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L1 = L L Lr = (−∞,+∞) L 9^1 /'*<( 1 /'(i < r

,Li+1 ⊂ Li

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r + 16)& ( #$B,m:'#I9 6)/ 7#49 6)/

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x ∈ S,'10; 1 ( #

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(x(i)1 , . . . , x

(i)k )

, 6 j@Ê© HI#8!$# 6)6)/'6 #f:'#

Li* #49 *32 3#4 ( # 6 r & ( # * o0 6)6 #

[x(i)j , x

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6 # 10H7> * #:'# 6 ":'# 6 r & ( # * o0 6)6 #

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Θ(n)!

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# ÄÊLzzF & Æ P(Rn > K log n) = o(1)

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R =1 + maxx∈S `(x)

,<# (6 #$: &Aµs24* #4 ( `(x)

310 ( :'#$o- *+& > 6 #$P 2 10H 24(+*+&FEN/ #$:'#y9? * H ©4(+* #1/2

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x ∈ S,P(`(x) > m) = (1/2)m , !$# EG/'& #4 (+* & #

P(R ≥ m) ≤ n.(1/2)m Q9^1-5 (

m = α log2 n, &)6 o & #4 (

P(R > α log2 n) ≤ 1

nα−1,

!$# EG/'& HI10 (+* # EN/ # ( 1 /'( #!$10 ( ( #K > 1

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E(N) =∑

m≥0 P(N >m)

QgN1 /'*m ≤ log2(n)

,' 1 / JHMu1 * 10 P(R > m)

9? *1 L # ( ,'9^1 /'* m > log2(n)

, 1 / /'(+&)6)& 310 6 r & 2 PN 6)&)(32

P(R > m) ≤ 1/2m−log2(n) Q /( 1 ( 6 ,' 1 / Bop10 RÅ#4 / 9'9|1-5 ( 9? * # #4Hn9 6 #

n > 2 SE(R) =

∑

k≥0

P(R > k)

≤ log2(n) +∑

m>log2(n)

1

2m−log2(n)

≤ log2(n) +∑

m≥0

1

2m

≤ log2(n) + 2

2

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#$ (: 24( # * H & 2 #9? *

R∑

i=1

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i=1

f(Ii(k))

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E(f(I)) ≤ 2,

Ã4ª-Æ %ÃL¯I = [−∞,+∞[

#´ª0Æ & Æ Lzx5ªÃ8Ç ± ªE(f(I)) ≤ 3.

1 &)(I6 r & ( # * o- 6)6 #7!$10 & : 24*32 ,?# (

i + 1310w & op#B / Q1 &)(

y06 f>^1 * # & .Î 24*+& # /'* #7:'#

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~y0Q

%)+, CJ)j$ #. ÑM 6 #$ ( ÎÐ0! &)6 #<:'#"op1 &)* EG/ # 6 #´ 10H7> * #:'# 6

f(I):'# 6 r & ( # * o- 6)6 #

I,-#$ (2 PN 6 / 9 6)/ 9^# (+&)(& : & !$#

j( # 6?EG/ #yj ∈ Li+1

QI * ,N!5O?0! / Z:'#J!$#$ 246F2 HI#4 ( 24( ( :'#< & op#B / / HI1 & i9? * !$10 (+*+/ ! (+& 10m, &)6

9 * 10>?> &)6)&)(321/2

:srt (+* #=:'#= & op#B / (+*+& ! ( #4HI#4 ( / 9 24*+& # /'*<~i# ( :'10 !=:sr¶9'9? *+( #4 &)*=~

Li+1 H & ~9? *+(

yk = +∞ , EG/'& #$ ( : Li+1

9? * !$10 (+*+/ ! (+& 10mQgN * !$10 2$EN/ #4 ( ,f(I)

#$ ( : & (+*+& > / 2 !$10HnHI#/ #o0 *+& > 6 #P 2 10H 24(+*+&FEN/ #%:'#J9? * H ©4(+* #

1/2,sx3Ç ± ²-¯ ÄÅ¯ÐÇ ± ± 5 ~ (+* #% / 9 6)/ 2 PN 6 # ~

k L 6 rt#$+9 24* !$#:sr / # ( # 6)6 #o0 *+& > 6 # 6F2 ( 1 &)* ##$ ( , EG/ # 6)6 # EG/ #31 &)(86 Ïo- 6 # /'* :'#k,vHMu1 *32 #9? *

2R 6 rt#$+9 24* !$#

:sr / #o- *+& > 6 #P 2 10H 24(+*+&FEG/ #7:'#%9? * H ©4(+* #1/2

10Ï!$10 : &)(+& 10' 2 # S Q¦ # (+( #=? 6)C 3#J:'1 &)( (+* # (+*3© 6F2 P ©4* #4HI#4 ( HI1G: &A?2 #: 6 #d!B0§1I = [−∞,+∞[

RÅ!-rt#$ (@D~@ : &)* # &

I#$ (\6 r / &FEG/ # & ( # * o- 6)6 #d:'#" & op#B /

R S QI# µ # ( ,09 /'& EG/ # R#$ (\6 #"9 * #4H & # * & op#B /M~ #<!$10 ( #4 &)*

/ ! / 246F2 HI#4 ( :'#S,N10Z /'* 9? * !$10 (+*+/ ! (+& 10

f(I) ≥ 2QN !$#J!B0B,

f(I)#$ ( ( 1 / u1 /'* §:'10H & 2

( 1!3O?0 (+&FEG/ #4HI#4 ( 9? *M/ #o- *+& > 6 #;P 2 10H 24(+*+&FEG/ #:'#9? * H ©4(+* #1/2

,§!$10 : &)(+& 10' 2 # ~ (+* # /HI1 & 2 PN 6 # ~

2 L 6 rt#$+9 24* !$#7:sr / # ( # 6)6 #o- *+& > 6 #7 6F2 ( 1 &)* #7#$ ( :'# 3Q

2

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# $ÄkÆ ± fx4zF"!

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O(log n)!

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O(log n),s# (EG/ #q!3O? EG/ #I310HnHI# ( : / !3O #4H & :'# * #$!3O # * !3O #:'#

k / Ì 10H7> * #I#$39 24*32 :'#

6 O(1)

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10H7> * # 6F2 ( 1 &)* #:'#o0 *+& > 6 #$< 6F2 ( 1 &)* #$<:'10 ( 1 / §mr¶$op10 EG/ r / #=HMu1 * (+& 10:'#$´#$+9 24* !$#$ L10Ì #n9^# /'( 9?0: 2 : /'&)* #I:'# ( # 6)6 #$O C 9|1 ( O © 3#$ EN/ # 6 rt#$39 24* !$#I:'# 6 w510HnHI#I31 &)( ,^9? * # #4Hn9 6 #-,HMu+1 *32 #9? *d6 #%9 * 1: /'&)( :'#$d#$+9 24* !$#$ S Q1 &)(

Yi6 #´ 10H7> * #<:sr & ( # * o- 6)6 #$o & &)(32 ~ O? /'( # /'*

i:'# 6 r¶ * > * # L 10qU &)(EG/ # 6 rt10I E(Yi) ≤ 2

,0# (Yi ≤ n

Q 1 &)( 2 PN 6 #4HI#4 (Y =

∑

i Yi6 #" 10H8> * # ( 1 ( 6 :sr & ( # * o- 6)6 #$ o & &)(32 / !$1 /'* :'# 6 * #$!3O # * !3O #-Q

G=1 / J$op10

E(Y ) =+∞∑

i=0

E(Yi)

=

+∞∑

i=0

E(Yi|Yi ≥ 1)P(Yi ≥ 1)

=

+∞∑

i=0

E(Yi|Yi ≥ 1)P(R ≥ i)

≤+∞∑

i=0

2P(R ≥ i)

≤ 2 log2(n) + 2∑

i>log2(n)

n

2i

≤ 2 log2(n) + 4

Uf p lf neponlm nqm,n p X f! p f"=r# $&%ln r# ep'" (*f) lf=n* + Y , r pm f#n P(X ≥ x) ≤ P(Y ≥ x)- !. ! x )/4mEr n k p 01 :f"2 mon epbf3 f +m rJy4qw t5 X 6 7 Y , 6 7):p r# 7):pbnqm'*f) 08m9 n:; n epbfr+lp n f sp m6 r - n$ - *En*:p mop? r , <3 neponlm r X ′ 6 Y ′ , @ (A ( rmbp r0*r - $6 epB r0" X 6

Y , 6 7 *mom r0" m bfn p? - n+ * X ≥ Y "CDbp m'8 , B!:r - !"!EF!lm p65$6 + Jf* epbf> m') - 36<* ep'"b*

Ò ! #""$#% & #'('(*)+,RvN1 /'* K * HI# *EN/ # 6 rt10®

E(Yi|Yi ≥ 1) ≤ 2, &)6 / K ( :'# * #4HM *3EN/ # *EG/ #

Yi#$ ( / 9 6)/ 2 PN 6

/ 10H7> * #:'# 6 <: / :'# * & # *d& ( # * o0 6)6 ## H & 2~ O? /'( # /'*i + 1 L !-rt#$ ( :'10 !-,': ( 1 / 6 #$<!B0,/ #8o0 *+& > 6 #n 6F2 ( 1 &)* #q ( 1!3O?0 (+&FEG/ #4HI#4 ( :'10H & 2 #f9? *=/ #7o- *+& > 6 #fP 2 10H 24(+*+&FEN/ #q:'#89? * H ©4(+* #

1/2Q S 2

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O(log n)Q

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u#$ (n24(+&FEG/ # (32

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v,k(u) < k(v) L /

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v,k(v) < k(w)

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k(u),-#$ (

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v,10n

k(v) < k(u) Q ÄÅ«3+ªUÈ¨#$ (8/ * > * #Z> & ? &)* #M:'10 ( !3O? EG/ #310HnHI# (7& ( # * #ZHM & (+& #4 ( :'# /.®& .ÎÐ1 * HM (+& 10 ªnÈa«U¯ÐÇ0«L¯: & (+& ! ( #$Z / #x4zF

k(u),# (/ #qÈa«U¯ÐÇ0«L¯ ÄÊ

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S ⊂ K × PÆ ± ± ÃBM¬Lz)E ± ¯d²NMx5Ç0ÆBÈ|z)UÃ

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# Ò P a 2 HI10 (+* 10 d9? *´*32 ! /'*+* #4 !$#% /'*

n = |S| 6 9 * 109 *+&F24(32 Q N1 /'* n = 0, 6 79 * 109 *+&F24(32 #$ (

(+*+& o & 6 #I &)6 # .& ( # / / &FEN/ #8 * > * #> & ? &)* #!$10Hn9 6 # ( U d310HnHI# (d& ( # * #-Q / 9'9^1-310 w:'10 ! 6 9 * 109 *+&F24(32 RÅ:srt# & ( #4 !$#Ï# ( :sr / & ! &)(32 S o * & #9|1 /'*( 1 /'(

m ≤ n,§# ( 51 &)(

S/ h#4 3#4H8> 6 #:'# ( &)6)6 #

n + 1Q 1 &)(

(k, p)6 r 246F2 HI#4 ( :'#

S:'#Ì9 *+& 1 *+&)(32 H & & HM 6 #-,J# ( 31 & #4 (

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(k, p),:'10 (86 #

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k, # ( !$# 6)6 #$=: / 31 / @ * > * #7: * 1 &)( ,

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S<,-# (6 #<31 / @ * > * #<: * 1 &)( , / (+* #B9I9|1 /'*

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l rt109 24* (+& 10 5m s? (k, p, S)#$ (s& Hn9 6F2 HI#4 (32 #§#4!$10HnHI#4 HB ( 9? * # µ #$! (+/ # */ 7?^v (k, S)

