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Card(E) = nz72/[

|E| = n.

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|∅| = 0.yzwqQA ⊂ B s KNT\ZRML` |A| 6 |B|

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∣ = |B| − |A|w

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(i, j) ∈ [[1, n]]2jH0T0]iH

i 6= j s Ai∩Aj = ∅ s KRT\ZNML` |A1∪· · ·∪An| = |A1|+ · · ·+ |An|w

YwKN`WX0SX0MPKNT|A ∪ B| = |A| + |B| − |A ∩ B|

wRw

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n∑

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16i1<···<ik6n

|Ai1 ∩ · · · ∩Aik| =

∑

I⊂[[1,n]]

(−1)|I|−1

∣

∣

∣

∣

∣

⋂

i∈I

Ai

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ayQ9R=<46PE0>&B

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b^>ILZ H I0QDH!SjE]~T_jkQ<SZREJQDKRT ( n

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@%>eB?2/[:B _ >D8:B?>D457EF>GB@%>[[1, n]]

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(x1 + · · · + xk)n =∑

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(

n

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[[1, n]]U>D=BAEO >D8Bm>&4u57E0>p@%>&BuQ:9R=?;?60>&B (

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p− 1n&EFn&4d>D8o;<BN@%>

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p− 1

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i∈I

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A ∩ B.

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;<>DE0EF>eZo[7>d~-U.

Ω ∈ A]

L7.∀A ∈ P(Ω), A ∈ A =⇒ A ∈ A

!Bm;<9/576FE06F;?nIQ:9/=SD2/4dQ7E0nD4d>D8o;9+;?602/8 ]h:.

∀(A,B) ∈ P(Ω)2, (A,B) ∈ A2 =⇒ A ∪ B ∈ AB`;9R5760E06;<nIQ:9R=[787602/8 ¦:8:6F>l.

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ww

∅ ∈ A Yw∀(A,B) ∈ P(Ω), (A,B) ∈ A2 =⇒ A ∩ B ∈ A Rw$ZN]~MFjlZN]~jlHu!KREJQ<T<T\Hd¦:8760>

Ai, i ∈ IO? X!T\X0EcH!SjDÒYH

A s ZRS K ⋃i∈I

Ai ∈ AH0j ⋂

i∈I

Ai ∈ Aw

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C = Ai, i ∈ I[787>!9/460E0EF>Zo[7>&E0SD2/8:Zo[7>3@PO n&/n&87>D4d>D8o;B^@79R8B

P(Ω).b¥O 9RE0g/&57=<>

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L7.NQ2U[7=;?2U[%;?>I9RE0g/&57=<>A′ z C ⊂ A′ =⇒ A ⊂ A′ .

m b=%ni= $ >=?; B Bd uH!jkjlHJKRT [0MPH4HPWQv`'jH w

B 8%8 ; 'i; @:@e 'bf?g B BTS bO 9/EFgUD57=<>C>&87g/>&8:@%=<nD>CQ:9/=3[:87>Y!9/4d6FE0EF>AS\2U8o;?60>D8o;;<2/[:B3EF>GBS\2/4dQ7E0nD4d>&8U;9R60=?>GB@%>GBn&/n _87>&4>&8o;<BI@%> S\>D;m;<>!9/4d6FE0EF>Uz^9R608:B?6Zo[7>;<2/[%;<>&BAE0>&BA[787602/8BV>D;I6F8o;?>&=<B?>&S\;?602/8:BV¦:8:6F>GBI@%>&BJnDUnD87>&4d>D8o;<BI@%>ET9!9R4d60EFE0>I>\;Y@7>JE0>D[:=<BNSD2/4dQ7E0nD4d>D8o;<9/6F=<>&B&.m b=%ni= $ >=?; B BYXrqbZRQ<j

