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C8f>&8:Bm>&4u57E0> >&Bm;JKN]J^ T_]a` OWX!SbZNEc[!MPKW[!TdHuB&O 60E^>GB`;N¦:8:6^2/[f@%nD8:2/4u5:=<9/57EF>U. E e ' bf?g h@:BD@iB Xy;?;?>&8U;<6F2U8^zUSD2/8o;?= 9R60=<>D4d>D8o; (eSD>YZo[7>CQ7=<n\;<>D8:@%>&8o;y@%>C8:2/4u5:=?>&[% E06F s=<>&B&z:`'jkM'QDI!jlZ `!H0S:`'] [78f>D8 Bm>&4u57E0>J¦:876^8^O >&Bm;CQ:9UBN@%nD872U4u57= 9R5:EF>UzsB?>D[:EF>&4 >&8o;Y9R[ Q7EF[ BC@%nD8:2/4u5:=<9/57EF>U. m b=%ni = $ >=?;o@:BD@:BdprqbZRQ<j As B s A , . . . , A OYH!` H0S:`0H0E4[0T\H!`utuSiQv`0w xYw |∅| = 0. yzw q{Q A ⊂ B s KNT\ZRML` |A| 6 |B| w |Ww q{Q A ⊂ B s KNT\ZRML`

{ A

= |B| − |A| w }{w q{Q A ∩ B = ∅s KRT\ZNML` |A ∪ B| = |A| + |B| w q{Qz^iZN]~M jlZN]~j (i, j) ∈ [[1, n]] j H0T{ 0]iH i 6= j s A ∩ A = ∅s KRT\ZNML` |A ∪ · · ·∪ A | = |A | + · · ·+ |A | w Yw KN` WX0S X0MPKNT |A ∪ B| = |A| + |B| − |A ∩ B| w Rw |A × B| = |A| · |B| w T_]a` zX!SbX!MPKRT\H0E4H0Sij |A × · · · × A | = Y |A | w F 8 = & h@:BC@:B\ > %ZNM'E ]~T\H ON] I0M'QD[!TdH OWHV .ZNQ<S ILKRMPX s ZN] !ZNM'E ]~T\H O Q<SbI!T_]a`'QDZRSi H zI0T_]a`'QDZNSY qbZRQ<j A , . . . , A OYH!` H0S:`0H0E4[0T\H!`utuSiQv`0w T\ZRML` 1

n

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j

n

1

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n

|A1 ∪ · · · ∪ An | =

n X

k=1

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X

16i1 <···<ik 6n

|Ai1 ∩ · · · ∩ Aik | =

X

\

(−1)|I|−1 Ai .

I⊂[[1,n]]

i∈I

n

?n @:B V= & <; =

! "$#% '&)(+* ( ,-( ,%(. /0( , * " /! "$# ! ,

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FE

TP H!Si`!H0Ec[!T\H OYH!`

|E|

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1

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p

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]~S H K ^W^ T_QDILKRjCQDZRSo`']~HM G0HLI!jkQ H :w 3 S `L] ^W^:ZN`0H !]u Q<T ' ' @:B B\p T\H0E E4H OR] [LH0MC zH!M qbZNQ<j H YQv`'jlHV]~S H0SijkQDH0M k ∈ N j H0T 0]iH ^:ZR]~M jlZN]~jfy:∈EF →s |fF (y)| = k kjlZN]a`.TdH'`FX!T\X0EcH!S{4 jD` OWH F ZRSij)T\H$EcG!EcH S ZRE4[0MPH k O KRSijlX ILXLOYH0SijC` Yw? T\ZNML` |E| = k · |F | w e ' bf?g h@:B B\ bc>AEF>&4 4d>e@%[f5'>D=<g/>&=yQ'>D=<4d>\;Y@%>A )2U=?4 9/EF6TBm>&= E09 872/;?602/8f@%> 2YS ©:2/6F Bm[ SDS\>GB?B?6F 5 .

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n! (n−p)!

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@%>

[[1, p]]

|F | = n

F

E

p

F

f

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f

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n! (n−p)!

f

f

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Q_ ` &-B LK(* M "!)#

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[[1, n − 1]]

n!

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n!

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n

k n−k

n

k=0 k n−k

n

n k

∗

∗

1

1

k

1

1

k

k

k

j

1

j

k

X

(x1 + · · · + xk )n =

i1 +···+ik =n

n xi1 · · · xikk . i1 , . . . , i k 1

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Ai

S

Ai = Ω

i∈I

Ω

Ω

A

A

A

A∩B

B

A

Ω

A

Ω

P(Ω)

Ω∈A

∀A ∈ P(Ω), A ∈ A =⇒ A ∈ A

∀(A, B) ∈ P(Ω)2 , (A, B) ∈ A2 =⇒ A ∪ B ∈ A A

Ω

∅∈A

∀(A, B) ∈ P(Ω), (A, B) ∈ A2 =⇒ A ∩ B ∈ A {Ai , i ∈ I}

A

S

Ai ∈ A

i∈I

A

Ω

C = {Ai , i ∈ I}

A

A(C)

C

C⊂A

A0 C ⊂ A0 =⇒ A ⊂ A0

Ω

A

Ω

A0

Ai ∈ A

i∈I

(Ω, A)

A

T

A ∩ A0

P(Ω) C

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,

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A

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n

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T

T

σ

Ω

P(Ω)

Ω∈A

∀A ∈ P(Ω), A ∈ A =⇒ A ∈ A

T

(An )n∈N

+∞ S

An ∈ T

n=0

T

Ω

∅∈T

∀(A, B) ∈ P(Ω), (A, B) ∈ A2 =⇒ A ∪ B ∈ A

T

(An )n∈N

+∞ T

An ∈ T

n=0

∀(A, B) ∈ P(Ω), (A, B) ∈ A2 =⇒ A ∩ B ∈ A

σ

(Ω, T )

T

T

Ω

σ

Ω

Ω

T

σ C

T

T0

σ

C = {Ai , i ∈ I} T (C)

T ∩T0

Ω

P(Ω)

C

σ

?n B V n n b= 8

@

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T 0 C ⊂ T 0 =⇒ T ⊂ T 0

σ

σ

σ

C

Ω

C

(

T (C) =

[

A D⊂C

)

.

m b=%ni= $ >=?; B BC@6Zrq ZNQ<j Ω ]~S ]~SiQ 4 H0ML` OWX!SbZNEc[!M KY[0T\H w K σ KNT [!MPH H0Sz zH!SbONMPX ^iKNMcT H0S:`0H0E4[0T\H OYH!` X 4 X0S H0E4H0SijC` X!T\X0EcH!S{j KRQ<MPH'` {{ω}, ω ∈ Ω} H!`'j P(Ω) w e ' bf?g B B 2\ b^>yQ7EF[ B Bm2U[7 />&8U;Gz EF2U=<B<Zo[7> Ω >&Bm; ¦:876%2/[d@%n&872/4 57=<9/57E0>/z E O >&B?Q:9US\>yQ7=?2U5:9R5:6FE060B<9R5:EF>yZU[:> E O 2U8 SD2/8:B?60@7nD=<>D= 9u>GB`; (Ω, P(Ω)) .

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4 C, N" ( , & F "F !/! !, , A∈D

, I

(*# &-B (*# >-(? )B E@) @ P

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Yw s

Rw s

w s

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(Ω, T )

∀A ∈ T 0 6 P (A) 6 1 P (Ω) = 1

(An )n∈N

σ

P

S

An

n∈N

=

+∞ P

P (Ai )

n=0

(Ω, T )

P

(A, B)

P (A∪B) = P (A)+P (B)

∀A ∈ T P (A) = 1 − P (A)

∀(A, B) ∈ T A ⊂ B =⇒ P (B \ A) = P (B) − P (A) ∀(A, B) ∈ T A ⊂ B =⇒ P (A) 6 P (B)

∀(A, B) ∈ T P (A ∪ B) = P (A) + P (B) − P (A ∩ B)

(A1 , . . . , An ) ∈ T n

P

n [

i=1

Ai

!

=

X

(−1)|I|+1 P

I⊂[[1,n]]

\

i∈I

Ai

!

=

n X

k=1

(−1)k+1

X

16i1 <···<ik 6n

P (Ai1 ∩ · · · ∩ Aik ).

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F 8 = & B B ) ¥=<2/Q7=<60n\;?nI@%>IE0604 6F;?>I4d2U872R;<2/87> xYw qbZNQ<j (A ) ]~S H$`']~Q<jlH I0MPZNQv`L`0KNS{j H4O? X 4 X!SbH!EcH0SijC`!w TdZNML` P [ A ! = lim P (A ) w yzw qbZNQ<j (A ) ]~S H$`']~Q<jlH OWX I0MPZRQv` `0KRSijlH O? X 4 X0S H0E4H0SijC`0w? T\ZNML` P \ A ! = lim P (A ) w 748:9 <; >=?; B B\p £¤2%@% DE0>¥Q7=<2/5 9R5760EF6TBm;?> @% > V2/E04 2Ug/2U=?2+ Y8 H'`k^iKYILH?^ M ZY[LKY[0Q<T_Qv`0Xy>&Bm;^[78A;?=<60Q7EF>D; (Ω, T , P ) 62 5 (Ω, T ) >&Bm;N[78f>&B?Q:9/SD>AQ7=?2U5:9R5:6FE060B<9R5:EF>Uzo>D; P [78:>I4 >GBm[:=?>I@%>IQ7=<2/5 9R5760EF6F;?nIB?[7= (Ω, T ) .

F 8 = & B B\ !{Vn\¦:876F;?602/8 @^O [78:>IQ7=?2U5:9R5:6FE06 ;<nIBm[7=N[:8 B?¬%B`;< D4d>eS\2U4 Q:EF>D; qbZRQ<j (Ω, T ) ]~SrH'`k^iKYILH ^ MPZW[LKY[0Q<T_Qv`0KY[0T\H s H0j C ⊂ T ]~S ` 6`'j !EcH ILZRE.^bT\H!jcOYX0S ZRE4[0MPKW[!T\H O? X 4 X!SbH!EcH0SijC` H0Sz zH!SbONMPKRSij T QDw<H w T = T (C) Yw?qbZNQ<j P : C → [0, 1] j H0T<T\H !]iH P P (A) = 1 w T\ZRML` P `0H ^bMPZNT\ZRSz zHJOWH EcKNSiQ 0MPH ]~SiQD 0]iH H0S ]~SbH ^ M ZY[LKY[0Q<T_Q<jlX P : T → [0, 1] w V=

= B HB S qbZRQ<j Ω ]~S H0S:`0H!Ec[0T\H tuS{Q ZR] OWX!SbZNEc[!MPKW[!TdH w T\ZRML` s ]~S HVEcH!`L]~M HFOYH ^bMPZY[LKY[0Q<T_Q<jlXF`']~M H!`'j OYXCtuS{QDH OYH E4KRSiQ 0MPH4]~SiQD 0]iH ^iKNM T\K OYZRSiSbX H O ]~SbH K ^W^ T_QDILKNjkQDZNS P : Ω → [0, 1] jlH0T<T\H 0]iH (Ω, P(Ω)) P w P (ω) = 1 n n∈N

n

n+1

n n∈N

n n∈N

n+1

n n∈N

n

n→+∞

n∈N

n

n n∈N

n∈N

n→+∞

n

n

n

A∈C

ω∈Ω

, I

B E@ @ 2P*# # &-B &)! & !) +(CB # !)

. rs2U6 ; [78f[78760 />D= By¦:876 . 1 8fS\2U8:Bm6T@% &=?>JE O >&B?Q:9US\>AQ7=<2/5:9/576FE06TB?9/57EF> { O 9/Q7=? GB EF>fS\2U=?2UEFET9R60=?> L%. h7. wsz#Q 2U[7= @%n\¦:8:6F=d[787> 4d>&B?[7=<> @%> Q7=<2/5:9/5760EF6F;?n B?[7= z 60ENBm[ d;p@%> @%nD¦:8760= Bm[:= E0>&B B?6087g/E0>\;<2/8:Bd@%>fBm2U=m;<> Zo[7>fET9 B?2/4d4d>fBm2U6 ; nDgo9RE0> ( . 2/6TS\6N[:8 &S 9/B Q:9R=?;?6TS\[:EF60>D= ;?=< &B 604 Q'2/=?;<9/8o;A~ 748:9 <; >=?; B [B X b 9 ^bMPZY[LKW[!Q<T_Q<jlX ]~SiQ !ZNM'EcH !2/[AXL 0]~Q<MPXl^:KRM'jCQ<jkQDZNS dBm[:= G> B`;dET9HQ7=<2/5 9R5760EF6F;?n @%nD¦:8760>IQ:9R=I~ (Ω, P(Ω)) P 1

Ω

P

(Ω, P(Ω))

(Ω, P(Ω))

∀ω ∈ Ω, P ({ω}) =

1 . |Ω|

F 8 = & B B[Z !ª72U=?4 [7E0>J@%>Abc9/Q7ET9/S\> qbZNQ<j (Ω, P(Ω), P ) TP H'`k^iKYILH ^bMPZY[LKW[!Q<T_Qv`0X ]~SiQ !ZRM'E4HF`L]~MVT ]~S{Q< 4 H!ML`utuS{Q Ω w? T\ZNML` s ^iZN]~MFjlZN]~j A ∈ P(Ω) S ZNEc[0MPHJOWHJILKN` !K 4 ZRMPKY[0T\H!` |A| P (A) = = SbZNEc[!M HJOWHJILK6` ^iZ6`L`'QD[!TdH'` . |Ω|

, I I

! #1(R; @ (*# !-P*D" DE(C @ (*# 748:9 <; >=?; B BD@]\ rs2/6F; (Ω, T , P ) [78f>DQ 9/S\>JQ7=<2/5:9/5760EF6TBmnUz%>\;YB?2/6F;

.

U- . 1 8 @%6F;YZo[7> >&Bm; S X YT_Q WHLKW[!T\HDz:2U[ ^ MPH!`0 !]iH0 lQ<E.^iZ6`L`'QD[0T\H z:Bm6 .

L7. 1 8 @%6F;YZo[7> >&Bm; ^ MPH!`0 !]iH0 ILH!M'jlKNQ<S Bm6 . h:. C87> Q7=<2/Q7=<6FnD;?n >&Bm;d s= 9R60>c^bMPH'`0 0]iH$` ~MPH0E4H0SijNB?6 E O nD UnD87>&4 >&8o; 2AET9HQ:=?2UQ7=?60n\;<n >GB`;pB<9+;?6TBm !9R6F;?> 5 >&Bm; Q7=?>GB?Zo[7> _ SD>D=?;<9/6F8^.

P (A) = 0

A A

A∈T

P (A) = 1

?n B V n n b= 8 V-W n B BC@:@ b¥O 2/57;?>D8o;<6F2U8 @PO [787>eB?[76F;?>I608%¦:8760>e@%>A ¥60EF>e@:9R8:B [78 ;<6F= 9RgU>VnGZU[:6FE06F5:=?nA=<nDQ'n\;?n ( ¥60EF>I2U[ ª:9US\>A>&Bm;C[78fnD UnD87>&4d>D8o;NQ7=?>GB?Zo[7> _ 604dQ 2oB?B?6057EF>Uzs4d9/60B Q 9/B 604 Q'2UB<B?6F57E0>/.