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m, 8'ss (k, S)3#7ÎÐ &)( #4y# µ #$! (+/ (=/ # * 1 ( (+& 10Ì#4 (+* # 6 #f310HnHI# ( :'#8! 6F2

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EG/ # 6)6 # 6 1 & : &Aµ|/ 5#!$10No & #4 : * &)( L 6 6 1 &m/ & ÎÐ1 * HI##$ ( 5# /'6 #4HI#4 (J6 n9 6)/ d & Hn9 6 #

=* !$# ~M6 I9 * 109 *+&F24(32 :sr / & ! &)(32 , 6 # (+* #B9y10> ( #4 / 9 *3© / #f /'&)( #8:sr & 5# *+(+& 10 =# ( :'#f:'#$ (+*+/ ! @(+& 10 q #: 2 9|#4 : EG/ #Z:'#$ (+&)* Pp#$I 6F2 ( 1 &)* #$n:'#$f9 *+& 1 *+&)(32 I:'#$q10>.u+# ( EG/'& «+UÃ4ÄÊ ± Äf: 6 r¶ * > * #-,# ( 109?0B,a & :'# 6 rt1 * : * #8:'#f!$#$ & 3# *+(+& 10 B,a & :'#$ 2 op#4 (+/ # 6 10>.u+# ( EG/'& 10 (24(32* # (+&)*32 =9|1 /'* #

ÒL ! #""$#% & #'('(*)+,9?0J (+* # *324& 24*32 BQ Ï9? *+(+& ! /'6)& # * , 6 # (+* #B910> ( #4 / #$ ( ¯Ð²N ± ÄÅ¯ & Æ ~ !$# 6)/'&vEG/ # 6 rt10 /'* &)( 10> ( #4 /#4 & 24* (6 #$! 6F2 : 6 rt1 * : * #n! * 1 & 55 ( :'#$=9 *+& 1 *+&)(32 EG/'&\6 # /'* 10 (24(32 03 & P- 2 #$BQ^ !$# (+( # &)(+/ (+& 10m, / ! / # * 1 ( (+& 10mr¶ 6)& # / 6 r & 3# *+(+& 10 5#qÎÐ &)( # 0! ( #4HI#4 ( !$10HnHI#Z: / * > * #Z> &A@? &)* #:'# * #$!3O # * !5O #-Q ¦ # (+( #%9 * 109 *+&F24(32 #$ ( #$33#4 (+& # 6)6 #4HI#4 (J6 fHI4HI# EG/ #!$# 6)6 # EG/ #% 1 / d$op10 J: 2 u ~HI#4 (+& 10' 2 #$9^1 /'*d6 #$d & 9 6)& ( B,'# ( 9 *32 3#4 ( # 6 #$<HI4HI#$=$o- ( Pp#$BQ

ÎÐ &)( ,?9? * !$# EG/ # 6 #$d9 *+& 1 *+&)(32 %310 ( !3O 1 & & #$ & : 2 9^#4 :'#4HnHI#4 ( :'#$! 6F2 B, 6 # (+* #B910> ( #4 / ,| &6 rt10f # * # (+& #4 (\EG/ # 6 #$! 6F2 ,mrt#$ (v*+& #4n:sr¶ /'(+* # EG/ r / I * > * # > & ? &)* #< 6F2 ( 1 &)* #iRÐ * > * #´> & ? &)* #§10> ( #4 /#4 & 24* (J6 #$! 6F2 d: / 1 * : * #f 6F2 ( 1 &)* #-, ( 1 /'( #$ 6 #$d9^# * H /'( (+& 10 24( (2$EG/'& 9 * 10>?> 6 #$ S Q9? *+(+& ! /'6)& # * , ( 1 /'( #P * :'# /'* 9 * 10>?> &)6)& ( #!B 6 ! /'6F2 #w /'*f6 #$I * > * #$8> & ? &)* #$I 6F2 ( 1 &)* #$,\9|# /'( (+* #(+* +9^1- 2 # ~ 1 (+* #HI1: ©46 #:'# (+* #B9 BQ2/.8.43 @AJ9<;=A8= 6 ;=7 @@=?>= ;=7 ?6 @6 = ;=7 6 ;<= ; ; = ; @@9g@ 9L4j6 = ; 7

6 #J!B0 EG/'& 1 / & (324* #$53#-, 6 #J!$1 ( :sr / # & 3# *+(+& 10#$ ( HMu1 *32 9? * 6 O? /'( # /'* :'# 6 r¶ * > * #-,EG/'& #$ ( :'10 ! 6 fP * :'# /'*JEG/ #% 1 / d:'#4op10 J? 6)C 3# * Q 33#4 (+& # 6)6 #4HI#4 ( ,a 1 / J 6)6 10 d: 2 HI10 (+* # *J6 #( O 2 1 *3© HI#7 /'& o0 ( @ ª8³ª0Æ.ÄÊLÆG«

hn² $ Æ ± ª0«B¬L«+7¬L¯ ± ª0¯«3%²N«3+xL³.4«xL³'7ª0zF+ª-Ä¾Ç-¯«3²N n

Ã4ÇD M4Ä ÃQ#aL«L¯ EÉ(hn) = O(log n)

M 6 # .& ( #79 6)/ & # /'* =H 24( O 1G:'#$: &Aµs24* #4 ( #$=9^1 /'* 9 * 1 / op# * !$# ( O 2 1 *3© HI# L !$# 6)6 # EG/'& #$ ( 9 *32 3#4 (32 #& ! & #$ ( #$53#4 (+& # 6)6 #4HI#4 ( !$# 6)6 #9 * 109^1- 2 #: ÊQlm# 6 #4HnHI#7Ò.Q f 1 / "ÎÐ1 /'* &)(d/ #:'#$3! *+& 9 (+& 10# 9 6)& ! &)( #-, #4wÎÐ10 ! (+& 10:'# 6 rt1 * : * #:sr & 5# *+(+& 10:'#$

! 6F2 B,':'#$= !$ (+* #$J:sr / #! 6F2 : / * > * #> & ? &)* #7:'# * #$!5O # * !3O #-Q, @ |Ç0¯ÅÄ

σÆ ± È'L«fÆ.Ä¾ª-ÄÅ¯ÐÇ ± ²NLÃ

n ± ÄÅ¯Ð4«5Ã 1, . . . , n

#%4ÄÃ$Ç0¯ ÄT

z%$ ª0«B¬L«3¬L¯ ± ª0¯«3²NZ«3L°xL³.4«xL³'nÇN¬4ÄÊ ± Æ ± ¯ ± Ã$L«+ª ± Ä z)UÃf ± ÄÅ¯Ð4«5Ã7²pª ± Ã%z%$ Ç0«²-«+ σ1, . . . , σn

!z)Ç0«UÃQ#^È.Ç0Æ«

i < j#σi

UÃLÄ Æ ± ª ± xQ)4ÄÅ«3q²N σjÃU¯ 4Ä Ã$LÆzFM ± Ä§ÃU¯

σi = min k` : 1 ≤ ` ≤ i$Ä

σ` > σjÇ0Æ

σi = max k` : 1 ≤ ` ≤ i4Ä

σ` < σjRÅ#4:sr¶ /'(+* #$ ( # * HI#$B, 6

i@Ê© HI#7! 6F2& 24*32 #7:'#4o & #4 (/ !$ (+* #7:'# 6

j@Ê© HI#8 & # ( 5# /'6 #4HI#4 ( &

/ ! / #! 6F2 !$10Hn9 *+& 5##4 (+* #8# 6)6 #$<mr¶ 24(32%& 24*32 #7Bo0 (J6 i@Ê© HI# S 1 &)(

Ti6 r¶ * > * #10> ( #4 / #4Zmr & 24* ("EG/ # 6 #$§! 6F2

σ1, . . . , σiQ &

σi#$ ( / !$ (+* #=:'#

σj,. 6 1 * , 6 1 * <:'#%310 & 3# *+(+& 10;:

Tj−1, 6 f! 6F2

σj /'&)("6 #%!3O #4H & :'# 6 * 0! & # ~

σiQ ¦ #%!5O #4H &

#$ ( # 0! ( #4HI#4 ( !$# 6)/'& EN/ r &)6 /'* &)( /'& o & : Ti L 9? * !$10 2$EG/ #4 ( ,m & σj

24( &)(& 24*32 : Ti, &)6

3# * # (+* 1 / op# * &)( !$10HnHI# 6 7:'#σiQjI * ,s: / * > * #n> & ? &)* #n:'# * #$!3O # * !5O #-, / #fÎÐ# /'&)6)6 #n#$ ( 31 &)(

6 89 6)/ "P * :'#! 6F2%EN/'& 31 &)("& .Î 24*+& # /'* # ~ 510Z9 ©4* #CRÅ & !-rt#$ ("/ #=ÎÅ# /'&)6)6 #=PN / !3O # S ,?31 &)(§6 f9 6)/ ´9^# (+&)( #EG/'& 31 &)( / 9 24*+& # /'* # ~ 3109 ©4* #!RÅ & !-rt#$ (d/ #ÎÅ# /'&)6)6 #: * 1 &)( # S Q / 9'9^1-310 =HM & ( #4? (EN/ #σi

31 &)(=6 n9 6)/ =9^# (+&)( #7! 6F2 ,a9? * H &v6 #$i9 * #4H &F©4* #$, EG/'& 51 &)( / 9 2L@

*+& # /'* # ~σj

R 6 r¶ /'(+* #f!B0B, 1σi

#$ (d6 q9 6)/ JP * :'# EG/'& 31 &)(d& .Î 24*+& # /'* # ~σj,?3# (+* &)( #7:'# 6 qHI4HI#

HM &F©4* # S Q 6 1 * B, 6 1 * %:'# 6 r & 3# *+(+& 10®:'#σj, &)6 #$ (& : & 5!$# * ?> 6 #n:'#

σi( (EG/ rt10 # 6 #q!$10Hn9? * #

EG/ r ~ :'#$ ! 6F2 & 24*32 #$§Bo0 (σi

@ !$# EN/'& #$ (6 #d!B0 :'# ( 1 /'( #$ 6 #$ ! 6F2 EN/'& 5# (+* 1 / op#4 ( /'* 6 #d!5O #4H & :'# 6 * 0! & # ~

σiQ N\ * !$10 2$EN/ #4 ( ,

σj /'& o * 6 #fHI4HI#n!3O #4H & EG/ #

σi6 1 * :'#q310 & 3# *+(+& 10m,s# (

3# * q:'10 ! & 24*32 !$10HnHI#:'#$3!$#4 : ( :'#σiQ

2

# ÒN1 /'*J/ #! 6F2

s = σj,'9? *+(+&)(+& 10' 10 6 rt#4 3#4H8> 6 #8:'#3#$ !$ (+* #$#4 Ë !$ (+* #$JPN / !3O #$Í RÅ!$# /.

