Basic Concepts from Information Theory 11

!"#%$'&()*
+,+.-/10,+,23)4,51687496:;4,5<>=@?A41?CBEDF?ABHGD
I JKILJKI M(NOLP Q8NRFSEQ8TUOVQWXRFNOPLY
ZX[1\]U\#].^1_,[a`cb9].\Ade].f1gihUjlknm,]okpF].^qkn].rohsknm1hbt]9geu1hvwcxytzL{|nz}F|~F~~#|nz€,a‚1ƒ"h_,hF^1[.k#h
„)\…[af†‡jˆx‰z€Š‡‹xŒŽ†Hz,ŠH‹xŒ,Š,~
†AF‹
‘m1h“’”X•H–A—™˜š'[9›œj‚_,hF^1[.k#h_lž†KjŸ‹‚ [1\¡h¢,u1dibt]9gihF^£k#gi¤lknm1hhF^£kn\A[a¥£¤“[9›œk™m1h¦v#hk[9›§¥,\A[af,]£f1d¨gidik#dihv
ynŒ { |©Œ } |~F~F~|©Œ  a‚,_,hF^1[£k#h_>ž†iŒX‹ª‚.divs_,h™«^1h_]9v"›H[.gigi[tƒvF¬
ž†iŒ‹­x ®°¯ Š Œ,Š9g¨[£±ŽŒ,Š
†H².‹
ž†KjŸ‹­x ®
¯ „\A[afL†Kj!x‰z,ŠH‹,gi[.±"„)\A[afL†Kj!x‰z,ŠH‹~
†H·.‹
³F´Kµ.¶
1
H(p)
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Z di±au,\Ahq.¬‘m1h¹¸,m,].^,^1[1^qhF^.kn\…[a¥.¤
‘m1h(ºF–A—9»™»n¼A’F”•H–C—½˜š¾ž†iŒ¿|nÀ9‹[o›¿]_,div#kn\Adef,u1k#di[a^ ÀÁƒdiknmÂ\Ahvn¥,hpk"k#[U].^1[.knm1hF\ _,div#k™\Adef,u1k#di[a^UŒÃ_,h™«^1h_ [9b.hF\
knm1hvn]£`Uhhb.hF^£kvn¥,]9phd¨v±.dib.hF^°f£¤
ž†iŒ¿|nÀ9‹x‰ÄÆÅaÇe® gi[.±"ÀÈxÉ® ¯ ŒŽ†ÊzX‹,gi[.±)À,†Hz‹~
†KËa‹
³9µo¶
‘m1hUÌ(Í1ÎÏÎeÐÑ.ºÒ¼©Ó’FÔHÐÎi’–sÕ.ÔKÖF’–K×a’F”ºn’1‚1[a\–A’FÎiÑ.•HÔKÖF’’F”•H–C—½˜šqd¨v_,h™«^1h_Øf£¤
Ù>†iŒUÚXÀ9‹x‰Ä Å ÇÛgi[.±†ÜŒXÝ.À9‹ÞȈx ¯ Œ †HzX‹€g¨[£± ŒŽÀ8†Ê†HzXzX‹‹
†Hß.‹
x ³9ÄÆµ.Åo¶ Çe®àg¨[£±¾ÀªÈ®ŸÄÆÅaÇe® gi[.±ŽŒ8È
†Há.‹

1^ hÃ`UhF]ovnu,\AhU[9›)knm1hU¢,u,]9gid¨kA¤ [9›¾];` [a_,hgdiv(m1[9ƒ ƒ"hgigdik(¥,\Ah_,dipk#v¹]q^1hƒ¦vn].`Â¥1g¨hU[9›)_X]okn]¸,u,¥,¥,[.vh
ƒ"h(m,]b.h(]Uvn].`Â¥1g¨h¦x † { |~F~F~|s‹"vnu1pmqknm,]9khF]9p½m
Šdivvn]£`Ã¥1gih_°de^1_,hF¥,hF^1_,hF^£k#gi¤‘m1hF^
ŒŽ† ¹‹x Š { ŒŽ† FŠH‹
¸1de^1ph°knm1h gi[.±Ø›Êu,^1pk#di[a^div;`U[a^1[.k[a^1dip°].^1_œde^1pF\AhF]ov#de^1±X‚"ƒ"h pF].
