"Y\;------:o----~--------------------------~;-r~---;~---:;! -----lDlT?~-----;n• \ -----tT--T""l-)'1~----~~-\~----,),HiT\'~:ivJ -·s-r fi __ _

zs-

____ _1\ V\ (Liler lila_, , I e . . P- .::~~a_g ___ -~~---- -~ I /' .

. ,---.,

-·-5-_.f._Yl_ESQ _._:_[ CJ ~=-'--f.~.L.c·::.:.:.l¥'-.S .•.. _

__ . .. __ &D:?JJ tAaL-LJlt ;: . s_~:~r ----· -~-~~ ----·---- ---- --· ···-- ---------- -·- ·-.

. _ ... .. -" Al.s_o ce c..c~/ L _ ±.h..d _a. mo "'-"-Bj_ __ :J.f:_n!'.Ls:IJ.,~ ... _ .... _ _ _ _:f.u ~'~ clii!!t'_ .. Lli1Lf; ue.Ly-- _d_~ it: s t>? ;_,L es ..... the. __ ct _ _s: 5 o. cj_g{e J ...

·-·· ________ . ___ c{ __ ,StL' b_cJ-fLort____ __ __

····---·-----· ·- ·- .. ·-··----·-·-··---··- -----·····-·······-··- Y\ .

Ihe.<:;_{S'YI:.\ .. - ... i:f ... _1; "' L~cJ~f<'".cit>il_L . a".J- . )"'ccifX: I

____ ( b ~~/_. LJ ______ ±_b_e~L ___ f'i' f-t: )_~--- Itiiv-BJ. --- .. ----·-- -- -----------·· ····-- . .... ... . -r i=' ~~~ - . ----/1-- ~ ·- ./-Et.J~\ _ -· ~ -tX~)-, _ V\ -· _ ( +X0~

P-'-""'L _____ 1{:J: E~--) ~-- F 0;(e_ _ --:_ 1J f-ie. .. t· ~-1 L-1

····-------- ~~~=~~------ ·····--=--irEJ.\l.-(~2 0------- ···--·-------··--·

i_::' th

.. -------~---- -~---------···--

-f-.Xq__~p/e_ _: _____ ~f~_i_ Xl /\.- 0AJ-1.L\Ll!}_ X-z_~_§ltt1L1~--~)_, _ _ _____ _ _ _L_~_t__ Y ==- X-t-.4-- Xz., As)umr- X\-t-Xb '?::~'?(···· i_~Jepe~_d-~_.,i!

___ : /he~ .... L'-1~~~=~x,L~l~x~(~) .. -{~-\_tTl ~tf··· . -=t\~)~+e ~ ... GAf!J.i_ \ l rr- ;- e J

CXQ Vl1{)/t: (6.'1.~)..

------------

Le-t: Y:::X, t-V Vv,'f 1- V ,_ BIJJ(n ~) ··-··-·· .. ... L .... ,6Z. -- __ Vf It-- ... - . I

X'- ~ BIN( Wll ~) As5 !AIM(? X,/ /z q t'{" iv1 d{-fE- ti/dr::· 111i, n

lhe"' ;v/yftF ;11/X {t) 1- (-!:.) =- (t c/' +(1- 3)) 1 l ~(_ ~e~ +-(/-j)')M

_ (1-_ -t ~ ) n+ VVI 3e +~ """ f3I N (_ V\ + M/ ~)

26

ExcrVI1f.IE ', .. I-t xi/ Yz.. ct\t'

( t. lfS) ..

fvtdt:pevtJ.eni vv;-f~rt Y, /V Poi( s)

X._ .A- Po_r_(t=t)

TheV\ MY,+xJt) = f"'x,ft) 1'1yjt) = e_"~ r-') e ''r (e..-_,)

-- (s-+l"7)( e ~-1) e

'V Por( z z..)

ExaWlrt~.= r~ X, ,x"l.- ctrf' !'vrdfpt~-1dfni VV!'f"' (_t,!,{,f) "

X, AJ N (~-t,, cr, 2.) x"L-,-v N (~l.,az. 1 )

..

He ... f'/1/,t{.._(t):: !'1y,(i\l1y,ft\ -~~ -t +Oi?-t;z Pz. t + trz.,.-t.-/z.

e e.

h.t-.<~.) t t ~. ~ta-.• )tlz "'- N {~-t,-t!-1, 1 ~z.+c,--z.z) :; e

Zf-

'- --~--- - --

.. __ 2<:-tppQ~e _______ xi_~-~--~ d _____ L_ .. ____ _

'vJe CaV1 t~;vlk qbov-1 X X ; . v as -··-······· ••.• ···-······. •••••••· -· ···-···-··· ····-·--·-·-· ------ • -. • c I 1 ·-'---/- . .._ f ~ .....

___ 9::c __ _ _ _s.ctw..p_k____ -~±_ ___ _ SL=L e _ n _ t .Lo ¥""" .. _ . -~- . _ -)?of LL/Y.L oYl _ .

... . . ____ 'lf~l· LirL cL's _-fr_~lJ.Lj__i_o_!Ll_ F r - - - -- -

. ······---~-Becg ~ . .s..e ..... Q {' _iv.LJ~ pe.ttdr-. ,,.· _( r::f . . .... It'!..~ l/v· ow_/ d . . n e f cl ····-----------------:±_~--- b_e .$_Qmf-lt.-:L~ _____ ~ryl':-Jh ..... ___ \.~f_l12.LE.t!'If'tl_i._

____________ __ _H_Q_~~y_e_r1 _ _w_hf:_v'l _ ~t0e ~i~e Qf j_b__e_ f{)fi.~la.i·aVJ ~·s

___ t~-~JE?- / .. fJtfh ..... ±he _ <:: __ he) 1 c cJ a:£ _ Sr:z ~y~l~~./ tv i lt1 _

. .. __ )j.s_._ __________ b__;_; l_ho.Ycj___ .. L£' plaL.E'Y14/EH_f __________ J,S _________ ~.VLC.O.rJ~:p ~ f..L_EkJ. ~4~

_E>fq_~/e ______ C.oJIJ~LJfL __ g f-Gf~.tl:r/t'rh'l ~f _ ~~ _ _b_q_l/~----__ b_Ll __ red ___ qytc{_ _hg_l_f :b I ld e,

_ -~ _ :}_9_11Y/f& ..... --~~-_-lhQlf± __ cep.lq_ce_VVL f v~-{ _a_~.~..d_ _ le~ f ______ it_: ~----L .. ____ Lf __ Jh.~ ____ _t __ ~---···· b C(_LL. ... Ji__ /Je>d;/

. --- l_ (] _______ __ 0_/'J,v q ....... .

---·------

··· _ _ _ __ $2_.-v}pl~ . ~rvi -Lh C~flqce me VJd QLV'Ld ... If'+

~ ~ fo i ~9L:h,e ·+h L-.-- bq If 15 Red/

-------~-~-------··--·--t-0----~-----------~~-------------------------------

zz

rr(x 2_:: 1 / x, = 1) = ~ ~ ~ c: IF (X._" 1 l 5o Xi.. r5 a 'f' llloi ,·Vl c:lt-p-tv·cLt:vtf r h~-f

. f {Yz 00 / / Y, ;: I } oc i : fP (( z_ " \ I

.. 5c fi /j are /v;defe;tc/c·vti,

.. fJc;wl eo"" 5,· d €, 0- f of w I Q_ I t'uV' (.) -f s,'~e- l.n.

.• nfVI1

(\- \ .

(f CJ z ;:- ( ) X\ : I ) =- ~\ / tP ( X~ = I ) : ;z_ .

Na-f/ce 'I

fhqf h -1 -2y.- I ----:71-

2 a.s Y\ 5eJs krye.

''

.. l'o~A Cqn fh;nl< qbovi itu·5 Q.S fhe L5

.. bec.o~iVL_j le$s- depeuJev1t CLs; n Je-ls /cn:Je,

.. I Vl p \a. c f / c e ~ e_ n ~ ct (" !_j a /t-v a j .s

.. ivldffft-~cievtce w keV1 5ceVVJf/,·J -\'rovV\

, i W i 7 l, 0 vt i- rep lace WI of' VI -t ,

a. >" s- c..~ 1'11 e a /_:: r:J!. pof14./qt/oV1

'i

ZCj

__ _S~~'j_QJ,L__h.._"-e_r:qn_~~?__q~e_~a_{_:;_[;y; _n ... _ -~------------------- trDM ____ _Q ___ ~u /q fU:La _____ 'vl./.i Lh ____ _di.s_i_r,· bu ft'!:.L~------~-------- _________ _

·e v v v rv ··cl l-----------------------.l-.--'---------4--1+~~-r---A-n-------~~:__ _______ L ___________ ----------------- ------- --- --------------------

-~~ 5 -/ fOJ')~_k;~_t_ ______ _l:S _____ -1 h e _______ j_CL!~Vl-j__ _ ___ .cle__kt S 1_4--_r_u__ V7 c j,. d_M_-~----- __ ___ _ ___ _ __ _

s I - ~ )/- . L--- ...,. - ) -+-------------------- ------- --------·­-·--~----~----·--·-----·---------~, ~ ~-.t

=~~=--~ -L~~X~: ~~~re~~~-~~fhe K ~- ...•• / ~~~t ~b~:r:~:i'o~= ~ ________________ \rcVY1 ____ ±ht: _______ --$Q-lt'Ylf-~"T----- __ __________ _____________ _ ____________ ----------

----------------------------+~"- ~ f-"'--LiLc...t.tkL7 __ x/lJ'L i5 __ _ ihe ~.,_j( f'.J t_ _ gytd_ _ -----------------~~-----J:S ______ i_h._e ______ jQ_SJ_e_s__±__._____ __________ _ _ _ ___ _ _ __ _ ___ _ __

___________ _________ _ ___ bb_±e___ _f_k_(J_i_ ______ ~ _____ _i_..s _____ ~ _ __ Cql'ldovY~__ __lLqs_i_a_h__Le__. ______ _____ _

___________________ ___ I~---fca.ctl c e 1 .J .• -~ h~~L____ _ ~e __ _ i_q_ke ______ g___ ---S~fLe.1.

____________ ---------~{_ ~;v:e. _ --8--e ±__ _ __is _______ ±he C>_utha_vY\e _________ (j, , ._., ,_-M-"f---- __

~ -- - -- __ o_{__q _ _._qill.io.M _$_q_,.J_pk _(X, -t X•---t-" • ' I L) . --- -- ----

~~q__~_le~Le ~ l= @ d~~~~d2 F

__ _____ _____ __ _ _____ b_g s.e d __ 010 ____ .ik ___ -jos ~- _ .9_-f ________ a_ ____ f_q_LL ______ c..o;v'L. __

_ __ _____ _ ______ Q~E:t -~_f? ___ ±C?_ss the co,'v~ (_;-£ .. ;-t .:Jg_~_<..h_Q_I') ____ b~-qJ~) _____________ _ ±-1 &::>lA \_,o 1_,·!1 I et"P '>~ ::;.l ------------------- ·---------- _.____ -Vl~L-2 .. ______ Jl_\L.\:_ _____ . ~L. _ _ _. _ t:\ ______ _lL_l.,__ __ .. r-... .. ___ _ .. . _ ______ ------- _________ -----· _ ------ _____ --· --------- -·-· __ ------- .... __ _

-----------+------- -------------------------------------- -------- ----- --------------

. :_r:: + ,·s

becQ use

,I-f q

. X[~ ( 1 . z 6

no+ qttr·r'" {f' -1-o X IS a. ,. v .•

s~j

c ch '.., ,·s to 55 ed i { it e q ds o V'l

0/t-v Q

' ih t.--

V\ -1 1'VVI e s

~o5"5_,~

x·= J_

~r..-11 "

t~fvt (~ 1 xz_ I /

j~, ) l 's 0. ICI VI d dVVI

. :Sa vVl f I~ -\\ b VV\ a

. ; vr VL cL (x, 1 x .,_ 1 • .. 1 x ~"~) f3 £ R ( -k) pcpw I~ JidYJ

J.J -/he oc,~Ycovv.te • 'l

'! ReV1tlfMbe, -r.v.5 are fi_[{Acf,·cJvts X (}I)= l

X(T)= 0

bv(f 0 lA-I (_~y'VI e s are jvs-l Co~~'-.s -1-ct Yl fs •

5o

Exq V11f/e . , If

.. 3et " c-\'

'j OtA Jos 5 q Co1'~ 1:J -f !'vtt F 5 YC)tA VV/.Oj

( X, x"lo- X.s x"( x..,-) ::::: ( I, I / 0/ I I I I I, Q J a.s f he oufcoiM~

-+ Lte co. V1 doW~ Sa l'"1:.ie.

Example:. Svtpfose xi /V po.r Cs) . . , TheV\ (XI 1 • •• 1 XnJ ,·5 q raV~dovVJ .. X n;V) ;: VVlax tX In, I XIII~

SaWLp ((? o-f S'1'-te Vl,

IS ~ lf V,

.. .5'1ffose- }jov. 3e-f q111 ou-lcovwe ot (~,5,1 1 3,'1). The"' X.n:n :=:. rvtq x tx, 1 , •• 1 Xn ~ == IV! qX ~.~I 51 7 1 1,. t...( J = <f •

. .

. i

.. -----. ..... . ... ...... .. . ...... --. . . ···-.. , 3->. J~>"~---z ~ a·--·---~-~t-------------. -· 1A .1·· ···}oQ rd . Jr:n··-·~r~~~~---,-~~y·-·0f·····\;o_~ r b lh?;t5? ~ .1~ l ~-(·!·v~ij··-

--·--·---"" ·-···-· -----------· -----·-· ------------·---·

·-··-·--~~--~--··~~ ... . .• --~~--~--·· .-.~--....... ·~·--~~·~~-?.-'.0]]_·······~····· -·········-··--·--·~·----~·····~·--··· .••.. ····-··~··-··~·-~···-·~~ -··-·-=·-·--···-····--··· .......... .. --.............. ---·

.. li~~~j~~do(ttlc?xJ~~x) '(lj =-<~::;•A'Jy-ua·y~---~-~= . . .. fo zczx·fi';()~-.... ·r-y;-b·---~X 2-'>rt··trx1'X} -~ -~-zTr··. . . ... ----

---·--··· . -....... ······-----·-·--...... ·------r--(}z-c-z~-r··xn····· r~-~ ··:z.x7·-~x·--r--c.2x71xJ$--=-~---·--·-·· ....... ----·--·

....... ~-= -_ _ ... ·. ·--········ .• . .. or~~ ~fO-( l'K''KJ 1· Ft' ,~~ "-y-\A~T rp o~dti~--=---~--•.••• ······--·-cr-rz-ro-r··------d-,.-vvt····---···--·-q-rocr-··------~~-z-tr;_J·:p-rxl?---·---cr·r:z)· -----"d-snb:5~---···· ·-···--·--------·-········

·--·------7--=·of -~ r· ---·· frJ ~' ·-·--sy------V7 oTf~'i;zx)~-~. ·s:i:J.oJf _____ 'd_ cr;· ------f-1:5"\-ff --·· ··-.;;l 0 {\!---· -----···--------·-· ····

.s 1 -S'J7r->:tfcF_s ___ T3pTcr-------&~-T--------·-}-o-·--.{j:sTii:J"?-----·· · ----------

---------------+ lA_'Qf· ~Vf1 IAJ~ + C :t I }fJ -rn QVJ lA! t \/\0} . -----cr.-:s-•yy· --------------------------0 wory-dj JVV'-·rrs--wop·LAb1-··-o··-s7·--~; 'X) j1 ·-vioTOdlfl···

I~

Prove this for discrete random variables first. This provides great intuition for the continous case.

32__

·--------·------ -- ·- ------·-~---------------- --- -----· -- --·· - ·----------- - ~------------------------- --------------

. _ f)!:1_¥e k, ll 13 ~-~_L __ _j~_)__ = Z_x 1.fx 6- (<)I ' 2 J _______________ _ __ {~~-~~_)_____ Fifl d_ +~e jo,'v11- c16'Y15'Jy _ of 7 hf> orc:Jec _sj_q_i.s_ ltcs7 ___________ _

____ tL31 JL'.:J 1 .lfu-.-------------------- ------ -·--·- - . ---·--· ------·········--

------------·-·······--··-····--~~J-~?o+~~- :: ____ f~_j_S;~.f(J.;t -~-}! __ : ___ ii~(_ ~ (_ ~1 -- ..... --------

----···-······--···-----···------·------------·-----------·-·-··· = ~1; ~I ~--~-----i~O_S__fr-~~~7~---~-\ L--- ·------·---------

_________ _ . . _____ N o~_L:L~_e__ __ ~ ~."-Li ______ ±.Le_ .. d [5j.l~b-""j~_~f_,~ ___ 5_c:!_-j,~._ ____ _ ___________________ _

----- ----· -- -__ jk_~..m£_s)j_g_l4-r-___ :~ ... d~------c..c.t.JCL. - --· Lrll.~T[_g_f_e ____ --i.o__-y-e-1-.--t_h_g_ --- ---·-·

___ _ __ .m.cu.ry.l.vL~{ ... _______________________________________________ .. ___ ..... ___________________________________________________ ------------------------ ___ _ ___ ------------ _________ . _

·--------- -------- . ----- ···-·· -- -ro·---'j--'2.--------------------- ---- ·-·· --------------- - ----- ---. -----------------·--·--- ---------------- ------···-- --- -- -------- ... ---- ·- --- --- ..