Ω]~So]~SiQ 4 H0ML`Ft SiQDw KNT [!MPH H0SzzH0SONMX4îKNM TP H!Si`!H0Ec[!T\H OWH'` X 4 X0SH0EcH!SjD`X0T\X!EcH0SijlKNQ<MPH!`

ω, ω ∈ ΩH'`'j

P(Ω)w

,σ 0 DC@B(*# "& 2B @ &)#%>NO P"P*!)(*;=(*!+2#

bc9f8:2R;?602/8 @PO 9RE0g/&57=<>>GB`;6F8:B?[pB<9R8o;?> @79/8:Be@%>p8:2/4u5:=?>&[%SD9UBDz SD9R=I>&EFE0>p872U[:BI>&4Q3&S©:>p@%> S\2U8:B?60@%n&=?>&=@%>GB[787602/8:BN608%¦:8:6F>GBN@PO n&/n&87>D4d>D8o;BD.V-W n B BYZ 1 8d;?60=?>(AQ:6FE0>C2/[!9US\>C[78:>C6F87¦:876F;?nY@%>)2U60B&.

A>&Bm;¥EO nDUnD87>&4>&8o;Y~R2/8 2U5%;?60>D8o;9/[4d2U6F8:B

[787>u)2/6TBYQ:6FE0>/.'2/[7=Y;?2/[7;n ∈ N∗ z2U8 @%n\¦:8:6;AEO nDUnD87>&4d>D8o; An

~2U8¤;?60=?>uQ2U[7=VET9pQ:=?>&460D=<>I)2U60BYQ76FE0>9/[n _ 6F&4>C;?60=<9/g/>/.UXC608:Bm6 z A =

+∞⋃

n=1An

. 1 8p9I@%2U8:S;?2U[%;3608o;?n&=P3D;(e@%nD¦:8760=[78 B?¬%B`;<D4d>Y@PO nDUnD87>&4d>D8o;<B¥Bm;<9/57E0>Q:9/=N[787602/8:B608%¦:8760>&B&.'13>&E094d2R;<6FU>JET9d@%n\¦876;<6F2U8¤Bm[760+9R8o;?>d~748:9 <; >=?; B BD@]\ rs2/6F;

Ω[78 [78:6FU>D=BD.C8:>

σKRT [0MPH

T@PO n&/n&87>D4d>D8o;ByB?[7=

Ω!9/[:B?B?6P9RQ7Q'>DE0nD>jkM'QD[!]

>&Bm;C[78fBm2U[:B _ >&8:Bm>&4u57E0> T @7>P(Ω)

;?>&EFE0>eZo[7>d~-U.

Ω ∈ A]

L7.∀A ∈ P(Ω), A ∈ A =⇒ A ∈ A

]h:.N#2U[7=;?2U[%;!9R4d60EFE0>e@%nD8:2/4u5:=<9/57EF>

(An)n∈N

@Ô n&EFn&4>&8o;<BC@%>Tz +∞⋃

n=0An ∈ T

.

m b=%n 8 8 B BD@:@ q ZNQ<jT]~SHjCM'QD[0]O X 4 X!SbH!EcH!SjD`$`']~M Ω

ww

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

k`LjlKY[0Q<T_Q<jlXîKNM]~SiQDZNSJtuSiQDH Rw$ZN]~MFjlZN]~ji!KREJQ<T<TdHcOYX0SZNEc[0MPKY[0T\H

(An)n∈N

O X0T\X!EcH0SijCÒWHT s +∞

⋂

n=0An ∈ T

w w

∀(A,B) ∈ P(Ω), (A,B) ∈ A2 =⇒ A ∩ B ∈ Ak`LjlKY[0Q<T_Q<jlXîKNMQ<SijlH!ML`0HLI!jkQDZNSJtuSiQDH

V== B BC@ H'`σKNT [!MH'`F`0ZNSjOWH'`KNT [!MH'`0w

748:9 <; >=?; B BD@ C8H!`CîKYILH?^bMPZY[LKY[0Q<T_Qv`0KY[0T\HC>&Bm;^[78uS\2U[7Q7E0>(Ω, T )

zG265Ω>&Bm;c[78I>D8Bm>&4u57E0>¦:876 z&9/Q7Q>&EFn

[78760/>&=<B&z%>\;T[787>

σ _ 9RE0g/&57=<>J@PO nDUnD87>&4d>D8o;<BNB?[7= Ω.:bc>&BnDE0nD4d>&8U;BC@%>

TBm2U8o;N9/Q7Q'>DE0n&BX 4 X0SH0EcH!SjD`l.