EG/'& 310 ( / 9 24*+& # /'* ~s,0:'10 (

s9'9? *+(+& #4 ( / 31 / @ * > * #dPN / !3O # S # ( Ë !$ (+* #$ : * 1 &)( Í&R & .Î 24*+& # /'* ~

s S Ls = σi : i < j, σi ≤ σj, σi = maxσ` : ` ≤ i, σ` ≤ σjRs = σi : i < j, σi ≥ σj, σi = minσ` : ` ≤ i, σ` ≥ σj

N1 /'*r < s

,r ∈ Rs

& # ( 5# /'6 #4HI#4 ( &?6 8! 6F2r#$ (§& 24*32 Bo0 ( !5O?0! / #:'#$§! 6F2

r +1, . . . , s L6 rt1 * : * #=:sr & 3# *+(+& 10 24( ( 6F2 ( 1 &)* #-,!$# (§2 o 2 #4HI#4 ( 3#d9 * 1: /'&)( Bop#$!J9 * 10>?> &)6)&)(32 1s−r+1

QG=#JHI4HI#-,9^1 /'*

r > s,r ∈ Ls

& # ( 3# /'6 #4HI#4 ( &r#$ ( & 24*32 Bo0 ( !3O?0! / #%:'#$"! 6F2

s, s+1, . . . , r− 1,G31 &)(

$op#$!9 * 10>?> &)6)&)(32 1r−s+1

QÏ!$10 2$EN/ #4 !$#-,?10Ï

E (|Rs|) = Hs − 1

E (|Ls|) = Hn−s+1 − 1

I#4;: 2 : /'&)(=EN/ # 6 #% 10H8> * #As = |Rs|+ |Ls|

:sr¶ !$ (+* #$J:'#s,.o 24*+&A #

E(As) = Hs + Hn−s+1 − 2 ≤ 2 ln(n) + O(1).

v1 /'( #LÎÅ1 & B,^!$#f & Hn9 6 #f!B 6 ! /'6 :srt#$39 24* !$#f #89|# * HI# ( 9?0%:'#8!$10 ! 6)/'* # EG/ (~I6 rt#$39 24* !$#q:'#6 O? /'( # /'* :sr / * > * #M> & ? &)* #w:'# * #$!3O # * !3O #Ìv#4 # µ # ( , 6 O? /'( # /'* :'# 6 r¶ * > * ##$ (f6 # Iª 0¯qÆ:'#$

As, 6 #$ EN/ # 6 8 #w310 ( 9?0q: /( 1 /'(q& : 2 9^#4 : ( BQJN1 /'* K * HI# *f/ # & 2 PN 6)&)(32 :'# 6 ;ÎÐ1 * HI#

P(h ≤ k) < ε, &)6 1 / <ÎÐ / : * &)("/ # & 2 PN 6)&)(32 : /(ÊC 9^#

P(As ≤ k) < εn L 1 * , 6 r & 2 PN 6)&)(32 :'#y * p1BoR 6 q3# /'6 #9'9 6)& !B> 6 # &s6 rt10mr¶ & Æ' 6 rt#$+9 24* !$# S #% 1 / d:'10' # * &)(d/ # ( # 6)6 # & 2 PN 6)&)(32EN/ #%9^1 /'*:'#$<o- 6 # /'* J:'#

k> & #4 / 9 24*+& # /'* #$ ~

lnnQN1 /'* 10> ( #4 &)*f/ # & 2 PN 6)&)(32 9 6)/ f9 *32 ! & 3#-, 1 / q:'#4op10 f 1 / & (324* #$35# *n~6 Ìz)Ç0¯%²N7Èa«+ÇN¬+ªN¬4¯°

zÉ¯ ÄÊZ:'# |Rs|R 6 r¶? 6)C 3#M#$ ( #$33#4 (+& # 6)6 #4HI#4 (q6 wHI4HI#M9|1 /'*

Ls S Q ¦§6 &)* #4HI#4 ( , Rs #I: 2 9^#4 : EG/ #

:'# 6 rt1 * : * #:sr & 3# *+(+& 10¹:'#$7! 6F2 1, . . . , s

R 6 #$8! 6F2 8 / 9 24*+& # /'* #$8mr & ( # * o & #4' #4 ( 9?0 S Qvlo- *+& > 6 # 6F2 ( 1 &)* #

1 + |Rs|R 6 #

+19 * 1o & #4 ( :'#

s, EN/'& mrt#$ ( 9?0=9 *+& #4Ì!$10Hn9 ( #f:

Rs S #$ (%6 #7 10H8> * #:'#n«3+x3Ç0«+²Ã%R 246F2 HI#4 ( 9 6)/ ?> &)6)&)(32 9^1 /'*I6

k@Ê© HI# & 5# *+(+& 10 :'#;!$10 (+&)(+/ # *Z/ * #$!$1 * :¨#$ (

# 0! ( #4HI#4 (1/k

RÅ!$# EG/'& o0Z 1 / * #$:'10' # *8/ #n#$+9 24* !$#n:'#Hs

9|1 /'*1 + |Rs| L ( 1 /'( #LÎÅ1 & ,s!$#$2 o 2 #4HI#4 ( M310 ( HM & ( #4? (n& : 2 9|#4 : ( R 6 #IÎ¾ &)(8EG/ # 6 #

k@Ê© HI# 246F2 HI#4 ( :'# 6 /'&)( #31 &)(f/

* #$!$1 * :Ì: 2 9^#4 : / &FEN/ #4HI#4 ( :'# 6 w!$10Hn9? * & 310 /.y246F2 HI#4 ( 79 *32 ! 2 :'#4 ( B, & : 2 9^#4 :'#4HnHI#4 ( :'#6 # /'* 1 * : * #:sr & 5# *+(+& 10 S Q=N\ * !$10 2$EG/ #4 ( ,?10; ± ÇDM¬L«3 |Rs|

² $ ª ± x )4ÄÅ«+LÃ;²-«Ç0¯ÅÄ Ãw²Gs#UÃ4Ä²0¯ÃLÄÅ«L¯Ð¬LÆ ;x5ÇD I;z)ªÃ4ÇD M²N

s− 1aª0«L¯ÅªG¬Lz)LÃf²N4« ± Ç-ÆGz z ¯ ¯ ± ²GÊÈ' ± ²Nª ± ÄÊUÃ #´²N"Èa«ÇG¬+ªN¬L¯ z ¯ÅÄÊUÃ 1/i

È.Ç-ÆG«iaª0«L¯Ðª ± Ä§²G 2

Bs!

f )M #JzF ± ÇDM¬L«+ |Ls|² $ ª ± xQ)4ÄÅ«3UÃ²-«Ç0¯ÅÄ²N

s#UÃLÄ%²0¯ÃLÄÅ«L¯Ð¬LÆ wx3ÇD MZz)ª;Ã$ÇD MZ²G

n − saª0«L¯ÐªN¬LzFUÃf²N4« ± Ç-ÆGz z ¯ ¯ ± ²GÊÈ' ± ²Nª ± ÄÊUÃ #´²N<Èa«+ÇN¬+ªN¬L¯ zÉ¯ ÄÊUÃ 1/iÈ.Ç-ÆG«

iaª-«U¯Ðª ± Ä´²N 2

Bn− s + 1

!y & ( #4? (JEN/ # 1 / ": & +9^1-310 ":'# 6 6 1 & ,9|# /'(@ (+* # / & Hn9 6 #=!B 6 ! /'6 :'#=o- *+& !$#% / K * @¾(@¾&)6

~ !$10 ! 6)/'* #fl rt# # * ! & !$# /'& o- ( 9^# * HI# ( :'#HI10 (+* # *EG/ r &)6 mrt#4#$ (<*+& #4mQ ± «3ÊÈa«3 ± ²Zz)LÃ ± Ç-Ä¾ª-ÄÅ¯ÐÇ ± Ã È|«++xU+²N ± ÄÊLÃ

RsUÃ4ÄzF ± ÇLI¬4«+8² $ ª ± xQ)4ÄÅ«3UÃ8²0«+Ç0¯ÅÄ Ã²Gz)ªx4zF

s²Nª ± ÃÆ ± ª0«B¬L«3q¬L¯ ± ª0¯«3f²N7«+3xL³.L«+xL³.Mª0z)3ª-Ä¾Ç0¯ «+q²N8Ä¾ª0¯ z z)

n# ²pÇ ± Ä z)LÃ8x4z)LÃÃ4Ç ± Ä 1, . . . , n

! ! UÈa«L¯2ML«z)ªwo- *+& !$#²G |Rs|

Ã4Ç0ÆGÃz)ª*$Ç0«If² $ Æ ± %Ã$ÇD M7²NDÈ' ± ²pª ± Ä<²N s!

ÒÑ ! #""$#% & #'('(*)+, ! 'yÇ ± ÄÅ«3L« & Æ'z($ Ç ± ª

Var(|Rs|) = Hs−O(1) z)

O(1)LÃLÄ\¯Ðx4¯vÆ ± ¯ 4Ç-« M m¯z^²pÇ0¯ÅÄ 0¯ÃLÄÊL«=Æ ± Iª B$Ç0«+ª-ÄÅ¯ÐÇ ± ¯ ± ²NDÈ' ± ²pª ± ÄÊfª0ÆNÃUÃU¯´¬4¯Ð ± ²N

s & Æ q²N nL!

! ´ÈpÈazÉ¯ & Æ'L«7z($ ¯ ± Fpª0zÉ¯ ÄÊ8²N\xL³.B¬ I-x4³. BIz)ªCaª0«U¯ÐªN¬LzFfª0zF+ª-Ä¾Ç0¯ «+ |Rs|#|È.Ç-ÆG«7ÇN¬$ÄÊ ± ¯ «Æ ± &aª0zFLÆ«

kÄÊLz z) & Æ P(|Rs| > k) < 1

n

! ± È.Ç0Æ«U«+ªZÃUÆÈpÈ.Ç0Ã$L«n/4 ≤ s ≤ 3n/4

!

N /'& EN/ # / & Hn9 6 #Z!B 6 ! /'6 :'#Zo0 *+& !$#Z #M 1 / f9^# * HI# ( 9?0f:'#!$10 ! 6)/'* #-, &)6 !$10No & #4 ( :'#3#* >? (+(+* #% /'* / # & 2 PN 6)&)(32 9|1 *+( ( ,G:'#HM &F©4* #=P 2 24* 6 #-,G /'*§6 #$´310HnHI#$´:'#Jo- *+& > 6 #$< 6F2 ( 1 &)* #$& : 2 9^#4 : ( #$f 6 rA¯ ± Fpª0zÉ¯ ÄÊq²N v³.4« ± Ç R op1 &)*J6 r¶' # # , & 2 PN 6)&)(32 Q S QM 6 #$ ( ! 6 &)* ,-:sr¶9 *3© 6 : & 5! / 3 & 10n9 *32 ! 2 :'#4 ( #-, EN/ #-,9^1 /'* :'#$!$10 ( ( #$

C1# (

C29'9 * 109 *+&F2 #$,

# (n033#P * :s, 10

ln(n)− C1 ≤ E(As) ≤ 2 ln(n) + C2.

Ï!$10 2$EN/ #4 !$#-,?10Ï.,':sr¶9 *3© 6 r & 2 PN 6)&)(32 :'# ¦ O # * 1 µ ,

P (As > (1 + δ)(2 ln(n) + C2)) ≤(

eδ

(1 + δ)1+δ

)ln(n)−C1

&m6 rt109 * #4 :δ053#P * :9^1 /'*dEN/ # 6 rt10Ï &)(

eδ

(1 + δ)1+δ<

1

2

R 9? * # #4Hn9 6 #δ = 1.4 S ,a10;10> (+& #4 (

P (As > (1 + δ)(2 ln(n) + C2)) <C ′

n2.