. • ..• ····-·············· =jt.Jc j,_J..= f- .. { {('f.J+tcr~ +1~~ J ... ~-!--d1'd 'f3- -~~- ~- --~ -~ =-. ----------·-·--·······-··-----~-------------~------=q)--------····-· ----- ·--·- --·-- ······-··· ---------- ... ------·--·--···--- ·-------··· ... .

~-==--- -~~----·· ________ --·-~-=.~s<03 .' i~ f£:~~:r~) d:1$~ _ - . ~ _ _ . · ~~-- -. ---- __ _1!-_ - .. ··------------ ------------ .. ····- ------------- ---·-

~-~---_ -----~- -~~~~---~---~~- -- -=-- sJJt;~t;:5[~C -~EZ1~_1]~--~--~-~~- __

__ _t_bJs ____ _re ~_C-11-f co V1 _ _k_ __ J-e..tJ_~_s:_g__(~~4_ _ _ g_s _ ~folD;; ~s __ ;_

- - - --- ~ --~-- ------- - - - -- -- ------------------- --------------.

1/t P!Jf<> W1 ~. If 'Xw_. , , , k.,,_ . __ ;.s. -".. ... L<ll'·-~" ""'- .. :5_"'-""' f /, :L o.m <;!

{jt 5.'"l. J (Ov'\ {i 1<\ L(~.2_ ___ d_:'.2_j_r_!_h_~--i~9_1\__-/ -th f' __ IQ_ ········-- -- ---------------· -- ---- ------- --

~ -1 ('1-K .. -- .. -·-~· t T;) ="{:~~ JFC~J" [~ FC:J)j. ~~fG)~ -~=~ = _ - ----- -- ---- ----- ---- ------------- -X--- --- ------------------------- ---------~----------- ------------------------- --

K~ ~"'~

---- -- -·-·~----------+---------------- ------·---~---------------------·-----~---- ---------~~ ----------------------

33

/heo\f'yt'\ ... FO('" an_j (no1 _)v5T CoVJfr'VIuou$) J-,·5+r,'bufiaYl 1 -fhe:_ (t,s.s) .. cdf 0 f +he k:0-- o\cle\ s-foits{c_ -t\cJVY\ q ravtdcy¥7

.. $qw'f/e I 5

Y\ • n- ~.

F XK~h (~) = ~ ft';)L (1- F('jl) •

ExoVI'Ip le~ ;, Let X, 1 ... 1 t ~ L: t- d FXr{J). HV1d The_ c_ df Qvtd the del/l,51"1-J of' x,~V\.

.. ~~:~ (~)-=

••

fP ( X:V) ~ j) = J- rP (X,: n / J )

:: l-JT!P(t>J) ~ ) - U-. F[j)J n

1- e -::Jr.

0

::;/0

j~O

:; fx,ob) = t~~n.- ~ ;~: , . No1e ·1 ~.~-f i1r(l.~.f?.Y!1.. 61S, 3 ... ~.::Yo~ /d. h o ~.- e be f~ ~'1 .. v.Vlple-q~qVIT) bl-{+ tMe c_df +ec~h/~L{e 15 eas-j .. +G use dj,ec+l:J• t i

, Ncde fhQ-f oL(' rf:Sll (+ Merle k~s t~,€'ot€ ~ 6. LJ, 2._ ··r·« ................... -...................... . ............... .

; '

---~~-lA _s_:p --------------~

------···· ~o-·; ~-o----------------·-----------------------------····-.. -------·-----------········-·---··--·······-·----·-· ~r=·-f1 ----------------------------------------------------·--. --

. --~(I '0;:. h n-.. )~~ = I~~i TxJaTj .~ t~=-='":Qdl ~~hJ~3r=-n--~ ------. -y-·····rrf ___ -.r*-r;~IA.oT-·-r.T-------p-~-r-~-"lf>-f1AI5 ____ \.<wx----f~.~j-----------Tj;·z_-;tJ-

---------·-------------------(Tr9-TJINn ····· -r-T5-~--vx---rT-r-frx·----l-dl ___ :a1~!M Pxg __ _

.. -+yn\n'iT1~1fV1CYJ ___ s r--AJ--;> 'a '1 ro,--)[ -Tro -:ror·-l..X) ~---t~~~t -----------------··---····----·······-_-····-············------=r~~-----x-·---o-r----~~!710YJ=hef!Tf~---r;,r---~-€--.s-c;7\ V' o ~-----·-····-······----·----·

--·----·--······ -----vT -----S-dT-f1:>r-Jn-l\----~u-~oJ ______ J-o ____ &5-Txa.-l1i-~l(----··· ------·

! 3S

{\} o -t e f hq-f t h ~ 1 ,,1M ,'f ,,.s no i a. eel f. . .. (_ r +5 Vl6 + ri J !.,+ - Co Vl --LVI (..{ 0 lA--S )

-Do~s i his

0\ -t-hq-f

J/h. E'Q P'l -Jl,eti i-f do(:)5VI'{

l;.. does Vlot

CcMve~e io (OVIvfJC:::

7 r. V . . C(

No . T + c o V\ v e 'J f' 5 to :, def,,Y7 i i l a V\ a--t c..oVl'"' e r:y e V1 e-12

need VLo+ ef uq l 1 (x; q + , po,·vt-l.s o-f' dJs CoVI -/-/VI L{; f_j.

0 ~ Ned~

'fhaf

/VI "ft,.e_

/,'I!Y! ~ (x) h~C() /llll

'!

fxarvtfle, (1.2.3),

Let Y, 1 ... 1 X"' be ~ \QJ1davv1 5ct~Ntf 1~ ffowt

P A rz ( l 1 J ) so + lrt cr + !=rx) :: \ l - , ~x >< / o,...

• r

';

.. '

He., f (:f)::(l-( /+~ 'C' ~x,:~ \ J

(_ (J Q 11..-v-' f/1

'j '?()I \-e-'j

~

~ '?()

'I 0 6/"'v'. 0 ol~.,cv'

; j

'

ExceWlpl~ (_ 7, z. Lt) ..

. '

LoV\ $"j de~ XVI; r') tiOVV\ 0 ( ) -&#0~. tAR l,l . .frx)-J+x)~

Frxl= r :• X: 70 I

r. ('1);: i(~ y ~ /o / x~o .

ft!n ltj 6. ---7

0 0/vJ •

• I Tht.(s xl'\ ~ V\ dtJES ~ (_QVty E'IJ~ to a. c v. ••. .b~_(:;_QJ6:~e__ .l~~co······-bx;!v.(~)- =- 0 ~--···- i.~

l\),-1e Jb.,.+ the c.d {' . <:.oy,yf;' <:J "7. ... rp··-.t.,.;se +o ~E' t•o

lS_ -~Ycte _ Jd'·fecet'1t _ . {ttQf f· <::. . .. Cqt!do~ . vctr;qiJ/e

C-:QYI v E Cj ;_VJJ .... J[) ~e rc ,·1;1 c-Ls ir; b 1{-!i 9 f1 •

0<orV11pl~ 7:~ z_ s-. ~

c, ... ;der · .. ~ X-,~ trovYI t~oe p nc ..,.,· ..... s eka..., f le

~Y.,/:'1} "' !P( ~ X-·~ ~)) := Fx ( nj} ~ 1\:n

~

ExQ ""t.p.le_ _ _]_t'"2.7- .

IV\'j \~

\l+~ j J :J 7 G"

G. 0/"'-'-'.

[:~(/~ 0

-

'j '"?(;)/

O/'w.

36

Fi'.~~d t~e (lvVl;i,·v'J 4;s-iribw-l,.oVl of l, ~ke\E> A~"'iid N(l<,cr'l}.

R._~_c:qf( +h~-t - .Z "-' N{~)- ~J~-- -u ~L~) 5 !P/l ~)) ; f( f~ ~ ~ 1; ~ ; ~ ( ~ ~~ ~

---') .. -0 1 I

J ~ ~/ Vj.::;t-(_ .J I

j (A._

__ Th~------~-----c_c::>vt verqes _____ ,:_VL ___ ~:s.__tri bt.l-lioV! f.o 6.~. ·-

T~eo\~M . Cr,'3,1)

.J" f M 9 .Vt1-e.~i Je_,_e r-Q.±:Lo-.5 ... fl.!YI. c tJcLn s: _fil¢.(t.:) ..:,./'1 (t.)

'31-

iV'\ <:::t r.etyhhor hoo_d . _o_f ____ Q.., the"" fJ,.e

qss<:>cict-fed .Gv,5 C:.CJJ(LvPcj.e L,V\ J,,s1ribl.11t'oltl: Y ~ y

E;xa Y"'f le (+. s. 1)

.h .. . . .•

~~d . '-=~t X ;'V 13£R ( p). L ~ t y Vi ==- L_x i rv 5J)J ( V\ I p)

i==-1

4 L e. t p :: p1\ = ~ ·

.The .Ill 11 (1=) =. \K t

Y~ L_YI e_ ~o-~~~

- [' +(e_t-~)~r. ~ e~(et;-i)·

.. - Thu <; Yr\ ~ _ ~ PQ-b(&J ,

"·- ... ·-· . -·~ -· ~-

Exa_~p/e ( 8e,Y\o~.Af~ L_k_NJ ... 1,'3,2.

~et ... Xi~- BER{pJ

/'1 K. (-!:_) - & e_-tf~ + I-f r --) ~pi:

/-

Nt)+e pt:

1 ~ct"f: e_ - I$ .th~ ;tiG F o .p P-

T~us - d

X~~ r I+ rf

cq.V\ _be _5bvvv-. (ex a W'lf Le . f.?, 3') fh.ctt

f.. = i., ~p . He~ n (t-J ~ e c/z ' -~.--~-.·.--.·-··.--- .. .'t . . N~ p(l-p}

d ·-Th Lt 5 Yn ~ ~0 I l) •

lbeoce_~ (The Cevt{(c::tl J,.,,..,_,;J lheoceVVJ)

3ct

( 1,3.1) Jf x~ ,.._ t:~d(l{,_g--z-J v,_,;Jb- /1, o-'- (_co/ fheV1

Pcoof:

X~-~ d ~ N{0/1) o­-~

t-o\ 1 h ,·_s fto~:\/- "'-..te

/'1\§ f of the > ~ r '-{ e (.1 c.(}>

·-

Let M{-L ): MJ~k(t)~

Q.>su~.e

e x--t'sis,

A ~ylor serif'S fot: vvlt) 3Lv.e5

t~rt~1 ft.._e

m (_t_) :::: h-\(O\ t VV\ '(o j t t- VV~~(:3t~) t 2 -l'ot' 501tlr Q! ~(-t.)!:: t. 2-

~----- --- - ~ --- - ~

- h-\10) + /1 ( t2!t;\ \ i 2 .... ... .. c ____ k~L>...S:.~:l .. .). ... ___ _ ___ ... - __, ... - __ Z, __

- ~co) * a-· -c.. t ["(£fF)) -o--: Jr Now,

. ·- ·---· ·--···-- .......... Lef ___ lY\ -_ . X~-~---------..

t:r" .---NV\

The V\ 112 fi\ ..:: ;Vl (-t) ;::. M (~) · ~ ~-A{) L(Y: -~) .

o-~

=c V11x.J~J ~ rx.-~l!-~)J

= Ith~~ ~ ~ lM(O) + ~t ~'(S~t~z](k~~

=[M Co') --~- -~- _ * _c:l{_ n)1- ~ '2..V'\ - t-1.

.. .

"""'1--te ce. ~) ~ 0

~[1 + ~ + ~]~

-t:/z_ ~ e__ +he _11<5F \or

.•.... ···-· .. --· ·-·····. . .. ,........ . .. N (o, I). a

39

------------------·---

ExC{ V\'f R /e (~.3.t.f)

- ~ . - . . -· - .

L e t v. A.- ~ ~-· d 1 1 N .r F ( 0 r) - - ----- _Ak --- ------- - -- y --- --- - ----- -l - -- --

-T~ e. V\ V - .L ~pfroy ~0 J) 1\tz.._ - 2.. ~ I

1'2..

Tl-te +ex+ ~4 e + -Ttt + -It,... fJ,j · "'(-// . ., -- :JJ .5 :S q - - I 5 qo "{ f tJ Xt VVJ _ _ Ci _

So c_ lose ___ f hr:t- j: __ 5"PMf2 C ovY?f 4. fe f' Q /yo f ,· 1 brn s

LotS e fh i ic Jenecerf-e Y/OIV"'LQL( der-f~. ------- s

'iO

' t.J

YY~cty

Wh1 noi ____ use f __ ~'(!l) i ~ :5-Jectd Z (sfe "theort?,.-, 6, ?,Lf)

Giher- <!o""-'-"" s; <:le 5 1..

No -le ·f h_tt -f Zt_-l = 6 qo~td 2-6, 1.-rz ...

Ts _fhqi_- -- g ? pr:o ble~ ,

ExqiNiple (r. t-f" ) -

5e.ppos-e 'jou. c {,;E>,~~d c.{t:?/ms shf' __ J11akes 50% _o£ __ her ________ fr-e_e __ f b r() kv'..S. __ _

51.1ffos_e 5h_e_ ~i/( Shoo -1-..s 2~-- f/me 5 5o

t bq t Yr.- f3IN.(ZD7 ±) represer1 fs -The tJYVlllbe(' o..P sho-fs _She h.t;l/ ___ rY:t:t>tkf>.

(_~ {c t1 {_q_{-e f~ qJ {P(Y ~ 1) = 0, T 4 _&'3 _fsfJ~VVL_qlf ____ ±h.~_f_ __ f ( t Z _C( }_ ;:, Cx. -~L~_3 __ 0.6726

- s II-

-:-\.AO:tr·Jd j 10")

b -·vrrl"'vt -;)f o };Vt,ifS~ d ~ + ~1\f".JTJt.vf? vQ;i~ · 'df\ -

The ore 111"1

(1,5.t}

'12_

j_-f 4-~-;·H-;X~ is '{ ra~r1da~ saJA-'Ifl€> froVV? a CdY11,'vzuouS d/sfr,'bwf/o!A;

{(xr) ~ 0 ~- s-hert> Xp 1s -ft,e p 0 pe cceVI j,'/-e1

q111 d k" ---7 p- ( ~ ,·j t, _ kn - n p Da~.tnded) _ ~

fheVl

Vv i +h t .. : 11 b qsy ""r*-"~trd tty rrDr ""': I meq n. Xp qYJd ... ___ _ Vqr iqKt c. e- C /n ..

IV! o 1her Vv ()i"" ds"' ~·=" - Xe c

tfY\

No-lF ft,crf . <:--z. 1'5 _ def/v,eJ Q S

c' = p(l-p)

[f()(f )1'-

d > N(a/ 1)

w~J C.q1'1 1L ~e 5Q j 1 hct·i

hlhy ~""''± .. JCxr) = 0 <:

V ~f\(A c~\7 1\ Kn: v, · · ······· l"-l0'f; ~) •

Exqrnp le : Let Xt- A- ~ .:_J ExP(I).

(::'{,),2.) Let kn_=[o/z_ ,-f VI is. eve....., ___ · n+ I ·f I I - 1 ¥\ is o!;J.d.

- - ..... 2.;_· -· --·--·-.