^ #u1v#k;]ov¹ƒ"hgigp[a^1vd¨_,hF\ knm1h gi[.±Ÿ[9›knm1div
¥,\A[af,]£f1d¨gidikA¤¬
¯
Îi—#טL–C—1ÐF† U‹%x Š { gi[.±ŒŽ† Ї‹
† .‹
x ¯ ŒŽ† 1‹,gi[.±ŒŽ† a‹
† .‹
µ ƒsm1hF\Ah Œàd¨vÁknm1hÒ ˜LÔϖ™ÔϺ™Ñ.Î Õ.ÔE»½•H–™Ô‡ÐªÍa•HÔK—.”

Œ † a‹x ¯Š { † ŠA| 1‹
† .‹
¢,u1dibt]9gihF^£k#gi¤.‚Lƒ"h(`Udi±am.kÁ]9giv#[Ãp[a^1v#di_,hF\knm1h¹]b.hF\]9±.hÁg¨[£±a¥,\A[af
 ¯ gi[.±ŽŒŽ† FŠH‹x ¯ ŒŽ† 1‹,gi[.±ŒŽ† a‹
†A .‹
Š {
µ ![.k#hknm,]9k"knm1hU]b.hF\#]9±.hÁgi[.±a¥,\A[1fqd¨vÁh¢,u,]9gk#[Ã`Ude^1u1vsknm1hpF\…[.v##v "©hF^£kn\A[a¥£¤;f€hkCƒ"hhF^ Œ°].^1_Ã$
Œ
‘m1h(˜,’F–ʘÎi&’ %9ÔϕHšq[9›¾] `U[1_,hgLŒ‚Lƒdiknm\Ahvn¥€hpkk#[ ]qv™].`Ã¥1gihUƒdiknm hF`Ã¥1de\AdipF]9g_,di v#kn\Adef,u1k#di[a^ Œ°div_,h™«^1h_
]9v
˜8’–H˜LΨ'’ %tÔK•Hš.†iŒX‹x‰)² (+*-,/Å&. 0 1325476 Å0 31
†A.F‹
![.k#h"knm,]9kgi[.±a¥,\…[afL‚.pF\A[£v##v "©hF^£kn\A[a¥£¤.‚.].^1_;¥€hF\#¥1gi&h 8adikA¤;].\Ah]9gigL`U[a^1[.k#[1^1d¨pF]og¨gi¤Ã\Ahge]9k#h_Âk#[[a^1hs].^1[£knm1hF&\ ‘m,]9k
divF‚X]9vÁ]Ã`UhF]9vnu,\…h[9:› 9™`U[a_,hg¿¢8u,]9gidikA¤ ;Âknm1h¤°].\AhÁh¢8u1dib9]9gihF^.k <Þ`Ã]9±.de^1hknm,]ok¾¤£[au;m,]b.hÁv#[a` hshb.hF^£k#v"_,div#kn\Adef,u1k#h_]9pp[a\A_,de^1±Ãk[¹v[a`UhÁ_,d¨vkn\Adef,u1k#di[a^Ì¿‚8]£^1_ ¤.[auƒÆ]£^.k
k#[qv#hF^1_Ÿ]Ãvh¢8u1hF^1phÃ[o›)knm1hv#h(hb£hF^.k#vÁk#[qv#[a`Uh[1^1h(hgiv#h;].^1_m1hÃra^1[9ƒvÁknm1h(_,div#k™\Adef,u1k#di[a^°=Œ s‘m1hUhF^£kn\A[a¥£¤
ž†iŒX‹ÁdivUknm1h ^au,`Uf,hF\[9›sf1dik#vUdikUƒdig¨g"kn]£r9h;¥,hF\hb.hF^£kÃ]9v#vnu,` dÜ^1±Ÿ].^Ø[a¥1k#de`Ã]9gp[a_,h.‚)]£^1_œknm1hq^au,`Uf,hF\[9›