N\ * !$10 2$EG/ #4 ( , 6 9 * 10>?> &)6)&)(32IEG/ # 6 r / #M / HI1 & :'#$no0 *+& > 6 #$7 6F2 ( 1 &)* #$

As31 &)( 9 6)/

P * : EG/ #(1 + δ)(2 ln(n) + C2)

RÅ!-rt#$ (@D~ : &)* # EG/ # 6 8O? /'( # /'*h31 &)( 9 6)/ <P * : EG/ #%!$# (+( #%HI4HI#

o- 6 # /'* S ,a#$ ( HMu+1 *32 #79? *C ′/n

QHI * ,h#$ ( ,?:'#8HM &F©4* #q!$# *+( & #-,aHMu+1 *32 #89? *

nQIyM:'10 !-,^#4

9^1-5 (f(n) = (1 + δ)(2 ln(n) + C2)

,

E(h) ≤ f(n)P(h ≤ f(n)) + nP(h > f(n))

≤ f(n) + C ′

I®Z:'10 !8> & #4E(h) = O(log n)

,|!$# EN/'& 9 * 1 / op# 6 # ( O 2 1 *3© HI#n# ( 9^# * HI# ( :sr¶K * HI# *EG/ #-,^#4HI1 C #4' #-,a!3O? EG/ #H & 3# ~ u+1 /'* 1 /* # EG/ ( # /'*</ (+* #B9 /'* / ;!$1 (

O(log n)Q

R%SR w]<ÁqÂ3ke»fkf]dºf]dgdk #Ä¾ªG¬Lz)²N³ªpx4³ª"FN RD³ªÃU³GÄ¾ªN¬LzFÌ#4 'P 6 & S #$ (Z/ #y (+*+/ ! (+/'* #y:'#:'10' 2 #$w9|# * HI# (+( (

:srt# µ #$! (+/ # * 6 #$ 109 24* (+& 10 7?^v? , 3 s # ( 7's s #4 ( #4Hn9 #$53#4 (+& # 6)6 #4HI#4 ( !$10 ( ( L#4®!$10 (+* #49? *+(+& #-, 6 rt#$+9?0!$# * # EG/'& #$ ( 9 6)/ & Hn9^1 *+( ( ,v# (&)6 #$ ( !$1 ( # /. :'#q9? * !$1 /'*+&)*6 rt#4 3#4H8> 6 #:'#$d10>.u+# ( d!$10 ( #4 / %: 6 ( > 6 #-Q

l rt#4 5#4H7> 6 #:'#$<! 6F2 <3# * 8 1 (32K,Bop#$!

m = |K| Q=Iw9^1 /'*+* 89? * # #4Hn9 6 # / 9'9^1-3# *"EN/ # 6 rt10

K = 0, 1, . . . ,m− 1 Q

g'()+ ( Ò-Ò2/.420. @@?>98;g@A

# & : 2 # 246F2 HI#4 ( &)* #9|1 /'* # µ #$! (+/ # *<* #$!3O # * !3O #$J# ( H & 5#$ ~ u+1 /'* #4 ( #4Hn9 <!$10 ( ( ,'!$10 & ( #~ ( 1G! p# *8/ # ( > 6 # & :'# '2 #M9? *f6 #$q! 6F2 RÅ#4¹:sr¶ /'(+* #$ ( # * HI#$B, / ( > 6 #B / :'# ( &)6)6 #

m S L 9^1 /'*!3O? EG/ #q! 6F2x, 6 Z!$# 6)6)/'6 #q:sr & : & !$#

x!$10 (+& #4 : * 31 &)( RÅBr &)6 mr C I9?0%:sr 246F2 HI#4 ( :'#8! 6F2

k:

6 rt#4 3#4H8> 6 #S S ,'31 &)( u

&u#$ (<6 r 246F2 HI#4 ( RÅ / 9'9^1- 2%/ &FEG/ # S :'#! 6F2 x

: SQ

l r & !$10Go 2 & #4 ( HMu+# /'* :'#;!$# (+( #9'9 * 1!3O #Ï#$ (M2 o & :'#4 ( &)6 5# &)(+/ # / & op#B / :'# 6 rt#$+9?0!$#H 2 HI1 &)* # * # EG/'& BQ"% 6 #$w!B0 (ÊC 9 &FEG/ #$B,

mR 6 ( &)6)6 #y:'# 6 rt#$39?0!$#:'#$! 6F2 S #$ ( :'#>^#B / !$1 / 9

/ 9 24*+& # /'*~f6 ( &)6)6 #:'#$#4 3#4H7> 6 #$ EN/ # 6 rt10: 2 &)* # * #49 *32 3#4 ( # * ,a# ( ,'#4 ( 1 /'(J24( ( :'#!B / 5#-, &)6 #$ (& Hn9 * (+& !B> 6 #I:'#n ( 1! p# */ ( > 6 #B / :'# ( &)6)6 #

mR 9^#4 3# *7~

m = 232 ,^!$# EN/'& !$1 *+* #$+9^10 :® / !B01 6 #$d! 6F2 J310 ( :'#$J#4 (+& # * J /'*</ #HM0!3O & # Lq> &)( S Q2/.420.21 @@?>98;=7%; 9 4 9 6 4 7 6; @ @+;

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1 / op# *+( #:'#F

#$ (</ #9? *+(+& #HI#$ /'* > 6 #8:'#E

∀B ∈ T , x ∈ E, f(x) ∈ B ∈ E 2 2 ± C$Ç ± x$ÄÅ¯ÐÇ ± ILÃUÆ«ªN¬4z)w² $ Æ ± UÃÊÈ.ªpx58Èa«ÇG¬+ªN¬L¯ z ¯Ã$ a0L«5ÃÆ ± UÃÊÈ.ªpx5nÄ¾ÇUÈ.Ç0z)Ç%F0¯ & Æ IUÃLÄ"ªUÈpÈ Lz)5qo- *+& > 6 #7 6F2 ( 1 &)* #!

" jM 6 *+*+& op# EG/ # 6 rt10 *32 3# * op# 6 # ( # * HI#8:'# Ë o0 *+& > 6 #q 6F2 ( 1 &)* #LÍ79^1 /'*J6 #$<ÎÐ10 ! (+& 10 HI#$ /.@* > 6 #$ ~ o- 6 # /'* «33Lz z)LÃLQ ¦ #%mrt#$ ( 9?0 6 #%9^1 & ( :'#o / #0:'109 (32& ! & Q D : 2 |Ç0¯ÅÄ Ω Æ ± ± Ã$M¬LzF ± Ç ±aB¯Ð²N & Æ LzAx5Ç ± & Æ #J4Ä X

Æ ± 84Ç ± x$ÄÅ¯ÐÇ ± ²GE ± ¯¾%ÃLÆG« Ω4Ä#B aª0zFLÆG«UÃ8²pª ± Ã%Æ ± UÃ¾È'ªpx5

(F, T )!

ª (+*+& > / #4'Pp#4 : *32 #9? *X# ± Ç-ÄÊ3 σ(X)

# UÃLÄvz)ªJÈaz ÆGÃÈ'$ÄÅ¯ ÄÊ=ÄÅ«L¯Ð¬4ÆIÃUÆ«Ω«3 ± ²pª ± Ä X MUÃLÆG«+ªN¬LzF"!

$ UÃLÄ´ª0ÆNÃUÃU¯z)ªMÄÅ«L¯¾¬LÆ ± FG ± ²0«33JÈ.ª-«z)LÃf ± Ã$ I¬4z)UÃ

X−1(B) = ω ∈ Ω : X(ω) ∈ BÇ

BÈ'ª0«x5Ç0ÆG«4Ä# Ã$Ç0¯ÅÄ z)ªMÄ¾ÇUÈ'Ç0zAÇQF0¯Ð T ²G

F#Ã$Ç0¯ÅÄÃU¯ÈazFM ± Ä Æ ± <È.ª0«4ÄÅ¯¾ & ÆG¯ z%$ ± FN ± ²0«+ !

6 #8!B0,?Î *32$EG/ #4 ( ,^1Ω#$ ( : 2 u ~ H / & :sr / # (+*+& > / F # ( 1

X#$ ( : 2 u ~ : 2L & #f!$10HnHI#

/ #o0 *+& > 6 #; 6F2 ( 1 &)* #-,σ(X)

9'9? * :F ( ? (+/'* # 6)6 #4HI#4 ( !$10HnHI# / #w31 / @¾(+*+& > / :'# F Q ¦ rt#$ ( ,#4EG/ # 6FEN/ #31 *+( #-, 6 rt#4 5#4H7> 6 #=H & & HM 6 :sr 2 o © #4HI#4 ( "9^# * HI# (+( ( :'#: 2L &)*´6 7o0 *+& > 6 #% 6F2 ( 1 &)* #XQ

%# /. o- *+& > 6 #$ 6F2 ( 1 &)* #$

X# (

Y,9 * #4? (6 # /'* o- 6 # /'* \: / fHI4HI#"#$+9?0!$#

F,310 ( Fpª0zFUÃ

± zAÇ0¯ÃL,^ & # 6)6 #$10 (6 MHI4HI# 6 1 & :'#f9 * 10>?> &)6)&)(32 ,m!-rt#$ (@D~@ : &)* #M & ,?9^1 /'*( 1 /'( #4 3#4H7> 6 #IHI#$ /'* > 6 #B

:'#F, 10 6 r 2 PN 6)&)(32

P(X ∈ B) = P(Y ∈ B)R QQP SM 6 #$ (& Hn9^1 *+( ( :'#8 1 ( # *EN/ #

X# (

Ymrt10 ( /'6)6 #4HI#4 ( >|#$51 & y:srt (+* #q: 2L & #$ /'*=6 #8HI4HI#

#$+9?0!$#w:'#Z9 * 10>?> &)6)&)(32 L 6 r 2 PN 6)&)(32 #4 6 1 & f # Ë 9? *+6 #LÍ EG/ #Z:'#$ 6 1 & q:'#Z9 * 10>?> &)6)&)(32 BQ ¦ rt#$ ( !$# EG/'&9^# * HI# ( :'#%9? *+6 # * :'# Ë o- *+& > 6 #7 6F2 ( 1 &)* #7:'# 6 1 &

µÍ5Q

l r 2 PN 6)&)(32 #4 6 1 & & Hn9 6)&FEG/ #-, #4;9? *+(+& ! /'6)& # * , 6 r 2 PN 6)&)(32 :'# ( 1 / 6 #$<HI10HI#4 ( : 2L & Q #J!$10 : &)(+& 10 / KM5 ( #zRÅ# ( 2 !$#$35 &)* #-,G9 /'& EG/ r &)6 Br¶P &)( :sr / #=31 / @ !$10 : &)(+& 10 S 9|1 /'*´6 r 2 PN 6)&)(32#4 6 1 & :'#:'# /. o- *+& > 6 #$B, #$ (ZEN/ # R QQP S 31 &)( o * & #9^1 /'*I( 1 /'(

B9'9? *+( #4? (~y/ #4 3#4H7> 6 #

:sr 2 o © #4HI#4 ( EG/'& ± FN ± ²-«+ 6 (+*+& > / :'#FQN # ( # 6)6 #9 * 109 *+&F24(32 #$ ( 9? *+(+& ! /'6)&F©4* #4HI#4 (<& (324* #$35 ( #

EG/ :F#$ (´/ #$+9?0!$#9 * 1: /'&)( RÅ!-rt#$ (@D~@ : &)* #= &

X# (

Y310 ( :'#$ÃUÆ¯ ÄÊLÃ=:'#o0 *+& > 6 #$< 6F2 ( 1 &)* #$ S L# 6)6 #Br 2 10 !$#8 6 1 * J & &

Ñ ""% ) & #'('(*)+$ |Ç0¯¾ ± Ä X = (Xn)n>0

4ÄY = (Yn)n>0

#J²NLÆ ÃUÆ¯ ÄÊLÃI²GCaª-«U¯ÐªN¬LzFUÃZª0zF+ª-Ä¾Ç0¯ «3UÃ Baª0zFLÆ«5Ãq²pª ± ÃÆ ± UÃÊÈ.ªpx5F!

X4Ä

YÃ4Ç ± Ä FNª0z)LÃ7 ± z)Ç0¯ Ã #ÃL¯4ÄÃ$LÆz) I ± Ä\ÃU¯2#aÈ.Ç0Æ«%Ä¾Ç-ÆÄ k > 0

4Ä Ä¾Ç0ÆÄk°ÊÆÈaz)$Ä

(B1, . . . , Bk)² $ ± ÃBM¬Lz)LÃ ILÃUÆ«ªN¬4z)UÃn²pª ± Ã F#§Ç ± ªMz($ Fpª0zÉ¯ÅÄÊ

P(X1 ∈ B1, . . . , Xk ∈ Bk) = P(Y1 ∈ B1, . . . , Yk ∈ Bk).

lm1 * EN/ #

X#$ (d/ #%o- *+& > 6 #7 6F2 ( 1 &)* #-,?10; 1 ( #731 / op#4 (

(X = a)6 r 2 o © #4HI#4 (

ω ∈ Ω : X(ω) = a = X−1 (a) .