The~ XK. . 15 i h€. 5 0 lt'"Vt[) (c: t'vte dic:ua. "' • r'\ I

' \A

T

v:.r ··T

\ z.) fo;

-z(~) (T-~l)i

\ ?) { o I = VtA 6 (z)hol _ =-~-(=

= 01 d- ~­IM

? -T w-

(::::

0

-xp · ·· C> \ = -z -t.-) I

~

---

LiL-f

···-·

De_\r'/11·-/_to_~ -~ __ _ .... .1, co"' v Its J ~s r'"> pr t. h~ 'oj./d) -f. o X j_f _ _ Jf({~ ~Jc) 2 £.) _ -:!_o__ _ v _ £_ 7 o_

T~eoceW\ : __ (oJ\_V€.\Jfrl.L~ J .-,

.C..C l"\ VfefJ€vt C€'

p to h_q h ,·1/'!_y it¥~ f 1.-e 5 d . --/ ·L { . _ t..S ____ r, i2 4 'a Y1 ) l· e ~ 14-1

x~ ~_x ____ :=--? X!\ __ ~ X,

TheQ~e M: If X ,·s Cf ~ovt ,-fqv1t _,c., theVl

. -- . 1P K_ A, <:. -=) __ X~ ) c

"-Jc;te +-~-gt

tb-f +oJ_ef4er th~s~ fhearevvts shov Xh -4 c. __ rf_f X~ ~c. -\o, a~)

~-n:SiQV1 t -~ •

Thea~ -em; ( TLb. z_) - -

_ Lg __ vv ---~ f _ _L_<;'t r J e N _l..f ~b-e cs

1-f ihe.Yl

XL ~ i ~ 4 c~~ q- 'L) ,__; th V-4K -·--·~·----~--· ----·--· -- -

~ -4 {'in ·J r:r./ o- 't. -- 1'[€)

JXooi' : _ g_{i;c,a.fl f he Che b :J ~ kte v . lnef u. q/-it:J --~

. _ .. wax-:?X)J2}<.o-2.) ~· t'c.~- __ {QL ~7QI 6._ /._co,

.Le_± __ l<.n_.~ f-J; cs--- . - - -· . - - ---·

The" lP(lx .. -1"-.) ') E- __ J ~fP Ox~-Al /;;. ~) L o-Z.. -[z..l"\ z.. ~0

- IP - cL Th lA j X~ ~ /w{ ==) X~ ~ /<. I' p

Theo r~ VVi

(:;' 6 * 3) ...Lf (YV\- rV\ J ~ 1\J(o/1) 1

c. ~

flrteVl Y~~ M.

Pfov;de 5cVY~e io~~+\A;~ioVI <:\b(H .... f- why 't-";s is --lrt.{e.

TheC)Cf'vz-?s 7. 5, I q n d

fhctf .~""·~· 61. ~ Xp

f, 6. 3 foJefher ~·f K"' p ar~d

---?' n

a. s 5 lA mf h Q.Vl s C)f + 111 e fit e 4 r e vvzs ha ld,

:)ho~-."" i he

45

lh e C)'\ e m If lP

VV1 --=, c. q V1 d ;t~. f j I .S coV~fi¥TUous ct'i c. 1

(1.T,z_) ihe V\ j(~) __,. j(<:),

proo~: Let £ 7 Oo__ (.hC?o,se ~ 7 Cl s:l. tx·- c I~<!) -;;> /~c~) ·3(c.)li~

fh~~ /P ((3(X.)- yc l t / [) ~ f (If" -.:.1 / .!r) ~ O,

0

N.R+e. 1 hct f the- above proof l;v'O\}c:-5 tc.r f'\'tV (f; VC<\;Q-h-

-\uvtc"'L't~vt s.

Lib

T k~()f_~~ - ... - - _f.s_?l::l VVI e> v tP

j\h ~ c_ qY1d ~ ~ d.

(1.1.3~-I h~V)

· tP l) ~ Yto\ ± .h Y~-- ~ cr C t bd

z) XV\Y~ 4 ~d

s) _&_ ~ 1 ·-- . ··- .. ------- ----------·- ----·-

c...

'-t) t ~ ± If c"\6

5) ~ ~ G ;f Y~-~ 7 0.

P~col~ A +riv_lq f opfflc ctf,·o,., c)f 1 h-eor-evY1 7, "::(, 2

_, I heo1::e V\/1

(i.]_,tj~ S/ L-tt.sk j,..5 T£teore.vn

As5(,( me X~~ c qvzJ ~~y

-r_~e"" c:l

1) ___ x~L! Y~ ~-- -~- +- r ~)

d. 'i~ Y~ ? __ ~ y

3) Y-/f., ... .L:!. .~ c.~ o

. ( 110)N ( f>

(t 'Q)~ ~ p '

t h~s{-h}$

( d-J 7 tl

((\{ 11)

(

d1

~..~~-~~·Jr ·· \.. J-l)dt ...

~ecre«\ (r.i.6).

Proof:

Lfo I f . \(., =~m ~ I'J.(o,. 1)

c: ____.. t-fv,

avtd t'f 5/(c.) e X15fS .. .JZ I~S V1¢V1,.... ee>r~ ctf J;VJ_,

th eV1 j(Yn) - j(~) d> N (o 1 i)

lc ~(~)~ ~et u..(j ) = [ d('J) - J (~~-

J- W\

0

I - j {M)

N c ·It- f hot f 4 1.5 co,, -/,-.,., fA o v s

'-,·-!~ ~A(VVJ)- 0

1\Jcvv,

j(rY)) . - '(If"\ .J = (c g~(~)\ " ~ I

__ YY1- ~ \). j 1 {M) r "(Y.,) - ~ 1"m)

y:fVVI/

'j .:: VVl •

~1 Wl

g.:f_ . fheatevV\ 1.J. 2 1 gm) +~ern) --~ i, Jf·~1)

The.., l)) 51"+5kf> tht9reVV1 1 'th_'t' r-es vt (1 ; s e s+q b /,·Jh e d,

D

Tl-te­

~f

A,~

+exf ~s~e 5 l~~~J~C~~)~ (_ ' IV.

The C£P<>J . ..-e _ t-J.zeace~.

q bs;:o/L( te' ___ Ytz!w es nee e95_q ')

Ycr +t-.e de #1 t; vw/n t:J-hr

here 7

Ckqefer- __ S: ~~--fr:tfJ-f-ILc_s t::lnd dSCl.mfL'Vt Cf Pt5-fn'bui,·oJ;?~ - J

Th/s chqf{er def>n_es Q /:>fcrfi5-lic' _ q~J

J f:IMOVt5-!rct '{f s 5o01~ p_rap-~e~c --//e 5 of "ttte

$qw,f/e rvzeqVII )"/ _q....rd ihe Scz14'1jlle Vqr,t:rr'<-e/

5~ -

De f: _ A 15-fqf ,, s f,-c' (s_a, _{u.,'iioVJ of ~ Sqwtple.

.\

fht.ts \-t l~ c:t Tqvtd~c:)(,~Vl vqrtqble

is =;{./de ly de ie r ,..,/,_, e c:LI. b) Q-r . ____ CqVI~d.a.w1 S q 1/14 P e.

fxq""fle! The ~CI!""f}e_ .. IYJE'C?n X"

K, = t(t/ X,_, ••. ., l.J _where

,_

whose 04.i-cowte

i-he o <..tfc<!J vvt ~

IS cl e f,' Yl f- J e.t s

t (x, ./ X "t. I f ,, J ~ xt:\)~ .:::. X, + x~ +- x~ t-, f , + X.,.. V\

so ExaVV~p fe: Tl e sg W?f le . v. a. ' 1 t::{ VI, ~ S 2 I S

Theo re ft'(\ (~. 2 f ()

Def:

.S 2. = t{J', _ _, ,t I II • , / fn) wh-ere

t (x, / X z ,. . · • • 1 X.,., ) -

1'\

X.. (xi-- x") z. ~:,

n-\

If Xi "- i. ;_ d (~I o- 2) ) fhett~

E(f:) =/'(_ C!VId vqr (A"~ J = ~~

Pto v E' +h' , IS fheor e 1-'V\ f ,,

1t s-lq f,5t'c) I; ,·s ~ l.fn bict sed f'cr ,<f E(i) = 6

0\ fctrq Y'¥!1!? ffl r/ &/

N~i~ +~~-f The qbc-_v_e J/tC?cre"""'1 5-lq--fes thctf X j s ctY) . {Anbiq s:ed es f,-I'Vl.a fc; r- -for A.

ThGts OV\ averctc.Je / ~ . J

,;/( be .l'-l1 _ b~,.tT qJ be ~lA; t@ d,·f-fere rtf__ trov.,

oL£/coVIte s X

p a- r fl c k ( q r

,;t(.

o-f' -)(

~-

fWC!_;j

Hovv-ev-er J th~ vqric,wce 5htiAk..s QV! Y1

irtc:reosesl' S'<jjf!>S-ft"') _ Q $me,/!.,, "'"d 5rna!!t:: r c hQvtc.P fhcr.-f X o~ c bicu'.-, ''""'j q""

X f~ed IS fete {roW\ /-1.

Sl

-~- -~- ---- ----- ---·· -·------------·-·--- ------ ----- ----- ---

E Yq""'R /e . Ld XJ, ... ,X.. ... "- ..: icl i3 E R{p) ('$, 2.,2 J ... ·····- _ ·A· -- . ----

L~t _p _____ ,t_ __ _ be +~e SC~~fl~ · prcpGr--1/e;.,.,,

n~"" . ti lfJ-=: p a-. J va, (P)=' r (1-p)

A

NQie . f ~ otf ... p •'::> q"' v.., bi'Ctse d est/~" l"r 1o.C py

Tlteo,-em (~.2,2)

J:f ~~~iL'd {Aitr-z_)) +he.Vl

e cs ~) = a- z.

VQ( (S ._) ~ ~'~ - (~~~) <>- -t "' 7 I

h

~h-e, e s'-1 = E ()'-A ) Cf '·>" ft:t e '-1 t!1- C..f: 11 { r Cf / Yl1 o Vf.1 @ II'! -~ of' J, ,

Pfoof' t ~ut-f_ E: 6 '-) .::: 0"" ~ : ..

-E -S ~ = E ·[;ttJi ;·f)"'"] = _L_ -[ 1\(Xi. 'Z. + - 'Z. -2 . ..... .... 0_ ). .. . .. ... . ..... . . ( .. -1) E L x X. X) n-\ ~'

= ~~~ E~~ x,~y hx~J = ~ [z~~ = h ECK~\1.

= i;[ h(cr~+A(~)- ~ ~E{XtX~')J = 2=il"(~~+-.')- ~(n(~~·)t0,- ")H~ J

d -,--= . { .. = t 1-t.)] '; .

---~-~E---h'(kl1}±~~~J~.R >~Jl-~ -

s-s --· - . --- -- .. - ... - ··-·· ·---·-·- ·-···-- --- ----- ----·------

Tl1 e J !5 }_r_~-h~ft'oY-L _ __o-f __ JA. sk+ ,'5JJc, _ ,·5 _ _ ~-_g II ed q

" Sqm.fl t'ny ..... D.i~lri.h r.dLrutt '' , - ... -

Ei"' <L Jb.~ S<?mf. (iJ.. d;s.·fr;b,dJ•.., Cl{ ,t J

w h _e r e __ Yi- _ ~ _i ~-d _ __ 1\) {k1 cr 2) _

E I • _y q Y"?f re ._ _

. /"1f(-t;) = /'l f Xi 0) :: i_::,

t ... ,~\tl.)n "" ~ t--c-rtz. rr ct~J _= . ~." " . •

1.-.::l

- e ~ t -~"-(~t t'"i

Th~.-~..s X~ N (,((/ ~)

Ex q ""fk : f.:. ""' (~.3.f)

i vt4 e_p -e 111 cL en t N (~i~ ~~)_

_ __ _ ____ ~~~~ J~~ clr~J6 b ~.~ i lo~ o--\' " Y-= ~x.

-- ,-:,-.. ·--

J'1,t & ) " g--1'<\ .. it) -..IT?"' t .. "' ... "~ t. \_ ----~--· ~----A,_ rrt - 1 -

- -- - --- ~~I

--~-~<:.) -rt"'- cc:r·2. -- ")~~ '"%: £. e - --

T~l.cs Y ~ !'J(i~-~~~- t_ c-,. 2). ,.. ... ,

·~

S'1 Exg Wlp&. _ _ ___ _ _____________ __ ______________ _ __ _ ___ _ ___ _________ _ ___ _ __ __________ ______ _ _____ _ [<g. 3 _, JJ _ _ ___ .I_t_ __ ~ ___ c..Lq~~-d ___ t~g:t _______ thf' J;_{f: t ,·VI'1_~-- ___ 9_5' __ _

et ... _ c.~ r _1g!~tl __ _ .hL~Yl __ d ______ .Q.f ______ _hq +ieij------ __ J~S______ _

_ N_(6.0-;-S-~-). ___________ _ _ _ _ __ .

le 1: X\ 1 Xz.r ,_.~ 1 fz..s-- _ _be C( rqnda.,.,_ 5oY"'lf le .... ±r9V"' these _h_~++~ ri~S ..

.X "" 1\J ( 60 1 ¥s} 1 f the c/ .. ,.., ,:r ft<t ~

If the c( a J~VYI _ IS -1r'-fe 1 _ fher1

fP(l >c) - IP T ~ = \- ~;o)

--· --- ' . -? w~~-t \jq_{~,t~ o-\' c. wow ld . roqk~ . tkr is • r,- ·' c-6_6 = -~.o5-= -}.l'-15".

6 --~-

-:) •• c:: - 1. 6'~ 5 (~} t_{O __ -_- - 5~. 02-6

Whqf ltvOL< /J 'jDJA C.0¥1 dude !J x = S{ 1

fP (K- /' 56) __ == l_ ~~~~---e~)- ~ J _ ~ ,_; 3:?, =- l _ Tl~~. ~ ~1- - -- --- - - If-\ - -- ____ !,_ ,. J ~- - - ')

. - - - - ._lr;} . ... ........ .. . . .. .

--- ::: t- .-o6-c5'·r·--=-----~·16f'lb ___ _

S)

A 5f" c~.,( c g se ,-L _±he <;).,..,""'.. __ dfs -fr:i h"--Ito vL ''s

_ GAM(2>~} ;~L "-(vJ.

T~eorE'Wl (~:s.z)

Let_ '(~_;?_~(v),

- . - . .. . \ . ··)'Y'z t :: .. - ..... The"' f!lyCL 6 -zt

=: (\ - 2 t) -V/2.

~ (Y) --=- __ y

~(Y~ _=_ v (~_-t?J

VC{\(y).: v( v+2) - v z = Z v

TheoreM I_f .X~-~ttM(e,k)_~ t-t-t~VI. ~ ~ Jt-' 2~k) ( ~~ "3,_3) e

. - - - . .. ( ) ( I Y" ( I )" fn.o~~ -- _ 1"1~x(t) _- tfx-l¥ ::: 1-S(!f)j = 1-2-t ··

___ Th~s ~ _ "'"' GA~( 2 1 k) ,-./- ;X-z(Zk). _e_ -- .. -

l=xawrfle (~.3.5)

Theo,eW\: (11.'3,t.f)

proot:

56 . S_1.1pp Q.S.e __ ··--'j-CD4 .. _ h9...v.e_ o. Y\ i~ ~ fct b /~ +ha--l

+ell S !j-9-~ _ tit~+_ ~:JQ(Yl;: /. Q 6 a "'Jj .... fh Ctf

a S9Jctr- ... p_gne ls. [,·{e +in-te_ t·s .. GAM. (31 "2. ).

For- adyer4i '~""'J' v.-e/d /t} .. ~ rolo of S'~Ja.r pct:vtf..l5 ~-.,'11

6 c I OYL J e c.. F)ncf c .

lc Sqy ft., eJ.t lt:t.s t ;:. J ect r.s_

. I 0 = IP(X ~ c_) . = fP ( ¥ .~ ~)

), '1C e ¥ ,.__ ~ 2 (i·-1) ! ~ = X.2• 1o('l) = /. 06

Ttt us/ .C.. ::: ~ ( /. 6 6) := /. 5 '1 ~ €ot\ s.

- ----

5 '-! ppo s e x~ AJ f rtdep. % ~ (VL ) . i'\. . .... . . 1\ .

The v1 ~X~ A- :Z,?-(~,~) 1.-, I

(

lfv• ... \. . . . j. z.. , - . "" - ~.--1

1-2-t • 0 AJI - ft - r. ~-- ) )":./z

I y t:_,.r±) - ... _·· ;11y,, r± ).;:: rr ___ . ... . ... L- ~.::. ._ L' 1 -zt

. . . a.:::. I

T!teorettn; .If l:__lv N(O, D 1 fhe.n t_' A./ ~~(I ) (8',3.5) .

) -X~

-·· . t-l' - .. fx2 j_ e. /'L d _l ~-1_\X'l-proo~: /1~, ft) = t: l~ ... -: s(::' .. .rz:1f.. ... X T£11 ~ e d)(

.tR ..

Co\olfq\ j (ce, 3. 2.)

.r~ Xi /\..- iicl N ( .t(' """z )/

CQII"l th~

+tt~~ #,~t;:-r """ Zzt"')

a~d Cxir 1'- ??~{!)

fhf? C~.bov E..

distr:J, w ~h()~ caro /fa ry_ be I.( sed o.f ·. 5 ~ = Z(Y; -fL.

..... 11- ,

J)epevzdevtce o-t' -lervY?s IS q rrobleM?rJ

fo '7 ,

f,· ... J

The ore vvt : I-f' )'; / , , , / }'.., ~ C ~ d A) ( ,t.t 1 c- 2 ) f h € 11

58

(<6,3.6) 1• -

'

i ~ Je:'f€>14 de"' -t of [~-X I i://?, ... / Y~j -• X IS

2: f IS r'lld ff~VI clent of 52

3 : VI - l . 5 2 "- :z_ z_ (.,. - J ) . a-Z.