4 * L# ! (. ( J# ( # ! * , ( * " ( @

, Q_)

B E@ @ 2P*# "!)> +!)!-(R (R#

1 8HS ©7>&=<S ©:>e4 9R608U;<>D8:9/8o; (p U2/60=VSD2/4d4d>D8o;VET9pS\2U878:9/60B<B?9/8:S\> @%> SD>D=?;<9R6087>GBC608% )2/=<4 9+;?602/8 BVB?[7=YE0> =?nGBm[7EF;<9R; @PO [787> >\ %Q'nD=<6F>&8:S\> 4 2%@%6F¦:> EF>GBYQ:=?2U5:9R5760E06 ;<n&B&. 2/[:=JSD>DET97z'2/8H@%nD¦:876F;JET9pQ:=?2U5:9R5760E06 ;<n S\2U8:@%6F;?602/878:>DE0EF>d@PO [78 nD UnD87>&4d>D8o; A B?9US ©:9R8o;NZU[:>IE O nD UnD87>&4 >&8o; B >GB`;N=<n&9RE06TBmnU. 748:9 <; >=?; B BD@ r%2/6F; (Ω, T , P ) [:8 >&B?Q:9/SD>CQ7=<2/5:9/576FE06TB?9/57EF>Uz A >D; B @%>&[% n& /n&87>D4d>D8o; B¥@%> T ;<>DETB Zo[7> 87>IB?2/6F;CQ:9/B Q7=<>&B<Zo[7> 604 Q'2UB<B?6F57E0>/.%XYEF2U=<B&z%E09 ^bMPZY[LKY[0Q<T_Q<jlX4I ZRS ORQ<jCQDZRSiS H0T<T\HcOWH A H0S B !2/[fE09 ^bMPZY[LKY[0Q<T_Q<jlX B OYH A `0KWI iKNSij B y>GB`;I~ _

PB (A) =

P (A ∩ B) .

P (B)

1 8 ;?=<2/[7 U>Jn&gU9/EF>&4 >&8o; B?2/[7 U>D8o;NE09 872/;<9R;?602/8 P (A | B) . m b=%ni= $ >=?; B B rqbZRQ<j B ]~S X 4 X0S H0E4H0SijVS ZRS ^ MPH!`0 !]iH0 lQ<E ^:ZN` `'QD[0T\H w T\ZRML` P OWXDt SiQ<jV]~S HcEcH!`L]~M H OYH ^ M ZY[LKY[0Q<T_Q<jlXJ`L]~M$T H!`k^:KWILH ^bMPZY[LKW[!Q<T_Qv`0KW[!T\H (Ω, T ) w e ' bf?g B B 1 8 [7;?60EF6TBm>VBm2U[7 />&8U;¥ET9I=<>DET9+;<6F2U8d@7n\¦:876TB<B?9/8U;¥E0>&B Q7=<2/5:9/576FE06F;?n&B3S\2U8:@%6F;?602/8787>&EFE0>&B Bm2U[:B ET9 )2/=<4d>IBm[760 +9R8o;?>d~ P (A ∩ B) = P (A | B)P (B). ¥8¤> '>D;&z'2/8¤Bm>eB?>D=?;VB?2/[7 U>D8o;C@%>&BYQ7=<2/5:9/576FE06F;?n&BNSD2/8:@76 ;<6F2U8787>DE0E0>&BCQ'2/[:=YSD9/E0SD[7EF>&=NEF>GBCQ7=<2/5 9R5760EF6F;?nGBC@PO 6F8o;<>D= _ B?>&Sl;<6F2U8:B&z%>\;C872U8 E O 608s />&=<B?>/z%SD9/= EF>GBNQ7=<2/5:9/576FE06F;?n&B SD2/8:@%6F;?602/8:87>DE0EF>GBNBm2U8o;NB?2/[: />D8o; Q7E0[:BCB?6F4dQ7E0>&*B (dSD9/E0SD[7E0>D=G. 1 8 />D=<= 9I[78fQ >&[ Q:EF[:B E02/608 [787>Ag/n&87nD= 9RE060B<9+;<6F2U8p@%>IS\>D;m;?>J )2U=?4 [7EF> ( [787>A608U;<>D= Bm>GSl;<6F2U8 @%> n n& /n&87>D4d>D8o; B )2U=?4 [7E0>A@7>&BNQ7=<2/5 9R5760EF6F;?nGB S\2/4dQ'2UB?nD>GBDzs;<©7nD2U=? &4 >eL7. K7. h B

, Q_) JK!)>)PC (*!)>) !) (

ij8o;?[76F;?60 />&4d>D8o;&zP@%>&[% nD UnD87>&4 >&8o;<B >D; Bm2U8U;A608:@%nDQ'>D8 @79R8o;<BAZo6#E09 S\2U878:9R6TB<B?9/8:S\>u@7> E O [:8H87> 4d2s@76 ¦:> Q:9UB ET9 Q7=<2/5:9/576FE06F;?nf@%>fE O 9R[%;<=?>Uz¥@%2/8 S B?6 >D; . b O 608:SD2/8s /n&876F>&8o; @%> S\>D;m;<>d@7n\¦:876F;?602/8 =<n&B?60@%>d@79/8:BAEF> !9R6F;uZo[^O >DE0EF> 8^O >&Bm;A +9/E09/57EF> Zo[7>dBm2U[:BJ[787>d©s¬sQ 2/;?©7 GBm> @%> 872/8 Zo[:9/B?6 _ 604 Q'2UB<B?6F5760E06 ;<n/. 1 8fS\2U8:B`; 9+;<>ASD>DQ'>D8:@:9R8o;YZo[7>I@79R8:B E0>eSD9UB @%>I872/8fZo[:9/B?6 _ 604dQ 2oB?B?6F5:6FE06 ;<n/z%E0>&BC@7>D[% n&gU9/EF6F;?nGB .P13>D;m;?> nDgo9RE06 ;<neQ 2U[7 +9R8o; 3D;?=<>uS\2U8:Bm6T@%n&=?n&>eB?9/8:B B?2/8o;Yn&Zo[760 +9RE0>D8o;?>G B ( E O n&gU9/EF6F;?np~ ©s¬sQ 2/;?©7 GBm>Y@7>YZo[:9/B?6 _ 604 Q'2UB<B?6F5760E06 ;<n/z/S/O >GB`; >&EFE0>VZo[7>Y872/[:B Q7=<>D872U8:B¥S\2U4d4 >V@%n\¦:8:6 ;<6F2U8p@%>CE O 608:@%n&Q >&8:@79R8 S\>u~ 748:9 <; >=?; B B r%2/6F; [78 >&B?Q:9/SD>uQ7=<2/5:9/5760EF6TBmnUz >D; .Pb^>&BJn& /nD8:>D4d>D8o;<B >\; B?2/8o;FQ<S OYXl^iH!SbOYKNS{jD`eB?6 .

748:9 <; >=?; B B\p r%2/6F; [78 >&B?Q:9/SD> Q7=<2/5 9R5760EF6TB?n/zc>D; . b^>GB n& /n&87>D4d>D8o; B Bm2U8o; E ]~jk]iH!T<T\H0EcH!S{juQ<S OWX ^iH!SbOYKRSijC`uBm6 ~ A

B P (A | B) = P (A)

P (B | A) = P (B)

P (A ∩ B) = P (A)P (B)

(A, B) ∈ T 2

(Ω, T , P ) P (A ∩ B) = P (A)P (B)

(A1 , . . . , An ) ∈ T n

(Ω, T , P )

A1 , . . . , An

¥8fQ:9/=m;<60SD[7EF60>D=Gz%>D8fQ7=<>D8 9R8o;



∀J ⊂ [[1, n]], P 

J = [[1, n]]

~

\

j∈J



Aj  =

Y

j∈J

P (Aj ).

A

B

Bdp F = 8 = ' 9 =?;

' :; g W m b=%ni= $ >=?; B Bd rqbZRQ<j A , . . . , A OYH!`JX 4 X!SbH!EcH0SijC`VE ]~jC]iH0T<T\H0E4H0Sij Q<S OYXl^iH!SbOYKNS{jD`0w? TdZNML` 1

@Yp

n

n \

P

Ai

!

=

n Y

P (Ai ).

1 8 />&=?= 9f@79R8:BeEF> Q:9/=<9/g/= 9RQ7©:> B?[760 9/8o;uS\2/4d4d>D8o;uS&9RETS\[7E0>D=IET9fQ7=<2/5:9/576FE06F;?np@PO [787>p608o;?>&=<B?>&S\;?602/8 EF2U=<B<Zo[7> E0>&B¥n& /n&87>D4d>D8o; B 87>CB?2/8o; Q:9UB 4u[%;<[7>DE0E0>D4d>D8o; 608:@%n&Q >&8:@79R8o; B' )2U=?4 [7E0>C@%>&B Q7=?2U5:9R5:6FE06 ;<n&B¥SD2/4dQ'2UB?nD>&B&z+;<©7nD2 _ =< D4d>eL%.MK%. h l.

748:9 <; >=?; B TB S r%2/6F; [:8 >&B?Q:9/SD>¤Q7=<2/5:9/576FE06TBmnUz¥>\; [787>H !9R4d6FE0E0> 6F87¦:8760> @PO nD UnD87> B?2/8o; E ]~jC]iH0T<T\H0E4H0Sij Q<S OWX ^iH!SbOYKRSijC` Bm63Q 2U[7=e;<2/[%; z EF>GBen& /n __ 4d>D8o;<B&. bc>&BunD /n&87>D4d>&8U; B 87>&4 >&8o;<B B?2/8o;V4 [%;?[:>DE0EF>&4 >&8o;A608:@%n&Q >&8:@79/8U; BDzP@%2U8:SuB?6 Q 2U[7=J;?2U[%;AB?2/[ B _ >D8:B?>D4 57E0> tuS{Q @7> z i=1

i=1

(Ω, T , P ) Ai , i ∈ I

(Ai )i∈I

n∈ N

A1 , . . . , An

J

I



P

\

j∈J



Aj  =

Y

P (Aj ).

j∈J

, Q_) I JK!)>)PC (*!)>) !) ( ( B @ & # 748:9 <; >=?; B B[X r%2/6F; C >\; C @%>&[% B?2/[:B _ >&8:B?>D4u5:EF>GBV@%>

. 1 8 @%6F;eZo[7>dEF>GBIS\ET9/B<Bm>GB >\; B?2/8o; 608:@%nDQ'>D8 @79R8o;?>GBCB?6PQ 2U[7= ;?2U[%; >\;N;<2/[%; z >\; Bm2U8o; 608:@%n&Q >&8:@79/8U; BD. 1 8f@%n\¦:8:6 ;V@7>JET9 443D4d>I4 9R8760 D=<>AE O 6F8 @%nDQ'>D8:@:9R8:SD>I4u[%;<[7>DE0EF>e@^O [78:>A !9R4d6FE0E0>e@%>IS\ET9/B<Bm>GB .

F 8 = & B B[ZAqbZNQ<j H!j OWH!]Y cH0S:`0H0E4[0T\H!`VOYX0S ZRE4[0MPKW[!T\H!`VO? X 4 X!SbH!EcH0SijC`!Ew 3 S `'] ^Y^iZ6`0H$ 0]iH H!j `0ZNS{j)I :KWI!]~S$ !ZNM'EcX!` O X X0S H0E4H0SijC` OYH0]Y JOWH!]Y JQ<SbI ZRE.^iKNjkQD[0T\H'` H!j 0]iH H0j `0ZNS{j%Q<SbOYXl^:H0S OWKNSijC`0w T\ZRML`$TdH'`FjkM'QD[!]a` H!Sa WH0S ORMPXLH'`4 H!j `0ZNS{j X WKRT\H0E4H0Sij Q<S OWX s ^iH!SbOYKRSijlH'`0w F 8 = & B BD@6\AqbZRQ<j OYH!`VH0S:`0H0E4[0T\H!`VOYX0S ZRE4[0MPKY[0T\H!`FO? X 4 X0S H0E4H0SijC`0Ew 3 S `L] ^W^:ZN`0H !]iHVTdH'` `0ZNSij I :KWI!]~S !ZRM'E4X!` O? X X!SbH!EcH0SijC` OYH0]Y OWH!]Y Q<SbILZNE ^:KRjCQD[0T\H!` s H0j !]iH `0ZNSij$E ]~jC]iH0T_ T\H0E4H0Sij Q<S OWX ^iH!SbOYKRSijC`!wu TdZNML`44 T\H'`4jkM'QD[!]a` H!Sa WH0S ORMPXLH'` `0ZNS{j X WKRT\H!EcH0Sij E ]~jk]iH!T<T\H0EcH!S{j Q<S OWX ^iH0S OYKRSijlH'`0w V=

= B BC@:@ qbZRQ<j ]~SbHb !KNE Q<T<T\H O? H!Si`0H!Ec[!TdH'`.OYX0S ZRE4[0MPKY[0T\H!` O X X0S H0E4H0SijC`0Rw 3 S `'] ^Y^iZ6`0H 0]iH T\H!` `0ZNS{jJI :KYI0]~S !ZRM'E4X!` O? X 4 X0S H0E4H0SijC` OWH!]Y OYH0]Y Q<S ILZNE ^:KRjCQD[0T\H!` s 4 H!jJ 0]iH H!`'j ]~SbH !KNE Q<T<T\H OWH I0T\KN` `0H!` EJ]~jk]iH0T<T\H!EcH0Sij Q<S OYXl^iH!SbOYKNS{j H!`0w T\ZRML` T\H'` jkM'QD[0]a` H0Sz zH0S ONM X H!` s s `!ZRSij X WKRT\H0E4H0Sij EJ]~jk]iH0T<T\H!EcH0SijuQ<SbOYXl^:H0S OWKNSijlH!`!w V=

= B BC@ rqbZRQ<j OYH!`JX 4 X0S H0E4H0SijC`FEJ]~jk]iH!T<TdH!EcH!S{j Q<SbOYXl^:H0S OWKNSijC`0w% T\ZRML`FQ<T H0S H!`Lj OYH EcG!EcHJOWH jlZN]~jlHu !KREJQ<T<TdH ZN] w s Z ^iZN]~MFjlZN]~j s 4 ( , #?&) !, # F& . ( , N * " ( J#%" 1

2

A1 ∈ C 1

A2 ∈ C 2 A1

T

C1

C2

A2

(Ci )i∈I

C1

C2

C1

C2

C1

T (C1 )

C2

T (C2 )

C1 , . . . , Cn

Ci

C1 , . . . , Cn

T (C1 ), . . . , T (Cn )

(Ci )i∈I

Ci

(Ci )i∈I T (Ci ) i ∈ I

A1 , . . . , An

(B1 , . . . , Bn )

i ∈ [[1, n]] Bi = Ai

Bi = A i

,

- EB ; & ( >)(*# B E@ @ 2P*# 2 2 (*#

1 8fBm>IQ7ET9/SD>I@79R8:B [:8 >GBmQ 9/S\>JQ7=<2/5:9/5760EF6TBmn (Ω, T , P ) .

748:9 <; >=?; Bdp BD@ r%2/6F; . 1 8 @%6 ;dZo[7> >&Bm;u[78¨B?¬%B`;< D4d> Zo[:9/B?6 SD2/4dQ7E0>\;u@PO nD /n&87>D4d>&8U; BeB?6 E0>&B nD UnD87>&4d>D8o;<BN@%> C Bm2U8U;N872UC8 ⊂604dQ T2oB?B?6F5:EF>GBDz7@%>&[% C(d@%>&[% 608:SD2/4dQ:9+;<6F5:EF>GBDz%>D_;YBm6 P P (A) = 1.