%#dHI4HI#-,G &X

#$ ( / #o0 *+& > 6 #= 6F2 ( 1 &)* # *32 # 6)6 #R ~ o- 6 # /'* : R S ,G# ( & I #$ ( / & ( # * o- 6)6 # *32 # 6

EG/ # 6 !$10 EN/ #-, 10 1 ( #(X ∈ I) = X−1(I) = ω ∈ Ω : X(ω) ∈ I .l r LÃ¾È'L«+ª ± xUn:sr / #o- *+& > 6 #7 6F2 ( 1 &)* # *32 # 6)6 # X

,. 1 (32 #E(X)

, #$ ( : 2L & #9? *

E(X) =∑

a

aP(X = a)

=∑

ω∈Ω

X(ω)P(ω).

¦ # (+( #=: 2L &)(+& 10Zmrt#$ ( o- 6 > 6 # EG/ #J: 6 #J!B0´1X

9 * #4 :M5#$ o- 6 # /'* §: / Z#4 5#4H7> 6 # &1 / : 2 10H8> * > 6 # L : 6 #d!B0 !$10 (+* &)* #-, 6 310HnHI#J#$ ( * #4Hn9 6 0! 2 #d9? * / # & (32 P * 6 #=:'#dlm#4>^#$+P / #-Q #J9 * 109 *+&F24(32 # .(+* 4HI#4HI#4 (</'(+&)6 #-,.# ("EN/'& # 2 !$#$3 &)( #% / ! / #%O C 9|1 ( O © 3#9? *+(+& ! /'6)&F©4* # /'*´6 #$o- *+& > 6 #$ 6F2 ( 1 &)* #$ , #$ (<6 z ¯ ± +ª0«L¯ÅÄÊ8²Gz%$ UÃ¾È'4«ª ± x5w |Ç0¯¾ ± Ä X

4ÄY

²NLÆ Caª0«L¯ÐªN¬LzFUÃª-z)+ªpÄ¾Ç0¯«3UÃ²NE ± ¯ÐUÃJÃUÆ«zF )MUÃÊÈ.ªpx5"Èa«+ÇN¬+ªN¬L¯ °zÉ¯ ÃB #§4Ä\Ã4Ç-¯Ð ± Ä λ

$Äµ²NLÆ Z«+5Lz Ã & Æ Lz)x3Ç ± & Æ'UÃ"!z)Ç0«UÃ=z)ªCaª-«U¯ÐªN¬LzFfª0z)3ª-Ä¾Ç0¯ «+ Z = λX + µY

²GE ± ¯¾È.ª0« È.Ç0Æ«7Ä¾Ç0Æ.Äω ∈ Ω

#Z(ω) = λX(ω) + µY (ω)

a0L«U¯ EÉ(Z) = λE(X) + µE(Y ).

R Q Sl aª-«U¯Ðª ± x57:sr / #=o0 *+& > 6 # 6F2 ( 1 &)* ##$ (´/ # EN/ (+&A !B (+& 10:'#%579 * 109|#4 & 10 ~ 9 * #4 : * #:'#$

o- 6 # /'* 246 1 & P- 2 #$:'#510;#$+9 24* !$#M' & !$# (+( ##$+9 24* !$##$ (µ = E(X)

,?10;9|1-5#Var(X) = E

(

(X − µ)2)

= E(X2)− µ2

R 6 r 2 PN 6)&)(32 :'#$:'# /. # 9 * #$3 & 10 *32 /'6)( #7: / ÎÐ &)(=EN/ # 6 no- *+& > 6 #f 6F2 ( 1 &)* #X − µ

#$ ( :srt#$+9 24* !$# /'6)6 # S Ql7o- *+& !$##$ ( 31 / op#4 ( 1 (32 #

σ2 , 6 6 # (+(+* # σR EG/'& #4#$ ( 6 * 0! & #!B *+*32 #9^1- &)(+& op# S * #49 *32 5#4 @( ( 6 1 * 6 r +x3ª-«LÄÅ° Ä I5È q:'# 6 qo- *+& > 6 #7 6F2 ( 1 &)* #-Q

" 1 /'( #no- *+& > 6 #n 6F2 ( 1 &)* # *32 # 6)6 #fmr¶9?0JÎÐ1 * ! 2 HI#4 (7/ #f#$39 24* !$# & #%R 6 24*+& #-,s1 /6 r & (32 P * 6 #-,§9^# /'( : & op# * Pp# * S , # (n/ #o- *+& > 6 # 6F2 ( 1 &)* # EG/'& / ##$+9 24* !$#9^# /'( #w9?0ZBop1 &)*:'#Ïo0 *+& !$#-Q<lm#$9 * 109 *+&F24(32 2 10 ! 2 #$:'1 & op#4 ( (+* ##4 ( #4 : / #$;31 / *32 3# * op#:srt# & ( #4 !$#-QN 6)/ 9 *32 ! & 2 HI#4 ( , 6 rt# & ( #4 !$#%9|1 /'*"6 # Ë HI#4H7> * #PN / !5O #LÍ#4 (+* & # 6 rt# .& ( #4 !$#=9|1 /'*<6 # Ë HI#4H8> * #: * 1 &)( Í5QV h5 +@ m &<:p r# f$ *r5*r - 68n f$ r , :n f:rjm@$,nqr5 r - n$ rjp f !Ef:p r

$*" #"c " ig #"i -Ò 39 24* !$#%# ( o- *+& !$#= #510 ("EN/ # 6 #$§:'# /. 9 * #4H & # * ( # * HI#$":sr / #% /'&)( #=:'# IÇDM ± Ä Ã%:sr / #o- *+& > 6 #7 6F2 ( 1 &)* #-QHIÏ: 2L &)( ,'9|1 /'*J( 1 /'( #4 (+& # *

k > 0RÅ# ( 31 / *32 3# * op#7:srt# .& ( #4 !$# S , 6 # IÇDM ± Ä² $ Ç0«+²0«+

kR * #$+9^#$! (+& op#4HI#4 ( , IÇDM ± Ä§xU ± ÄÅ«3n² $ Ç0«²-«+ k

9? *

µk(X) = E(Xk)

µck(X) = E((X − µ1(X))k)

I * #$!$10'?:F ( : µ1(X)

6 rt#$+9 24* !$#-,?# ( : µc

2(X), 6 qo- *+& !$#-Q

..1 86 :;; 86 @ ;MD (+/'&)(+& op#4HI#4 ( ,s:'# /.2 o © #4HI#4 ( %1 / :'# /. o- *+& > 6 #$= 6F2 ( 1 &)* #$%310 ( ¯ ± ²NÊÈ ± ²pª ± Ä Ãf &m/ # & @ÎÐ1 * HM (+& 10q10> ( #4 / #d /'*6 r / m, #§HI1: &A #´9?0 6 !$10'? & 55 !$# EN/ # 6 rt10IJ:'# 6 r¶ /'(+* #-Q ¦ # 6 =3# (+* 0: /'&)(

ÎÐ1 * HI# 6)6 #4HI#4 ( :'# 6 fHM &F©4* #7 /'& o- ( #-Q fLÆ ;ba ± I ± Ä Ã A

4ÄB

Ã4Ç ± Ä§²0¯ÅÄ Ã & : 2 9^#4 : ( #§ÃU¯§4Ä Ã$LÆz) I ± Ä§ÃU¯

P(A ∩B) = P(A)P(B).

I8 1 ( # * EG/ r / 2 o © #4HI#4 ( :'# 9 * 10>?> &)6)&)(32 <1 / P-,B#$ (& : 2 9^#4 : ( :'# ( 1 /'( /'(+* # 2 o © #4HI#4 ( Q fLÆ aª0«L¯ÅªG¬Lz)LÃª0zF+ª-Ä¾Ç0¯ «+LÃÏ«33Lz z)UÃ

X4Ä

YÃ4Ç ± Äf²0¯ÅÄÊUÃ & : 2 9|#4 : ( #$ #7ÃU¯q4Ä

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$ÄJ# zFUÃfba ± M ± Ä Ã (X ∈ I)

4Ä(Y ∈ J)

Ã4Ç ± Ä ¯ ± ²NÊÈ' ± ²pª ± Ä Ã !´È'Ç0ÆG«fÄ¾Ç0Æ.ÄÊUÃ&aª0zFLÆ«5Ã

x4Ä

y#§Ç ± ª

P (X ≤ x4Ä

Y ≤ y) = P(X ≤ x)P(Y ≤ y).

RÅ 6 #n!B0: * # 6 #q9 6)/ %P 2 24* 6 9|1-3 & > 6 #-,s1 6 #$o0 *+& > 6 #$8 6F2 ( 1 &)* #$9^# / op#4 ( (+* # ~ o- 6 # /'* : %:'#$#$+9?0!$#$ EN/ # 6 !$10 EG/ #$B, 6 r & : 2 9^#4 : !$#nBrt# 9 *+& HI#f9? *%6 #7ÎÐ &)(%EN/ # 6 #$nÄÅ«L¯Ð¬LÆGÃn#4'Pp#4 : *32 #$9? *6 #$:'# /. o0 *+& > 6 #$f 6F2 ( 1 &)* #$310 (7& : 2 9|#4 : ( #$B,m!-rt#$ (@D~@ : &)* # EG/ # ( 1 /'( #4 3#4H8> 6 #I:'# 6 r / ##$ (<& : 2 9^#4 : ( :'# ( 1 /'( #4 3#4H8> 6 #7:'# 6 r¶ /'(+* # S 6 #!B0§9? *+(+& ! /'6)& # * 1 6 #$ o- *+& > 6 #$< 6F2 ( 1 &)* #$" #J9^# / op#4 ( 9 * #4 : * # EN/ r / 10H7> * # & 1 /: 2 10H7> * > 6 #q:'#7o0 6 # /'* R 6 #7!B0 (ÊC 9 &FEN/ # 24( ( !$# 6)/'& :'#o- *+& > 6 #$% 6F2 ( 1 &)* #$ ~ o0 6 # /'* n ± ÄÅ¯&L«3UÃ S ,6 q: 2L &)(+& 10Ï#$ (d2 PN 6 #4HI#4 (%2$EN/'& o- 6 #4 ( # ~n6 q /'& o0 ( #I M fLÆ aª0«U¯ÐªN¬LzFUÃZª0zF+ª-Ä¾Ç-¯«3UÃ

X4Ä

Y#B%aª-z)LÆ«UÃZ²pª ± ÃnÆ ± ± ÃBM¬LzF*E ± ¯dÇ0Æ²NL°

± ÇDM¬L«+ªN¬LzFV#Ã$Ç ± Ä & : 2 9^#4 : ( #$fÃL¯§4ÄÃBLÆz)M ± Ä ÃL¯ #|È.Ç-ÆG«fÄ¾Ç0ÆNÃ x

4Äy²pª ± Ã V

#§Ç ± ª

P(X = x4Ä

Y = y) = P(X = x)P(Y = y).

l r & : 2 9^#4 : !$#M:'#Io0 *+& > 6 #$7 6F2 ( 1 &)* #$8#$ (/ #q 1 (+& 10 & Hn9^1 *+( ( #-,v# ( w:'#$!$10 2$EG/ #4 !$#$EG/'& #310 ( 9?0<o * & #$J#4;P 2 24* 6 |Ç0¯¾ ± Ä X