hb.hF^£k#v k#[sf,h"p[a`Â`¹u,^1dipF]9kh_¹divge].\A±£>h Ž‘m1divdivhv#v#hF^£k#de]9gigi¤¹k™m1h)pm,].^,^1hgap[1_,de^1±knm1h[a\AhF` ›©\A[a`'dÜ^£›H[a\#`Â]9k#di[a^
knm1h[a\…>¤ ‘m1hhF^.kn\…[a¥.¤q[9›)]›©]9de\p[.de^àd¨vž†iŒX‹ x ? @<HkÁkn].r9hv"[a^1hUf1dik ¥€hF\k#[£v#v"k#[Ãp[a`Ã`Uu,^1dipF]9k#hknm1h[au1k#p[1`Uhv
[9›)]›©]9de\"p[.de^L@¬ ›H[a\m1hF]9_,v‚Á›H[a\"kn]9digiv Z[a\A] 9A›©]9de#\ ;U›H[au,B\ "©v#di_,h_ _,d¨h.‚
ŒŽ† Cs‹xŒ † D(‹xŒŽ† EU‹ŽxŒ †HÙ;‹x Ë 
†AF².‹
].^1_
ž†iŒX‹ x¦® gi[.± Ë  x‰²~
†AF·.‹
F ^q3h G;pdihF^.kÁp[a_,hk#[Ãp[a`Â`¹u,^1dipF]9khknm1h([au1kp[a`Uhv"div
CH
x >1|IDH
x ,9J| Ex¦ ,|Ù x¦.
†AËa‹
²
[9ƒvnu,¥,¥,[.vh(knm,]9kƒ¾hÁde^1v#k#hF]9_m,]b.h(]Ugi[a¥1v#di_,h__,dih.¬
ŒŽ† CÁ‹x ² |Œ † D(‹x Ë  |)ŒŽ†EU‹ŽxŒŽ†HÙq‹x 
†AFß.‹
‘m1hF^àknm1hhF^.k™\A[a¥£¤;[9› knm1divs_,dih(div
ž†iŒX‹x ² ˲ · · xɐ.~ .ß
†AFá.‹
‘¿[Ãp[a_,hknm1h[au1k#p[1`Uhv[9›knm1div_,dih.‚Xƒ"hpF].^ u1vh
C H
x ,| D x¦ a| E x¦. a|ÁÙ x¦£.
†A.‹
‘m1divdivs]˜–A’ %ún—Õ.’ ¸1[X‚€ƒ¾hÁv#hF^1_Ÿf1dik"›H[a+\ C m,]9g ›knm1hÁk#de`Uh.‚X²Uf1dik#v"›H[a\+D ]¢8u,]£\Ak#hF\[9›Žknm1hkdÜ` h.‚,hk#p>e‚
›H[a\].^ ]b.hF\#]o±.hs[o›¾> .ßUf1dik#v ‰m,]9k"d ›ƒ¾hvhF^1_°knm1hhb.hF^£k#v"ƒdiknmàknm1hƒs\A[a^1±U_,div#kn\…dÜf,u1kd¨[1^ Ÿ¸,u,¥,¥,[.vh(ƒ"h_,hv#di±a^ k™m1hp[a_,hƒdiknmqknm1h
_,div#kn\Adef,u1k#di[a^
Œ † Cs‹xŒŽ† D(‹x  |ŒŽ† E¹‹x Ë  |ŒŽ†ÊÙ;‹x ²
†A .‹
f,u1kƒ"h(u1v#hd¨kÁ]9pp[a\A_,de^1±qk#[
ŒŽ† CÁ‹x ² |Œ † D(‹x Ë  |)ŒŽ† EU‹ŽxŒŽ†HÙq‹x 
†A .‹
‘m1hF^àƒ¾hƒd¨gigvhF^1_
® ¯ ŒŽ† 1‹,gi[.±Œ † a‹x ·² Ë· ²  x‰²,~ á.²£ß
†H>² .‹
f1dik#v [a^¦]b.hF\#]9±.>h ‘m1divœdivœ] g¨[1u1v#¤¦p[a_,