A∈C

Y@ ?n B V n n b= 8 XC608:B?6 zPET9 @%6 nD=<>D8 S\>d9 />&Su[78 Bm¬%Bm;? D4d>dS\2U4 Q:EF>D;A=<n&B?6T@%> [:8760Zo[7>&4d>D8o;A@:9R8:BJE0> !9/6 ;eZo[7> E O [787602/8 @%>GBVn& /n _ 87>&4 >&8o;<Be8cO >GB`; nDgU9/EF> ( Ω ZU[cO (¤[:8 >D8:B?>D4 57EF> 87nDgUEF60g/>G9R57E0> Q7=< &B !E O [787602/8¨>&Bm;uQ7=<>&B<ZU[:> _ S\>D=?;<9/6F8:> >\;u872U8 S\>&=m; 9R6087> \.

F 8 = & Bdp B ª72/=<4u[:EF>I@%>GBNQ7=<2/5:9/576FE06F;?n&B3;<2R;<9/EF>GB qbZRQ<j ]~S ` 6`'j !EcHF 0]iK6`'Q< ILZNE ^ T\H0j:tuS{Q%ZN] OYX0S ZNEc[0MPKY[0T\H s jlH!T_`V !]iH T\H!`VX 4 X0S H0EcH!S{jD` A SbH `!ZRSij{^iK6` ^ M H'`0 0(A]iH! l)Q<E ^:ZN`L`LQD[0T\H!` s H0ju`0ZNQ<j B ∈ T w T\ZNML` i i∈I

i

P (B) =

X

P (B | Ai )P (Ai ).

. $G U( o$ N>\;<=?2U[7 />&= [787>IQ7=<2/5:9/5760EF6F;?n S\2/8:8:9R6TB?B<9R8o; EF>GB Q7=<2/5:9/576FE06F;?n&BNSD2/8:@76 ;<6F2U8787>DE0E0>&B CX 608:B?6 zs2U8 Q'>D[%;CSD9/E0SD[7E0>D= >D8 2C@%6TBm;?6087g/[:9/8o; B?[760 9/8o;y[78 S\>&=m; 9R608p872/4 57=<>J@%>ASD9UB 5 !B?[760 9/8o; Zo[7> >&Bm;C=<n&9RE06TBmn lz%Bm6 S\>&BNS&9/B Q:9/=m; 9RgU>D8o;yE O >D8:B?>D4 57EF>I@%>GBNSD9UB Q 2oB?B?6F5:EF>GBD. i∈I

P (B | Ai ) Ai

P (B)

P (B)

,

- EB ; & ( >)(*# B E@ @ 2P*#% ";= "#KPC(*#

F 8 = & Bdp B !ª72U=?4 [7E0>A@7>&BNQ7=<2/5 9R5760EF6F;?nGB S\2/4dQ'2UB?nD>GB jlH!T !]iH P (A ∩ · · · ∩ A ) 6= 0 lOWZNSbI A ∩ · · · ∩ A S H!`'j ^iK6` ^ MPH!`0 !]iH0 qbZRQ<j Q<E ^:ZN` (A`'QD[0T\H, R. w . . , AT\ZNML` ) ∈ T

1

n

n

P

n \

i=1

Ai

!

1

1

n−1

n−1

= P (A1 )P (A2 | A1 )P (A3 | A1 ∩ A2 ) · · · P (An | A1 ∩ · · · ∩ An−1 )

= P (A1 )

n Y



i−1 

\

P A i

Aj  .

$G U( o$ 1y9/E0SD[7EF>&= @%>GBuQ:=?2U5:9R5760E06 ;<n&B @PO 608o;?>&=<B?>&S\;?602/8:B @PO n& /n&87>D4d>D8o; BDz#872R; 9R4d4d>D8o; @79R8:B E0> SD9UB @^O [78:> ?B [:SDSD>&B<Bm602/8f@PO >\ %Q n&=?60>D8 S\>/.P13>\;?;?>A )2/=<4u[:EF>e@%n& /2U6FE0>A;?2U[%;?>uB?2/8f[%;<6FE06F;?neE02/= B?Zo[7>AE0>&BYnD /n&87>D4d>&8U; BCS\2U8:B?60@%n&=?nGB 87>eB?2/8o;NQ:9/BC4u[%;<[7>DE0EF>&4d>D8o;C6F8 @%nDQ'>D8:@:9R8o;<B&.7 ¥8 S&9/BN@PO 6F8 @%nDQ'>D8:@:9R8:SD>I4u[%;<[7>DE0EF>Uz:>&EFE0>I87>e@%6F;C=?60>D8¤@%>IQ:EF[:B Zo[7>AE09dQ:=?2UQ 2oBm6F;?602/8fL%. q:. W7. i=2

j=1

,

I - EB ; & ( >)( "(*#

F 8 = & Bdp B !ª72U=?4 [7E0>A@7>exy9 ¬U>&B qbZRQ<j ]~S ` 6`'j !EcH 0]iK6`'Q< ILZNE ^ T\H0j tuS{Q ZR] OYX0S ZNEc[0MPKY[0T\H s j H0T 0]iHV^iZN]~M4jlZN]~j i ∈ I s P (A ) 6= 0 w qbZRQ<j j(A∈ )I w T\ZNML`

i i∈I

i

P (B | Aj )P (Aj ) P (Aj | B) = P .

P (B | Ai )P (Ai )

$G U( o$ 0E >3B?¬%B`;< D4d> =<>DQ7=<n&B?>D8o;<> Bm2U[7 />&8o;c[787>3EF6TB`;<>3@%>ySD9/[:Bm>GB^Q'2/[7 +9/8U;#9R4d>D87>&=cE O n& /n&87>D4d>D8o; EF2U=<B @PO [787> n\;<9/Q > Bm[760 +9R8o;?> @%>eE O >D sQ'nD=<60>D8:SD>/.:ijE#>&Bm;V9/EF2U=<BNgUnD87n&=<9/EF>&4d>D8o;N !9/SD6FE0>u@%>u@7n\;?>&=?4d6087>D=VE09pQ7=?2U5:9R5:6FE06 ;<n Zo[^O [787>CS\>&=m; 9R6087> S\2/8 BmnGZU[:>D8:SD> 9/6 ;¥E06F>&[^z/B<9/S ©:9/8o; Zo[7> ET9JS&9R[:B?> 9A>&[ E06F>&[^z/S/O >GB`; _ ( _ @%60=<>3ET9JQ7=?2U5:9R5:6FE06 ;<n S\2U8:@%6F;?602/878:>DE0EF> !2/8 =?>GBmQ'>&S\;?>N@79R8 B#SD>yS&9/B#E O 2U=<@%=<>¥;<>D4dQ'2/=<>D E l.U13>&B @%2U8787n&>&B Q'>D=<4 >D;m;<>D8o;&z+gU= US\> (dET9u )2U=?4 [7E0>A@7>exy9 ¬U>&B&z7@%>I=<>D4d2/8o;?>&= EF>A;<>D4dQ:B&z7>D8¤@%nD;?>D=<4d6F8 9R8o;CE09 Q7=<2/5 9R5760EF6F;?neZo[^O [787>eSD>D=?;<9/6F87>IS&9R[:B?> 9R6F; >&[pEF60>D[ B?9US ©:9R8o; ET9eS\2U8:B?n&Zo[7>D8 S\> .U 2/[:=3S\>D;m;<>Y= 9R6TBm2U8^zUSD>\;m;<>C )2U=?4 [7EF>Y>&Bm;y9R[ B?B?6 B?2/[7 U>D8o; 9RQ:Q >&EFn&> !ZNM'E ]~T\H OYH.^ M ZY[LKY[0Q<T_Q<jlX OWH'`JILKR]a`!H!`l. i∈I

(Ai )

B

Ai

P (B | Ai )

Ai

B

B

N

N N

{J9R8:BNSD>eS ©:9RQ76F;?=<>/z7B<9R[7 4d>D8o;?602/8fS\2U8o;?= 9R60=?>Uz @%n&B?6FgU87>A[78f>&B?Q:9/SD>AQ7=?2U5:9R5:6FE060B?nAZU[:>DETS\2/8 ZU[:>/. / " , & ( , ( # V"F& 0"F /0( , " / "$# '&)( , (Ω, T , P )

I, E

P ! "! #

748:9 <; >=?; BC@:BD@ Y8 KNT\XLK Si]~EcX!M'QD 0]iHA>&Bm;C[787>e9/Q7Q7E060S&9+;<6F2U8 X @%> Ω @79/8:B R z%;?>&EcZo[7>d~ >&Bm;C[78fBm2U[:B _ >D8 Bm>&4u57E0>J@7> Ω Ω % Q 2U[7= ;<2/[%; a ∈ R z X (] − ∞, a]) ∈ T @%2/8 SCE O n& /nD8:>D4d>D8o; X (] − ∞, a]) 9/@%4d>D;y[78:>YQ7=<2/5:9/5760EF6F;?n l.

C8¤9/EFnG9 8s[74dn&=?6TZo[7> 9UB?B?2%S\60>I@%2/8:S (dSD>D=?;<9R608:B n&EFn&4 >&8o;<BC@%> [787>eSD>D=?;<9/6F87>I +9/EF>&[7=N8s[74dnD=<60Zo[7>Uz74 9R6TB ;?2U[:B E0>&BNn&EFn&4 >&8o;<BN@%> XΩ 8^O 2U8o;CQ:9/By )2U=<SDnD4d>D8o;C@PO 6F4 9RgU>AQ:9R= X Ω.

748:9 <; >=?; BC@:B Y87> 4 KNM'QDKW[!TdHJKRT\XLKNjlZNQ<MPH MPXLH!T<T\ H ) . 97. =&.o>D8 9/57=<nDg/ n >&Bm;3[:8 9RE0n&9I8s[74dn&=?6TZo[7>C;<>DE ZU[:> ]79/[%;?=<>D4d>&8U;Y@76 ;Gz%;?2U[:B EF>GB nDE0nD4d>D8o; BC@%> Ω 2U8o; [:87>I6F4 9/g/>AQ:9R= X . Ω =Ω m b=%ni = $ >=?; BD@:B 87%SbH !ZRSijkQDZNS H!`'j ]~SbH KNM'QDKW[!T\H KRT\XLKNjlZNQ<MPH MPXLH0T<T\H `'Q H!j `!H0]~T\H0E4H0Sij `'Q ^iZN]~MFjlZN]~j Q<S{j H0M 4 KNT<T\H I s X (I) H'`'j X]~S : X!ΩTdX!Ec→H!S{Rj OWH TdKcjkM'QD[!4 ] T w = >=%; BC@:B 1 8I@%2U8787> ET9 872R; 9+;<6F2U8J>&4dQ7EF2+¬UnD> [:B?[7>DE0E0>D4d>D8o;cQ'2/[7= S\>D=?;<9/6F8 B nD /n&87>D4d>&8U;G.&rs2U6 ; (a, b) ∈ >\; x ∈ R ~ R % b¥O nD /n&87>D4d>&8U; {ω | X(ω) = x} >GB`;C872/;?n [X = x] ] % b¥O nD /n&87>D4d>&8U; {ω | X(ω) < a} >GB`;N872/;?n [X < a] ] % b¥O nD /n&87>D4d>&8U; {ω | X(ω) 6 a} >GB`;N872/;?n [X 6 a] ] % b¥O nD /n&87>D4d>&8U; {ω | a < X(ω) < b} >&Bm;C872R;<n [a < X < b] z%>\;<S/. e ' bf?g BD@iB\p 13>&=m; 9R608:B 9R[7;?>D[:=<Bp@7n\¦:876TB<Bm>&8U; Q:9R=? )2/6TBd[787> . 97. =&. SD2/4d4d>¤[78 9RE0n&9 8s[74dn&=?6TZo[7> @%nD¦:8763B?[7= Q7=?>GB?Zo[7>d;<2/[%;<>&BuEF>GBe +9RE0>D[7= Be@%> Ω z SRO >&Bm; _ ( _ @%60=?>pB?[7= Ω ;?>&E3ZoX[7> P (Ω ) = 1 .cXC608:B?6 z Bm>&[7E [:8 B?2/[:B >&8:B?>D4u5:EF> 87nDgUEF60g/>G9R57E0>d8^O 9/@%4d>\;eQ 9/BI@%> +9RE0>D[:=eQ:9R= . 13>\;m;<>pS\2U8o U>D8o;?602/8 Q'>D=<4 >D;u@%>pS\2U8:B?60@%n&=?>&= E O 2U5%k`_ >\;Y@%nD¦:876^@79/8:BNE O >D %>D4dQ7E0>AB?[760 +9R8o;YS\2U4 4d>A[787>I . 97. =&.0.X V-W n BD@:Bd 1 8 E09/8:S\> [:87>p608%¦:876F;?n @%> )2/6TBe[:87>pQ760 &SD>/z^>D; X S\2U=?=<>&B?Q 2U8:@ 9R[ =<9/87g @%>pET9R8 S\>D= @%[ Q7=<>D4d60>D=C ¥60E0>u2/57;?>D8s[^. ¥8H9U@%2/Q%; 9R8o;YET9pS\2U8s />D8o;<6F2U8 Q7=<n&SDn&@%>&8o;?>/z X >&Bm;V[787>u '. 97. =G.Fz7Q:=?>&8:9R8o;YE0>&BY 9/EF>&[7= B 0

%

0

−1

−1

0

−1

2

0

0

%n B 8 = 8

. N>D4 9/=<Zo[7> yZo[7> E O n& /nD8:>D4d>D8o; 238^O 2U5%;?>&8760=¥ZU[:> @7>&B# !9/SD>&B 5 zRZo[76%>GB`; Zo[:9UBm6 _ 6F4dQ'2UB<Bm6057E0>/z 4 9R6TB X(Ω) = N Q:9UB 6F4dQ'2UB<Bm6057E0>/z%8^O 9/@%4d>D;CQ:9/BN@PO 604d9/g/>AQ:9/= X . m b=%ni= $ >=?; BD@:BHS qbZRQ<j X ]~S H 4 w<KYwdM0w<wu TdZNML` {[X = x], x ∈ X(Ω)} H'`'j.]~S ` 6`'j 0EcH ILZNE ^ T\H0j ZN] 0]iK6`'Q< ILZNE ^ T\H0j.H!S KWOYZL^ jlKNSij T\K ILZNS 4 H0SijkQDZNS OWH TdK MPH0EcKNMP 0]iH |zw<xYw< 748:9 <; >=?; BC@:[B X b 9p;<=?6057[ 9/B<Bm2%SD6Fn& > ( E09 . 9:. =G. X >GB`;IET9 ;<=?6057[ >D87gU>D8:@7=?n&>uQ:9/=JE0> Bm¬%Bm;? &4 > ZU[ 9/B?6 _ S\2U4dQ7EF>D; {[X = x], x ∈ X(Ω)}.7 Y2R; 9+;?602/8 )=<n&Zo[7>D8o;<>I4d9/60B Q 9/B [78760 />&=<B?>DE0E0> ~ A 2U[ T . @6X

∗

X

I, E - "!) M +! >)( B PC ) B2 2 "!