4ÄY

²NLÆ caª0«L¯ÅªG¬Lz)LÃfª0zF+ª-Ä¾Ç0¯ «+LÃ7¯ ± ²NÊÈ' ± ²pª ± ÄÊUÃ"! ± ª

E(XY ) = E(X)E(Y )R Q S

Var(X + Y ) = Var(X) + Var(Y )R Q Ñ S

< ""% ) & #'('(*)+$ |Ç0¯¾ ± Ä X 4Ä

Y²N4Æaª0«U¯ÐªN¬LzFUÃª0zF+ª-Ä¾Ç0¯ «+LÃ¯ ± ²NÊÈ' ± ²pª ± ÄÊLÃ ! zAÇ0«UÃQ# & Æ'4zzFUÃ & Æ'Ã4Ç-¯Ð ± Ä\z)UÃ/$Ç ± x$ÄÅ¯ÐÇ ± Ã f

$Äg ²NE ± ¯¾UÃ=ÃUÆ«zFUÃ8 ± ÃBM¬LzFUÃ²Giaª-z)LÆ«UÃ²G

X$Ä

Y#\«3UÃ¾È'3x$ÄÅ¯4a0M ± Ä #

f(X)4Ä

g(Y )Ã$Ç ± Ä ¯ ± ²GÊÈ' ± ²Nª ± ÄÊUÃ"!

l 1 (+& 10:sr & : 2 9^#4 : !$#w:'#Z:'# /. o- *+& > 6 #$q 6F2 ( 1 &)* #$RÅ1 / :'#M:'# /.2 o © #4HI#4 ( S Br 24( #4 : & Hn9 6 #4HI#4 (=~f/ ; 10H8> * # EG/ # 6 !$10 EN/ #I ± k

°ÊÆÈaz)$Ä(X1, . . . , Xk)

²Ncaª0«L¯ÐªN¬LzFUÃª-z)+ªpÄ¾Ç0¯«3UÃUÃLÄx5Ç ± ÃLÄÅ¯ÅÄÅÆ ;²Ncaª0«U¯ÐªN¬LzFUÃ¯ ± ²NÊÈ' ± ²pª ± ÄÊLÃQ# ÃL¯§4Ä Ã$LÆz) I ± Ä´ÃU¯2#^È.Ç0Æ«8Ä¾Ç0Æ.Äa+x$ÄÊLÆ«I²N k

± Ã$M¬LzFUÃ²G aª0zFLÆG«UÃ(A1, . . . , Ak)

#Ç ± ª

P(Xi ∈ Ai(1 ≤ i ≤ k)) =∏

1≤i≤k

P(Xi ∈ Ai)

± 4ªDq¯ zzFd¯ ± E ± ¯Ð (Xi)i∈I²Naª0«L¯ÅªN¬4z)UÃª0zF+ª-Ä¾Ç0¯ «+LÃ%UÃLÄ\x3ÇDÈ.Ç0Ã$3²Gaª0«L¯ÅªN¬4z)UÃ

ª0zF+ª-Ä¾Ç0¯ «3UÃ7¯ ± ²NÊÈ' ± ²pª ± ÄÊLÃQ# ÃL¯"!5O? EN/ #MÃ4Ç-ÆNÃL° $ªDq¯z z),E ± ¯¾z($ UÃLÄA! & 3 & 9^10 ( 1 /'( :'#M /'&)( # / #Z# *+* # /'* Î *32$EN/ #4HnHI#4 ( !$10HnH & 3#v!$#Mmrt#$ ( 9?079? * !$# EG/ #

3o- @

*+& > 6 #$§ 6F2 ( 1 &)* #$§310 ( :'# /.n~ :'# /.f& : 2 9^#4 : ( #$ EG/ rt# 6)6 #$ 310 ( & : 2 9|#4 : ( #$BQG%#"HI4HI#-,k+1o- *+& > 6 #$ 6F2 ( 1 &)* #$9|# / op#4 ( (+* # ( # 6)6 #$ EG/ #"!3O? EG/ #

k@¾/ 9 6 # ( #$ (/

k@¾/ 9 6 # ( :'#ó0 *+& > 6 #$ 6F2 ( 1 &)* #$

& : 2 9^#4 : ( #$ RÅ10w: &)( 6 1 * EG/ # 6 #$§o- *+& > 6 #$"310 (k~

k& : 2 9|#4 : ( #$ S 5 EG/ # 6 rt#4 3#4H8> 6 # 6 #31 &)( Ql rt# #4Hn9 6 # 6 # 9 6)/ \ & Hn9 6 #"#$ ( !$# 6)/'& :'#

ko0 *+& > 6 #$:'#"# * 1 /'6)6)&'& : 2 9|#4 : ( #$B,-:'# 9? * H ©4(+* #

1/2, 6

k + 1@Ê© HI# 24( ( & Hn9 6 #4HI#4 (J6 # /'* 310HnHI#HI1G: /'6 1 .Q

|Ç-¯Ð ± Ä X1, . . . , Xk#

k¬L¯ÅÄ Ãª0zF+ª-Ä¾Ç-¯«3UÃw¯ ± ²GÊÈ' ± ²Nª ± Ä Ã aª0«L¯ÐªN¬LzFUÃ²G =L« ± Ç0ÆGz zÉ¯%²NÈ.ª0«+ªD 4ÄÅ«3UÃ

1/2#¯ ± ²NÊÈ' ± ²pª ± ÄÊLÃ L! 'Ç ± ÄÅ«3L« & Æ z($ Ç ± È'LÆ.Ä\x5Ç ± ÃLÄÅ«LÆ¯«3 #²pª ± Ã<z)C )MLÃ¾È.ªpxU #

2k−1aª0«L¯ÐªN¬LzFUÃf²N4« ± Ç-ÆGz z ¯´²N<È.ª0«+ªD 4ÄÅ«+ 1/2#´²GLÆ%B²NLÆ ¯ ± ²NÊÈ' ± ²pª ± ÄÊLÃ !

.. 3 =4? @@?#6A986e9 =7 4 866 9 6 4 ;:9 9e;=7N1 /'*w6 #$Z: 2L &)(+& 10 EG/'& op10 ( /'& o * #-, 1 / 1 / Z9 6 DH$10 *32 51 6)/ HI#4 ( : 6 #Ï!B0Z1 6 rt10

#!$10 : &)(+& 10' # EN/ #9? * :'#$ 2 o © #4HI#4 ( RÅ1 / o- *+& > 6 #$ 6F2 ( 1 &)* #$ S ²NMÈa«+ÇN¬+ªN¬4¯zÉ¯ ÄÊyÃLÄÅ«L¯ÅxBÄÊM ± ÄÈ.ÇÃL¯ ÄÅ¯ a4Q M 6 #$ ( 9^1-3 & > 6 #:'#7: 2L &)* :'#$J!$10 : &)(+& 10' #4HI#4 ( %9? *d* 9'9|1 *+(%~ :'#$do- *+& > 6 #$% 6F2 ( 1 &)* #$EG/ # 6 !$10 EN/ #$ L !$# 6 n31 *+( : / !B0: * #:'#!$#$d 1 ( #$BQ / / |Ç0¯¾ ± Ä A 4Ä

B²NLÆ ba ± M ± Ä Ã #sªEa0+x

P(B) >0!ª9 * 10>?> &)6)&)(32 :'#

A50!3O? (

B#JÇ0Æ9 * 10>?> &)6)&)(32 :'#

A!$10 : &)(+& 10' # 6)6 #4HI#4 (I~

B#UÃ4Ä²NE ± ¯ÐÈ.ª0«

P(A|B) =P(A ∩B)

P(B).

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µB²NE ± ¯Ð<È.ª0«

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t,

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E(X) = S′X(1)

E(X(X − 1)) = S ′′X(1)

E(X · · · (X − k + 1)) = S(k)X (1)

Var(X) = S ′′X(1) + S′

X(1)−(

S′X(1))

)2

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n ≥ 0

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p1# (

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p1p2Q

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Q .21/.1 5C4j6 4 9L=6A ;

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(Xn)n∈N

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p,'10Ï

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p

Var(G) =1− p

p2

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SG(t) =pt

1− (1− p)t

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s,t# (

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t s) = P(X ∈ [t, u])

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s = 1,a!-rt#$ ( #4;Î¾ &)(J/ #7o0 *+& > 6 #7P 2 10H 24(+*+&FEN/ #q:'10 (

6 #%9? * H ©4(+* #7#$ (1− P(X = 0)

Q .21/. 3 5C4j6 ?6 6e@9e;

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# (pRn ∈ N

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n

k

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p,(Xi)1≤i≤n

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∑ni=1 Xi

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pQ

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p:'#$´:'# /.I6 1 & §> & _ H & 6 #$" #J310 (

9?0 6 #$<HI4HI#$Q &

X#$ (</ # ( # 6)6 #o- *+& > 6 #7 6F2 ( 1 &)* #-,?10

E(X) = np

Var(X) = np(1− p)

lI 24*+& #%P 2 24* (+*+& !$#7:'#9 * 10>?> &)6)&)(32 =:'# 6 q> & _ H & 6 #8#$ (S(t) = ((1− p) + pt)n

[ f 8n -- m m'"mon(f"* 8n epbf `

n

k

´ r p kbfJm'@$66:($ p f 0lp f"* p nqm n!k!(n−k)!

z)z )+ ) # % & #'('(*)+ $ P .21/. 5C4j6 6; 4j687E74

#aª0«L¯ÅªG¬Lz)q²G dÇ0¯ ÃUÃ4Ç ± :'#%9? * H ©4(+* #λRλ > 0 S ,.o0 /'( k

Rk ∈ N S $op#$!9 * 10>?> &)6)&)(32

pk = e−λ λk

k! &

X#$ (</ #o- *+& > 6 #:'#&N1 & 3310:'#%9? * H ©4(+* #

λ, 10

E(X) = λ

Var(X) = λ

lI 24*+& #%P 2 24* (+*+& !$#7:'#9 * 10>?> &)6)&)(32 =:'# 6 6 1 & :'#&N1 & 3310Ï#$ (

SP (t) = eλ(t−1)

l 6 1 & :'# N1 & 3310¨#$ (n& Hn9^1 *+( ( #!B * # 6)6 #9'9? * :F ( 31 / op#4 ( !$10HnHI# Ë 6)& H &)( #LÍ;:'# 6 1 & BQN\ *# #4Hn9 6 #-,| &

µn#$ (6 6 1 & > & _ H & 6 #n:'#79? * H ©4(+* #$

n# (

λnRÅ1

λ#$ (%/ *32 # 6 9^1- &)(+& Î L µn

mrt#$ (> & #4#4 ( #4 : / : 2L & # EN/ #M &

λn ≤ 1 S , 6 1 * µnË ( #4 :®op# * +Í 6 6 1 & :'# N1 & 3310 :'#n9? * H ©4(+* #

λ,

6 1 * EG/ #n( #4 :op# *

+∞ RÐ / 5#4 6 #%9 6)/ d & Hn9 6 #9^1-3 & > 6 #n 9|1 /'*<( 1 /'(k,µn(k)→ e−λλk/k! S Q

.21/. 2 5C4j687 A 6 4 = ?;=7N\ *6 rt# 9 * #$5 & 10 Ë 6 1 &./ & ÎÅ1 * HI#LÍ5,p10n: 2 & P- #<#4fÎ¾ &)( :'# /.8(ÊC 9^#$:'# 6 1 & ,- /'& o0 (EN/ # 6 rt#4 3#4H8> 6 #

31 / @ u30!$#4 ( #$ (8/ #4 3#4H8> 6 # & 1 / / & ( # * o- 6)6 #KRÅ1 // 9 * 1G: /'&)( :sr & ( # * o- 6)6 #$B,\#4¹: & HI#4 & 10 / 9 24*+& # /'* # ~ P S QN1 /'*§/ #4 3#4H7> 6 # & :'#J!B * : & ? 6

n, 6 6 1 & / & ÎÅ1 * HI#=#$ ( !$# 6)6 # EN/'& (+(+*+& > / # ~ !3O? EG/ # 246F2 HI#4 (

:'# 6 rt#4 3#4H8> 6 # 6 qHI4HI#9 * 10>?> &)6)&)(32 , EN/'& #$ ( :'10 ! 2 PN 6 # ~1/n

QN1 /'*§/ & ( # * o- 6)6 #[a, b]

, 6 6 1 & / & ÎÐ1 * HI#J#$ ( !$# 6)6 # EN/'& (+(+*+& > / # ~ !5O? EN/ #=31 / @¾& ( # * o0 6)6 #[c, d]

,/ #§9 * 10>?> &)6)&)(32 9 * 109|1 *+(+& 10' # 6)6 # ~ 5 6 10'P / # /'* L !$# (+( #´9 * 10>?> &)6)&)(32 #$ ( :'10 ! 2 PN 6 # ~ d−c

b−a

QlM 6 mrt# .& ( #9?0d:'# 6 1 &s/ & ÎÐ1 * HI#7 /'*</ & ( # * o- 6)6 # 10;>^1 * 2 Q .21/. 5C4j6; 0; 4 ; 9 6e; 9A9e;

#aª0«U¯ÐªN¬LzFª0zF+ª-Ä¾Ç-¯«3LÈ.Ç ± ± ÄÅ¯ÐLz zF:'#I9? * H ©4(+* #λRλ > 0 S 9|1 /'* :'#4 &)(32 RÅ: 2L & # /'*

R+ S

ρ(x) =1

λe−x/λ.