No1e ~ Q )z '"' "\ z

thqt 2; X,; -A = l._((xc -X)+ (X -A.)J t.....-1 o- L - L

.;:: I a-

::: t~;.-X)' +(X-R.)2 +2(x;X::ftX,-X•-rAi)l i:-, L~ o- z_ J

:/[ (Y, -xt.) ~ ~ (x -;t,c)z. + o li:, o- 2.. o- l..

Th u5 fhe joi, t clensdy of' Xt, · -· / X, - (x~- ;(( y~-

' 'S

. V\ I z c:r'Z.

+&.,x,~--·,x~) ==IT e . _ , M rr cr-z ~.--

=(~"~>a-; ·exp[b [(x;-R) 2]

s-1

I) {?~~c-~ · exp[fo-~ ( ;~x0 -x)c 1- n~-~l•)]

1\Jva,.v t Oh s ;Jer +hE> j ot'P1 + ~ r ct n r fe;~r vvt ~ t/tJ Y1 ~

Y,:=X) ~=t-Y ~' i ~z~ J,.. .. .,, .,.,. ..

"' TtteY1 1 x,-x ::: -2/x; -x) = - ~ ::1;

..:~z. L-;:2.

s~ !;.(x;.-xt = (- ,_{ 'J>) 2 + ;f2..c!j~)~ t-- I t. -

60

Novv fy(~. ~'•*•/ ':1") - 1:1\ -(? 11') "/z. o-, ' expltL(~1·r{t~·}o(,,-~)i]

)'\

Afso; x,= ~,-Z':J~ i=z

and 'Ji, = :J; -r j I

lh~..t -5 J IS

q5 q prod'fct { l.l nc ..J ;·o.., of

c o vt 5' -let Y? i a y1 J .f y C q..,

of q flincl/cm of' -o., /y ~~

c~ly (':J.,_ ~ ~3/ ,,,, !f,.)~

T h u 5 Y ,_ / s i'Vlc:J~pevr devt f o +' Y,

C. e. 'iz- f iJ /., df'ft>"' d M-f ,.-f Y) -

f:, ~.: 21 )I It • /I'?'

b~ 1/1..--r ,·+4~.-,

qnJ a

"vh J doeS" of ~;_-X

ih/s M~Cil'? 1hof r /'5 tl?defFH d~rlt -fCJ, C;:.. /1 Z I' 31 , , .. 1 ;, ( e veVJ ,; = I ) 7.

(]:) l s ,... () loV' rrovey?,

To prove_@ J n c.de -fhq f 5 z_

o"£ ~X~- X / ~.-: r, 2,. ~,) r1] h-hlc h

. 15 q JI..{V!Cf;()V! avtl'j

,·n J e fe;1 de""~ t of' ore

X.

For~> ~nc,// ttu,f ttJc -ft)'- ::: "' (j-.« )"' ~ {~.:-f)' ~ -~ - -~ ~ ~~

'1/, = 0- r) 5~ =:: t. (X.- -4) ,_ - "'(l-.« t ~ v'L -v, C"'.l. (..- l J_

o-·. a- '2.

•

'l,-Q

'(1-LA)-z, ?{. -v ~5(1-lA) tn'11 ..

-.t+~~ (~'"W r :I:.!.) -<?/~~ w ==' (+J ~w &= 7./.,.\ )

( 1-) 'iA w <:'!)'A w ;:; (+fA HA vJ =\1Y VJ .. d I •

19

Exq~fle :

R~cqfl

.'rJe Cctv\

t')lc~. ke

6Z.

. 5yfosx Lf i5 . c/cr,M-1 e d iA"L __ a_ . cer -1-cr<n

~c; nd o-f . . 1~1; 'J. ... h." s ____ .'1: .. L .f'e {,'w, e ... 1-LQ-t l.S ... U.(£ -I . 6) .

5"1.1 (7fos.e . Is irl..{_e

6 f 2 5"

we will o ss v r-n.e

. LAVJless s ~ lro_V"!

f h C( -f- iht"s

fs < 5''-1, ( 5 " .

o \A. c 5 ct-.. rn ~~~

.If . the c..fq/vrz (s -lr"l e / k..../bt¥f IS

+he fro bcrb,l,!J . Jt, cr1 .~;v. ~-i + err-on eo~-A_sl~ '( _.

w ,'/( rej~ct

rr(s 2 f_ 5't.65) :c· tr /(YJ..LJ)s< <- ~(5'1-65 J' . 0 36 )

- JP (xz(l"f ) L ; f c/9 ;,., is

(_ ~ (t;<f. 65 J) ::: o. 0 s-

irLfe

±~crT i~N(;{, ~·) ~hrv'"" . X1, ~·~. / t"' r-- ~ ~cL _ N 0/ a-~).

~As-e. X .. as c:tvt es-fiyV(qfor tQc A q nd -1(;. i "'' fe '" E' 111. c e r-- ·-- -- ~ .. i- ---' ·-~'------- " ---· , .. - ··--' --------- _;')_

a-. me ct~s vve dovt 1t Q bil. .., t 8 / bed: nof I< riPt.vil'l.J

knoa.v 10. clis-fr,'fo~../{i()fl o-f:' _f. k!5 -G <' IN1 M Cf /L}_7 in .(ere~' c e s Thl5 COvt {d p cev evti

a_~ ()vt 1 ... /d. 1

65

We L-v;ll see_ f~cd X----_ 8 _. _ _ hQ.S Q._ . d ,'s fr; h 1.-l 1£o 111

5 __.. .. rr;...

thqt is not iY~{hr f'vzc t!' £1 .. by r::;r-.

Q e~i t.1 ; 1; on : It' __ -L ..-v l\)(01 !} q VI d V ~"- ~ <t(v ) are ihdr'fe>"'Jfni>

theV\ ~ A-/ t(v) .

ff

_\,J he \f' v-..,,' 1 [,

-l:tv ') t E" p r e s e v1 Js ct ~'+- c:1 t's -/, 1, h~ -l~·o "'' jl1 d eJ r-e e s c f -\ r-e edoV'-1

DeS'c ,,'be the sh4ee of' ih~ dEY1$'I'Iy f'-fY!C ·{,(}VI i:r Srt:tQ(/ _ C£nd fqrye.. _ vctlt-1 es of V.

-h 0 \ ___ eoreWI • Yf:.\\. LT) . :;:; )L_ . - ··· -- v~ '2.. -~ hf' Iff --~A- ifv)_ a~d V /2.

('3,0,2) proc:rf>: E-(T)::: 0 h'j S:JMtn eity.

~_(T')_ = f !_. __ ~-------. r.. _' \ ..... ; '*"} E(V):: E~_.·) . c.~rr .. . . ..... \v - - - "tz. -I - Xtz. . ··:... .... t.l.-1- v jj_ . L ··X e. \

- V EJ - X "12.. v cl.X 2 -. -v. . .... - rt2~ .

- v/t)r(Tt_ I v3- \ -><lz LZ Jii~zilf(~2)K ,._ · e

- v (1) r(~) -2 •.2. ---

(

\(~)

- vGJ r(y) tfJrt~)

= _;;t.-. v-2.

b'-/

Jx

S"J'vcce rcx+l)= xr(j.~. c i,f, r ('X) .= (Y- I ) r (X- I) )

Nett 1heti -\or pos;i-·"e ir~4rJers ~, r(") :~- \) ~

Th eo'~VV"x (~,Y.3)

[-K ~ t:(n-l) s 7:..

'P{c>ef ~ FRoM t"ke"re fh1 0·~3" b 1 f 1 S

( i-/t< 't. ..,. .;,;;; .J:(n-t)~~{ - .G.

---tG

c:r-4.

-(.Y\- \)

1' f Y; ,_~· Cci )J ( ;t( r tr')

ar-e r'Yl Je fe'Y!d f v1 t'

• tJ

Def: Tf VI/\.. }?2-(v,) 0.v1d

ir1depert der1i J +hen

{!;)

~) .-...., f ( v,, v'" J

{s--"'· /,' V :£ 2

"l, - C11z.) ~-re

wke re .. F(v,,vl.) represeV'!-15 ctV\ 11F- d ,·5 +r; bvit'oVll, ~,;v ,·1 frt

11 I \1 V1 V1vv>1erq1or qYid

I! I I ,, V2.. c:LEJtlomi;,qJor de j re e s cf

-fr~edom~

Desu:be hoW' He dcn5tfy f,..,M l1o"' looks fo..-Va dou.) \/ct / u e s cf -f· l-te pqr q //he fe r J,

The c ,. €v1"'1

(~.4.s)

v'z. E{X ):: ;::1 l.../her(' XA.- F(v.,vL)

YQ_' (X) = Z v~ 2 ( V1 + v.,_ - LJ v, ( v. -z y- (v~-11)

Proof of pod j_: .E(X} = ~1=(1/.) E(~,) ~ Vz E(-&.) = ..;,: 2

N.,fe 1!ta1 we '"51' -!he c,t/c"(,Aio.-, c:S E({,.} ~ro.,., U..€

proo~. c)1 f1€cc:eVV'L _<K .. Vf, 2 .

..

ExC(IIYJp/e: Le..t ,'l,_ h~ il-te 5"!.!J_ percevtf,/e --~~--£ F{v,/ ~).

66

FilAd t~e 'i51o of' F(vz., v,) ,·vt -lerWI s

of' ~.

Nofe l

-1'\.,..-x Tkq-1 ,- f 'X~ F ( v, / v ~) 1 fheY\

F( Va 1 V,) •

0

• • oy==f(X?K):: IP({-z ~)~ {-!P(f'-~) 0:=-) .~s = IP(* <~). Jh ... s -i Is fl,e "' ~ rercer~ ·Me of F( v~, II,) .

. .

Ex:qwtffe.:_ Dev€fop ihe idea o-t' q ca~tde11ce 0 0 i 14{er v <1 /, ( IJ.-;~e "''·d :f_i-Je J €11etlll s:fE>(ls)

~~u~,X~. ~ c:~d ~(.tr,/<'IL)

~ I .~I Ya1 "' .; ~d N L J.(r.l a-,. 2 )

. -~'

5wppose ~-ve w .. ~T -lo es+t'.,_,qfe c::r.-2.

I -<Si. 'Z. •

--------" ----------

.. (Jt-1) s.z "-' _J? '(l6- \ J

..

o-:'Z. - l

_. (z,- ') s~z. -"--' r-z.( z,- ') _

...::-)

-- c::r: -:z._ z._

S,' a;"Z... S'L.._ cr; z.. ..,_. F(' 'i, z.o) .. ...............

C'i. ----

.. =f u. l{) = f(L, (l),zo) L ?f:ri," ... ~ {_.,s('<;;Z-o)) .

=If (.t-13 L ~~,:.,_ L. z. Lo)

=-f(.Y3 s; /._ ·~·· ~ 2.ZD~ ._ 5, O'j ..... ,

. ... -----·· --,·--~-·- --· ·-" ----~- ·"

.. 5'tfase> )a v. do fl.e exper-iYY~ e 11-t a.~d c_g b.Jd:.~

z. 2 z._L-l s, ::. ~_z. ---- _j . ---·----- ------------------- --- J- - t row1 t£.t~ OtA+CaWl e -- -- --- - - -------------------------- -- -4 --

···-

··- hJhai ___ Cq Vl __ --- ~ f sory Q bo~ T (~3f£})2.2o(~))

.. : ~ La.iL_,li,LJil} ~

-----.--------~ -- - ---

6rg .. Thf> q,have ,'w 1f'r Vt? / ,"_s q f{J~ eo~~~.~·den. c~-·---.. Jr1tervtt L __ .feu:· __ ~~o-~~~~-~---~~------~~---~--~~---~--"'--~-~-----------O"jz.

Is ~ 7.. ditferef'l+ th t.\V\ cr: z. 1 z.. f

:: F;·,5t q Yt5 Vv e c

.. deve/cp 1t-.f ft,,'s

id.eQ tra VV1 -the CI .> 'the,

of o. h J p o-! ~ e s t'.s -f e s t.

.. 5+eps -for c~q-//~ ~ CL

j_) f1L)<

qo/lJ

0-.Y\ exrreS51QII1 ,·1>1 VD lvin. J o.. .Sfq-/isJ/c

the t.~lflKttCJ~NVI pc:traW~e -1--e~ o-f /v-(-1-erest.

.. 2...) ltJr;f€ ct. p , o bq /;,'/,'ly e /'"' ~ liw1 ' 1 ... .(~/,

The> a bavf' ex pre S"5 itJV7,

.. 3) So/ve ik~;de IP {o\ the pqr-~ Wleler

of \tr+e\esta

:· ~) Wc1f-e ctc> \,v"YI th~ rqnda~ it1 ~f?\vq {

:: 5) P/uJ !'VI the o 1..1 +co IIVl~P +C) o~+q;Vf tlttf' c_J:,

...__.,..- ---........ --.__....... - --- ~-

6~ S+eps -fa, do,,~ a h_ypo-lhesi.s it?st;

L) choo_s~ a nu// h j(O fj, P 5"/s '~ fJo.

2) clto~ se .C\ s-fq1/5t/c.. ~has!~? d/s/r,'/7v-lol'1 ~~ /cv-rrJ~,.:vV? t..t4rler J..lo 5uch fhq-f

the clr'sk;bf.(i/f.ln /s f'1-1iie cl:-fi'~r€'Y~i if fl~ ,--:; . n~f true,

~) c koos€. y 0..., +;t"" .I e nor . r- .. -1 e fill!,} is cq((ed -fhe "Fifh·r /eve t) \ e~ fhe 5,-~e G-f ihe +P5-I::

Y) D e-1-er mine 7 he c I /~/c ct I re; ,"Q V\'

5) Do +hF Pxperi,-,e.-t"t <>V/J pi";! ft,p a..,f.:c..-r~ 11.,-4. t~e- .s~i/s-1-(-c .... _

. (:) Rejec 1 Ho ;f . s+., 1is-l,c. . ~lis ' ,,

the . o u1C()wt(:'

the ... c r; ·he e~l of fite

r ey ,-o"'.

C~opie, ~ - - Po,v'\ + E s+trv~ q "f,on 1-o

Oc.~qs5; ot~q~ ~~vJ:J{ _ alfow-

a. .f e "" , .. <:zXi o>:"l s c r q s s"' 7, /;.,.s us fo c OY>t f ff' le!J de-le r ,.,; n e i he

d~s-/r; bc.~fi 0111 of' ~ fQnJoW7 Vq f ;· ~ble~

Ex: .. ~ Lss o..

be the Cot'VI s- -/t'vY!G' S' Q v?d /'2 f' X

n !..( 1-11 be r o-r heads. If 'the

.. Col'~ rs ~t'r;; X ~ B:cN (S:/-i )

.. UoVv eve\ 'v\.../e o-t'-le Yl CQ,., Y/0 -f·

.. the exqc-{ dis lr :bv1io v1 c but Vv.€

"krw~...~ the -fcrm i!J c..A' di.dr • bvflows .; 1 i caVV?e S" froV¥1.

cle--krW1/"1e V'-1cJhf fhqi

For e)( ct YVIf le / q.. P<'Jf.'-'1'?-(,o,. ,·s

~e m,iyhi k ~o~ t h{J.f r1or mao~ d;s-fr; b~.~Jed b ~.-~-f

.. no-f K tto "v' fJ., e pa rq Yl'lf'> fer s/ ~ qYid q-~

-1- i Vl1 e.s / ":"' e vvt "') nc;f k ~w c.~;; 'I J,;";; the J,~s-f<, b,..ri/()Vt of' 0< popCA/eti7'ov11 b ""':1

0-lh€\ • i

.. Cl bov-f

.. ~r.,; e cq Vl stil / esft'mqfe pctrQWie4et.s St..Lch

.. qs -1 Itt e ¥11-e. Ct Yl Ct vt d me di'co,.L

.J'Y\ fhr's

.tq~;~ e-h-~pler

,--_5 k VI 0""" (\I

V\...-f> ae s: .1 ~.-~ vrt e -fhe J,·s1 r; b '-'-{o 1;1

.. I

-~., .....

f/

.Let X_, ... 1 X., be q r o" cl o..,-, Sq "':I! /e toY¥\ {;a 0 -1/o!-k:'>Lo~ fo -~~~~ t --~/({

tate Y1 b '} t J . . - _"'&ILL_}

. [ -f&; G) 1 G E9.-3_

... N.ci5? .. .for

1 hc:t-f . .5]_. is ft,e sei o-f> pos.r/b/~ 1/Q/L<PS

.. be ±he: PQ[QJ::ne ie r qrtd . '/11«1 e Cq v1

VE> c -~o; ~v qJ LLe d.

.. Fo\ e x q .v-nf. le : - ·£ .I - ()(_:.. ..... }"-. e z.-o-2.