748:9 <; >=?; BC@:B[Z r%2/6F; X : Ω → R [:87> . 97. =&.0.cbc9

X

)2U8:Sl;<6F2U8 @%>p=<nDQ:9/=m;<6 ;<6F2U8 @7> >&Bm;uET9 )2/8:S\;?602/8^z 872/;?nD>AgUnD87n&=<9/EF>&4 >&8o; z:@%n\¦ 876F>IQ'2/[:= ;?2/[7; Q:9R= . `0K. !ZNSbI!jkQDZNS OYHJMPXl^:KRM'jCQ<jkQDZNSbw? T\ZRML`F F 8 = & BC@:BD@6\AqbZRQ<j ]~SbH 4 w<KYwdM0w s H0j xYw H'`'j ]~S Hu !ZRS I0jCQDZRS I0MPZRQv` `0KRSijlHF`']~M yzw H'`'j I ZRSijkQ<Si]iH ONM ZNQ<jlH H0S jlZN]~j ^:ZRQ<Sij OYH |Ww KYORE4H0j OWH'`FT_Q<E Q<jlH'`JH0S s H0j w s X!M'Q t KNS{j T\H!` ^bMPZ ^bM'QDX!jlX!` OR] F 8 = & BC@:BD@i@ KYONE Qv` XLI!Q ^bMPZ !]iH0E4H0Sij s j ZR]~j H !ZRS I0jCQDZRS 4 j :XLZNM 0EcH |zw<xYw<x H!`'j T\K !ZNSbI!jkQDZNS OYH MPXl^:KRM'jCQ<jkQDZNS O ]~S H ILH!M'jlKNQ<SbH 4 w<KYwdM0w<w " , * ( , " & * !*, & #%( , x∈R

FX

X

X

FX (x) = P (X 6 x)

FX

FX

R

FX

R

±∞

FX

lim FX (x) = 0

x→−∞

lim FX (x) = 1

x→+∞

F :R→R

I,

P ! "! #

b^>GBA@%n\¦:8:6 ;<6F2U8:BI@%6 ' &=?>&8U;A[78 Q >&[ B?[760 +9R8o;IEF>GBJSD2/8o;?>D s;?>&BJ>D;JE0>&BA5'>&B?2/608:B&. b 9 @7n\¦:876F;?602/8 B?[76F +9/8U;<> >GB`;AET9 4d2/608:BNS\2U8o;?= 9R60g/8:9/8U;<>/zs>\;N>GB`;Y9/4 Q:EF>&4 >&8o;YBm[ pB<9R8o;?>JQ'2/[7=N;<2/[%;<>AE09 ;<©7nD2U=?60>/.

748:9 <; >=?; B BD@ Y87> 4 KNM'QDKY[0T\HcKRT\XLKNjlZNQ<MPHJMPXLH0T<T\HJORQv`0I!M 0jlH ) . 97. =&. @P.%>D8¤9/57=?n&g/n >GB`;N[78:>I '. 97. =G.s;?>DE0E0> Zo[7> B?2/6F;N[78f>D8:B?>D4 57EF>J¦:8:6^2/[¤@%n&872/4 57=<9/57E0>/. { O 9/Q7=? GBAEF>GBA=? &g/E0>&BA@%>pS&9R= @%6F8 9RE06 ;<n SD2/8:SD>D=<8:9R8o;AEF>GBA>D8:B?>D4 57EF>GBJ¦:876TBA2/[ @%n&872/4 57=<9/57E0>&B&z'2U8 2U5%;?60>D8o;I604 _ 4dn&@%6T9+;<>D4d>D8o;e~ m b=%ni = $ >=?; B B rqbZRQ<j H0j OYH0]Y 4 w<KYwdM0w<OWw s ]~S MPXLH0T{ !]iH0T\ILZNSb !]iH s H0j ]~S H KL^Y^bT_QDILKNjkQDZNS OYH OYKRS:` w T\ZRML` s T\H'` 4 w<KYwdM0wuOWXDt SiQDH!` OYHJT\KcEcKNS{Q !M HF`']~Q 4 KNSijlH `!ZRSij H!SbI ZRMPH ONQv`0I!M !jlH'` xYw

yzw

|Ww }{w w X

X(Ω)

X

Y

λ

f

X(Ω)

R

X + Y : ω 7→ X(ω) + Y (ω) λX : ω 7→ λX(ω)

XY : ω 7→ X(ω)Y (ω) f (X) : ω 7→ f (X(ω))

I,

- "!) M +! >)( B PC ) B2 2 "!

%r 9/[% @79R8 BV@%>GBVS&9/B 2YQ 9+;?©:2/E02/g/6TZo[7>&B 5f=<>D8 @79R8o;V[:87>e;<>DE0EF> @%>&B<S\=<6FQ7;?602/8 604dQ 2oB?B?6F5:EF> ) Q7=<n&B?>D8:SD>u@%> Q 2U6F8o;<B @PO 9/S&S\[74 [7E09R;?602/8 @79R8:BdE0>&Bp 9/EF>&[7= B Q'2UB<B?6F57E0>&BdQ:=?6TBm>GB Q:9R= z Q:9/= >\ %>D4dQ7E0>¤Bm6 lz E0>fg/= 9RQ7©7> X

X(Ω) = Q

B h= & :; \g ; B B B B @6Z @%> E09f )2U8:Sl;<6F2U8 @%> =<nDQ 9R=?;?6F;?602/8 @PO [787> . 97. =&. @P. X >&Bm;u[787> )2/8:S\;?602/8 >D8¨Q7E09R;?>G9R[% Q7=<>D8:9/8U; @%>&Bu 9/EF>&[7= B S\2U8:Bm;<9R8o;<>&By>D8o;?=<>IS ©:9/Zo[7>ISD2/[7Q7E0>I@%>I +9RE0>D[7= BNB?[:SDSD>&B<Bm60 />GB3Q'2UB<B?6F57E0>&B @%> X .

I,

I 9: " >-( )B E@) @ P >NO & !-( Q , QBC >:

748:9 <; >=?; B B r%2/6F; [787>C . 97. =&. @P.0./bc9AE02/6:@7>CQ7=<2/5:9/576FE06F;?nC@%> >GB`;¥ET9A )2/8 Sl;?602/8 %@ nD¦:8760>IQ 2U[7= ;?2U[%; Q:9R= . {V>J4 9/876F &=?>Jn&Zo[760 +9RE0>D8o;?>Uz%Bm6P2U8 @%6TBmQ'2UB?>A@PO [:87>AnD8s[74dnD= 9+;<6F2U8 @%>GBynDE0nD4d>D8o;<BN@7> zsET9 E02/6P@%> Q7=<2/5:9/5760EF6F;?nI@%> >&Bm;CE09d@%2U8787nD>e@7>JET9dBm[:6 ;<> ;?>&EFE0>eZo[7>IQ 2U[7= ;?2U[%; z . e ' bf?g B B 2U[7=IQ 2U[7 /2U6F=IQ 9R=<EF>&=A@7>pE09 E02/6 @7> z^60E !9R[%;u9 +9/8U;A;<2/[%; 9 U2/60=J@7n&S\=<6F;eE O >D8:B?>D4 57E0> @%>GBN +9RE0>D[7= B Q7=<60B?>&ByQ:9/= . `0K TdZNQDwb T\ZNML` H!`'ju[!QDH0S OYXCtuSiQDH$H0j F 8 = & B B\pAqbZNQ<j ]~S H 4 w<KWwdM0w<OYw)H0j X x ∈ X(Ω)

f : X(Ω) → [0, 1]

X

f (x) = P (X = x)

(xn )n∈N

X(Ω) n ∈ N pn = P (X = xn )

(pn )n∈N

X

X

X(Ω)

X

P

f : X(Ω) → R X f (x) = 1.

X

f (x)

x∈X(Ω)

e ' bf?g B B\ {V60=?> Zo[7> S\>\;?;?> B?2/4d4d> )¦:8760> 2/[ @7nD872U4u57= 9R57E0> J>GB`; 576F>&8¨@%nD¦:8760> S\2U8:Bm6TBm;?> (¤@%60=<> Zo[^O >DE0EF> >&Bm;dSD2/8s />&=?gU>D8o;?> Q<SbOYXl^:H0S OWKNE E4H0SijJO? ]~S H X0Si]~EcX!M KNjkQDZNS OYH z SRO >&Bm; _ ( _ @%60=?> 608:@%n&Q >&8:@79/4 4d>&8U; @PO [78¨2U=<@7=?> @%> B?2/4d4 9+;<6F2U8^z#>\; ZU[:> ET9 +9RE0>D[:= @%> ET9 B?2/4d4d> 87>f@%n&Q >&8:@¨Q 9/B 872/8 Q7EF[ B @%> E O 2U=<@%=<> @7> B?2/4d4 9+;?602/8c.U13>&S\6:>&Bm;¥ /nD=<6F¦:n >D8pSD9UB @%>CS\2U8s />D=<g/>&8:S\>y9R5:B?2/E0[7>U.Ri`S\6 z/S\2U4d4 >NET9ABmn&=?60> >&Bm; (V;?>&=?4d>&B¥Q 2oBm6F;?6F !BDz >DE0E0>V>GB`;N5760>D8 @%n\¦:8:6F>IB?6'>D;CBm>&[7EF>&4d>D8o; B?6P>DE0E0>V>GB`;CS\2U8s />D=<g/>&8o;?>CQ'2/[7= [78:>Vn&8o[:4 n&=<9R;?602/8 @72/8787n&>I@%> . ¥8¤9/@72/Q%; 9R8o;NE09dS\2U8s />D8o;<6F2U8 @%>IET9 =?>&4d9/=<Zo[7>Ah7.0-/.MK%z%2U8 2U5%;?60>D8o;I~ m b=%ni = $ >=?; B HB S 7%S KNT\XLK H'`'j%]~S H 4 w<KYwdM0w<OWw `'Q?H0j `0H0]~T\H!EcH0Sij `'Q w m b=%ni = $ >=?; B YB XrqbZRQ<j ]~SbH H0j `0K. !ZRS I0jCQDZRS OWH MPXl^iKNM'jkQ<jCQDZRS w? ZR]~MFj ZR]~j s x∈X(Ω)

Ω

X(Ω)

X : Ω0 → R

P

P (X = x) = 1

x∈X(Ω0 )

X

v.a.r.d.

x∈R

FX

P (X = x) = lim+ FX (y) − lim− FX (y) = FX (x) − lim− FX (y).

748:9 <; >=?; B [B Z Y8¨4d2%@%>f@PO [:87> . 97. =&. @P. X >&Bm; [:87> +9/EF>&[7= x ∈ X(Ω) ;<>DE0EF>fZo[7> ET9 Q:=?2U5:9R5760E06 ;<n >&Bm;N4 9+% 6F4 9/EF>U. C87>A;<>DE0EF>I +9RE0>D[:=N8^O >GB`;NQ:9UBy )2/= S\n&4 >&8o; [:8760Zo[7>U. P (X = x) ( J#%,-* ( " & * y→x

I, I

y→x

y→x

#K PCB ! ( (*&)B ;= " (*!)!)(

748:9 <; >=?; B BD@ r%2/6F; X [787>N . 97. =&. @P.0. b OvH!`k^:X0MPKRS ILHC@%> X >&Bm;&zRB?2/[:B =<n&B?>D=< />y@%> I ZRS 4 H!Mk WH0S ILHVKY[!`!ZRT_]iHI~ E(X) =

%

%

e

X

xP (X = x).

bf?g B B b O >GBmQ'nD= 9R8 S\>JQ >&[%;C87>IQ:9UB >\ %60Bm;?>&=I~ Bmn&=?60>I@%6F U>D=<g/>&8U;<> ~ X(Ω) = N >\; ∀n ∈ N z P (X = n) = 6 .

Bmn&=?60>IBm>&4 6 _ S\2/8s U>D=<g/>D8o;<>e~ X(Ω) = {(−1) n, n ∈ N } z%>\;eπ~ ∀nn ∈ N , x∈X(Ω)

'

∗

∗

2 2

n

∗

∗

P (X = (−1)n n) =

6 . π 2 n2

%n B 8 = 8

2\ ,I I

B !) ( >) #K (CB #K "!

47 8:9 <; >=?; B B r%2/6F; X [787> '. 97. =G. @P.y9/@%4d>D;m;<9/8o; [787> >&B?Q n&=<9/8:S\> S\2U8s />D=<g/>&8:S\>J@%>IET9 B?nD=<60>/z%ET9 +9/=?6T9R8:SD>J@7> X >GB`;I~ X

V (X) =

E(X)

. XCE02/= BDzyBm2U[:B =<n&B?>D=< /> @%>

(x − E(X))2 P (X = x).

b¥O n&S&9R=?; _ ;`¬sQ >J>&Bm;Y9RE02/= B @%n\¦:8:6^Q:9R=I~ . e ' bf?g B B b 9 9/=?6T9R8 S\>JQ >&[%;C87>IQ:9UB >\ %60Bm;?>&=N443D4d>eBm6PE O >&B?Q n&=<9/8:S\>A>\ %6TB`;<>/. x∈X(Ω)

σ(X) =

I, I I 9: Q , QB* >N

p V (X)

{J9R8:BNSD>\;m;<>eBm>GSl;<6F2U8^z >GB`;C[:87>I . 9:. =G. @^.%>\; [:87>e9RQ7Q7E06TSD9+;<6F2U8 @%> @79/8:B . m b=%ni = $ >=?; B Bdp !bc2/6#@%> H0S:`0H0E4[0T\HcOWH'` 4 KNT\H0]~ML` ^iZ6`L`'QD[!T\H!` OWH H!`'j H!`Lj OYZRSiS XLH ^:KRMJ F 8 = & B B\ !@%> ;?= 9R8:Bm )>D=?; H WQv`'j H `'Q$H0jV`0H!]~T\H0EcH!S{j `'Q KY[!`0ZNT_]~EcH0Sij s H!j OYKRS:` ILHJILK6` V=

= B HB S ZN]~MFjlZN]~j w s V=

= B YB XrqbZRQ<j H0j OWH!]Y V !ZRS I0jCQDZRS:` OYH OYKNSi` w TdZNML` f (X)

X

f

X(Ω)

R

f (X) ∀y ∈ Im(f ), P (f (X) = y) =

f (X)

X

Im(f )

P (X = x).

x∈f −1 (x)

P

E(X)

g(x)P (X = x)

x∈X(Ω)

E(g(X)) =

P

g(x)P (X = x).

s H0j `0K T\ZNQ ILZNS 4 H0MC zH

x∈X(Ω)

(a, b) ∈ R2 E(aX + b) = aE(X) + b

ϕ

ψ

X(Ω)

R

E(ϕ(X) + ψ(X)) = E(ϕ(X)) + E(ψ(X)).

F 8 = & B B[Z V2o>&8760g _ [7¬sg/©:>D8:B qbZRQ<j X KWONEcH!jkj KRSij ]~S H 4 KRM'QDKNS ILH w? T\ZRML` H!j ∀a ∈ R, V (X) = E((X − a) ) − (E(X) − a) .

V (X) = E(X ) − E(X) V=

= B BC]@ \ ∀(a, b) ∈ R s V (X + b) = V (X) H0j V (aX) = a V (X) w 748:9 <; >=?; B BD@:@¨rs6 E(X) = 0 z+2U8 @%6F; Zo[7> X >GB`;¥S\>&8o;?=<nD>¥]+B?6 V (X) = 1 z+2U8 @%6F; Zo[7> X >GB`; =<n&@%[76F;?>U. {V>Y4 9R8760 D=<>CgUnD87n&=<9/EF>YB?6 E(X) >D; V (X) >\ %60Bm;?>&8o;&zoE09I . 97. =&. @P. X = X − E(X) >GB`; S\>D8o;<=?n&>C>\;y=?nG@%[76F;?>/. σ(X) 1 8 E O 9RQ7Q'>DE0E0> 4 w<KWwdM0w<OYw I H0SijkMPXLHFM X OR]~Q<j HJK6`L`0Z I0QDXLH X .