&X

#$ (</ # ( # 6)6 #o- *+& > 6 #7 6F2 ( 1 &)* #-,?10

E(X) = λ

Var(X) = λ2

¦ 10HnHI# 6 #$ 6 1 & P 2 10H 24(+*+&FEN/ #$B, 6 #$ 6 1 & # 9^10 #4 (+& # 6)6 #$ o 24*+&A #4 ( 6 dÈa«+ÇUÈa«L¯¾4ÄÊ%²N 'ª0« PÇEaZ EG/ # 6 EG/ #31 & #4 (d6 #$ *32 # 6 <9^1- &)(+& ÎÅ

s,t# (

uRÐ$op#$!

t < u S ,?10

P(X ∈ [t + s, u + s]|X ≥ s) = P(X ∈ [t, u])

( ,!$10HnHI#M: 6 #M!B0P 2 10H 24(+*+&FEN/ #-,v!$# (+( #I9 * 109 *+&F24(32 :'# * p1oÌ!B * 0! (324*+& 3# 6 #$ 6 1 & 8# 9^1 @ #4 (+& # 6)6 #$9? * H &m6 #$ 6 1 & J:'#9 * 10>?> &)6)&)(32 /'*

R+ Q

D ""% ) & #'('(*)+$l8 & H &)6 *+&)(32 #4 (+* # 6 1 & ´P 2 10H 24(+*+&FEN/ #$"# (´6 1 & §# 9^10 #4 (+& # 6)6 #$" #=Br¶ *+* ( #J9?0 6 ~ Q &

Gn#$ (§/ #

o- *+& > 6 #; 6F2 ( 1 &)* #P 2 10H 24(+*+&FEN/ #:'# 6 1 &1/n

,"z)ªyzAÇ-¯²GGn/n

x3Ç ± aL« FN aL«UÃz)ªÌz)Ç0¯UÈ.Ç ± ± ÄÅ¯¾Lz z)²N<È.ª0«+ªD 4ÄÅ«319|1 /'*J/ #: 2L &)(+& 109'9 * 109 *+&F2 #8:'# 6 q!$10Nop# * Pp#4 !$#f:'#$ 6 1 & d:'#9 * 10>?> &)6)&)(32 EG/ r &)6

mrt#$ ( 9?0 EG/ #$ (+& 10:srt# 9 6)& ! &)( # *"& ! & L : & 310 § & Hn9 6 #4HI#4 (<EN/ #-,G & Gε#$ ( / #o- *+& > 6 #= 6F2 ( 1 &)* #P 2 1 @

H 24(+*+&FEG/ #d:'#<9? * H ©4(+* #εÈ'4ÄÅ¯ÅÄÐ,N 6 1 *

εGεË * #$33#4H7> 6 #LÍ ~=/ #do0 *+& > 6 #d# 9^10 #4 (+& # 6)6 #=:'#"9? * H ©4(+* #

1Q

S8- cfYgd]dÂ3`+_aYehj7_a`3Â3ke .43/. [email protected] 9 6 ;<3 @ =4

l r & 2 PN 6)&)(32 :'# * p1o9|# * HI# ( :'# Iª B$Ç0«+4« 6 n9 * 10>?> &)6)&)(32qEN/ r / #7o0 *+& > 6 #f 6F2 ( 1 &)* #È'ÇÃU¯ÅÄÅ¯4a0Ç0Æ ± ÆGz z)731 &)( Ë (+*3© = /.@ :'#$3 / Í:'#310#$+9 24* !$#n M |Ç0¯ÅÄ

XÆ ± !aª0«L¯ÅªG¬Lz)ª0zF+ª-Ä¾Ç-¯«3cB aª-z)LÆ«UÃZ²pª ± Ã

R+ #4Ä<Ã$Ç0¯ÅÄ µ = E(X)

Ã4Ç ±UÃÊÈ'L«+ª ± x5"! z)Ç0«5Ã7Ç ± ª#^È.Ç-ÆG«7Ä¾Ç-ÆÄ§«+5LzmÃ4ÄÅ«U¯Ðx$ÄÊM ± ÄmÈ.ÇÃL¯ ÄÅ¯ λ#

P(X ≥ λ) ≤ µ

λ.

R Q¶Ò S 1 &)(

Y6 f 1 / op# 6)6 #o- *+& > 6 #7 6F2 ( 1 &)* #8: 2L & #9? * .9^1 /'*<( 1 /'(

ω ∈ Ω,

Y (ω) =

0 &

X(ω) < λλ

&X(ω) ≥ λ

M 6 #$ ( ! 6 &)*dEN/ # 6 rt10 ( 1 / u+1 /'* Y ≤ X

,'!$# EG/'& #4 (+* :FF #E(Y ) ≤ E(X)

Q I * ,E(Y ) = λP(Y >

0) = λP(X ≥ λ)Q=IÏq:'10 !

P(X ≥ λ) ≤ E(Y )

λ

≤ E(X)

λ.

2

#4HM *3EN/ 10 EG/ r / #q /'(+* #7Î¾DH$10:sr 2 ! *+&)* # 6 r & 2 PN 6)&)(32 :'# * p1o#$ (=6 M /'& o- ( #a9|1 /'*=( 1 /'(*32 # 6 (+*+& ! ( #4HI#4 ( 9^1- &)(+& Î

λ,

P(X ≥ λµ) ≤ 1

λ.

.43/.1 [email protected] 9 6 ; l r & 2 PN 6)&)(32 :'# * p1o;#$ (J/'(+&)6 #-,aHM & =# 6)6 #7 #79|# * HI# ( 9?0:'#7HMu1 * # *=6 n9 * 10>?> &)6)&)(32qEG/ r / #

o- *+& > 6 #Z 6F2 ( 1 &)* #M51 &)( ± °²NUÃ5Ã$Ç0ÆGÃ:'#I310#$+9 24* !$#-Qv%#n9 6)/ B, &)6 *+*+& op#nÎ *32$EN/ #4HnHI#4 (8EG/ # 6 rt1031 &)( !$10.Î * 10 (32~ :'#$"o0 *+& > 6 #$d 6F2 ( 1 &)* #$J:'10 ( 10#$ ( !B9?> 6 #:'#%!B 6 ! /'6 # * #$+9 24* !$## ( o- *+& !$#-,9^1 /'*I6 #$ EG/ # 6)6 #$ 6 r & 2 PN 6)&)(32 :'# v!3O #4> C !3O # µ RÅ1 / D-,1 / ¦ O #4> C +O #4o^, : Z5P * 9'O & #& ( # * ? (+& 10? 6 # S #$ ( 51 / op#4 ( 9 6)/ J9|# * ÎÐ1 * HM ( #89^1 /'* HMu1 * # *J6 q9 * 10>?> &)6)&)(32 :srt (+* # Ë 6 1 & .Í:'# 6 rt#$ @9 24* !$#-Q

rHn er # 8n f n p r , ln f:r:f( mon f - n +ep' 8n , bf *flnqf"$58Jm n -"- *m'&"! by w | #+w %$'& )(+* , .-)$ , 0/T21,h5(m$6 .:p (r+* - 6bflf68n@ n B pD$6 ( ( f =n -"- ep r$* ( ( f p f"$ m0;$)3)ep m mop'"5:n flrDlf;$ ( f 54)6 d 798 ε ,': F - bf":r+ren f:r2%" r p? 6s 4%;qx <- y .

."$ ()+$ .)+#

M |Ç0¯ÅÄX

Æ ± aª0«L¯ÐªN¬LzFnª0zF+ª-Ä¾Ç0¯ «+#<² $ UÃÊÈ'L«+ª ± x5 µ4Ä<²N aª0«U¯Ðª ± xU σ2 ! z)Ç0«UÃ8Ç ±ª#^È.Ç-ÆG«7Ä¾Ç-ÆÄ§«+5LzmÃ4ÄÅ«U¯Ðx$ÄÊM ± ÄmÈ.ÇÃL¯ ÄÅ¯ λ

#

P (|X − µ| ≥ λσ) ≤ 1

λ2.

R Q S ¦ 10 & : 24* 10 6 <o- *+& > 6 #§ 6F2 ( 1 &)* #

Y = (X−µ)2Q N\ * : 2L &)(+& 10m,!$# (+( #§o0 *+& > 6 #´ 6F2 ( 1 &)* #

f9^1 /'* #$+9 24* !$#σ2 ,'# (d6 r & 2 PN 6)&)(32 :'# y * p1Bo^,?9'9 6)&FEG/ 2 # ~

Y,':'10' #7# 0! ( #4HI#4 ( Q .Q

2

l r & 2 PN 6)&)(32 :'# v!3O #4> C !5O # µ 9^# /'(72 PN 6 #4HI#4 ( 3# *32$2 ! *+&)* #Z31 / 6 ÎÐ1 * HI#M /'& o- ( #Ïm9^1 /'*( 1 /'(λ > 0

,P (|X − µ| ≥ λ) ≤ σ2

λ2.I#4Z: 2 : /'&)( & HnH 2 : & ( #4HI#4 ( :'#$ HMu+1 * (+& 10 § & H &)6 &)* #$§ /'*§6 %9 * 10>?> &)6)&)(32%EN/ #

X31 &)( Ë 6 1 &

/.@ :'#$3 / +Í%1 / Ë 6 1 & ;#4Ï:'#$331 / Í:'#310;#$+9 24* !$#I.9^1 /'*d( 1 /'(λ > 0

,

P(X ≥ µ + λ) ≤(σ

λ

)2 R Q SP(X ≤ µ− λ) ≤

(σ

λ

)2 R Q S .43/. 3 [email protected] 9 6 ; ; =84

l r & 2 PN 6)&)(32 :'# ¦ O # * 1 µ #$ (f/ # & 2 PN 6)&)(32EG/'& #ZBr¶9'9 6)&FEG/ # ( # 6)6 # EG/ # 6)6 # EG/ r ~/ #Mo- *+& > 6 # 6F2 ( 1 &)* # EG/'& 3#89 *32 3#4 ( #f51 / 6 MÎÐ1 * HI#8:sr / #q310HnHI#q:'#8o0 *+& > 6 #$:'# "# * 1 /'6)6)& ¯ ± ²NÊÈ' ± ²pª ± ÄÊUÃR 9?0MÎÅ1 * ! 2 HI#4 ( :'#HI4HI#;9? * H ©4(+* # S L &)6 #$ ( 9|1-3 & > 6 #:srt#4 :'10' # * :'#$IP 2 24* 6)& U (+& 10 ~ :'#$310HnHI#$:'#o0 *+& > 6 #$ & : 2 9|#4 : ( #$:'# 6 1 & J: &Aµs24* #4 ( #$B,?HM & 6 r & : 2 9^#4 : !$#8#$ (</ #7!$10 : &)(+& 10& Hn9^1 *+( ( #-Q M: |Ç0¯ÅÄ