h Ø]oknm1hF`Ã]9k#dipF]9gigi¤.‚Ãk™m1d¨v d¨v &h 8,¥1gÜ]odÜ^1h_“f£¤ k™m1hdÜ^1h¢,u,]9gidikA¤
‹ * Œ† a‹€gi[.±¾À,† 1‹‚,]9v"ƒ"
h ÛgigŽ^1[9ƒ‰_,div#pFu1v#v * Œ † a‹€gi[.± ŒŽ† 1
IJKILJ @TªQaV SHWLQ KSERnY
¸,u,¥,¥,[£v#h knm,]9k div;] p[a^£b.&h 8 ›Êu,^1pk#di[a^ ¤Ÿ_,h™«^1dik#di[a^knm1div;`UhF].^1vUknm,]ok(d ›¾Œ { |~F~F~Œ  div ]¥,\A[1f,].f1dig¨dikA¤
_,div#kn\Adef,u1k#di[a^ knm1hF^
!€¯ z€ŠŒ,#Š "%$ ¯ †Hz,ŠH‹.Œ,Š
†H²,F‹
Š
Š
<©^à[.knm1hF\"ƒ"[a\…_,vF‚1d ›j divs]Ã\#].^1_,[1`cbt].\…dÜ]£f1g¨h.‚knm1hF^
†ÊÄ;Ç j°ÈH&‹ $¡Äq'Ç †KjŸ‹ÞÈ8~
†H².².‹
‘m1div›Ê]9pk"divra^1[9ƒs^ ]9)v (8’”,»’F” * »sÔK”X,’ +FÍ1Ñ.ÎiÔϕHš ^1hÂ[9›knm1h;`U[.vk(u1v#h™›Êu1g ›Ê]opk#v(de^ knm1div¹f,u1v#de^1hv#vÃdivknm1hÃp[a^1pF]b1d¨kA¤[9›knm1hÃgi[.±1].\Adiknm,` ‚¿ƒsm1dip½m pF].^Ÿf€h
&h 8,¥,\Ahv#v#h_ dÜ^àbt].\…d¨[1u1v›H[a\#`Uv .-v#de^1
± /.hF^1vhF0^ Ûvsde^1h¢,u,]9gidikC¤.‚Lƒ"hpF].^qƒÁ\Adik#h
®Ù†iŒ¹ÚLÀ9‹ x ¯ ŒŽ†Hz‹,gi[.± ŒŽÀ,†H†ÊzzX‹‹
†H².·.‹
³9µo¶
$ g¨[£± ¯ Œ †HzX‹ Œ À,†H†HzzX‹‹
†H²9Ëa‹
³9µ.¶
x g¨[£± ¯ À,†Hz‹
†H².ß.‹
³9µ.¶
x
†H².á.‹
·
!
¸1de^1phUg¨[£±;divv#k™\Adipk#gi¤ p[a^1pF]b.h.‚1knm1divsvnm1[9ƒvknm,]9kÙ>†ÜŒ Ú Àt‹ Uƒdiknmqh¢,u,]9gidikA¤ d ›¾]£^1_ [a^1gi¤ÃŒÃx‰À)
¢8u1dib9]9gihF^.k#gi¤.‚Lƒ"hpF].^qhF`Ã¥1gi[9¤qknm1h(b.hF\…¤ u1v#h™›©u1gde^1h¢,u,]9gid¨kA¤
z® gi[.±@†ÊzX‹
†H²>.‹
2
1
0
-1
-2
-3
log(x)
x-1
-4
0
-v#de^1±Ãknm1divF‚dik"›H[.gig¨[9ƒvknm,]9k
0.5
1
1.5
2
2.5
3
Zdi±au,\Ah(²€¬ F 1u vh™›Êu1gde^1h¢,u,]9gidikC¤
¯ ŽŒ †1‹,gi[.±)À,†1‹¡¯ ŽŒ †1‹,gi[.±ŒŽ†a‹
†H²>.‹