2

2

2

2

2

2

∗

I, I Q_

";=(*!+ 2# ( ;= ";=(*!+ #% (*!+ 2B P*#

47 8:9 <; >=?; B BD@ rs2/6F; X [:87>Y . 97. =&. @P.o>\; k ∈ N .obc>Y4d2/4d>D8o;N@PO 2/= @%=?> k @%> X >GB`;GzoB?2/[ B =<n&B?>D=< />C@%> S\2U8s />D=<g/>&8:S\>J9R5:B?2/E0[7>d~ X ∗

xk P (X = x) = E(X k ).

mk (X) =

b^>V4d2/4d>D8o;NS\>D8o;<=?n&>V@PO 2/= @%=<> >&Bm;&z%B?2/[:B3=<n&B?>D=< />C@PO >\ %6TB`;<>D8:SD>J@%> 9R5 Bm2UEF[7>d~ x∈X(Ω)

k

µk (X) =

X

x∈X(Ω)

E(X)

>\;NB?2/[:B =?nGBm>&=? U>Y@%>AS\2/8s U>D=<g/>D8 S\>

(x − E(X))k P (X = x) = E((X − E(X))k ).

N

N

1 8 n\;<[:@%60>J@:9R8:B3S\>YS ©:9/Q76F;?=<>C[78 SD>D=?;<9R608 872U4u57=<>Y@%>VEF2U60B39RQ7Q:9/=<9/60B<B<9R8o;¥@79R8:By@%>C8:2/4u5:=?>&[:Bm>GB B?6 ;<[:9+;<6F2U8:B&. bc9dQ:EF[7Q 9R=?;V@%>eS\>GBCE02/6TBC9RQ:Q:9R= 9R6TB?B?>D8o; EF2U=<BY@PO >D %Q n&=?60>D8:SD>&BC@7>A;?60=<9/g/>&BY@%>I5 2U[7E0>&BC2U[¤@%>I;?60= 9Rg/>GB*( Q760EF>I2U[ !9/SD>/.

!, !,%, ( , * #% '&)" ( (

_- E 9: " &)! EB ;=(

b¥O >\ %Q'nD=<6F>&8:S\>JS\2U8:Bm6TBm;?>&(e;?60=?>&= [78:>V5'2/[7E0>V@:9R8:B3[787>J[7=<87>VSD2/8o;?>&8:9R8o; 5'2/[:EF>GB 8s[74dnD=<2R;<nD>GB3@%> +9R=<609/57E0> >&Bm;CEF>I8s[74dn&=?2d@%>IET9 5'2/[:EF>A;<6F=<nD>U.7XC608:Bm6 z72U8 @%nD¦:876F;u~ 748:9 <; >=?; BC@:BD@ r%2/6F; . C87>J . 9:. =G. Bm[76F;3ET9eE02/6'[7876F )2/=<4d>J@7>YQ:9/=<9/4 D;?=<> B?6 >\;YB?6 ~ n

1

(

n

.%b 9

X

n ∈ N∗

X

n

∀k ∈ [[1, n]], P (X = k) =

X(Ω) = [[1, n]]

1 . n

1 8 872R;<> X ,→ U(n) . m b=%ni= $ >=?; BD@:B rqbZRQ<j X ,→ U(n) w? T\ZRML` E(X) = n + 1 H0j V (X) = n − 1 w 2/[7=A;<2/[%;I>&8:Bm>&4u57E0> ¦:876 E zc2/8H8:2R;?>dQ7E0[:BIgUnD87n&=<9/EF>&4d>D8o; 2X ,→ U(E) Q'2/[7=I12[:87>d '. 97. =G.^Bm[760 +9R8o;IE09 E02/6 [7876F )2/=<4d>IBm[7= X z7SRO >&Bm; _ ( _ @%6F=<>J[787>I . 9:. =G.s;?>&EFE0>eZo[7>d~ 2

∀x ∈ E, P (X = x) =

1 . |X|

_- E 9: " >-( (CB !) "&)

b¥O >\ %Q'nD=<6F>&8:S\>pSD2/8:B?6TB`;<> ( ;?60=?>&=e@79R8:Be[787>p[7=<87>pSD2/8o;?>&8:9R8o;e[787> Q:=?2UQ 2U=m;<6F2U8 @7> 5 2U[7EF>GBe57ET9R8 S ©7>&B&. 1 8 872/;?> ET9d +9R=<609/57E0>J9/EFnG9+;?2U6F=<>JnDgU9/EF>)( Bm6P2U8 ;<6F=<>A[787>I5'2/[7E0>A57ET9R8:S ©7>Uz%>\; B?6F872U8^.

1 8 Q'>D[%; 9R[ B?B?63@%nD¦:8760= (HE O 9R6T@%> @%>p;<6F= 9RgU>&B (HQ76FE0> 2U[ !9/SD> 9 U>&Sd[787> Q760 &S\> @%n&B?n&Zo[760E06F57=<nD>Uz#@72/8o;uET9 Q7=<2/5:9/5760EF6F;?nI@%>IQ760E0>J>GB`; ] Q7=<>D8:@f9/EF2U=<ByET9 +9/EF>&[7= Bm6P2U8 ;<6F=<>AQ760EF>Uz7>\; B?6P2/8 ;?60=?>J !9US\>/.

XC608:B?6 z =?>&Q7=?nGBm>&8o;?>JEF>eB?[:SDSD &BN@79/8:B [787>I>D %Q n&=?60>D8:SD>I9 ¬U9R8o; [787>IQ7=<2/5 9R5760EF6F;?n @%>eB?[:SDSD &B&.

1 8f@%n\¦:8:6 ;V@72/8:S ~ p

1

X

X p X

[X = 1]

0

1

0

p

?n B F = g g 748:9 <; >=?; BC@:B r%2/6F; X [:87>e . 9:. =G.7>D; p ∈]0, 1[ . 1 8¤@%6F;VZo[7> X Bm[:6 ;Y[:87>eE02/6c@%>exy>&=?872U[7E0EF6c@7>eQ:9R= 9 _ 4d \;<=?> p )>D;C2/8 872R;<> X ,→ B(1, p) 3B?6¥~ ( P (X = 0) = 1 − p \ > ; X(Ω) = {0, 1} P (X = 1) = p {Vn&B?2/=<4d9/60B&zs2/8¤B?>A@72/8787>A[78f=<nD>&E p ∈]0, 1[ z%>\;C2/8 872/;?> q = 1 − p . m b=%ni= $ >=?; BD@:B qbZRQ<j X ,→ B(1, p) w T\ZRML` E(X) = p H!j V (X) = pq w !, !,%, ( , * ( # '& " ( , , ' ( ,%,? N, " ( &)(. !,%(

_-

9: " @ !- ";< ( 8!- "; @ B ( >-( # &) *#

rs2U6 ; . 1 8 =<nDQ' \;?> )2/6TB [787> >\ %Q n&=?60>D8 S\> @7> xy>D=<872U[7EFE06 z >D; 2U8 872R;<> EF> 872/4 57=?> %@ >fBm[:S&S\ GB 2/57;?>D8s[:B&.¥ 9R= >D %>D4dQ7E0> 2/8 S\2U4dQ%;?>¤E0>¤872/4 57=<>f@7> ¥60EF> 2U5%;?>&8s[:B @79R8 B [787>HB?[:S&S\>&B<B?6F2U8 @%> TE 9R8:SD>D= B )608:@%n&Q >&8:@79/8U; B 9 U>&SV[787>IQ760 &SD>e@%n&B?n&Zo[760E06F57=<nD>A@%2U878:9R8o;C ¥60E0>e9 />&SV[787>IQ:=?2U5:9R5760E06 ;<nJ@%> .

P. rs2/6F; {Vn\;?>&=?4d60872/8 BYET9pE02/6 @%> .'| 2/[7;A@PO 9R5'2/= @Pz 2/8HQ'>D[7;A9 U2/60=Y@%> ( B?[:SDSD &B&z'9/6F8 Bm6 @%2U8:S . b¥O nD /n&87>D4d>&8U; >&Bm;N2/5%;<>D8s[fBm6^2U8 9 >&[ Bm[:S&S\ GBD.:rs2U6 ;Gz%Q 2U[7= ;?2U[%; z E O n& /nD8:>D4d>D8o;I~72/57;?>D8:6F=N[78¤B?[:SDSD &*B ( ET9 _ 6F &4d>J>D %Q n&=?60>D8:SD>/.7XYEF2U=<BA~ n ∈ N∗

n

X

0 n k

X

k ∈ [[0, n]]

n p X(Ω) = [[0, n]] i ∈ [[1, n]] Si

[X = k] i

[

[X = k] =

I⊂[[1,n]] |I|=k

 

\

\

Si ∩

i∈I

i∈[[1,n]]\I



Si 

^b >GBNnD UnD87>&4d>D8o;<B z z:B?2/8o;N4u[%;<[7>DE0EF>&4d>D8o;Y608:@%nDQ'>D8 @79R8o;<B&z7>D;CEF>GBNnD UnD87>&4 >&8o;<BN@%>eE O [787602/8¤Bm2U8o; @%>&[% (d@%>&[% 608:S\2U4 Q 9+;?6057E0>&B&.:XC608:B?6 z Si i ∈ [[1, n]]

P (X = k) =

X

I⊂[[1,n]] |I|=k

 

Y i∈I

p·

Y

i∈[[1,n]]\I



(1 − p) =

X

pk (1 − p)n−k =

I⊂[[1,n]] |I|=k

n k p (1 − p)n−k . k

b^> S\2s> pS\60>D8o;C5760872U4 6T9RE#S\2U=?=<>&B?Q 2U8:@ (dE09 Q'2UB?6F;?602/8¤@%>&B Bm[:S&S\ GBDz >&Bm;YET9dQ7=<2/5:9/576FE06F;?ne@PO 2/5%;<>D8o;?602/8 @7> >GB`;CET9 Q7=<2/5:9/576FE06F;?nI@PO 2/5%;<>D8o;?602/8f@%>&BCn&S ©7>GSDB 9R[% 9R[%;<=?>GBy;?60=<9/g/>&B&.

S\>GB B?[:S&S\ &B >D; 1 8f@%n\¦:8:6 ;V@72/8:S ~ 748:9 <; >=?; B BD@ r%2/6F; [787>e . 97. =&.0. 1 8 @%6F;VZo[7> B?[76F;YE09dE02/6 5760872/4d6T9RE0>e@%>eQ:9/=<9/4 D;?=<>&B )>\; 2/8 872/;?> yB?6¥~ >D; 265 13>&SD6^@%n\¦ 876 ;V576F>&8f[787>AEF2U6c@%>AQ7=<2/5:9/5760EF6F;?nGBD.% ¥8f> '>D;e~ m b=%ni = $ >=?; B B rqbZRQ<j w T\ZNML` H0j H WQv`'j H0Sij s H!j H!j 1 8¨Q'>D[7; 2U5%;?>&876F= SD> =<n&B?[7E ; 9+; Bm2U6 ; (HE O 9R6T@%> @PO [78 SD9RETS\[:E @76F=<>&S\;&z Bm2U6 ; >D8 =?>&4d9/=<Zo[:9/8o; Zo[7> >&Bm; [78:> B?2/4d4d>V@%> 9/=?6T9R5:EF>GB¥@%>Ax3>&=?8:2/[7E0EF6 6F8 @%nDQ'>D8:@:9R8o;?>GBDzo>\;y>D8 [7;?60EF6TB?9/8o;3E0>&B¥=<n&B?[7EF;<9R;<By@%[ S ©:9/Q76 ;<=?>VBm[760 +9R8o;&. pr

k

(1 − p)r

r

X

(n, p)

X

X ,→ B(n, p)

X(Ω) = [[0, n]],

n k n−k ∀k ∈ [[0, n]], P (X = k) = p q , q = 1 − p. k n X n k p (1 − p)n−k = (p + (1 − p))n = 1. k k=0

X ,→ B(n, p)

E(X) = np,

E(X)

V (X)

V (X) = npq.

X

n

9

B F = 0g

g

$ $

_-

9: " DEPC ";=P 2B $,&-(

' $

2(*;= #%>:OQ (R!" 2( >)&A B (*;< (*B # &) *#

1 8 SD2/8:B?6T@% D=<>u4 9R608o;?>&8:9R8o;A[787> Bm[ SDS\>GB?B?602/8 608%¦:8760>d@PO >D %Q n&=?60>D8:SD>&BA@%>dxy>D=<872/[7E0E06#608:@%n&Q >&8:@79/8U;<>&BA@%>dQ:9 _ = 9R4d \;<=?> p . 1 8fS\2U8:B?60@% &=?> X E09 +9R=<609/57E0>J9/EFnG9+;?2U6F=<>AS\2/=<=<>&B?Q 2U8:@79R8o;N9R[ =<9/87gd@%[fQ7=?>&4d6F>&=CBm[:S&S\ GBD. 9R=y>\ %>D4dQ7E0> X >&Bm; EF>A= 9R87g @%[ Q7=<>D4d60>D= Q760EF>J2/57;?>D8s[ @79R8:By[78:>JB?[:SDSD>&B<Bm602/8 @%>VET9R8 S\>D= By9 />&SY[787>AQ7=<2/5:9 _ 5760EF6F;?ne@^O 2U5%;?>&8U;<6F2U8 @7>IQ76FE0>InDgo9RE0> ( p Q'2/[7=CS © 9/Zo[7>J;?60=<9/g/>U. {Vn\;?>&=?4d60872/8 BAE09 EF2U6¥@%> X .^b^>GBJ 9/EF>&[7= BVQ'2UB<B?6F57E0>&BI@7> X Bm2U8o;V;<2/[%;<>&BAE0>&BA +9RE0>D[7= BJ>D8o;?60 D=<>&BABm;?=<60S\;?>&4 >&8o; Q'2UB?6 ;<6F U>&BA~ X(Ω) = N . rs2U6 ; k ∈ N .:XYEF2U=<B&zs>D8f[%;<6FE060B<9R8o;CE0>&B 443D4d>GBN872R; 9+;<6F2U8:B ZU[:>IQ7=?nGS\nG@%>D4d4d>D8o;&z ∗

∗

P (X = k) = P (S1 ∩ · · · ∩ Sk−1 ∩ Sk ) = P (S1 ) · · · P (Sk−1 )P (Sk ) = pq r−1

1 8f@%n\¦:8:6 ;V@72/8:S ~ 748:9 <; >=?; B B r%2/6F; X [:87>I . 9:. =G.F. 1 ¤8 @%6F;YZo[7> X Bm[:6 ;CET9dEF2U6^g/n&2/4dn\;<=?6TZo[7>A@%>eQ:9/=<9/4 D;?=<> p z7>\;C2U8 872/;?> X ,→ G(p)z Bm6¥~ >D; ∀k ∈ N , P (X = k) = pq .

X(Ω) = N , 13>DET9d@%n\¦ 876 ;C5:6F>&8 [:87>IEF2U6^@%>IQ7=<2/5:9/5760EF6F;?nd~ X P (X = k) = X pq = p = 1.

1−q m b=%ni = $ >=?; B B qbZRQ<j X ,→ G(p) w? T\ZRML` X KWONEcH!j ]~S HJH'`k^iX!M KNS ILH H!j ]~S H 4 KRM'QDKNS ILH s H0j 1 H!j V (X) = q .