X1, . . . , XnÆ ± JÃLÆG¯ÅÄÊ%²N n

aª0«L¯ÅªN¬4z)UÃª0zF+ª-Ä¾Ç0¯ «+LÃ%²N =L« ± Ç0ÆzzÉ¯ & : 2 9^#4 @: ( #$ #Xi

ªDIpª ± ÄvÈa«+ÇN¬+ªG¬L¯zÉ¯ÅÄÊ pi²N8ÃLÆ.x3x$UÃMªEa+x

0 < pi < 1#d4Ä´Ã4Ç0¯ÅÄ

X =∑n

i=1 XizFLÆG«8Ã4ÇL M"!

|Ç0¯ÅÄ´ Fpª0zFM ± Ä µ =∑n

i=1 piz%$ LÃ¾È'L«+ª ± xUn²N X

!z)Ç0«UÃÇ ± ª#^È'Ç0ÆG«fÄ¾Ç0Æ.Ä «33Lz

δ > 0#

P (X > (1 + δ)µ) ≤(

eδ

(1 + δ)1+δ

)µ R Q SRvN1 /'* 9 * #4 : * #!$10 5! & #4 !$#:'# 6 r & Hn9|1 *+( !$#8:'#!$# (+( # & 2 PN 6)&)(32 , &)6 !$10Go & #4 ( :'#% 1 ( # *JEG/ #

eδ <(1 + δ)1+δ 9^1 /'*%( 1 /'(

δ > 0,^# (=EG/ #8!$# (+( # EG/ (+&)(32q( #4 :op# *

06 1 * EN/ #

δ( #4 :op# *

+∞ L 9|1 /'*δ = 1

, 6 # * 9'9|1 *+( #$ ( 9'9 * 1 .& HM (+& op#4HI#4 ( :'#0.68

Q & & ,0 &µ:'#4o & #4 ( P * :s, 6 =>|1 * #< / 9 24*+& # /'* #

: 2 ! * 1 &)( # 9|10 #4 (+& # 6)6 #4HI#4 ( o &)( #$op#$!µQ SlI9 * # / op#8:'# 6 r & 2 PN 6)&)(32 :'# ¦ O # * 1 µ , > & #4 EN/ r¶035#7! 6 03 &FEG/ #-,?#$ (/ 9^# / !B 6 ! /'6 ( 1 &)* # L # 6)6 ##$ ( :'10' 2 # & ! & 9? * 31 / ! & :'#!$10Hn9 6F24(+/ :'#-Q

1 &)(t/ *32 # 6 9|1- &)(+& Î EG/ # 6 !$10 EN/ # R 1 / J9 *32 ! & 3# * 10 J!$#7!3O 1 &A/'6)(324*+& # /'* #4HI#4 ( S QHN\ *! * 1 & 35 !$#7:'# 6 fÎÅ10 ! (+& 10Ï# 9^10 #4 (+& # 6)6 #-, &)6 o & #4 (

P (X > (1 + δ)µ) = P

(

etX > et(1+δ)µ)

.

9'9 6)&FEN/ 10 6 r & 2 PN 6)&)(32 :'# * p1oM / HI#4H7> * #=: * 1 &)( RÅ /'* 6 o0 *+& > 6 # 6F2 ( 1 &)* #etX S L &)6 o & #4 (

P (X > (1 + δ)µ) ≤ E(

etX)

et(1+δ)µ.

Ñ ""% ) & #'('(*)+$N1 /'*2 o0 6)/ # * D,D

E(

etX) , 1 / * #4HM *3EN/ 10 EG/ #-, 6 #$o0 *+& > 6 #$

Xi24( (=& : 2 9^#4 : ( #$, &)6 #4

#$ ( :'#HI4HI#7:'#$<o0 *+& > 6 #$etXi

Q

E(

etX)

= E

(

n∏

i=1

etXi

)

=n∏

i=1

E(

etXi)

lI 24*+& #%P 2 24* (+*+& !$#7:'#HI10HI#4 ( :sr / #o0 *+& > 6 #:'# "# * 1 /'6)6)& #$ ( # (+* 4HI#4HI#4 ( & Hn9 6 #M

E(

etXi)

= piet + (1− pi) = 1 + pi

(

et − 1)

.

(+&)6)& 310 HM & ( #4? (6 r & 2 PN 6)&)(321 + u < eu ,"o- 6 > 6 #Ì9^1 /'*( 1 /'(;*32 # 6

u > 0,JBop#$!

u =pi

(

et − 1) L &)6 o & #4 ( 6 1 *

E(

etXi)

< epi(et−1),# ( :'10 !

P (X > (1 + δ)µ) <

∏ni=1 epi(et−1)

et(1+δ)µ

=e(e

t−1)Pn

i=1 pi

et(1+δ)µ

=e(e

t−1)µ

et(1+δ)µ

¦ # (+( #Ï:'# * &F©4* # & 2 PN 6)&)(32 #$ ( o- 6 > 6 #;9^1 /'* Ä¾Ç0Æ.Ä *32 # 6 9^1- &)(+& Ît L 1 / Z9|1 / op10 Z:'10 !!3O 1 & &)*

!$10HnHI#=o0 6 # /'* ,!$# 6)6 # EN/'&|* #4 : 6 #JHI#4H8> * #: * 1 &)(´6 #9 6)/ ´9|# (+&)( 9^1-3 & > 6 #-QJ # 24(+/ :'# * 9 & :'#%:'#=!$# (+( #ÎÐ10 ! (+& 10w:'#

t*32 o ©46 # EN/ # 6 7ÎÅ10 ! (+& 10w#$ ( : 2 ! * 1 & 35 ( #-,.9 /'& "! * 1 & 3U ( #-,'$op#$! / H & & H / H (+( # & (

#4ln(1 + δ) L #4 * #4Hn9 6 DHB (

t9? * !$# (+( #o0 6 # /'* ,'10Ï10> (+& #4 (J6 r & 2 PN 6)&)(32 ' 10 ! 2 #-Q

2

T T & + Enqf ep - + m f"* it .w q|.w &;4'& :|Qt mon>bnep nlmn m,n p? X r+bf 6 0 f r# n*:r+bm" (*f) - n rkbn8n f epr p X f r# - n r58#r+ -"- + 2 *ef"

\\

QP GfQ "1 /'6 #B / , ´«+ÇN¬+ªN¬L¯ zÉ¯ ÄÊUÃq²N7z($ ¯ ± FN ± ¯¾LÆ« # * HM'm,:L Q ¦ 1 * HI#4m, ¦ Q lm# & 3# * 310m, Q & op#$ ( , ± ÄÅ«Ç²0Æ.xBÄÅ¯ÅÇ ± BMz($ ª0z FpÇ0«L¯ Ä ³f¯ & Æ / 1G:s,+P) Ñ R& ± ÄÅ«Ç-°²0Æ'x$ÄÅ¯ÐÇ ± Ä¾Çz FpÇ0«L¯ÅÄ ³fÃL, M N * #$5B,P) S Q 8qQ O?1p ( ., Q O / !5O B, ª0z)x4ÆGz'²NUÃÈa«+ÇN¬+ªN¬L¯ zÉ¯ ÄÊUÃ ?x3Ç0Æ«5Ã # NL«+x4¯ÅxUUÃ=4ÄÈa«+ÇN¬LzMUÃdx3Ç0«L«U¯ FNUÃ / 1:s,P) RÅ5#$!$10 :'# 2 : &)(+& 10 S Q Ñ8qQ O 1p ( ., Q O / !5O B, ´«Çx5UÃ5ÃLÆNÃÃ4Ä¾ÇBxL³ª0ÃLÄÅ¯ & Æ'LÃ ´«+ÇBx5UÃUÃUÆGÃ8²N <Ç-¯ ÃUÃ4Ç ± # xL³ª ± LÃ²N 'ª0« PÇEaI4Ä

Iª0«4ÄÅ¯ ± Fpª0zFUÃ \x3Ç0Æ«5Ãq$Ä"GL«x$¯Åx5LÃqx3Ç0«L«L¯ FNLÃ / 1G:s, L.Q¶Ò 8qQ dQ% /'( Om, ³. «4Ä<Ç ÇLJÈ|ÆÄÊL«´«+Ç%F0«+ªD q¯ ± Fp,aop1 6)/ HI# ., ^q¯ ± ÆML«L¯Åx5ª0z z FpÇ0«L¯ÅÄ ³fÃ

:': & 310 @ #$ 6 # C ,jP) R (+* 1 & &F© HI# 2 : &)(+& 10 S 8qQ dQ% /'( Om, ³. «4ÄÇ ÇLJÈ|ÆÄÊL« ´«+Ç%F0«+ªD f¯ ± F-,op1 6)/ HI# ., |Ç-«LÄÅ¯ ± Fª ± ² ^+ª-«xL³G¯ ± F

:': & 310 @ #$ 6 # C ,jP) RÅ3#$!$10 :'# 2 : &)(+& 10 S Q Q Ï1 ( & ,=NQ P-O?$o-m,<ª ± ²NÇDq¯+² z FpÇ-«U¯ÅÄ ³fÃ ¦ H7> *+& :.Pp#8J & op# * &)(DC N * #$3B,+P) -Ò.Q 8l Q?=#4o * 1 C #-,Ç ± °DÆ ± ¯ $Ç0« «ª ± ²pÇD aª0«U¯Ðª-ÄÊ FN ± L«ªpÄÅ¯ÅÇ ± G9 *+& 'Pp# *@ # *+6 P ,P) <.Q Q #4 #B#$B,#NvQó-I=1 * 3!5O 1 ( , |Q ( 10 #-,7ª ± ²N¬3ÇBÇP¹Ç( §ÈpÈ|z ¯¾+² «IUÈ|Ä¾Ç%F0«+ªUÈ?³I ¦ ¦N * #$5B,P) <.Q

0Ò

E '(*'#)j ( *

P-QQP /'& ! 1 *+( QQQQQQ7QQQQQQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQQQ P-Q : /'& ! 1 *+( QQQ7QQQQQQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQQQ P-Q : & ¦§/'( QQQQ7QQQQQQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQQQ .QQP 1 /'* !$#7:'# "# * 1 /'6)6)& Q7QQQQQQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQQQ P .Q 1 /'* !$#7:'#%> &)( 6F2 ( 1 &)* #$ÀQQQQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQQQ P .QQP :'l C # 6 #$! ( QQQ7QQQQQQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQQQ Ñ QQP ¦ 6 ! /'6 :'# * 0! & #!B *+*32 #7HI1G: /'6 1

pQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQ

Ñ Q #$ ( :'# &)6)6 # *@ > & QQQQQQQQQ7QQQQQQQQQ7QQQQQQQQ7QQQQQQQÑ

L

)+, %) #

6 Pp1 *+&)( O'HI#9 * 10>?> &)6)& ( #: 2L & 9? * >? :'#7 6F2 ( 1 &)* #-, : 2L & 9? *J6 q31 /'* !$#-,+P

¦ O # * 1 µ& 2 PN 6)&)(32 :'#-,:

: 24* :'10H & 5 (+& 10m,P<P#$+9?0!$#9 * 10>?> &)6)& 2

& 1 / : 2 10H8> * > 6 #-, P 10: 2 10H8> * > 6 #-,

#$+9 24* !$#-,=Ñ2 o © #4HI#4 ( ,H PÎÐ10 ! (+& 10;HI#$ /'* > 6 #-, & : 2 9^#4 : !$#

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