ƒdiknmàh¢8u,]9gidikA¤ d › .] ^1_ [a^1gi¤;d ›ŽŒÃx‰À))‘m1d¨vÁdiv f€hpF].u1v#h
x‰¯ À,† 1‹Ž® ¯ ŒŽ† 1‹%x ¯ ŽŒ †1‹ ŒŽÀ,††11‹‹ ®
†H²>.‹
¯ ŒŽ†1‹,gi[.± ŒŽÀ8††aa‹‹
†H·>.‹
‘m1div`UhF].^1v"knm,]9kknm1hÁhF`Ã¥1de\AdipF]9gL_,d¨vkn\Adef,u1k#di[a^°]og¨ƒ]¤1v)±.dib.hv"knm1hgi[9ƒ"hv#k)pF\…[.v#v#"©hF^£kn\A[a¥£¤.‚.gi[9ƒ"hv#k¥,hF\#¥1gih&81dikC¤.‚
].^1_ knm1h±a\AhF]9khv#k"g¨[£±a¥,\A[af IJKILJ WXRQ,N9PŽNQ1R ,RFSeO¿W§OVR L
Q S Q,N XQW oQ
/.hF^1v#hF0^ vdÜ^1h¢,u,]9gidikA¤°vnm1[9ƒvk™m,]9kknm1h u1g¨gef,]9pr "Lhdef,gihF\_,d¨b£hF\A±.hF^1phUdiv ]9giƒ]¤avÁ^1[a?^ "C^1h±a]9k#dib.>h J<Êksdivde^£k#hFB\ "
¥,\Ahk#h_ ]9vknm1hs]b.hF\#]9±£hÆ^1u,`Uf,hF\[9›Žf1dik#v"knm,]9k"].\Ahƒ]9v#kh_ f£¤ÃhF^1p[1_,de^1±Ãhb.hF^£k#v›©\A[a`¦]_,d¨vkn\Adef,u1k#di[a^UŒqƒdiknm
]Up[1_,h¹f,]9vh_ [a^ ] 9#ƒs\…[a^1>± ;U_,div#kn\Adef,u1k#di[a^)À <Þ^qk#hF\#`Uv[9› h¢,u,]9k#di[a^1v‚,ƒ"hpF].^qpF]9gipFu1ge]9k#hUknm,]9k
ٝ†iŒ¹ÚLÀ9‹ x ¯ ŒŽ†ÊzX‹,gi[.± Œ À,†H†HzzX‹‹
†H·,F‹
³
x ¯ ŒŽ†ÊzX‹,gi[.±Œ †HzX‹ ®œ¯ ŒŽ†ÊzX‹,gi[.±)À,†Hz‹
†H·.².‹
³
³
Ë
‘m1hF\Ah™›H[a\Ah.‚
°® ¯ ŒŽ†Hz‹,gi[.±)À,†HzX‹x'ž§†ÜŒ|nÀ9‹x‰ž†iŒ‹ Ù>†ÜŒ¹ÚÀt‹
†H·.·.‹
³
h[af1v#hF\Ab.hk™m1hÁ›‡[.gigi[9ƒde^1±;p[1^.b.hF^£k#di[a^1vF¬
gi[.@± x
†H·9Ëa‹
Œ
Œg¨[£± x
†H·.ß.‹
‘m1h _,d¨b£hF\A±.hF^1phÃdiv^1[.ksv¤8`Ã` hkn\Adip¹].^1__,[1hv ^1[£ksvn]ok#divA›‡¤ k™m1h(kn\Ade].^1±£g¨hUde^1h¢,u,]9gid¨kA¤>[9ƒ"hb.hF\‚,dikm,]9v
`Ã].^£¤q[9› k™m1h(¥,\A[a¥,hF\…k#dihvs[9›]Ã`Uhkn\Adi>p ZX[a\&h 8,].`Ã¥1gih.‚
Ù>†iŒUÚXÀ9‹ †H·.á.‹
ٝ†iŒ¹ÚLÀ9‹H
x Œ;x‰À
†H>· .‹
<HkÁ]9giv#[;m,]9vs^1diphUp[a^£b.&h 81d¨kA¤ ¥,\A[a¥€hF\Ak#dihvF¬