. rlZrro-; rLr=!R/ a-"-::>oJ

.. ..2. Is cql/ec/ ihe /,-• \I

f<::J.CC!i"l't~ f@r Sfq'- e.

Ve--frvz;J,·on! It s-fech$j,·'- LAs-ed -fo es-f-ivngfe 1-he vt::tl~e .. of 1-he Fc;Hqvvze1e\ / e1 l>\ q ~1..-(,;JC-f.t'oVl.

t f_ . l"' ..-:v{_e) - c_ I I d /; _/ ' . ..../ \\ .. nececn:) ./ ........ ) 1s .qcte ...... <2'"1. e.S!tY'Yl<:tnu-

-.. o-t e ~r 1'tf2) re5f(=>cft"'vej; .. J •

.. Thf>. Ot.-1-fco.dfe . D.-E .... t~ e. estim.cr.ff!2.C _t's cc:rlLed .. ... av:1. _ e.s..f..lVl<!.afe~ . Ih..\.-t.s aVl .. .esfJ~ q los- 1'.S Gt .

.. r:a.ndoll\4 variqble ovtd. ~·fs ... Q(,{-lcome ls q"l

.. es~,'mCl-fe.

1-2 A

.. l--Je ~.;vi /( of +e Vl use G ~~o rep C"es e ~-+- a Y\

.. <2-s-li'wz.ce..fo r or QY1 es 1 I'Y'Vlct fe o-f' f /.1{::> p q rt:rl11f1~ r; ,. e) e 1 J ~ /1_ "". Q VI d ~7- C.( re ec;: -1 Jy1/t ct!&Jf $ / es ••hrne>fes

. . "{' ~ q n d o- 2 res peci!'ve )',

-f,·,s+ VVIe ft-zoJ -fer C..{"e_<=t --lt'vr_y qY/ esi·Y'1-1ofor c q 1/ ed fhe /r t11f> 7 hod o-f VV?oYI?& rl is.''

'.lhE'

. 'cs

A rneihoJ o-\ rvtovvteVl-/-.5 es-/iwtf:fJf.), 01f1E) IS

.. der1'ved bj se-/-//v-.j the f'f'-'!ctflo~ t"'LOVY1PI1{5

.. e f (A a ( -1- o .s q JNl pIe yVt o vvt e .,., -l.s q ..-1 d f he n

.. scsI v I'~ -for + he. pq r C( wt e ie r.

E >eq m f /e. {r,Z.IJ ..

.I.f lfve' c:t \ ~ fl"l ieo· r Ps1@J I;·; 1he p~ lr; v /~ {o ., m~ctn/ ~ 1 c...-e se-t

1"\

A.= LX:· i~r - "'

(nCJ solv>'J ;e!v/red)

o"'d i·oss Q J-1~-f cV) i?tP F~QrCtmetPr +c de VI o-+e -f hri1 ,·f 1:.S c::r, es1iYV'Lcilor ·,

~

M - tt ~·~ I

~ ......

. Tf iVtec~ o~e f"Vtt;)\e

<:>~ es-1 I M or{; "'·J ~€ .. . )

.. VVto VVl e 111. -t, ct fl\· J so o r'l ,

par ct vvt e -1-e- r..s 1""

rvrvv~ -1-o f~e

Y1f!ed

Se>co,.·td

. . .. _ ..... -·--·-·· U7 X

-·. /UC.X

0

e .(;l ~--· -= (~~~ ~ x) 3-cu-~-

.,lAolpq.uf~'P ... f:v.c,-.>~,t .F!J<b'ftt .. di-J e> s 3 W W d/\~l1 1::> a'1-). ds n ~:>-::> ~1 0: 1M o}'·-·

.. tAO,Ifh 1/Jf~,' p o'LfJ ra·J.' :J~S . J.1Va/12:7lj .. ;;t"""1··· 1-b"Lf f d ?/fOfV.

-·--------..

ozu

0>7L 0 --6l~

d --(o~ .:-)x'Ow-

' .

, ~ 15 -;~".t r :0 ~ ( ·~ Vf

xq--=:t:v :::: <e G · · 'Z-t./\I 'Zv

~7: X+e·· -z _s. -z-'Zv

-

t-5

SlYLc e IP (X? 0 ) Is ~ { L.tr?C J/<u' of e~ 7t.

.. qytc{ k---e q.fr.~~--- _h_~ v~ e.r:.i/r::nt;¥1acs ~\ ih.cs~

.. ,4~ es-1,-,-n~{"s .. . f~r /P(JZ()) c.qn be o.b.Yct/nec/

. b3 fi"YJ'"J ,., . 6 '7t f",. EOl_, ~;

~­.. _fp(Xz_a)

.. Lhere

•··

j_

e i ~se ... ., ..

., 1z.L6 1

.., ? . /t -0

ees <l b-ic, 1Y1ed h@~P? ~

--',_ '"''~-·

h} ~ J I, -;] . ho · &· Joo, 1 z ~ b)., ..

-·-··=-

I 2 I f' }?o, 'J 1 • 'J I , h) I zr---:·

(I 'l) \0 11) \ I '0} (oo) d • <

D Z.)' I IX .;·,

)_~F~s,.,o)·

s-, ;;; '~ ~ sd10~ J d-d \75 ..

'[t ( F ~0 ~ -f?'l i.:~ 1 }11 ~-'-"0 d>j ht 0uo i.Ab;) X .. ~,,..,,..,

lXE -· ~-_}Qj _jww ..

d~_L S'l

Sl f' ~>".

4 (df?J~ 9 -v ( d /1 )rff8 ( s·z 'b) d;d w bx3

Jt

7-i

No-+e -H'tq-f i~ -p==-·2J +~e"' .lo,.,.o) is -the ot.tfcoM~ wd h the ~ 1~f s."l __ p'\o b~ h; /,'/y.

'r f p = • 8_, -- t h f .. rl ... - L t, I) .. h .. ~_s __ :f'-t:> b. 7 ~e.s+ f r ~ bet 1; /,'1;·

.. ~oweve\ _. k.-towij. .. Vqrio.\.otS

Vve c:t '\: e Ylc.d \ ,, + tt ¢='-' s; 4 (.{ q ~fiovt f? . . .<? dd___ _c cxn p<Li ;';' pro b~ b;/; hu o u""( c o.V"' E>-S. _

.. r~~tr1e-ctd / v-...-e V\1<=1Ylf +c jt.it:SS. _the

~~ the?. fqrql1'tt" +er b R.sed _ CV? 1 t-,~ o-f +~t~ e>< rerime,-,. t.

~f

~-~

Vc:t.J!.) ~ .

Ot.tfC' G f\1\ e

.. s~ppose .. + ke

... L s (tl o)~ 0~(-lc.C;vY~e .... ~f ihe f::'>cfer,'Mei(J t-.

Which parqme+er woo.,~Jd prov:dE' ±he h:Jhest . .. pmb<r b;/;4_; of ;3e+l•3' -1 he c.,-(c o>+te (/., 6} S ··-·

.. We c q ;'1 fo.r.vw '-(lc:t '1- e_ _ a.. _ f~!!~_i;(JY! ~..v1.ose da_vvrQ/1'1 _ ...

.... ___ )..s. ...... '1 h~ je:{ oj' .. po~_5;j;Je ___ ~QJ..t_~es. . . _q vzr:L . W~Ps- e i • .l.h _/ <£)_ • . _ __;_~ Cqnje .. 15 f. t' ---p-~u:R.J'1:(;,J'E' .. C ......... 7lfq_c.. .. ~-.- -

-------------- ......... ----

. " _ \ • 3 i-f~-l4) f: [C!?A) ,()J_Q)J; . --·-· --~-----/2--- ~_.__L _______ LE C\(~~-4') E lL4-D}J / -- .

.. z i-\ /itiX'l\-=CLt)o

------------------- ------------~-,----

1.

I fO J; W "- Co.4 fd de . (t .... Sitfn'Jqc p l<> hie rll 'IA/dJ,

l· _Covri in u o vt S ('. V: S J b.lrlf 1-f w ot..tfd n '/. J!vt_r.tk._e_

l;- Se ll'lSE' -f-o Q ~ k J /I flv lJ /c. ~\ pq FQ-V1ctf'J~s:. --W D.l..,f.fcl j d i j ' ' : -','I r---e JJ t . p provJ· e _ :he ni_yhe.;·/ ~\ oa~ b, L'!_J oi __ Je_u/~---- _ ~e .

I ~ bse r "e d . a. c.; -I.e""" e."" . be Let 4.S e ±h. e p rotx-b j'!J ... 1S .:Cere {or ~11 pqco_rneiers •..

1

1 l r"' .s 1 ead / "" e r C' f I q C(> the Je f ~ "'p r oJ:x, b,'j;f)" i vv ,· -/ h '' I! k e (; h .o1d " "'- " d t.v e c cu> c.o"' -1 1., •-H?

:~ b _:1 d e.~~ Y7 r' vt j -ft-t f' /,, k-E'(,· h c c d of . a. Y1 C 1.-1 'fc ""~f fl I a(" -he -< · ·f -'!'.;, . (">,·-/ .a I Jed _1_ JJIO o ·1 M 0 l4 .... • .. ... . J ().V1 . a.~ r. . .;HJ ..__yq '4 ~JL .. QT. ('J'\...._ .1.£/C.O .,.._

lj TL_.- -f h? }, keftho,d 1's '? .f'~.enc fiovt of ike

it pa~-a me -fee. •I

. \

~J_V1 -1~e {qs--f E'X qvnfle h..-e .5cr~ i~1. for q

-1 .(; ><' e l o<-1-f e <>>M E. l", / x..,) it.~ I i keGhco d +("''" -{;.,_, (vq s

-j r/. . r f.+X~ Z-)(,- )<._ . { d J ll ::JL)(,,x.._; P ') - L_P (1- P) . r E r .2r·-37 .D- , .. i . _ _ ___ 0 <dL~-.

; ['. I

----tN<?.ie __ ±h.~t--L ____ Since ... x.,f t~ rs. Q. __ Jix.ed o~..ticavte/ L~-

___ -~} i-1 f5 f~ ---if~gj_-t_ {Q~J_)7 (t~_a}r11.,_~1')J-y w~- _ d~~i:f __ ~eed__ . ________ _ _ __1"ljc::. spec_j{J -· 1ne _____ Vqft.te <:>4 .Jhf __ jc>_l'_..fj__ .Jeg_.s_tf_j __ ..

_.:[«:._ _ e:.-f hE c . ':Jq_{tA..Ps Q-f _(x_tr.K'?~- .. -- -- -- - --

t

.Jk~Nty~~) . \IV~ s.h..ouid Sf<:L;.f_y p -~ _L~!J + Re1-1ze m be'> 1/ve

£ () • ·' lA o'\: "' -t_V! ~~ q. . ___ _ 4!'1 c L a . . f.J ,

. ii.s Vc:t{ 4(? ... ±or . ct.\e

. !

.. C:..J=2Vl5Lderfh).

.\! lj i I ·~I 1'

:/~ , I ,. , i' €' d.. ' . • 1lh~ m tho o-t mq XIW1.41'V? ,/f..ef, hood .t..S

l! +o C.. hoos -e the. p cr rcz rvze -fe, vq {'-< e fh_qf II · j[ Wol.( {d p~Ov;de +t.rte h]he~-f //kef/hood ~f .. Jj i he ohs€Cv-Ld C:::n •. ,_-/c_ Owt e QS the esft1t11 c{:/~

'\~f o{ it1e pet \q v11 e ·f.e r, !''! !': "

1-: Th ~ e E: Q~j t'Y\ Q X + (x, I xl I . ' . x., j e) 1- e ESl. I• r r

-1-Cf

Ll . - ... . ~- ~-

i.J IS •·j ' .

!A INl ~ k",'VV! ~ m lik.eLihoatL .. e=st/~ ~ic~- .(fotLE)--: t

. !JN,ole ' +~1 ; f YYI" 0 ;ek f.Cl f<t"' e fer s ' J .. llke{!h/)eJd 1 .fkte0 il1-P- 111---c ... 1.-s

m_ qx_-t'nt z i. ~- .ih~---- __ J1<2f ... un-'J .u e~-----------

r·:!J::L_ ~--- ct (5~ __ p-o..ssth.Le ... _ t_~gf ___ flc _____ gr_JJ4-¥11£~- .. . .. _______ _

I. ' ±"'- + ~ . ' ----.:--LI(l _______ e_ ___ p_etLQM.re .e-.r Sf-qc.e7 ____ ~&...-7 ___ aJ_qJtl.mL-Ee.-s ----------

. ____ _ -~: __ ±he_ ____ J,'t<.e{;_ho,d. ___ .. Il4-- t1t;s_ C.ct~.J"--1'1.0 l'i .. LE ... ex_,:'5i$-.----11' , I

c~:pT~=-r~:pp~:e u 'I, ~ ;:, ·; iiJ =-e~-=~ ~u-- n. ··--·-

[q,z,6)_jr- .......... . __ . ___ m------ ~--J_k. llke(;bood .I~¥.LC-kf\ __ / _L(A_)_7 __ ,_s__ __________________ .. -----·-··-------

I

l• -------·-----·---------8---- ---- · ·A-------------------- ----- -----· --------·--------·---------- ----·- --- ... -···-·--- --·----------- ----- -·· -----·--------------------

_· L( A); 11 -;.. 4-ti. . - ------ -- -- .. -- . _,, ______ -·---- -- .. _Jj --~

!J t;t X· I -- ··+.!··-·--·---· --- . ... - .. ----~-~

B ..

·---·- . ---------- -·-·-·-- -------------------- - ·- -·--- - ---------· ----------------------------------· ----------------------------···

__ ______________ . ___ ll..b/f_ w-~~-t ... ___ -f~ _____ {J..Yts:L ____ A ... ih_qi ______ V\II.:QXl~L~~ _ ..l.(l) ____ _ ~ ____ n(<>c g~;v:~~~ ~rt d =c\~~n _ (O<ts~eL_j~~- ................. .

r

.... ·--··· ·---····---· ·---······--.. --· ------------------·-····--· --······-... ······--------------·----. . -·-·--. ··------------· ----·-·····-····----· -... -·-· -........ ··---. --. .. . .... .I. -. --. --

--------------------------------------------------------------·--r --~~-----~w----~~-+---~-1----x-; --------------------------------------------------------·-·· -----------· -------------·---·-----------·---------------------··· .. -----------------------·-------------------------. ----------1 --------------------

················ -. ~t1mrr·-:Ya7Ti(;f)Xi>W ~-f"'ft ____ ,jOdA: ····· ----· -------of----~raw· -·(jr·,.-.;)01'11:) j--J ?1-r;; I;---· asoo,;r-:~ · 1"'1o-~ --------

-. -t .... 1

------------------------·-u------_____ .. _____________ ----~--·

-----) ··-· -· -· X -1 :::2. · --·t

-----~--------------:-~ -__ -7~~ ----_ --r---~-:. --

--.... ----·------y. .. , /--1 0 ---. ..... ··-·· ···-·· --1---. ·+···

-,~)( 7 --··------·-·-----..... -... .. \.11-

l . l·

i'

~--f -

. I I

-I

Theo'E'W) ( C(,2.1) I

I

I l I

i·l

Tl-.e In va.tlqvtce Prop€( ~

It e 's -H~ 1'1 L- E c-{ 8 I fhe <-<.(G') 1 s i ~ e ~ L E o-\ ld( $).

i 4

Pr cO~ i-l 0 { Cct se Vv he(' -e LA :I

,£ 1- fo-- (:

~~

11 L(e 'l := L(u'1 (u(e))) = L{u.-'(0)) __ ~lere Q_-= ~(&) II ,J

1·1 A

T~ej I Ll+:> Cs Mq)(/ml~ t cl at e f

~~~ A ~ L e~ ?! ~ a. '5 Wtc; x L( <-<-- 'tO)) __ so 7 h.(}{ -I {[

ll lsi:\ 1 he . ~L E. fa, 't::.td_((:;\~ . ....... ... . _

1~1 _ T h~ pr~o1' . Cs

! I - A··H .

.. ((.-ii - Li(~): .. . Jl

! I i·l

CoV>"'p~ie ___ if t.......:-e .C:..Ctrt --~Ita~ ih.a.i

. LS·(J§5)J -- L(t9_j.~== Q -••-••-• - - -

ti --·· . - - - - - -. -- - -- . --- ..... - -. -- - --- -- -- - . -·····- ··-· Ex..a.";fle U let X 1. ~ .• 1 . t ~'- _l.Jd ___ E_xJ2_j_Ai _.

, I

( ~2 . ) ij - " -x-vA - - - - - - - - . . . . .. ·--·-~_•ltLfJ).~ .. Tr fe______ __ ... _____ A_2_eL

I. • I . L:l I· I 1"1 . . .. .. . - ···--····· ---····- -- . -· .. - --

ii 6 o/'v ·1t·-· .. -------· ---··-- -····-··--·-·· ... - . . -.... o. ....

k . ( --~- --· .. ..,., ··- -·- --- . ··- . -- . - ····- -··- ····--- ~ : i ..._ _ x~ _ f .., ...... _ il - - 0 - _ -V\ - G.-A~_ . 11 io3 L{)) ~-=l_c_~ - T --- JI:Lj)'+- --- ·· · jj L-- I /'\

i

h..o -z "?;..$:) -z ~e ~

o = ·2cw~ !XJ? ~ 4·· UT u~-:.:. ·re·t . . --· ... ------. .. .. ····-····· . + ,_s;:; z . !

. . 15. "' )r (:_ ,., u ::-:>"Kt-c -0 _:_ (:;;,.;= I X) "?Z-1-::::' ~-t ... -···«·····------··-····· ---------·----·-···-···· ---·-•«•««··--··--·--··--·-· .. --. ------· ····-··t .. ····---------------

2Z

• z~? i ;;lV-)'91: _:. (~~foorJ: ~ 7~ .. ... l ---- ------·-------·--· ----. ...... ---------. ---· -------··-· -----·--·-. --------·---.-·---------·-·· t--

( ~~=!~;; -·.· c~~liz)bo I --r}? -Sr7bor:co P"0t . ' i

I

. . . d ~l_lL ····~ (z;gtJi)·7 V'Jqll ..

' · d r' · ·~·· j<>J · 9ft,/ "'11 t 0-:-.. -, Y/,--

: [ 1{1 .("! J C;Jw! $

· Jos · ··~1w d~+

~-7W .dlf+ ~~-. X . -

!:',1 . ~X~ -\' 6 'I. -.. -~--

(

' ' i . 0 --~x2···+-·f -lA -( r . . --. --_,I

-i ( . .. Y . -.. -· . Yf~ -o-.:·· ~xz 4· v-:··· ={W7J'ol YH

-·---. _________________________________ ......_ __ _

j

l I

<65

lA..- ,· t 0 __ x ____ 21'11 ce . -~~ _ ~l ce_q{y __ s ~-Lo/ ---l Ke:>pla.c e. A

l t~ t i·- q . . /'1. m '-'tsf .. b~. .X Q_f ct ~- jp_~qj m.q.)(,~I.(Wl; _

j .l 1

I -~--- +--1., .. '1 5:c(x.~j)z -: o

~-.., z -/ -no- -t- L(x.:-x )<! :- G

9 a- 4 =- L. (f (_ --r )<-

r"'\

N CJ '"'""' ciVte ¥vt us+ v e r i -('~ 1 h. ll( t i ~ ,·5 Wl ctX ,VJ.t,'-ie s .J.. •

.. The_ b,·va n~+e ~e eoVJ.d d~f,·v~i,'ve ..... ftt?s( ,\;; -Jh/s:

~x1 )

_,f:J 'J

D = d.e-t

J'jx

/0 i rnp/,_- t"S / d cq l ex1fEhfc.t 111.

D )0 q~d l'_)CJ{ ? 0 ,'w!fhes loc..ql Wlt'h"

.. D2_Q _ qn_d . -~~}(- .?.6. ~

1·Ynpkes. /Gcql mQ)f,

-~t I A - "" :i( - )Z.. --~-~X}--~~ =--___ X~=-~-- do , · + L. J • r ... _ I· t d IIJtt. t?XI.&.U.-le>- "'le t~fll'l o.c. . .

"'

q\_~ __ j __ 6~ I flc._'\f>_ ;t1L~,s._

~~jJJo.±r .. f.bcr.f. . .. ihR_ ;V/_Lf ... -\a'\ l. I

.tr ' L'S .. (lgJ _ . t.h~ (11/!1£,.

Actually, the MM and ML estimators are the same in this case.

lA - ··-------------·· ·-·-···-------·-····-~----.. -·-----------··--·-----··-·-·-·-

~-X :::-4t -:'X? ::: G (

a:n·,l5 . ······ !

! ' ' ' ... ~-

'

o -= 7jv---~---:x~----+ ---g ~X--(_:::= -

{oz{} 1"1'X'?~~T (F + ~---(~ybo 1. IAJ -.foce1"''x7 J~$'t ( (-ae_~X)-(9)b"/.J7

.. -·--~· ' -.

7c"2X ~ ;;; t : {e/?Z!xJ-j -(?L-)<) ----:

-l1-1 -, TV( r:-~-~-'!l' --;;,sor)/3 .,

'! i)

i)Foc J'l L-

..r,'x~d e/ 11

i l

l.s /ctr-Jes-l

liTh 11s X11n .J I

l.l t~J PI'JJ''J

l-1

' ilte IS

Jh/5 If';

,·+ t's c I e 0-~ f_h_e:, -L ,.,._,heY! .;e =X,:, •

~LE._ for /(~

fo d./ dlit I ""-e see

J·.-·1 x--. "'. 1s r1. /',!,

fl?e /'11.-E .fo, e ~ l. !-_)

u i:

<b)

tk-~-t

W'e no~~<><ve !"'-" Wleih•ck f..r _c:k-f,·.,;":J esf:*':-k•s, a,d

t.ee<;\ 5'"~e c f, 1eria to dB-fee...,, n e 1/'vh_;~"' tS bt?s"i~

Pe i' ,·, ,· i' o Jj .r + ~ (r ) ~ ;--_@) V 0f_S2. _, T ,·s q n u" /;,'«sed ~ - ~-(Cf,~, l) t~ ~5+,k~for- o .f 1{€:)).

~- .... - .. - -- .... .. - -

~~~~f)'" -f:Js-"ff"'Se . X ~uJ N~/- r-} - ---- --- ---------------------

.. .. hi~----~C( ve -SE'_~/11 .. . i~L.cr.f:_ .. £(X_}=6_ ____ V& __ §_f{_._ - .. -- --------IhJA~-- X l's q'v] ... ui'Lht'~secf_ es:iL~L---~~£-~_!_______ --------·

---- - f --- - - .. - ---- ----- ------ -- ------------ ___ -~.•.·.!_Le_{ _j~_J-~-;1!gi! _cr3~A;._ f-:::e;-1;:~ _af 5!/-'</_') ___ _ _ _______ _ __ -fi-Ilt~ 5.. r_~J-- -----------·~ .. ( __ _) __ __ ________ [ ____ £)__ __ _ ____ __ L{._'{£_ . ·-- __

j.

~ . . . . ... . . ... . . .. . ----- .. ........ - ..... ··-· ... - ---· ···--- . -···--- ....... - ··------------··- . .

--~-~[''if-r l£q_I}' ~f~'i·7So} . He:d';;~Jl' ±~~ ,.;:s M _______ · lu1---~ ___ e.s-LMC( o . .. \' ____ .. ]h.... . ______ .... pe ___ niL'f..L._ -··--· _ _ _______ _

f J t:

-----__,.~-------------~---------------~~

~6

7~.;~~fe- · ~e=zJL Tfqi __ [£ _ _XA.-_~f\_11 ( 6 1 /<}, tAf'Kl. _ -~ ~ -- 'JL ~ ~- "'- --~ __ (zi!) ..

~ e -l----~- ~ -- ~-~ ~ -~~-~---~--- ~~.'

J_ ~ __ Re~rdL __ tflrd ____ EXJ~(e_)_~ §A~{~L._L} ___ _ I

'-'" .

Tl-ru 5 if x~ '\.- ~(.d_ ,.EX~P(e)l Th~.r? 2-X;, ,._ ~ ~d g ~(z_) ~ ........ e

a~d L ¥/- ~.21:2v,)

z. -59 ~ fh-cr t. ~ X_ ,_ ~ ('- "')

f"lt-1 -\ ' ~ q Vld e c e; x l = 2.~n

59 tltgf fff1~ (!) C/11 d _ ~~A'~ _ J~ _ ~~"'~ U!"!_b/t:~SE-d ' t?stl~~lor -6c e. (Noie 1 ked' X is qlwaJ5 Ullbi4jed to( £}(~

.--.----

s .. ,fo se tl ov th .. i :1 °"' v..-a .. -t .. -&,,. . . e.rt·rn q -k I ------ ~~~~--~~ --~-

~---e- · ~ S';_nc,g ____ x___ -~'_$ ___ t~E> /~1kE .. ±.o.c .e, f. t's

_f.he~-- A. I.- 1-=. ___ ~-5:.~.!:- .. i . .- ----~-Uo~ ev~-~--------~ ; --~---~~-~~=~-~-~~--~-~-~be _5 ~-~~::(· :~q~-~-iia_~-~~ ~~~.3.£ ___ 1r:~t4.for~th ... ~.3._~)

.. -:t-E(fTt: = l r("·') ~ ~ r 1

7l~Iff~ 4~\t-~n). ,::"s'Lfrs .. ~ .... -. b.'!~~J- ~d~~~g1~L ..

• 1 ~<>Li· Th~ __ l>L~-~L.L+f}=-~ :' ~(il - t) j ·~------~---~-----------·-· ' ···------~- ~ ---~----~--~--~~ .. . -~---· ~(--\ -) ~-±··---~-~- ~ 1 - "h-"1 ~

. ctvtcl 1 h,-s ''.t 51"'1 ({ // ~r /a. ry e 111.

j

j' 4 /sa/ /n fl, is co 5., / Vvl"

.) q..., "'Ill hiaseJ esf,~.qfor f" r IJtko ft1 LE 1y n- I , r:J n

Cq~~"

-L e

<Zf

c. rect f e"

b_y nty lltly /h'

~.-.. ~(~-J ) 1 ; 5 no-t Q n /1L E~ b 1.( i ,·..; 1 :s . u n }; i~secf !· Y\. -I· . X r ! I '4

:·i ' H ~0~ e · n,s in'ck

1j q.,_d _ r'l'l~ /l,p1 '':J ~ a c c ns f<r.d 1l J,_aJ,.L Lv e __

~QH'(,>, ±he §_<f?_:_~_Y:q~J=H~~=--as _ qtt

o-f ..sfct_cf~.y W/·ti-J ttze ll1LE

lt" ~ bJ~ ed_ -_E'sl,m Q -&;c _Q~ -tk ----r-<.-f'+/.,1• d<£1 ll'!!rt'<t n c ~-I

11 (,'.1-- ~ rtoc~ "'L_ pPf <LjQ-ILo .. ). == - -

f .-.--__ ~~-~=~-- =--_- ------1i

fJ:~d:~J;~r~ ~d~- J;=~~~j ~iu~ ~ --j•ss:1l:_- J.: C~€-Cl-/-:----+l-rll1 ______ 14 nk·· rued_ <i>5-li=~~ .il4-- -J.4;.s - _~,vc/. --- - ---H- ·-H·

II I I ! I II

fxqMHie {r, 1. ~D

l' I -1 ,, q I.

~·. ;.

~e f X~ "'- ~· ~ J eyp( ~). w~ h C/" f' ~ f' PVJ fhcrf X t$ V!n bt'ctseJ i':r e.

I+ CQYI he showVI fvts /1"1-) -{-hp c:df -1-ech"'tj 1.( e

thct+ x, .. h -v ~t r (%).

Tl-tu 5 t'\ 'Xt ;1'1 ,·s l( n b /~sec/ tor G.

0hic~ e51t'vnq for- is be ffe c ~ ( II ) _ ?..(9)2-- /'\2-

VQ.\ __ VIrt!VI- Yl ~~ -o/

v q, ( x) = ~ . _ . ____ _ _ •.· _ h€ p u e ~ illJVI <:l-r w h, 'c h Y!1JJ'Qs_e~ e_~_fth-fq ~ r J.L's he5t }<ta.ds +a 1htc dei't'v?/f,·~--~--- of tAe __ i

<6<6

k W niJar M / 'J ;tf,·~, ~~ u_m_ V,._,;., '"'e _ I,J.,j , « 1Ld __ &f,m~l..r" {J)g VIJ E) \ ..

liJ4~-~~;4im~~r yq,; 'i'l~)rs « UL'tv'ld_E r± '~ _i} __ Y•bi<t5'1:!_

~ji.W.d v_,.c{T ,.)~_ygL["J") \:.L§.;g _a"L "II _ _ __ . JLab__'-C!sEd ___ ~s_-l~.mJtlars/ __ T; _ .of _l'[_e_~._ __ _____ _ _

-.. ---== - 11-- ------- -.... .. --. __ -~± ___ S.e ~vn_s ___ desis_~Le __ h _ ___ _h_ilJle.. a _U __ }'Lll U ~ 1 ____ /w_i_ ___ h_Q'-\L ___ ~av li ___ _ . !j9\L ~ll.fl(\_ k...l'\oW .l{ -Y 1JM d ;cJ ~ ___ . ------.- - ------

'· --------- -- -u----------- --------- --------------------------- - ---- ---------- -- --------------------------- -- ----_11~-·q<!ME_c-: __ gag l.co~.yecJ3gy.,d" _ (c:__~_f_/f) ·/ __ IJ_ ,'f f'._x,5is__.

-Jiii€.S _______ sa_E·-.lo~-~L __ kQy_tl'l_,J_ 9Y\ _-the llacig.vtc-e_ ____ _0:_ gV\ YVJb..ig5e_d _ _ __

-~sL"tJ,il ~ R..-~f 'j-"-VS: . <? s.-IJ'a;aiP. r ls.1 ~ (L va.;~C,Jtl ce. t:.'l M. q I fa. . ~------1-e. ____ ----~----t-------t~---~--~Cl-'4--s _ _ -~---- __ 1L_A1_ _____ _ .E ___

-'2~9 ;l,.:'X')t ~01 't]::r •(i) j b/\ 7 'Z ~t.;>) ~J . ( s

(n)J"" (_l)J 1>11 7 z. ( ( YJ J-) po:) ( h

'(..J.../.3 ~ (ej~ d.JJ~M ( YU)3.. ::: (6J ~ (-z

( f1 ~"t'X} ::h -"JJ~NI O::(YI);J (1

=E

J{e~xttbo,e;]g -(lTJb7\ -7---· · (eJ,..J,~---·· --~····

·---........ -·---------. ·t;7fJ;J·-··--···--··-···-·--··ti7Jo7\ ·=,· . (.i~----~

~ r

··-········ 1-.. !

.i ... I ;

' ' ........ ·-t --

(.L}Jo~ (h)JbA

-r-: -)) -· · ( r1 , 1·~1·:.::::; cF a? u_,5J · . .. -t/ . 7 ~1. 'h f\0 5 L ·-:./ . f ~ ..

~ r.

(1 'n)M 5 = (~ _,~; J<>j" >rt-J.;J w_rf( ! .

(.J. 'n)l\o / -=-( ..L hJ3 r ~b ·chJ_) 0/\ · :=-c ,h)'~~r-c (\) ----~ d ~-'cJ[

()b

(nj_)~-:

(n=rJ"S =

r. l t . !

r l

--------------------···-----...........__ __ _

2

. . -.... ~ -·---. () ? X . . . .. . . _ji ----·---·· -. .. . .......... t !

·---·---· ----........ --------__________________ .. __ ................... ·-------~------e-----------·-:;::---·-----------·---.. --......... ·----!)()( v+-r-

............ _________ ................ ------..... ------...... ------~ ;o ............................... ..

lb

1

I

l . l1tl-~J· 2_ = E (X-e'{- ::. , e~ ------ .. . . ·l - . --- ··- ... .. 8'2-

.........-- -.. Ql.j -

1 -e2. ! .. d- - - -..

I! ·H fltt~5/ _-±l-Ie. Cf<LB -'5 I

l - &2 --I n/6Jz. V\

. "v'h,c ~ we 5 Q' 14/ vvas +he va.r icz,u::. e o-P X,

X is -~ _ llll/1 VUE +~c G. II v

ff~~Je .!l b~d .a

l -\<x)pi::

L{"'fV k< E -\"oc e ,, "' G£o(~)

p(!-p)X-1 X;;: 1;2t /

-~_·jj· - ..

- - .

0 O/ "'-- •

--r~l~jTkx)Fl J ='. l{~o3(pJ+/X-t)lo;{h·)}

t . · · .••.••••.. ·· .· -L_~ ,_ e~~y-~)) · · l i

.. -- ~]Ih-~~; _____ tb~-~~--~;~---a~L ____ g_ ___ __ (,{MY.u£_ ---~-Y.sf .. ... h.fd. ~ .... +e~. ~(·· ···--······ ......... .

... .. it·······- ....... . . .... - .. . 1 •

\ .. + b --·}--···-··--· .....

. . .1 ... -- . . ... -· ·--- -...... 4 °<" .. e 1 ,., ~ q/ ef11~.'J- _<;_it.d~

, l lc.::: I ·· ~V?d · c:r::.: 0 · · t; · -h~ve ·· aV1 un-.bi··;S"ecl .. es1/M qtol

crz_

"' 15 I

•. l I

!