E(X) = ∗

∗

+∞

k−1

+∞

k−1

k=1

k=1

p2

p

_-

I 9: " >-( #K

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r

p [X = k]

k ∈ [[r, +∞[[ k

X(Ω) = [[r, +∞[[

p r−1

r−1 pr k−r

(1 − p)k

r−1

k−1 r k P (X = k) = p q r−1

k−1

k−1 r−1

X

(r, p)

X

X ,→ P(r, p)

X(Ω) = [[r, +∞[[

∀k ∈ [[r, +∞[[, P (X = k) =

k−1 r k p q . r−1

+∞ X k=r

P (X = k) =

+∞ X k−1 k=r

r−1

pr q k =

pr = 1.

(1 − q)r

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r

r

X(Ω) = N X +r

X r

r X + r ,→ P(r, p)

X X ,→ J (r, p)

(r, p)

X

X

X

∀k ∈ N, P (X = k) =

X(Ω) = N

k+r−1 r k p q . k

1

r−1 k

r−1

q

k

pr

k

k

r

X ,→ J (r, p) E(X) =

N ∈N

X

rq p

V (X) =

p ∈]0, 1[ q = 1 − p N

rq . p2

Np

Nq

p

_- I 9: " , (*B DEPC ";=P 2B $,&-(

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n

X

0

n

n > Nq

n − Nq

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X(Ω) =

[X = k]

n

n

B F = 0g

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\ > ; X(Ω) ⊂ [[0, n]] P (X = k) =

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N n

Np k

Nq n−k

Np k

Np k

_- I

Nq n−k

Nq n−k N n

N −1

(*;= # >NOQ 2(*!+ 2( > & )B (*;< (CB # &) R#

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N k

N k

Np 1

Nq k−1

k

Nq k−1 N k

:

?n B F = g g

_- I I

(*;= # >NOQ 2(*!+ 2( > & r0 *;=(?# & *#

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"F /!( " & " L# /!"$#% C2U4 9R= 9R4d D;?=<>&B C2R; 9+;<6F2U8 ¥9/EF>&[7= B bc2/6 C876F )2/=<4d> xy>D=<872U[7EFE06 xy6F8:2/4d609/EF> nD2U4 nD;?=<60Zo[7> 9/B<SD9/E xy60872/4d6T9RE0>J8:nDgU9R;?60 /> ¬sQ'>D=<g/nD2U4dn\;?=<6TZU[:> Xy;m;<>D8o;?>e@7[ B?[:SDSD &B ;<6F= 9RgU>IB?9/8:B =?>&4d60B?> Xy;m;<>D8o;?>e@%[ B?[:S&S\ &B ;<6F= 9RgU>IB?9/8:B =?>&4d60B?> 2/6TB?B?2/8 k−r

r

r

r

∀k ∈ [[r, N q + r]], P (X = k) =

X(Ω) = [[r, N q + r]]

Np

r · k

r

N k

r ∈ N∗

Nq

k−r

.

λ ∈ R+ X ,→ P(λ)

λ

X

X

∀n ∈ N, P (X = n) = e−λ ·

X(Ω) = N

X ,→ P(λ)

X

E(X) = λ

1

r

V (X) = λ.

n ∈ N∗

U(n)

[[1, n]]

p ∈]0, 1[

B(1, p)

{0, 1}

(n, p)

B(n, p)

[[0, n]]

p

G(p)

N∗

(r, p)

P(r, p)

[[r, +∞[[

(r, p)

J (r, p)

N

(N, n, p)

H(N, n, p)

⊂ [[0, n]]

(N, p)

(r, N, p)

λ ∈ R+

P(λ)

λn . n!

1 n

P (X = k) =

P (X = 1) = p

n k n−k P (X = k) = p q k

P (X = k) = pq k−1

k − 1 r k−r P (X = k) = p q r−1

k+r−1 r k P (X = k) = p q k

Np Nq P (X = k) =

k

n−k

N n

[[1, N q + 1]]

Np P (X = k) = · k

[[r, N q + r]]

r P (X = k) = k

N

Np r

Nq k−1

N k

N k

P (X = k) = e−λ ·

λk k!

V (X)

n+1 2

n2 − 1 12

p

pq

np

npq

1 p r p rq p

q p2 rq p2 rq p N −n npq N −1

np

Nq k−r

E(X)

λ

λ

§ v

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B @ & # #1 PC( =&)! "&) (?>)( Q , QB* >N

m b=%ni= $ >=?;hp BD@:BC@ K !KREJQ<T<T\H O X 4 X0S H0E4H0SijC` [X = x] ∩ [Y = y] O X 4 X!SbH!EcH!S{jD` OWH!]Y OWH!]Y Q<S ILZRE.^iKNjkQD[!TdH'` s OWZNS{j T ]~S{QDZNS H'`'j X zKNT\H Ω w , E

(x,y)∈X(Ω)×Y (Ω)

H'`'j ]~S H !KNE Q<T<T\H

e ' bf?g p BD@iB ijE 8:>pBDO 9RgU6 ;A>D8 g/n&87nD= 9RE Q:9UBI@PO [78 Bm¬%Bm;? D4d> S\2U4dQ7EF>D;&zcS&9R=IS\>&=m; 9R6087>&BJQ:9/=m; BAQ >&[7 />&8o; 3\;<=?>A s6T@%>&B&.

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(x,y)∈X(Ω)×Y (Ω)

(X,Y )

X

(X,Y )

(X,Y )

Y

(X,Y )

k k∈K

(Xk )k∈K

, E 9: " "!*LK " !+ ( " #%;= B D" !- (*#

= >=%; p BC@:B\ 2U[7=39/EFE0nDgU>D= E0>&B3872/;<9+;<6F2U8:B&z/2/8 872R;<> P (X = x, Y = y) Q 2U[7= P ([X = x] ∩ [Y = y]) .

?n p B :g 8 = 748:9 <; >=?; p BC@:BTS b 9 T\ZNQ ILZRS G0ZNQ<S{j H @%[fS\2U[7Q7E0> (X, Y ) >&Bm;CE09d@%2U8787nD>I@%>IET9 )2/8:S\;?602/8 ~ X

p : X(Ω) × Y (Ω) −→ [0, 1]

; 9R8o; @%2U8787np@%>GBenD8s[74dn&=<9R;?602/8:B >D; @%> >\; ¦:87632/[ ] @%>p443D4d> Q'2/[7= \z/ET9J@72/8787n&>C@%>CET9AEF2U6:S\2U8+k`2/608o;?>N@%> >\; >&Bm;¥n&Zo[760 9/EF>&8o;?> (AET9J@72/8787n&>C@%>CET9IBm[:6 ;<>C@%2/[75:EF>&4 >&8o; z7215 ~ 608:@%>\ %n&> 7−→ P (X = x, Y = y).

(x, y)

(xi )i∈I

J

(yj )j∈J X Y

X(Ω)

Y (Ω) I

I =N

(pi,j )(i,j)∈I×J

∀(i, j) ∈ I × J, pi,j = P (X = xi , Y = yj ).

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i,• i∈I

L7.Cbc9 `!HLILZNSbOYH T\ZNQ EcKNMC YQ<S KRT\H @%[ SD2/[7Q:EF> >&Bm;NE09 E02/6P@%>AE09 +9R=<609/57E0>J9/EFnG9+;?2U6F=<> .% ;<9/8o;N@72/8787n @%> zoET9u@72/8787n&>V@%>JET9uB?>&S\2U8:@%>CE02/6'4 9R=<g/608:9RE0>C>GB`; n&Zo[760 +9RE0>D8o;?>'(uE09 [787>Yn&8s[74dnD= 9+;?602/8 @%2/878:nD>I@PO [787>eB?[76 ;<>/z7gUnD87n&=<9/EF>&4d>D8o;y8:2R;?n&> z7215 ~ ∀i ∈ I, pi,• = P (X = xi ).

(X, Y )

(yj )j∈J

Y

Y (Ω)

(p•,j )j∈J

∀j ∈ J, p•,j = P (Y = yj ).

P m b =%ni= $ >=?;hp BD@:BYZ 1. ∀i ∈ I, p = P p w p 2. ∀j ∈ J, p = e ' bf?g p BD@iBD]@ \ b^> Q'2/608o;#@79/8:B E09Y872R; 9+;?602/8 @%>&B EF2U60B 4 9R=<g/608:9RE0>&B^6F8:@760Zo[7> Zo[^O 2U8u9YB?2/4d4dn B?[7= ;?2/[7;?>&B E0>&B +9RE0>D[7= B Q'2UB<Bm6057E0>&BN@%>IE O 608:@%6TS\>eZo[^O 60EP=?>&4 Q:E09US\>/. e ' bf?g p BD@iBD@:@¨b^>&B^EF2U60BP4 9/=?gU6F8:9/EF>GB^B?2/8o;c@%n\;<>D=<4d6F87n&>&BPQ 9R=cET9 E02/6USD2/8+k`2U6F8o;<>/. o U ) #(+ # * " ## G &* bc9dE02/6#S\2U8+k`2/608o;?>e@%> (X, Y ) >&Bm;YBm2U[7 />&8o;CQ7EF[ BN !9/SD6FE0> ( @%nD;?>&=?4d6087>D=YZo[7>uEF>GBCE02/6TBC@%> X >\; Y . 1 8¤B?>uB?>D=?; @%2U8:SA@%[¤SD9RETS\[:E^@%>eS\>D;m;?>IE02/6^SD2/8+k`2U6F8o;<>JQ'2/[:=C@%n\;<>D=<4 6087>&=C>D8:B?[76F;?>AE0>&BNE02/6TBN@%> X >D; Y . m b=%ni = $ >=?;hp BD@:BC@ D MPZL^ M'QDX0j X4ILKNM KYI0j X0M'Qv`'jCQD 0]iH4O ]~S H T\ZRQ I ZR S G0ZRQ<SijlHl MPXLI0Q ^ MPZ !]iH KYOREJQv`0H 7%SbH$`']~Q<jlH (p ) I H0j J tuS{Qv` ZR] OYX0S ZRE4[0MPKY[0T\H!` H'`'j TdKcT\ZRQ O ]~S I ZR] ^ T\H (X, Y ) ` `'Q xYw ∀(i, j) ∈ I × J, p > 0 yzw P p = 1 w e ' bf?g p BD@iBD@ |c2U[%;?>GB^E0>&B#Bm2U4 4d>GB SD2/8:B?60@7nD=<nD>&B @79R8 BcS\>yQ:9/=<9/g/= 9RQ7©7> B?2/8o; (C;?>&=?4d>&B Q'2UB?6 ;<6 !B&.G{V2/8 S E0>D[7= SD2/8s />&=?gU>D8:SD> n&Zo[760 9/[%; (IE0>D[7=3S\2/8s U>D=<g/>D8 S\>N9R5:B?2/E0[7>Uz+>\; E O 2U=<@7=?>C@%>VBm2U4 4 9R;?602/8d6F4dQ'2/=?;?>NQ'>D[^.sXC608:B?6 z ;?2U[%;?>GB EF>GBCB?2/4d4d>&BNZo[^O 2/8f9 SD2/8:B?6T@%nD=<nD>GByB?2/8o;N5760>D8¤@%nD¦:8760>&B&. i,j

i,•

j∈J

•,j

i,j

i∈I

i,j i∈I,j∈J

i,j

i,j

(i,j)∈I×J

, E I 9: " # "!)> "! !-(* (*#

47 8:9 <; >=?; p BC@:BD@ rs2/6F; y ∈ Y (Ω) ;?>&EoZo[7> P (Y = y) 6= 0 . 1 8e@%nD¦:876F; ET9 4 KRM'QDKY[0T\H KRT\XLKNjlZNQ<MPH I ZRS ORQ<jCQDZRSi S H0T<T\HV@%> B<9/S © 9R8o; zR>\;¥2U8 872/;?> E09J +9R=<609/57E0> 9/EFnG9+;<2/60=?>yQ7=<>D8:9/8U; E0>&B +9RE0>D[:=<B Q'2UB<Bm6057E0>&B @%> X B<9/S X©:9/8o;yZo[7> YY ==yjzs>\;C@%2/8o; E09 E02/(X6 >&Bm|; Y@72/=8787j)n&>AQ 2U[7=y;?2/[7; x ∈ X(Ω) Q:9/= P (X = x | Y = y) .7bc9 T\ZRQ ILZNS ORQ<jCQDZRSiSbH!T<T\H4OWH X `0KYI :KRSij Y = j >&Bm; ET9dE02/6^@%> (X | Y = j) .

p B L; 8 n :; ?;

8 = Z m b=%ni = $ >=?;hp BD@:BC@Yp jlKNS{j OYZRSiSbXF]~S HFX0Si]~EcX0MPKNjkQDZNS (x ) OYH X(Ω) H0ju]~S H X!S{]~E4X0MPKNjkQDZNS (y ) OYH Y (Ω)s KRQ<S:`'Qu !]u ]~S Q<SbONQDILH j ∈ J s T\K4T\ZRQ ILZRS ONQ<jkQDZNS{S H0T<T\H OWH X `!KWI iKNS{j !]iH Y = y H!`'j OYZNS{S XLH ^iKNM i i∈I

j j∈J

j

∀i ∈ I, P (X = xi , Y = yj ) =

pi,j . p•,j

m b=%ni= $ >=?;hp BD@:BC@Y uH0jCjlHF`']~Q<jlH OWXDt SiQ<j [!QDH0S ]~SbHFT\ZRQ OYH ^ MPZW[LKY[0Q<T_Q<j X w 4 * , ( * " ( * ( "F&) !"F /!( , " / "$#% !& ( , pi,j p•,j

i∈I

,

"&- (*# >-( ' BC !)>)PC (*! >- !+ (R#

{J9R8:BNSD>IQ:9R= 9RgU=<9/Q7©7>/z >\; @%n&B?60g/87>&8U;C@%>&[% . 97. =&. @P.0.

748:9 <; >=?; p B BD@ 1 8 @%6F; Zo[7> H0j `0ZNS{j Q<SbOYXl^:H0S OWKNSijlH!` Bm63Q 2U[7=I;?2U[%; >D;uQ'2/[7=I;<2/[%; z7E0>&B nD UnD87>&4d>D8o;<B >\; Bm2U8o; 608:@%n&Q >&8:@79/8U; BD. m b=%ni = $ >=?;hp B B H'` ^ M Z ^iZ6`'Q<jCQDZRS:`F`']~Q 4 KNSijlH!`V`0ZRSij XL 0]~Q 4 KRT\H0SijlH'` xYw H!j `0ZNSij Q<S OWX ^iH0S OYKRSijlH'` yzw T\H'`$` 6`'j 0EcH'` ILZNE ^ T\H0jD` H0j `0ZNSij Q<S OWX ^iH0S OYKRSijC` |Ww H!` KRT [0MPH!` H0j `0ZNS{j Q<SbOYXl^:H0S OWKNSijlH!`!w m b=%ni = $ >=?;hp B B rq{Q H0j `0ZNSij Q<S OWX ^iH!SbOYKRSijlH'` s KRT\ZNML`J X

Y

X [X = x]

y ∈ Y (Ω)

x ∈ X(Ω)

Y [Y = y]

X

Y

([X = x])x∈X(Ω)

AX

σ

X

([Y = y])y∈Y (Ω)

AY

Y

∀(x, y) ∈ X(Ω) × Y (Ω), P (X = x, Y = y) = P (X = x)P (Y = y).

]~jkMPH!EcH0Sij ORQ<j s X0j KRSij OYZNS{S X OYH!`JX!S{]~E4X0MPKRjCQDZRS:` (x ) H!j (y ) i i∈I

j j∈J

∀(i, j) ∈ I × J, pi,j = pi,• · p•,j .