ٝ† {‡ŒL{ }AŒX} {#À{ Ž}À}‹ {nٝ†iŒL{ŽÚLÀ{n‹ Ž}™Ù>†ÜŒX}ÚLÀ}‹
†H>· .‹
›H[a\ { } x¦> ZdÜ^,]og¨gi¤.‚diksvn]ok#divA«Xhv ¾ÔK”,»™Òt’– * »Ôϔ’ +ÍaÑ.ÎiÔK•HšX¬
†H>· .‹
`ó ] 8 ŒŽ†HzX‹ ® À8†ÊzX‹ $ ¯ Œ †HzX‹ ®ŸÀ,†HzX‹ $ E Ù>†iŒ Ú À9‹
³
IJKILJ (O¿W SeRFSEO0W QXWXRFNOLPLY
‘m1h º™—.”XÕ.ÔK•HÔK—.”XÑ.Î ’F”•H–A—™˜š[9›Æ] \#]£^1_,[a`“b9].\Ade].f1gih “p[a^1_,dik#di[a^1h_ [a^].^1[.knm1hF\‚j‚Ldiv_,hF^1[.k#h_œž† …jŸ‹
].^1_ _,h™«^1h_Ÿ]9v"›H[.gigi[9ƒvF¬
ž† AjŸ‹ŽxÉ®°¯ ŒŽ†HzX‹¯ ŒŽ† zX‹€g¨[£±ŒŽ† #z‹
†KË .‹
³
‘m1h —.ÔK”X•)’”X•H–A—™˜šž§†‡j>| ‹"divÁ_,h™«^1h_>]9vU†Hf,hƒ].\Ahk™m1h(^1[.kn]9k#di[a^,]ogp[1^£›Êu1v#di[a^àƒdiknm pF\…[.v##v "©hF^£kn\A[a¥£¤,‹
ž†Kj7| (‹x¦® ¯ Œ †Hz3| a‹€g¨[£±ŒŽ†Hz3| a‹
†KËXF‹
³ -v#de^1±Ÿknm1hq_,h™«^1d¨kd¨[1^[9›sp[1^1_,d¨kd¨[1^,]9gs¥,\A[af,]£f1d¨gidikA¤.‚ŽŒ †…jŸ‹x Å Å0 0 !$1 1 ‚ƒ"h;pF]£^Ÿh&8,¥,\Ahv#vÃ]\Ahge]9k#di[a^1vnm1de¥
f,hkAƒ"hhF^ A[.de^£ks].^1_ p[a^1_,dik#di[a^,]9g hF^£kn\A[a¥£¤¬
ž†Kj>| ‹ x ® ¯ ŒŽ†Êz| 1‹,gi[.± ŒŽŒŽ†Hz¿†Hz| 1‹ ‹ ŒŽ†ÊzX‹
†KËa².‹
³ x ®°¯ ŒŽ†Êz| 1‹F†Kgi[.±LŒŽ† "#z‹ gi[.±ŽŒŽ†ÊzX‹A‹
†KËa·.‹
³ x ® ¯ ŒŽ†ÊzX‹,gi[.±Œ †HzX‹ ® ¯ ŒŽ†ÊzX‹HŒŽ† #z‹,gi[.±LŒŽ† "#zX‹
†KË.Ëa‹
x ž†KjŸ³ ‹ ž† AjŸ‹Žx'³ ž§ † ¹‹ ž†K#
j ¹‹
†KËaß.‹
[a\Ah±.hF^1hF\]9gig¨¤.‚

ž†Kj { |~F~F~|#j  ‹x ¯Š { ž†KjÂ$Š AjŠ ( { |F~~F~#|j { ‹
†KËaá.‹
‘m1divÁd¨vv#[a`Uhk#de`UhvÁpF]9gigih_°knm1hq&º %aÑ£Ôϔ–™ÍaÎi’›H[a\p[a^1_,dik#di[a^,]9g hF^.kn\…[a¥.>¤ ß
IJKILJ
LR "SÊWVÞON ,RFSEOW
>Í1•HÍ1Ñ.Î)ÔK” —.–3¹Ñ£•HÔϗ.” L†Kj 7‹±.dib.hvUu1v¹]°` hF]9vnu,\AhÃ[o›Æm1[9ƒ `Uu1pm°de^£›H[a\#`Ã]ok#di[a^Ÿdivvnm,].\Ah_œf€hkAƒ¾hhF^ØkCƒ"[