J"!Th~_j' ,~.f there r's .:::tVl ui'/PI'ctseJ. esfiYJtqfor

ilihe?f-f fu~s q Vctr/t:rrrcP (!?9vqf. Ia 1he cR l-- B/ lj - p !J ,-f vvru5+ he X. i I

rh~ofi?WJ f.t: T ~~ er;, es-ll~t:f~ o+' 1(6) """'h/c"' aC~t'eves 'fit~ C;!:.J..&/

(9,"3)) ~fft-ten OV/'J /,'.,~~• '\unc:f,'o~ ~~ 1'(6) will qJM;+ S"r.t.eh ctn PsY/Mqfof,

fki;VI; 1,'()4:! r -t- T, / /2- C( ct l4 ¥1 b /ose J ey_ltm,q,~,-__s d"""r 7Y&)' _ I. t /_ jl ( J _rr-- . q ('I _,.--r f~et'\ VIE' re C{T1've eli'"IC !&n c_y () T Ia . to vvrpq re

~~~ ~ ,·5 va\(T;.) • ,-1 ) I:J v a~ (T. . i

i

l

Def,·.,;JiaV1; M A_V1 1411 bict sed es-!Anctfo~> __ T;1 _ l<S _ J.~f'-Gc.ied.''_ ''f _____ _ 1} ~a c(T; J !::_ j_ m-fc.-c q':) Vln bLq_s_ed ~£-£m~it:lL 4 -~ Vqr(T.) _<.::~rtd _(;{")y _$__~~-·

-- --I \:

PJi!ld·.itm_!Jih_e e~-c itvu;.'f d "~ '"' k ... secf es:-b Wo <- _is. ils lk_ e I <Iii Ye e._ fJ!kJen J -- c OVVJf-g_re d_ Jc. ___ SlJ1 ~_f':{','.()ML -

W- -~-.d/tvt q -to '(" . __ . ___ _ _ __ _ __ _ _ _____________ _ I -t··-I

I'

·-···- n . --- -----·~----. -------

.. ---- r

-· ·-· -·---·-·· --

1-I

.. -#-- -- . ... -i

-----------------------------

l j

I I I

de111si-/) <> f es J t'V<A q lor i -\'., c 6 o/t-( • j

Ex<tmP--11 --

- · · ·• rl

e

_ JJeyz~, ·~y of' e s 4,/n.., lor z -tor E7

-~

e

..... £.s1i~J-or Z. is .wnhLa.seJ-;r __ b_'-lj is. tl . _b~iitoc? .

DR1'i!'ldiofllJ The _'Meg,. s/u -.r~ e r roc'' (i1S~ J o-V cu• --~~f,~g_lo_r ________ t'_$ _____ E-:(T- /(f})_}~-------------------···

_ .. xt ~ ~~ses1 ]: ___ JoLo.)JlJ. pr-t?~-C-- _the _ .e.sL'vr:utfor lh,;j~ th~ sm.cJif_c ___ f'!ls1i~. .. _________ _ l . .. . ... - ········ ..... ·1-;~~eKi~~~: .. ~~b•;~~~qiioe~.~:ior ~~;~~ h~,~~ ~!~

--------------

Theoce W\ {j _L115_(; = . \fPdl} :r{p,·"~(I)y- ___ _ ( r,1,z)_ --~~

I Q..ffd io": u rs it p • .,; b}e_ to J.i~d __ Q[l - e~t:~~p~:- i hat

111 • r { ,_, 7 lLJd/ltt.drV"l J _rn_r'nt.Yl11dnJ_ !"1S £ ' _ . _________ _

~CIS

YV\ _ esli!Jrl ctlor ih.oti _______ ,·§ ___ C!/""Vq_/s ___ _ S: -·~- f ____ ±he p .. Q.r. !:lMt-1 e )e r_ 1 5_ ~ _fh_!·s _ t:::sl~._~,. r.r.·.{r

_ --~h.Cl's __ the Smct/lfP_r_r[ dSEJ /;f4-_j ~~ £ _ i6~ _ . .-.. lcl./1 f f OJ c c:r J-21__(f_fe__r . 1':s. _ ~' tJ t 5· ~ ,. t•

- --· --

Fe,, "'"} f"'~ -/; r " ( .,c Y<> /" p 0 f LMe_ ps-L<?.m.eler/ th_~_ deye11 e.c~Je: __ es_L~ctft:Jc af __ j·_t,e_ S9!~Le. ___ __

_ .\/CJl"f .. e h'is __ ~/) .... A5...c __ o __ f _Q ..

-- .. ._

Jv'iy v1 o i i«s.e.. i h ,-_, _ "' s

--- .. .. . ·•- <(-,

fh.£? __ ~5."~l~:J.sr.fc2_f __ C_ .

Ex-a m_ple_·,

Cr.~~~i1_ . -~~--- ----tJ.QJ~----··- ±b.a..t .... faL. fex_·} _EJ,_?Z~-.:

K!:-7L

--------- --- -----t+S__'-1-fPr;se. ___ e. ___ t~. -~~C!.~_:'-· J:o be. j •---

----- ______ ----·-··------------·T+-t_h_e ______ /'!J..L.E ______ L~):_______·~--..:_- x," """ -

.. 1!fi! ___ t~e .. tt1E .. __ is __ _____ L_:: ____ x __ ~ .1. ______________ ______________________ _

1~

.. -·-··---·-.._---··-~-·---

.. - ----- -- - IT 1 c:.ruJ _ h~- s.HQ_\<-!0'1 __ __±b"L i-:A "- G .A!!JL ~ _J__~l-- __ _ __

-=-~ ~~- -·::.\~u~l~j--- L0e.. __ ~ ~:--=t~~~~e~-- __ _ __ _ _ _ ~=======- ------ ---- =· T-t- --~,. hp ~~c.~-~---i.h~t ... XJ:.~=Z!:: /\; .t:rr(i\ ________________ __

vtsl" 9 +t-e e-d-\ -fee ~t1 i 1-1.{ e.

~------··--· . .. ---·---------------···· ------·· ---·----·----·-·· ... ········ ··-.. ti ......... ·--....... -·---· ·····-····-·· t • . --· .. ·----·-·-·-·····---·-----------···----------------r··· .... ·-----... ---------·· ----

---------·--·---------·-·---·-·----------------------_____ ~? 14~ ~ s ~~ ~~o ~--:·----~~~a p\A ~ p_~.:~~1 LA n t ---·· _______________ _ VI _-

-----*!703!fr1fbksa······-+n-cr··--·--p·-asorq--·---s-:r-·-c,J! ~-~---~-~(~ _ ::x)? t ·--------------------------------------------'"-~~----u~'F6ln-JoJ---T10'14AT(iu-· 6---· ·c;;·r-··------~ _r:j-j-Q ---51 l-i·---~~ 1 --·,; ___ "d,~iriJ;:J-

---·-········-------·-··--··· -------······--·-······· -----------------·-------------·--------------------........... ··------·-------

.... --..•.....•. ------··-··--... ···--·--------·-· ........ . .. ------------------------------------·---···--· -----------------------

. -·-----····-----·· ····-· -·. ---------····· ....... ---···--·-···---·-··· -··· -----·---·-··------·-···-····-·----· ............. -·---·········--H·-·····--·------·----------

. ... --.... . ........... ---··t~%-c=dl-:r---------f-----··tJ·h "b 7 .J., (t;}).L ··· --TCYJ ------"'J:j;,·a.tsrs 1/t o :J~,-------~-----vl. _____ -=> "}10 w:{Sd --·--vy--; ---TAlJ!f!"Pl ~~-cr

" . ' ~ '--' ···--·-------·--------~-----~·-· -~···'·· ---···-·· -... ---~---~---------~ ---------------.-------.

! . ··-\A-------·-........ ----··--. ····-· ----------------------------· ......... .................. ---·· .... ··-·····-··· .. -

T == \A

------------· "ltu·~-1-iz+·D---~--·T · -~-

••·• --zK71-I-x@]; r~-=xr3"~ ~ .

. ·~-, .. _, . ----··--········---~---·----~---------.··---------·-·-··· -----~ ··-··-··----· ·····--. ··-··-····-~----~---

.f

·-. ----·------· -· -~---·-------·

-·····---·------------------··-------·· -------------·---··· ...... --~---·----······-····-·-----·-··--·-

1! i

t-zr~ 'J; ---J;M;A?Ja'"'1r---f=tiT~,------::1-cf"-. · --------· ------

~~-~ ~C~ -J-~r~;------=-·;(A ~-i c;;;J-b7' ___ --r:zA/3 --------·· --------·------·· ........ -. . . ---------i-~\1;~-(zA)~ = ( )\) Jb/\ ----·-a·?-j;:~-

--------------------------·--..... rt · · ·· · · -·· · l~tefX"3~X--;,·:rd'jM I[

---------------------(;rn---~ru-=":rxJ-~-H "\Wl~/3 sw ~'14f

9b \ I

------...... ...------------ --. -----------

err-

F)cC{ »1f_l§' H - .l'LX:._., ___ ,-5 _ <>~ ....... "<. n iJl'gseL ~~·--- .. ....... .. _ . --···-- __ _J _______ _f~5: _____ ~----------LfL.~ _______ 9f.xL _____ g)(f(e) _______ p_afult::lJioVJ/_ bvi

De_f:

DeF:

I JJ __ ~-s n oi (;oyzsi5tt'f'J i;

! ... t

I --~

Kf"c ct(l fhcrf va r ( V\ '/., :&t'l - ~--· .. J--- ·----

n~-f __ pe}ssi.ble. .. _if1t:tf _ 111 t:11 ~ t _e. __ -' r· -

l ___ f\:.V' ______ ~5.:J~M-ft<>iicq{Y_ __ (A,f1_br'qsed .. . es:li¥'J"Zt-?£c. .fr I I II (" r · · '~ . -f ... __ U __ ~_sy.m-p-,r;.":t.it:._q_lty--- e-:t:+.i~JeJa __ t,_ .L... __ _

'$__Q___ I 'i_ I '.J

(/"') ........ 1. ·--·

. . .. -----

~ Vq r( ·ft'' ) .. L ... . j_, ______ _ - ·-----------(~~-- .\ va r ( Tz. .... L

-- ·-----~ ·------ ------·····-----

v . "'~ffl'lfldc411y .M•L 6,f=:J L<") v £).__§__~_ . .. .... -· . -- --····

~ ~ 4i Ih~ ___ ~~ 'fPJf:?iL(._ r e> (ctJ,·~.re _ eff,·cJel(lcy __ gf __ ]:' __ 1 __ c;Qvy~pqc~. to .. . l.i') . r's hwr vote(~) --------- ...... .

r\~Q) Vet\ ~}\ • >> --- -- -- ll-l)..

f~eii\_JL~_hg, .. L _}lo_____ .e£Lm.ttft:?J ____ g,J(t's ts _____ ~-~~Lh___ ~~- _ L{a_}Jks_erf ___ _

g.~<-t. *e eis . J~e -.c; KL ~ ..... .. it .... 1S a ~f: <:.;, c rr; H''1:-,~:c, -··· -· -· ··· IuLcL JLQ;1 edui!Lct4.o c .. k:L LS.. . . ':) _'p..o:l. -~_r -e._':S!_L "t ~

.... fXfr}.ll'!,j!: .a.o ""'- t.h,i .... ··1· +~ _f!.s'J¥¥~f!s,iJ.c.d/y_e{t£ ievLL .... f., c i .. ---+---Jh Q<Jr'L .... ex-F-""-U-~~~J .... f~fi4.l~{£Il>tJ.~-------·----- ····- ··· · ··········----··----- ·- ·--·· -···· ---

lj - ····•· ·-· .•.. ··-·-··· ···--··-···--~ . . ·• .. ···-. ·····• ·--····-····-·-------··---------------···· --· ...• . ····-·······- ... .. . ··-····--···- --- ···---~-

/ L_.;. _fPc)__~ ..... _ 5.U_ 1 1:.s ... C o>dSi.s.k.t.. -~-- tfut.s..-. --· ----

-------_Jtl __ =~~~~~~~=====~~-------=~---~ 1 1'

-f l '

1co ~ I -f-GKL f3 -= ('f'f? t $ .., _.,.,_ ....

'

... n f ~ lo3 ~)~fl 2 ·

"E[t(~lo,(6} -fJ}~

-- .__...,........ G't

-- •... --......,.-···----. _t\l~[~i tf~· ]2

I ---8~ -------.--- -.- ·-· -·r··-x- =-e_-Jz. .. ·-· -h f...:.----'

....... ·- c- ez..

---

-

j_ Lc;.'f -

c:::::;:;J --~ . h §2_.

. -&'1-.. -

R~c;q// ___ _ fbrti ... -b. X /L. -J?-Z-Cz.~)_ G;

- ~ q ' / l ).. ::: jJ yz - Z .l \ 2.. ---· ·- --. cr·. - r::tf_l ·········~· f I · , _______ .. - .. -------·-· -- . -· ...

_ --~- ~Ju:~_h ____ {.£wWl __ T~eor::em _____ .K.3.~2·) __________ is ....

--· - p \.£_ ···-__ \ ~ ---------- ··----· -------~h=t-·r···zn-z;·e·-z;· j ·

·---- ·---·-----· ..... ····-----~----------------------···-······-- ···-··

t ------,,b.Z. ..... .

·:::, tJo?f:<_lJifd-1"'?!f"f} ..... t:s 0 .. >! . "~ t (., . .. -... -~ -1 ~ :-._:; --d·~-f bf --)1071 i;;) a 'J t7t tJ:n P7l . ---

J!fofdWA~o ·pVf'B ·· 1 g· /·vn:>diM ?~tFofJcu£;y;; · --·-·

'1{!'171 JbiM><Fi -kiJ~>''V'fJ.,..Rso .. s• "'(} n -·---7 1/i-Jf">i:> !Aa 5 -SJ -""(9--(z---J ~,1_1-u"r----·n f5~o r-fr _1 xa · -"~ {r· - v

~ ...... ------. -------.

----~---------' '

I pt-t b-e --·-V'I·o pv-cid.Jy· (··-< r

ee !

(9fX)f Cor e fi I --

. T -t{ ------···:_:.::_·

z9.\i ... ·• -\

------'h'?vdr~'JJ-cr--?!~stT---~~7-·--~ . -

·I I i

/00

C ~~p{e' /0 - ),.f'.f>cienJ £. ~f/e{PV!f''i'S

De+:

De~~

A s1criistic1 5 1

f(~ Is) does no-1 X\S

I( \\

/S ~"' {.f,-c i~ n i {or Jefevtd Ovl e.

G

Yo 14 ~ q Yl -f h ; 1"1 k o {' o._ S 1..( t..f>c ) {) v-1 t: 5' k (/s 1/c

; f

Co¥~--/q!n,.:J ~If of -fhp /ntcri'YlCI'fioVJ tr-oVv? ihP

SerJ-vtfle qbou-f tl-te UY7kl-1o~A_.,V1 r~)C, fqV~-?e kr.

Thq-1 is/ ,'-f yo'-'~ kMotrv J/,e> oufcoYV?e a-f ot

5vfftclen-t s-lc,{;t/c.) +hen knC)j,.iJ 1he, 0 I..A ·( r 0 WI e 0 -(" _, h p e r1 .;,. r p s Q' Y"1 fIe ci 0 p 5" J/1 t QJJ ony levtoL.-vlec!Je abo'-{-! e4

I-f s .--

(7

S Is vec{or-vqfu_ed_~ ihen t..-ve s-~ 1t?o,1

is 1ju,·V1 +!J st..(ff/c t'P vt t-'' te; r 8. N e; -1~ 1 h 0( 1 VVtCty C!fso be Vf>c{or- vc,l(..{eJ,

f Y. : T h f' e ,-fht> s cr $4'f / ~ /'t;; a -=:e--/ of .J tJ\'w -fly S'..lif, 'c •' ~ .,, 1 s fer i 'S ,

fx-: Th~ <>rde, s~ft'.s-lics Qre q11'o JOtVT-I!J ~H'rr;'c;f'..,t.

_Def: A s e-f o-f j '"'"' 1!£ s"fr/.,. o'Po~~ t sf"'ltsflc 'i If ;-f ,s • +u..,clioVI c f ever'J o-fh~> r 5vftc tf'l1-{ 5-{q/t·) tic 5,

. II II( IS" 111/ni/rlCI

5f>t o-P

fx:q VV1 f (e ~ J -~ X i's c;u f-\; r;r=ni f e;("

;;~fd/} Sc.rq-,c i r vt i) 71-,e Yl X

e:> Qv,d {J,,J~,. . .. /,.Y.:) Clr~ i .5 q -C-t n c f lot-~ o. f ('ft 1 X z ,. • · ' 1 't.,)

1 .. tf ttoi {h.e 6tt-, Pr \A-.- a J a- fOl,, VI d I

(9(

--

LA~Q U ?/ W /7 . <::91j ~ . .f. I? -;)q 0

S n 1/ ~ t:;_, "'-( ;.r ..$:'

, cJ J d-f .;~ .b.J .0 J u1{)! f. b '-'fA JOJ {".l ~~q h /! f2..1 C )J () t;_,,~-.,o "::>( "'"'/1 s lMc>dS +I

·r· r d 1 >.! ""'f •i]. ? ][= ;,r 1-"'7 • -' .di-"'i f lA p-»s of "! ~.'oe> 'r/ (l•tol) : .X-:J

I Ol

IOZ

Theo,eWI (!6.2. () The Fe:u .. fot·l~ct ft'oV1 C ,;Jf: Uc VI

s (~ Sr..t Ff>c i P Yl t_ tc:) \ GJ itt

tX (~;e)= J(s) e) h~) +cJ\ SOJ..-¥1 e J j h •