XC608:B?6 z%B?6 >\; B?2/8o;3608:@%n&Q >&8:@79/8U;<>&B) )>D; B?>D[:EF>&4 >&8o; @79/8:ByS\>ASD9/B lzoE0>&ByEF2U60B34 9R=<g/608:9RE0>&B3@%n\;<>D=<4d6F87>&8o;3ET9 E02/6^S\2U8+k`2/608o;?>U. X

Y

,

- ;< (R# >)( ' BC !)>)PC (*! >- !+ (R#

rs2U6 ; [787>A !9R4d60EFE0>IZo[7>DETS\2U8:Zo[7>I@%>I . 9:. =G. @^.F. 748:9 <; >=?; p B B 1 8 @%6F;IZo[7>dE0>&BJ +9R=<609/57E0>&BJ9RE0n&9+;<2/60=?>GB z Q'2/[7= z B?2/8o;A4u[%;<[7>DE0E0>D4d>D8o;I6F8 @%n _ Q'>D8:@79/8o;?>&BJB?6 Q'2/[7=Y;?2U[%;?>u !9R4d6FE0E0> ;?>&EFE0> Zo[7> Q 2U[7=Y;<2/[%; z z EF>GBYn& /nD8:>D4d>D8o;<B z%Q'2/[7= z7B?2/8o;N4u[%;<[7>DE0EF>&4d>D8o;C6F8 @%nDQ'>D8:@:9R8o;<B&. m b=%ni = $ >=?;hp B Bdp H'` ^ M Z ^iZ6`'Q<jCQDZRS:`F`']~Q 4 KNSijlH!`V`0ZRSij XL 0]~Q 4 KRT\H0SijlH'` xYw T\H'` 4 KNM'QDKW[!T\H!` s ^iZN]~M s `!ZRSij EJ]~jk]iH!T<TdH!EcH!S{j Q<SbOYXl^:H0S OWKNSijlH!` yzw T\H'`$` 6`'j 0EcH'` ILZNE ^ T\H0jD` ^iZN]~M s `0ZNSij E ]~jC]iH0T<T\H0E4H0SijuQ<SbOYXl^:H0S OWKNS{jD` s |Ww T\H'` KRT [0MPH'` s ^:ZR]~M s `0ZNS{j E ]~jC]iH0T<T\H0E4H0Sij Q<S OYXl^iH!SbOYKNS{j H!`0w (Xk )k∈K

Xk

(xk )k∈K

[Xk = xk ]

k∈K

Xk

σ

k∈K

([Xk = x])x∈Xk (Ω)

A Xk

k∈K

k∈K

k∈K k ∈ K xk ∈ Xk (Ω)

?n p B

2\ ,

I

8 =

:g

"! #6 2B & # OQ >)(?>)( Q , QBC N !)>)PC (*!)>) !" 2(*#

"PR!-(*;=(*!+ #

rs2U6 ; [787>A !9R4d60EFE0>I@%>I . 9:. =G. @P.F. 1 8 )2U=?4 9RE06TBm>CET9e8:2R;?602/8 @PO n& /n&87>D4d>D8o;y87>V@%n&Q >&8:@79/8U;NZo[7>V@^O [78 S\>&=m; 9R608p872/4 57=<>Y@%>&B3 +9R=<609/57E0>&B ~sB?2/6F; ]7@%60=?>eZo[7> 87>e@%n&Q >&8:@fZo[7>I@%>&BN +9/=?6T9R57E0>&B z%Q'2/[:= Bm60g/876F¦:>IZo[7> . rs[7Q:Q 2oBm2U8:BIZo[7>dEF>GBA 9/=?6T9R5:EF>GB z z^B?2/8o;I4u[%;<[7>DE0E0>D4d>D8o;I6F8 @%nDQ'>D8:@:9R8o;?>GBD.cbc>dEF>&4 4d> K7. L7. W Q >&= 4d>\; @7>d )2/=<4d9/EF6TB?>D=IE0> =<9/60B?2/878:>D4d>D8o;I6F8o;<[76 ;<6 YB?[760 +9R8o;p~cB?6 >\; Bm2U8o;u@%>D[% nD UnD87>&4 >&8o;<Be@%n&Q >&8:@79/8U;_ [7876TZo[7>D4d>&8U;Y@7>I 9/=?6T9R5:EF>GB B?9/8:B +9R=<609/57E0>JSD2/4d4u[78:> ( >\; z:9/EF2U=<B >\; Bm2U8o; 608:@%n&Q >&8:@79/8U; BD. F ' ' p B B\ rq ZNQ<j ]~S H !KREJQ<T<T\H OYH 4 w<KWwdM0w<OYw%E ]~jC]iH0T<T\H0E4H0Sij Q<S OWX ^iH!SbOYKRSijlH'` s H0j `0ZNQ<j H!j OYH!`$`0ZN]a`' H!Si`0H!Ec[!TdH'` OYH jlH!T ` 0]iH w T\ZRML`VT\H!` KNT [!MPH!` H0j `0ZNSij Q<S OWX ^iH0S OYKRSijlH'`0w 13> E0>D4d4d>HB?>¤g/nD8:nD= 9RE060B?>f576F>&8 B := ( [78 872U4u57=<>¤Q7E0[:Bp604dQ 2U=m; 9R8o; @%>HB?2/[ B _ >D8:B?>D4 57E0>&B @%>&[% ( @7>D[% @%6TB)k`2U6F8o; BN@%> . 1 8 2/5%;<6F>&8o;Y9RE02/= B3E O 6F8:@7nDQ'>D8:@79/8:S\>I4 [%;?[:>DE0EF>e@%>GB 9/EFgU D57=<>&B SD2/=<=?>GBmQ'2/8:@:9R8o;?>GBD. XC608:B?6 zR@:9R8:B#ET9VB?6 ;<[:9+;<6F2U8 @%nGS\=<6 ;<>3Q:EF[:B © 9R[%; Q'2/[7= E0>&B#nD UnD87>&4 >&8o;<B _ >D; z+60Es>\ %60Bm;?> @%>&[% B?2/[:B _ >&8:B?>D4u5:EF>GB @%6TB)k`2U6F8o; B >D; @%> ;?>&E0B3ZU[:> 87>C@%nDQ'>D8 @ ZU[:>Y@%>&B Q'2/[7= >\; 87>Y@7nDQ'>D8:@ Zo[7>C@%>GB Q'2/[7= .UXC[%;<=?>&4 >&8o; @%6 ;C~ >\; £ 9R6TB @PO 9RQ7=< &B#E0>NEF>&4 4d>Uz >\; Bm2U8o; 608:@%n&Q >&8:@79/8U;<>&B&z:@%2U8:S >D; B?2/8o;N6F8:@7nDQ'>D8:@79/8o;?>&B&.

4 # * (+* ( Z = g(X, Y ) {J9R8:BAS\>D;m;<> B?>&S\;?602/8^z'2/8 @%n&B?6FgU87>uQ 9R= >\; @7>D[% . 97. =&. @P.0z >D;JQ:9/= [787> )2/8 Sl;?602/8 @%> @79/8:B . 1 8f@%n\¦:8:6 ;CET9d +9R=<609/57EF>A9RE0n&9R;?2U6F=<> Q:9/=I~ 1 8f9f~ .

(Xk )k∈K

K0 ⊂ K

E

Xk E ∈ A(Xk )k∈K 0

k ∈ K0

Xk

Xk k ∈ K

E E

Xk

F F

E

F

K0

(Xk )k∈K K

K 0 ∩ K 00 = ∅

A(Xk )k∈K 0

σ

K 00

A(Xk )k∈K 00

K

K0 k ∈ K 00

σ E

K 00

F E Xk k ∈ K0 E ∈ A(Xk )k∈K 0 F ∈ A(Xk )k∈K 00 .

E F

K

A(Xk )k∈K 00

F

Xk

A(Xk )k∈K 0

X

Y g(X, Y )

R

X(Ω) × Y (Ω)

g

∀ω ∈ Ω, g(X, Y )(ω) = g(X(ω), Y (ω)).

g(X, Y )(Ω) ⊂ Img

B @ & # #1 PC(

, I

m b =%ni= $ >=?;hp B BC@ qbZRQ<j z ∈ Img w TdZNML` [g(X, Y ) = z] = m b =%ni= $ >=?;hp B B , I 9: " >-(

g(X, Y )

Ag(X,Y ) ⊂ A(X,Y )

[

[X = x] ∩ [Y = y].

(x,y)∈g −1 (z)

w

m b =%ni= $ >=?;hp B B KcT\ZRQ OYH g(X, Y ) H'`'j OYZRSiS XLH ^:KRM H0j ∀z ∈ Img, P (g(X, Y ) = z) = g(X, Y )(Ω) ⊂ Img ^ S ^:KRM'jCQDI0]~T_QDH0M s `'Q X H!j Y `0ZNS{j Q<S OWX ^iH0S OYKRSijlH'`J X

Z = g(X, Y )

∀z ∈ Img, P g(X, Y ) = z =

-/.

V-W n 4p B B

7L . h:.Yrs6

X

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

.

(x,y)∈g −1 (z)

z

P (X = x, Y = y).

(x,y)∈g −1 (z)

P P (X = x, Y = y) ∀z ∈ R, P (X + Y = z) = x+y=z P P P (X = x, Y = z).

P (X = z, Y = y) + ∀z ∈ X(Ω) ∪ Y (Ω) P (min(X, Y ) = z) = X ,→ G(p)

>\;

Y ,→ G(p)

z 9/EF2U=<B

y>z

X + Y ,→ P(2, p)

.

x>z

pB

g

'

, I I

#K PCB ! ( >)(

@

Z = g(X, Y )

F 8 = & p B B\p kj :XLZRM !EcH$OYH jkMPKNSi`< !H0M'j\ H!`C^iX0MPKNSbI H$OWH g(X, Y ) HP WQv`'j H `'Q%H0j?`0H0]~T\H0E4H0Sij%`'Q T\K `!X0M'QDH I0Q< K ^ M '`JILZNS 4 H0MC zHJKW['`0ZRT_]~E4H0Sij s H0j OWKNS:` ILH I KN` g(X, Y )

E(g(X, Y )) =

X

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

F 8 = & p B B\ <3 S `'] ^Y^iZ6`0H4QDI0Q 0]iH Ω H!`'j OYX0S ZNEc[0MPKY[0T\H s H0j !]iH TP H'`k^iKYILH ^ M ZY[LKY[0Q<T_Qv`0KY[0T\H I ZRS:`'QDOYX0MPX H!`Lj (Ω, P(Ω)) w?q ZN]a`$MPX'`0H0M 4 HJOYH ILZNS 4 H0MC WH0S ILHJKY[!`!ZRT_]iHc XL 0]~Q 4 KRT\H!S{j cT HP WQv`'jlH!SbI HJOYH T H'`k^iX!M KNS ILH ZNS K (x,y)∈X(Ω)×Y (Ω)

E(g(X, Y )) =

X

g(X(ω), Y (ω))P ({ω}).

ω∈Ω

, I Q_ (*;= (*# #1 P*B !) ( >-( X + Y >-( XY F 8 = & p B BTo S qbZNQ<j X H!j Y OYH0]Y 4 w<KYwdM0w<OYw KWONEcH!jkjlKNSij)]~SbHVH!`k^:X0MPKRS ILH wb T\ZRML` T H!`C^iX0MPKNSbI HFOWH

X +Y H YQv`'jlH s H0j E(X + Y ) = E(X) + E(Y ) w V= = p B BYX T_Q<S XLKNM'Q<jlXFOYH T H!`k^:X0MPKRS ILHl qbZNQ<j X H0j Y OWH!]Y 4 w<KYwdM0w<OYw s H0j λ H0j µ OYH0]Y MPXLH!T_`!w{ T\ZNML` w E(λX + µY ) = λE(X) + µE(Y ) e ' bf?g p B B[Z Xy;?;?>&8U;<6F2U8^z:@%>A4 9R8:6F &=?>Ag/n&87nD= 9RE0>/z E(XY ) 6= E(X)E(Y ) .

F 8 = & p B BD6@ \Aq{Q X H!j Y `0ZNSij Q<S OYXl^iH!SbOYKNS{j H!` H!j$KYONEcH0jCjlH!S{j.]~S H H'`k^iX!MPKRS ILH s KRT\ZNML` XY KWONEcH!j ]~S HJH'`k^iX!M KNS ILH s H0j E(XY ) = E(X)E(Y ) w

, I B !) (

B ! (?>:O &)!)( #K ";<;=(

748:9 <; >=?; p B BD@:@¨rs2/[:B =<n&B?>D=< />A@PO >\ %6TB`;<>D8:SD>/z%2/8f@%nD¦:876F;YE09 LI Z 4 KNM'QDKRS ILH @%> X >D; Y Q:9R=I~

cov(X, Y ) = E (X − E(X))(Y − E(Y )) .

e ' bf?g p B BD@ rs6 cov(X, Y ) > 0 z X − E(X) >\; Y − E(Y ) 2U8U; ;?>D8 @79R8:SD>&(F3D;?=<>V@%>V443D4d>VB?6FgU87>/z @%2U8:S X >D; Y 2U8o;N;?>D8 @79R8:SD> ( B?>eB?6 ;<[7>D=C@%[f443D4d>IS /;?ne@%>AE0>D[7=Y>&B?Q n&=<9/8:S\>U.sXY6F8 Bm6¥~ % cov(X, Y ) > 0 B?6FgU876 ¦ >AZo[7> X >\; Y 2/8o;N;<>D8:@79/8:S\> ( 3\;<=?>e@7>J +9/=?6T9+;<6F2U8 Q:9/=<9/EFE0 DE0>¥] % cov(X, Y ) < 0 B?6FgU876 ¦ >AZo[7> X >\; Y 2/8o;N;<>D8:@79/8:S\> ( 3\;<=?>e@7>J +9/=?6T9+;<6F2U8 2UQ7Q 2oBmn&>¥] m b=% n 8 8 4p B BD@ xYw cov(X, Y ) = E(XY ) − E(X)E(Y ) yzw ^ S4^iKNM'jkQDI!]~T_QDH0M s cov(X, Y ) HP WQv`'jlHF`'Q X H0j Y KWONEcH!jkjlH!S{j OYH!` H'`k^iX!M KNS ILH!`0w |Ww cov H!`'j ` NEcX!jkM'QD 0]iHJ cov(X, Y ) = cov(Y, X) w }{w cov H!`'j [0Q<T_Q<SbX KRQ<MPH w% ]~jkMPH!EcH0Sij ONQ<j s ^iZN]~MFjlZN]~jlH'` 4 w<KYwdM0w<OYw X s X s Y H0j Y lK' o T_Q<S XLKRM'Q<j X.^iKNMFMPKL^Y^iZNM'j cT\K$^ MPH0E Q !MPH 4 KNM'QDKY[0T\H

0

0

∀(λ, µ) ∈ R2 , cov(λX + µX 0 , Y ) = λcov(X, Y ) + µcov(X 0 , Y );

l[ T_Q<S XLKRM'Q<j X.^iKNMFMPKL^Y^iZNM'j cT\Kc`0H ILZRS OYH 4 KNM'QDKY[0T\H

∀(λ, µ) ∈ R2 , cov(X, λY + µY 0 ) = λcov(X, Y ) + µcov(X, Y 0 );

Yw q{Q X H!j Y `0ZNS{j Q<SbOYXl^:H0S OWKNSijlH!` s cov(X, Y ) = 0 w

?n p B :g 8 = m b=%ni= $ >=?;hp B BC@ q Q X H!j Y KYORE4H0jkj H0Sij ~] S H 4 KNM'QDKRS ILH s KNTdZNML` X + Y X zKNT\H0E4H0Sij s H0j

V (X + Y ) = V (X) + V (Y ) + 2 · cov(X, Y ).