\#].^1_,[1`“b9].\Ade].f1gihvUj%].^1_ <Êkdivknm1hÃ\Ahge]9k#dib.hÃhF^£kn\A[a¥£¤°f,hkAƒ"hhF^ knm1h C[£dÜ^£k_,div#kn\Adef,u1k#di[a^ ].^1_Ÿ¥,\A[1_Xu1pk
_,div#kn\Adef,u1k#di[a^ [9›knm1hkAƒ"[Ã\#].^1_,[a`cb9].\Ade].f1gihvF‚ Ô E’
L†Kj 7‹)x ¯ ŒŽ†Êz|1‹,gi[.± ŒŽŒŽ†Hz†Hz‹HŒŽ|3a†1‹ ‹
†KË.‹
³ <Hkd¨vÁv#¤,`Ã`Uhkn\…d¨p.¬
†‡j (‹x‰ž†KjŸ‹®Ÿž†K#
j ¹‹x‰ž† U‹®Ÿž† …j؋
†KË .‹
‘m1h(de^.khF\#¥,\Ahkn]9kd¨[1^q[9›¾`Uu1knu,]9gde^£›H[a\#`Ã]9kd¨[1^qd¨vk™m1h(]Fb£hF\#]9±.h(\Ah_Xu1pkd¨[1^°de^àknm1h(gihF^1±.knmà[9›¾]Up[1_,hƒ"[a\A_
›H[a+\ ±£d¨b£hF^ knm,]9k"j divsr1^1[tƒÁ^ h¹_,h™«^1hÂknm1h;º™—.”XÕ£ÔϕHÔK—.”XѣΠ¹Í1•HÍ1Ñ.ÎÔϔ ½—£– UÑ.•HÔK—.” [9›"kAƒ¾[q\#].^1_,[a` b9].\Ade].f1gihv(j%]£^1_ ƒdiknm\Ahvn¥,hpk
k#[;]knm1de\A_‚ ‚X]9v
†Kj Á‹ x ž†Kj Á‹Ž® ž†K#
j #"| Á‹
†KË .‹
x ž† Á‹Ž® ž† Aj| Á‹
†H>ß .‹
‘m1h ^1[.kn]9kd¨[1^ ›‡[1\p[a^1_,dik#di[a^,]9gÁ`¹u1k™u,]9gde^£›‡[a\`Ã]9k#di[a^ div u,^£›H[a\Ak™u,^,]9k#h f,hpF].u1v#h L†Kj 7 Á‹gi[1[ar.vUgider9h dik
kn].rohv¾kAƒ"[Ã].\A±au,` hF^.k#v‚.knm1hv#hp[1^1_°] 9#p[a^1_,dik#di[a^,]9g \#].^1_,[a`cb9].\Ade].f1gi&h ; ,f,u1k"knm1hF\A
h ÛvÁ^1[Ãvnu1pmqknm1de^1± ‘m1h_,h™«^1d¨kd¨[1^°±£hF^1hF\#]9gid hvk#[Ã].^ ].\#f1dikn\#]£\A¤;p[£g¨gihpk#di[a^ [9›)\#].^1_,[1`cbt].\…dÜ]£f1g¨hv¬
†‡j { |j } |F~F~~A|#j  (‹ x ž§†‡j { ‹ ž†Kj } Aj { ‹
ž†Kj  Aj  ( { |~F~~#|#j { ‹
® ž§†‡j { ¹‹ ®œž†Kj } Aj { 7| (‹ ® ®Ÿž†Kj  …j  ( { |F~~F~Hj { ¹‹†Hß,F‹
x ¯Š { L†KjÊ 7…jÊ ( { |F~~F~#|j { ‹
†Hß.².‹
IJKILJ LQ ,R PŽNO 9Q,TFTFSÊW §SÊWQ 0ÏSER™Y