I~ o-fhe ... r Vt--o-rds,.. ,:{'. ft,p jui,d </er~s;ly cf'. fhc:"'

5Cfrvrf/e Ca~ b~' f'a.c:-lo{r:-d t'fl-/r:::, a (t..(vtc-lioVI o-f

~ q 11d 8 on}j Q vtcf ~ -(' C4 ~~ C f/!) 1-1 Cl-f ~ Ol-1 ly J 1 hP 5 i<s .st.(ff'ic i-e 0 t,

Ex: 'RepeCif e>cqV11flr- (o,{,{ 1/V;f ~ f~~s fheo\Pv'Yl,

[/ tx· n-~K· 't~; 6 )-= f ~ (1- r) ~ 1 i_ XL eiC-1 ff}

l_ -.1

j(Zx:; r) h(~)

Ex(!o.z.3): Lei ;( ~ t:d rJn,i(o/e)

-f(-1; 6) "' Tf ~ .1/x; e(o/e)3 = ~!x,. > o, x.,. < eJ

Tl-t ...... s by 7hp ~rio\ ;·cctlo~ c r:-lerr()VIJ x,~r'/ ~:n Ctre

jciv~-//j 5'1-f-\!'c i'fr1i {or G.

_,-) ..., I, ' A . -4 I. . . ( ~'""'f'. . .. t ? C.y({.:-S''ftuvr. 1--\{c IV!~_) Vlcfrvil~;t't:!1 ?f/l\f(t(,_·1 f

Q<ACt;tr'oll1; wha-1 T~ Is --

,5 1tt~ rtti;li.¥1'1"( ~,;ffrc t'r:ni s{ft·')f f.c ? Xn'r" af5o blf -fhe- -'lerc-fcA;.Zci'(;oV\ cr;ifr;oll7,

' ·"""'

103

EKCrmp/e(/0.2.4} Lei f,; ~,;:J N(Hro-- 2)

{U \ -0;dt r. I/ 'Z..) :::: IT - e . z.o-2. .~)n/o- J Zrtcr"·

fl/z_ (-){~ [L"(x.:-~)2] ~ (2rtcr 2 ) e

(2 ) 111/z_ *' [Z.x.-l- -2-ALX~ -t YIMz._} - 'lit.. .,cr?.. -. rro- e6-.

Tht.{) 2~z ctvtd cZx A'/ cr-z.

ore j01'111t!J sceff;c;~""i -Gr

N C) it=> fhQ-t -fhe,t> IS

b e-f 1/vf? t-v; ftt~ q b ov ~ +~r' !V1L£s j a~d

cr OVl e - fa- OVI P .('t.tvt e- {leVI

Suf.f,t /eV'!-t 5/afttft'cS c:?~d

~~::rr ~i • Y\

£y: (to,z .)) Lef ,t ~ cccl {)/V.J. F{ ~ G-rt)

to~· i G) ::: j__ ~ x,. / G .J X. : ~ c.. 6 < I j

T k 4 s X:' I l:n "' re jot'" i fy 5" fr;u,c • .,t fu e. Q.,E-;{,o.,: A rf? t&,eJ VM•'hil#af Scrf'-f'levd '? (Yes)

iheo\etM (to. 5. () r-t s ,·s s~-P+ic ieV?T ~c)\ e Qvld e IS

Qvtf'5-lr'o-A:

+~(C' r_("' ;'! v E' ' - /1LE' th.e, ~ ,·s ~ tt.twcit'o11 cf S.

If e IS the L{V! ~~ve .A1LE 4vtd ts svfttcre~(, +t-te.., Is e necess--q''J rvit'v,·vYtc,( 5(.({:.\,·cieV?t ~

Ve'5 iht"" ·thf'OH''JW'I sier..f<'··s f4 is cr .\!.n·c·f;OV! o-r t?verj svf''/'.;r;~-f I ( I c: 5-ral lSI I c. '

--~ -----------------------------------r------------------------------- ---------- ------ -- -------------------------- ----- ------ ---------------------------------------- -· ---------------------------------------------------------------

1:

__ J1.eQL~VbU _____ l{ ______ _ T Is ---~--flb.!~<!-~.d _ .f'c;)r ___ e _____ qnd_ .. S . ~,:.J. ... __ _ (1_9~}~_:t)__Jt.-- ____ ,s:w_£-f~~( ie¥L _ _t ... _foe .. . e .. , _ th_r.!Cl ... --------·····-··-·--·- __ ... . ......

. ... . .. .. ~J- .................. ---··-------····--· ... .... ......... . .. . .. .. ... ....... ..... . . ... .... .... .. ...... .. ..... .. ... ...... ... .. .. . . . ~l- .... i) ______ L_*_ := .. .E;GT.LS.). ............ ..J.5. .... ~--VLa.6~:gse_cL_ ..

... _____ ..................................... I:_ ... ....... 2) .. ___ T ... ~ .J~s. ______ g_ .. _f(,{Y[C f;oY! . _ o.f'. ..5.... .. . . .. . ...... 3) .vac.(T~).!: y_gr(T) . .. ...V eJ_G.S2_.

I .

, . .... _q(J_cl . .. vqc(T~)<,_yg_rli) ~r ... s-o..~~- e . . wnl~s:~ .. T~-= T. h-.i 11, /P. _-.1._.

_ -· . /, is 111 e.orf:<tJcL _s: c,Js 1/ud _ Jhe Y"' ~. <> s e d ....... --~-es.ti~'l<tcr-lar ..... Vl.,.. .. i i:h .. .. . t.tqe ...... Si"''l~.&..s: .. t: .... ---~tt'.rJ'.q..n.ce . 1.'s:

............ ---~ ........ tUn..C. £~r:LVl ....... <:! .. £. .. .±he .... S ~--· ff;~LL..t'.el t_ .... S: ~q (; S . .i: C. •·

............ '"[1" .................................................................................. .. .D<:f: -····· -t A f~..,;y .-t. de ... s:JI; f.,., ckms' . Rx ; .. 6.}'

.. . . . I .is ... ''c.ort!.fle Te 11 . . I £ - . .. . - -. . . . .... ·-· . . . .... .. .. -~ t· .. . -- -. ... .. . . .. .. .. .. . . ... .. .. . . .. .. . . . .. . .. ... + _ E~iJ)f~-o :::2 IP(u/X}'tQ) -0

......................... ____ J...................................... ....... ................. .......... ... .. .............................. .. ................... .. ! e

_De£; _ ···-··· •: ~- •~+ ~iq_i~-1~~-,.-~L~-~ ·-~~- -'~ o !'nfkJ;, =··~~f.... ~· r~- :·. cl~"s:!Y . _ _ .,Lc<d.'d.id_Je _.<>.":j-- _l:kif:-"'! _ _k c _ _o£_~_g_-J-<lmf-leie _ fQ ~M1!J ............ .

_:_: __ I±j-~:;j~--~Jh"t _:~~ ~li=~- h~ke~ · :!:~~ ~ --------- ..... (9..rtl.pL-e-Lf.L ____ _____________ f?_.btJ~.-__fQ..I_/, ____ F X~r------~ tj_t-----~--~----- -·---·-···---·

_________ .J.£ ___ _Q_ ______ s..:ls1.Ls:.~G·..c... ___ _ h.qs ...... dis.i.c.Lb.Y.ii~~L- .. -f:1(,q,o-_~) ....

__ -~ ___ :rAiL·-;;; ~~k*-:; t~ -~ :~;;~1~ -:f~':;t tS . _rl_ J he f"f!Y (qi •":_

---------------------------. ---------------------------·---tv-=--\~--1= 1) dl -r~----------------------------------------------------------fb"""i-,-J;r-pJ~T>vro:J-----:J"ro---_:1-f"Ub ___ J_ _____ ~"Tdq?i1} '-~ 5' -------------

-----------------------df·a.rJlliArJ) ____ o _____ Jo ___ IA~r~YrAriJ---p-dSO,I~ L4 n··----d""'iAo ------------------

-----------~~~~0--LU>J ----'dJ(J~+----~ Fd7VI0'15 rn;t' COY"' f'"1ff:. ___ df-ory·--------------------

-----------------f L4 !:>

·ci?J bA_ -~:1..)-=n:~x-----------------: ---------------­~ {_I :;oq--_j_ -= :r--s n ~ / ---------------,::. --------------------------------

~

------------------------------------------4-e----J5------p~·so-:sq-pn----------------------____ -----------------·-p-"'-c·-S -j-o·---~o.tf:Jc;tny-------o----~--~---fd_l ______ __

~-.s; J-0 \A 0? f -:nA n_r· b Sf a :; l.-1 O,IJ 01\

_::;.Sci~Oj d"J?--/-o/J.'/VI Jr:JjbW.'fSa fd.>h,JflAVi

-+-<> '1-1-~'"""'oy 5 --{h ·~ 10 l} J/ .;:~ >1 :;1=)8 -o b ~

~

s; tQ IAO{j--:JW)'J 0 S',l (>lt =l (£ f.LA;;! '!J--j-liS puo ~1-.;)1Jw.o) 5;i S (-z

(~~:X)j-f?~----j( (1 j-1 i

t I I

--------------------r;-·--------------.'

I

l

~:"~~~ Y,. ~ c .:d . po_r{A\ . . . . . .. . . . . [J':J,.f} lt . .. . . )

·-·-- .. ±Lx _;__A)_~ [[ __ e~ __ .J ~~----- .1LYi_6[0/.JI 2_~-~-u~} -. ------ - Y,f ..... ·---- .. -~-'4.~----

I I --

.J. - e_nx xr.;:;

·• IT(iix. .~ :~: ;, ?; •• sf)_···

thlfs ___ ~l Is _§w f {t'c..ieVJi. __ for _ A.

-·-~ _Re_c_a.[{ ___ __ ±_hq{ ~X· . __ A-/ P .Q_L_(!'l)_).. __

)"ff"S€' E~u,(l)};o{) .. Y §CO w-here, Y ~ Po.Jte~ ·-·--------·~-- --. ---a.>---- -- ----------

- - -~ R_a_{ t's;l -- Z_ ~f:j). e~~--t;J~ -~-- 0 . ___ 'r/_ __ G / 0, :r=-c 1

--·--·-- -- ---------------~+-- .. --- . -------- --- -------------------------------·__j _ _. ___ ---- -------------- ----------------

- ----- ------ - -----·----- ---~-.--- ----- ---------------------- -- - ---·-··:....e- 'j

________________ -~-- Ak-t~ __ ib.qi _e. _____ ~ __ --/-0- ________ {q ,_ .. Lt. f; __ f_c;" _ __(I-1"~-J J} ..)

____________________________ W_e: .... sJ:;._~..fd. _____ .c_h~s-e. _____ co.e~'...cLe-~~tis.. ___ .tij) __________ f.o .... ____ r.t~ke__ __ __ _ _ _ _____________ ________________ the __ S-4-¥4----~.Es..O-~------ky_f _______ 1.be_ _____ e.J:Jiy_----a..e.££.:ci:t . .r-.z.is ... _______ __ ___ . _________ ------------------~----tL~ .. i ____ JLv2.LL ______ fl1 q ke ____ j __ i _____ :ct.r.a_ _________ 't(" ____ ~ __ ?a ________ g_r__~--- ---~cor~-~-~---- ______ _

Tb~s ______ j:}!_~------- PO.I55.Q_N --~.tVl i (J-__ l'-s -----~.9.X'!!.p/e{f_________ ______ ____ __ _________ _

_______ -------------~----~--=---!- t ____ --X--- ____ T ~---- __ Lf_f.t_1i.Q.5.e.d ... t~~--------f2 _________ ~:;_~d~ __ b.f~~e- _______ _ n

------ · ------ ·----- --·T·D-----,r·--------- ------------ ------------------------.. -- --------·- ------·------------ ------- --------- ----- -- ·---a_ v..Mv U£.

',I ·--JI

:1\ ·I - ,rhis <:q.,_ be ve{:;I,·d 1/Vdt, ±he LRLB..

j~ b'1 Tlre o~ e""' 1, 3, ( .) _ J cf A , · wi II no-1- qrJm/i .<=~..

a (l_bV!_-If"l.E¢:i_i {UJ/t(_-1/a~

C_f<J.,-E __ es/1~Q'hr.

~ExqV"lfle = E(_f z.) = ). z_+ ___ f

I; Tk~s £(X 2. - ~ ) "'- A L

H x-~- x n Yv,'/1 __ n_~_±_ q:_<;_b_~~rvc> ___ +~e. C.R &::_f3_~ __ _

It --- b"i sin c F a - t5 --Qn._ -- -I.<JLI:>i.<>.s:e.cl. _{1-uci.C.iO "' - ---

-if-----Q+ ~--- ___ c: o_~ pLeif>. ___ f..y_f{}e.a_f ___ s_j_q_'isft_:~--+---- _____________________ __

_ _ lL __ _ j_f r~ ct__ _t./.,A1k'u_£ __ _____ f_cts-- ____ l~.L _ ________ _ _ __ _ _____ .. ___ ___ __ _________________ _

_ _ _o~T ~-v{=j)~;~)_~-~~lt;-;~-~~(ej~p)) -~-~--- _____________ -_ ~

=~==~==~}§~~-~~~~~.j~~~=~~ --------- ------ --- I J ----------------- --- ---- ----------- ----- ---- --- ------------------- ------------------ -------------------- ------~------------- ---- --- ----- -------------

-- ----- --------++---------D---l:= ___ J__~-f-- f6c; e) ) -cl- --- ~S- Vll)-f Je fl fLL___.j__~h __ EL ---------' ' -f.f-- ' --------------------- --------------------------------------------===- +-=~I:i.(~)~;~- -- .,::~~;:T,--L-~__i_~r~Wy_-

-__ _ ___ _ _____ ____ _ ____ Jl ____________ C. o rd-t'n u o L<5 fk YL di.alds: _____ Q_-[_ _ _e _________ ------------------------ -------------------------------- ·

t------------------------------------------------------------------ ---------- "'-- ----- ----------------------- --------------------------·

~4 "" "-"-"~-Jk)_ f:i,) __ :L [ (X)_ a,., li" e'!L!y- ,.., J ep e"' <fuLL _£ __ r~.JAy_~ __

oV\ A. ~<<re~~:_r-; b) t~~) J'c~--~"cJr.Lv.iq_L_®LJi!l=-o/-- ____________ _ ____________________ lr~~clepel4devtt -----·------------------------------------- - --- ---- -------------------

+keV\ ike deiii~7·J~ Is Q IMe,..,ber e.f -tke Re,. . .E)(1'· ct~s-~.

I' .. -··--·-·· -------··-·····------··---------------··--· ····---·--·-· ·-·-·-----···-·--·--·-----------·-·· ----···-·-----...... J ·-

l .. -···-..... ····---···-·------. ·------------------····---------·-----···1----------· .

l ---------------·------···---.. ·-····------·-···---·-------·············· ---··-·-------~-------·······-·······-------··--······-t---·-··---·--------

==---------~~===-=== ... --~~:=--~ ~--~:-_-----_:=~=j :::_ . -.... -· I ··--·~ .. --··-----~-~---··--····-· ··-· ------··· """ .. ----.-----~----······ .. ·------····t ·-····

. ----·· .. -·-·-···------.. ·-·-·--······· .. ·-· . . .. ·-d

-·--.... ---t. "W+ >O:Yt= 2 )( f~-:r -. ---~-------·--·-··· ·---·----............. -·· -------~----·····-.. -------· .

,2~At~ -------···-···· ···-·· ---..... d·-··-. ---·-··-·-c·::o-l)?'(x-k ---· ~7 1::? ·Y_J

---f~ -x_) --r': -. --. . . < \ -I 1------------..

( ;;;o·'Jil/ff.. ·r-?:1 ~ ~-

J ·t .

·-r . \

~-.... -I l ·-r-.. I I

-~--}····· ------·•···

! 1'h'Ol) -a Jw.ox· -·· l 3

----------------------------------------------------------------------------------l------

That's not quite true. If the population is a member of the REC, then the sufficient statistic from a random sample is complete.