F 8 = & p B BD@Wprq{Q X H0j Y `0ZNS{j Q<S OWX ^iH!SbOYKRSijlH'` s H0j KYONEcH0jCjlH!S{ju]~S H 4 KNM'QDKRS ILH s KNT\ZRML` X + Y KN]a`L`'Q s H0j V (X + Y ) = V (X) + V (Y ) w m b=%ni = $ >=?;hp B BC@Y rq ZNQ<j X , . . . , X OYH!` 4 w<KWwdM0w<OYw K YKNS{jF]~S EcZNEcH0Sij O ZRMPONM H 2 X + · · · + X KYORE4H0j ]~S H 4 KNM'QDKRS ILH4H!j

1

1

n

V (X1 + · · · + Xn ) = V (X1 ) + · · · + V (Xn ) + 2 ·

X

n

cov(Xi , Xj ).

F 8 = & p B BD@ S qbZR]a`VT\H!`FE4G0EcH'` ^iZNj !`0H'` s `LQ X , . . . , X `0ZNS{jC@%>D[7 d( @%>D[7 Q<SbOYXl^:H0S OWKNS{j H!` 16i<j6n

1

n

V (X1 + · · · + Xn ) = V (X1 ) + · · · + V (Xn ).

748:9 <; >=?; p B BD@6X b^> ILZ H I0QDH0Sij.OYH ILZNM'MPX0T\KNjkQDZNS T_Q<SbX KRQ<MPH @%> X >D; Y >GB`;e~ ρ(X, Y ) = cov(X, Y ) . σ(X)σ(Y ) 4 J# "F 0/! L# * ( ,-/! 0, /!" ,?,% ( , 9R= Bm;<9/5760EF6F;?nUzU2U8 >&8o;?>D8 @ E09IS ©:2UB?>CBm[760 +9R8o;?>u~UB?6 X >\; Y Bm[:6 ; [:8pS\>D=?;<9/6F8d;`¬sQ'>Y@%>CE02/6 zU9/EF2U=<B X + Y 9R[ B?B?6 .

, Q_) `' 2 @ 2P >-(R# + # @ !- ";< (*#

m b=%ni= $ >=?;hp B BC@ qbZRQ<j (m, n) ∈ (N ) H!j p ∈]0, 1[ w$q{Q X ,→ B(n, p) H0j Y ,→ B(m, p) `0ZNSij Q<S OWX ^iH0S OYKRSijlH'` s KRT\ZNML` X + Y ,→ B(n + m, p) w H'`'jcT\K `0ZNE EcH OYH n 4 KNM'QDKW[!TdH'` OYH H!M'SbZN]~T<T_Q OWH ^:KRMPKNE !jkMPH p ^E Sh]~jC^i]iKNH0T<M'T\jkH0QDE4I!]~H0T_SiQDH0j M Q<S `LQOYXlX^iH!Sb,→OYKNS{B(n, p) s X j H!`0w ∗ 2

, Q_) `' 2 @ 2P >-(R# + #%>-( #

m b=%ni= $ >=?;hp B B rqbZRQ<j (r, s) ∈ (N ) s H!j p ∈]0, 1[ w q{Q X ,→ P(r, p) H!j Y ,→ P(s, p) `0ZRSij Q<SbOYX0 ^iH!SbOYKNS{j H!` s KNT\ZRML` X + Y ,→ P(r + s, p) w H'`'jFT\K `0ZREJEcH OYH r 4 KNM'QDKW[!T\H!` `']~Q 4 KRSij ]~SbH T\ZNQ. zX ZRE4X0jkM'QD !]iH OWH P(r, p) s X ^^iKNS MPKR^iE KNM' 0jkjCQDM I!H ]~T QDH!M EJ`']~Q jk]iXH!T<Td,→ H!EcH!S{j Q<SbOYXl^:H0S OWKNSijlH!`!w ps ∗ 2

, Q_) I `' 2 @ 2P >-(R# + # @ !- ";< (*#%!-P*DE 2 "(*#

m b=%ni= $ >=?;hp B B rqbZRQ<j (r, p) ∈ (N ) H0j p ∈]0, 1[ wbq Q X ,→ J (r, p) H!j Y ,→ J (s, p) `0ZNS{j Q<S OWX ^iH!S{ OYKRSijlH'` s KRT\ZNML` X + Y ,→ J (r + s, p) w ∗ 2

, Q_) Q_ `' 2 @ 2P >-(R# + #%>-( " #K#K "!

m b =%ni= $ >=?;hp B B bq ZRQ<j (λ, µ) ∈ (R ) w q{Q X ,→ P(λ) H0j Y ,→ P(µ) `!ZRSij Q<SbOYXl^:H0S OWKNSijlH!` s KRT\ZNML` w X + Y ,→ P(λ + µ) +

2

t N

( & ( (+( & "F !/! '#

, E

P ! "!

47 8:9 <; >=?; BC@:BD@ r%2/6F; (X ) [787> B?[76 ;<> @7> . 97. =&. @P.PB?[7= (Ω, T , P ) zP>D; X [787> 9R[7;?=<> @%6F;YZo[7> X ILZNS 4 H0MC zHJH0Sc^bMPZY[LKY[0Q<T_Q<jlX )2/[ ILZNS 4 H0MC zH `'jlZ I :KN`'jCQD 0]iH!EcH0Sij 3 />&=<B X B?6 ~ n n∈N

v.a.r.d.

.18

n

∀ε > 0,

lim P |Xn − X| 6 ε = 0.

n→+∞

, E JK!-PCD" 2P >)( % (*!) -; P 0 (C@ ( F 8 = & BC@:B !S X zKNT_Q<jlXcOYH KNM WZ 4 q ZNQ<j X ]~S H 4 w<KYwdM0w<OYw `']~M KYORE4H0jkj KRSij ]~SbH H!`k^:X0MPKNSbILH 872/8 8s[7EFE0> w TdZNML`

(Ω, T , P )

E(X)

jlH!T<T\Hc 0]iH X(Ω) ⊂ R s +

1 P X > λE(X) 6 . λ

F 8 = & BC@:B !S X zKNT_Q<jlX OYH QDH0S K NE4X0 I iH [ YI :H 4 qbZRQ<j Y ]~S H 4 w<KWwdM0w<OYwVKYORE4H0jCjlKNS{j ]~SbH H!`k^:X0 MPKRS ILH m H!j ]~S H 4 KRM'QDKNSbI H σ w? T\ZNML`

2

σ2 ∀ε > 0, P |Y − m| > ε 6 2 . ε

, E I 9: " @ (?>)(*# DEB ! >)# !- "; @ B (*#

F 8 = & BC@:B ZNQ !KRQD[!T\H OYH!` RMPKRS ON` S ZRE4[0MPH!` q ZNQ<j (X ) ]~S H `']~Q<jlH OYH 4 w<KWwdM0w<OYw$E ]~jC]iH0T_ T\H0E4H0Sij Q<S OWX ^iH0S OYKRSijlH'` `L]~Q 4 KNS{jJ]~SbH EcG0E4H T\ZRQ s K YKRSij4]~S H H!`C^iX0MPKNSbI H m H0jJ]~SbH 4 KNM'QDKRS ILH σ w$qbZNQ<j OYXCtuSiQDH ^iKNM (Z ) X +···+X . ∀n ∈ N, Z = T\ZRML` (Z ) ILZNS 4 H0MC zHJH0S4^ M ZY[LKY[0Q<T_Q<jlX 4 H0ML`FT\K 4 KNM'QDKW[!T\HcnILH0M'j KRQ<S H X WKRT\H m w T_]a` ^ MPXLI0Qv`0X!EcH!S{j

n n∈N∗

2

n n∈N∗

1

n

n

n n∈N∗

σ2 ∀ε > 0, ∀n ∈ N∗ , P |Zn − m| > ε 6 2 . nε

F 8 = & BC@:B\p iX ZRM 0E4H O ZRM OYH H0M'S ZR]~T<T_Q qbZNQ<j (X ) ]~S H `']~Q<jlH OYH 4 KRM'QDKY[0T\H!` EJ]~jk]iH!T<TdH! EcH!S{j Q<S OWX ^iH0S OYKRSijlH'` s `']~Q 4 KNS{j jlZN]~jlH'`$]~S H T\ZNQ OWH H0M'S ZR]~T<T_Q OYH ^iKNMPKRE 0jCM H p w%qbZNQ<j

n n∈N∗

∀n ∈ N∗ , Zn =

X1 + · · · + X n . n

T\ZRML` (Z )

n n∈N∗

?n B V=%; :; ?nin = W >=?;

ILZNS 4 H0MC zHJH0S4^ M ZY[LKY[0Q<T_Q<jlX 4 H0ML`FT\K 4 w<KYwdM0w I H0M'jlKNQ<S H X WKRT\H p w T_]a` ^ MPXLI0Qv`0X!EcH!S{j 1 .

4nε2

132/8 S\E0[:Bm602/8 ~ b 9 )=?nGZo[7>D8:SD>HBm;<9+;<60Bm;?6TZo[7> @PO [78 nD UnD87>&4d>D8o; ;?>&8:@ />&=<BdET9 Q7=<2/5:9/5760EF6F;?n¤@PO 9RQ7Q 9R=<6 ;<6F2U8 @7> E O n& /nD8:>D4d>D8o;&.

∀ε > 0, ∀n ∈ N∗ , P |Zn − p| > ε 6

, E Q_ !A( (*;= ( # # $,&-( m b= ' BD@iB\ = 3 S jkQ<MPH4x $ !ZRQv` ^ Q<T\HcZR] !KYILHcK 4 HLI ]~S H.^bQ ILHcOWX'`0XL !]~Q<T QD[!MPXLH OWZNS{j TdKF^bMPZY[LKW[!Q<T_Q<jlX O ZW[!jlH0SijkQDZNS OWHJ Q<TdH H!`'j p w 3 S ZY[0jCQDH0Sij$ !ZNQv` Q<T\H w ZNS{S H0MJ]~S Q<S{j H0M 4 KRT<T\H I j H0Tu !]iH T\KJ^bMPZY[LKY[0Q<T_Q<jlX 0]iH p ∈ I `0ZNQ<j `'] ^:X0M'QDH0]~MPH Y w Ww

rs2U6 ; Q 2U[7=y;?2U[%; z E09 +9R=<609/57E0>V@%>Axy>D=<872/[7E0E06 @7>VQ:9/=<9/4d \;?=<> S\2U=?=<>&B?Q'2/8:@79/8o;39R[ =?nGBm[7EF;<9R; @%>IET9 _ D4d>A>\ %Q'nD=<6F>&8:S\>Uz%>\;YB?2/6F; . rs2U6 ; .¥{uO 9RQ7=< &BdE0> ;?©7n&2/=< D4d>¤@PO 2/= @%> xy>D=<872/[:EFE06 z . ijECB?[ d; @%2U8:SfZU[:> z%Bm2U6 ; .%XY6F8:B?6 z 1 4nε2

i ε > 0 6 0.1

i ∈ [[1, 1000]] Xi Z1000 =

p + · · · + X1 000) P (|Zn − p| > ε 6 P (0.52 6 p 6 0.62) > 1 − 0.1 = 0.9.

ε 6 0.05

1 1000 (X1

1 4nε2

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P ! "!

n n∈N

n

n→+∞

n→+∞

n

n n∈N

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n

n n∈N

∀x ∈ X(Ω),

lim P (Xn = x) = P (X = x).

,

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n→+∞

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∗

N

N

,

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n n∈N

n n∈N∗

n

n→+∞

n

∗ +

λ n

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n

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n > 30 p 6 0, 1

np 6 10

n

n n∈N∗

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┬и N ┬и

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X:Ω→R

Ω

X(Ω)

{x1 , . . . , xp }

X

X(Ω)

R

x1 < · · · < x p

xi ∈ X(Ω) (a, b) ∈ R2 xi

ni

Ω

ni =

xi

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xi

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−1

xi

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j=1 −1

(]a, b])|

]a, b]

I

|X −1 (I)|

b

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X(Ω)

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xi ∈ X(Ω) (a, b) ∈ R2 n = |Ω| xi i i P P xi fj = n1 ni

fi

I

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j=1 1 n

xi

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j=1

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xi

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C =R

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(C B P*#K(*!+ 2 2 "!)# DEB $,&-(*#

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xi

xi

ni

xi

x∈R

X P FX (x) = i|xi 6x ni

FX

[0, 1]

R

−∞

+∞

]ai , ai+1 ]

[ai , ai+1 ]

ni

ni ai+1 −ai

ai

X

?n S B ' 8 & :;

>f%g

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x

x

]ai , ai+1 ]

∀i ∈ [[1, p]], ∀x ∈]ai , ai+1 ], f (x) =

i−1 X

fj +

x − ai · fi . ai+1 − ai

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0

1

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V-W n SbB Bdp z z .

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{(xi , ni ), i ∈ [[1, p]]}

x x=

g g=

1 n

p P

n i xi =

s i=1 p Q n

xi n i =

h

=

q q=

k

p P

1 n

ni xi

s i=1 p 1 P n

i=1

=

p P

i=1

ni x2i =

{(xi , ni ), i ∈ [[1, p]]} p P mk = fi xki i=1

m0 = 1 m 1 = x m 2 = q 2

f i xi

i=1 p Q

i=1

i=1

1 h

p P

xfi i

fi xi

s

p P

i=1

fi x2i

k∈N

S B m

0n >=%;

Z

'

P*>) !)(

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{(xi , ni ), i ∈ [[1, p]]}

Ω = {ω1 , . . . , ωn }

Ω

X(ω1 ) 6 · · · 6 X(ωn ) Ω

Ω

{(xi , ni ), i ∈ [[1, p]]}

m

m = X(ω n+1 ) n = |Ω| 2 m = X(ω n2 ) + X(ω n2 +1 )

n

m

C

{(Ci , ni ), i ∈ [[1, p]]}

FX

[0, 1]

R

1 2

FX

−1 FX

{(Ci , ni ), i ∈ [[1, p]]} FX (m) = 21

m

m

1 2

1 2

Ci =]ai , ai+1 ]

gi

ai

gi+1

ai+1

1 − gi m − ai = 2 . ai+1 − ai gi+1 − gi

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"F&)" #?&)( , * ( * !,C ( & ,% !

I

1 2

n 2

2B 0 , (

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E=

1 n

p P

i=1

ni |xi − x| =

p P

i=1

fi |xi − x|

.

1 2

?n S B ' 8 & :;

\

>f%g

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V-W n SbB B z z .

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σ=

V = σ2

µk =

p P

s

1 n

p P

ni (xi − x) =

i=1

s

p P

fi (xi − x).

i=1

k k∈N

fi (xi − x)k

i=1

µ0 = 1 µ 1 = 0 µ 2 = V

I

σ 2 = q 2 − x2

&) !+ (*#

µ2 = m2 − m21

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m>x

{(Ci, ni ), i ∈ [[1, p]]

q3 [q1 , q3 ]

q1 FX (q3 ) =

FX (q1 ) =

3 4

q3 − q 1

k ∈ [[1, 9]]

k ∈ [[1, 99]] k ∈ [[1, 999]]

k k

k

milk

k FX (dk ) = 10 k ck FX (ck ) = 100 k FX (milk ) = 1000

dk

1 4