LhkÁj
‚ ‚].^1_ ¦f,hÃ\#].^1_,[1`“b9].\Ade].f1gihvvnu1pm knm,]9kÁj­].^1_ ¦]£\Ah(p[a^1_,dik#di[a^,]og¨gi¤de^1_,hF¥,hF^1_,hF^£kU±.dib.hF^ U‚
Ô e’
ŒŽ† €|nz 71‹lx ŒŽ† a‹HŒ †Hz a‹
†Hß.·.‹
ŒŽ†Hz¿| | .‹ x ŒŽ†Hz‹HŒŽ† zX‹HŒŽ† 71‹
†Hß9Ëa‹
<©^àknm1divpF]9v#hj‚ q‚X].^1_ ¦].\…hvn]9di_ k#[U›H[a\#`c] Ñ.–#Ò9—.Öú %aÑ.ÔK”Ž‚X]£^1_ knm1divsdivÁƒs\Adik#k#hF^
j ® c® ~
†Hß.ß.‹
<HkÁde`Ã¥1gidihv¹k™m,]9k
® ® j †Kk™m1h Ÿ].\#r9[9b;¥,\…[a¥,hF\AkA¤ divk#de`U3h "C\Ahb£hF\Av#def1gihF&‹ F g¨v[X‚Xd › x ¿† ¹‹‚knm1hF^
ƒ"h(m,]b.h(] Ÿ].\#r9[9bUpm,]9de^ j ® c® ކ (&‹ <‡› j ®
c®
'knm1hF^àƒ¾hpF]£^qvnm1[9ƒ‰knm,]9k L†Kj 7‹ †‡j s‹ª¬
L†Kj 7"| Á‹ x L†Kj 7(‹ L†Kj #U‹
†Hß.á.‹
x L† jŸ‹ L†Kj 7 Á‹
†H>ß .‹
á
!
"
#
$
% &
%'
%'
'
)
('
*
,+
.
,+
/+
0
,+
,+
,+-
,+31
,+-
0
4
5
%
21
"u1kUv#de^1ph j
.] ^1_ ].\AhÃde^1_,hF¥,hF^1_,hF^£kq±.dib.hF^ ;‚ ƒ"h m,]b.hÂknm,]9k L†Kj (‹Ux F vÃ] p[1^1v#h¢8u1hF^1ph.‚
v#de^1ph¹`Uu1knu,]ogde^£›‡[a\`Ã]9k#di[a^qpF].^,^1[£ksf,hU^1h±a]9k#dib.h.‚
†Kj ‹ L† #jŸ‹
†Hß>.‹
¤°].¥,¥1gi¤1de^1± knm1divÃÕ.Ñ.•HÑU˜L–C—ºn’ª»n»ÔK”.×;ÔK”X,’ +FÍ1Ñ.ÎiÔK•HšÃk#[Ãk™m1h(pF]9v#hUƒsm1hF\…h ‰x †U‹‚,ƒ"h(pF].^àde^.k#hF\¥,\Ahkknm1div
]9vvn]¤ade^1± knm,]9kq”X— >Ñ U—.Ía”•— ¾Õ.Õ,ÎiÔK”.× ÔK• %Ÿ• %a’ Õ.Ñ.•HÑ ÔKÎÏÎÔ ˜–A—.֒ • %a’ Ôϔ ½’–A’F”Xº™’»(• %1Ñ.• ’ ºnÑ£”-UÑ9Ò9’
–C—?“• %a’ÃÕ.Ñ£•HÑ
4
%
Z [a\)]gi[.k`U[a\AhÁde^£›‡[a\`Ã]9k#di[a^ ÇÛv#dip™È[1^;]9gigL[9›Žk™m1d¨v‚avhhsknm1hf,[1[ar"Îi’U’F”•K»s— n” —.–3¹Ñ£•HÔϗ.” a% ’n—£–nš9‚,f.¤
‘m1[a`Â]9v"[9b.hF\].^1_ /.[9¤ ‘m1[a`Ã]ov¹† dig¨h¤.‚¿>,F‹'