N>&4d9/=<Zo[7>JZo[7>ISD>\;m;<>A872R;<6F2U8 SD2 8:S\6T@%>A9/>&SJS\>&EFE0>I@%n\¦876F>I@79/8:ByEF>AQ:9/=<9/g/=9RQ7©:>VB?[76F+9/8U;Gz7@79R8B3E0>ISD9UB3215Ω>&Bm;y¦876.

F '' B BD@ qbZNQ<jTH0j

T ′ OYH0]Y σ KNT [!MPH!``L]~M Ωw)T\ZNML`

T ∩ T ′ H'`'j.]~SbHcKNT [0MPHO X 4 X0SH0E4H0SijC`0wuH!TdKc`0HVWX0SX0MPKNT Qv`!H ]~SHFQ<SjH0ML`0HLI!jkQDZNS !]iH0T\ILZNSb!]iH s tuSQDH s OWX!SbZNEc[!MPKW[!TdHZR] Q<St SiQDHFSbZNS OYX0SZNEc[0MPKY[0T\H w748:9 <; >=?; B BD@Yp rs2/6F;

C = Ai, i ∈ I[:87>p!9/460E0EF> ZU[:>DETS\2/8ZU[:> @PO nD/n&87>D4d>&8U;B@:9R8:B

P(Ω).bc9

σ _ 9RE0g/D5:=?> T )8:2R;?n&>VQ9R=?)2/6TBT (C)

3>D87gU>D8:@%=<nD>YQ:9R=ET9I!9R4d60EFE0>C>GB`;ET9eQ7E0[:BQ'>\;?6F;?>

σ _ 9RE0g/D5:=?>JS\2/8o;<>D8:9/8U;C.%¥E0EF>IUnD=<6¦:>I@%2U8:S~

@ ?n B V n nb= 8

-U.C ⊂ T

]L7.NQ2U[7=;?2U[%;?>

σ _ 9RE0g/D5:=?> T ′ z C ⊂ T ′ =⇒ T ⊂ T ′ .m b=%ni= $ >=?; B BC@Y uH0jCjlH

σKRT [0MPH4HPWQv`'jlHw

e 'bf?g B BD@KS bc9σ _ 9RE0g/&57=<>C>D87gU>D8:@7=?n&>YQ:9/=[787>V!9R4d60EFE0>AS\2/8o;<6F>&8U;3;?2U[:ByEF>GB3SD2/4dQ7E0nD4d>D8o;<9/6F=<>&B3@%>&BnDUnD87>&4d>D8o;<B¥@7>VS\>D;m;<>N!9R4d60EFE0>/zs9R608:B?6Zo[7>N;<2/[%;<>&B¥E0>&B[787602/8:B>\;608o;?>&=<B?>&S\;?602/8:B¦876F>GB2U[p@%nD8:2/4u5:=<9/57EF>GB@%>&B

nDUnD87>&4d>D8o;<BN@%>IET9!9R4d60EFE0>A>\;Y@%>IE0>D[7=BNS\2U4dQ7EFn&4d>D8o;<9/6F=<>&B&.m b=%ni= $ >=?; B BC@6Xrq ZNQ<j

C]~S`6`'j !EcHIZRE.^bT\H0jYtuSQZN]JOWX!SbZNEc[!MKY[0T\H.OYH

ΩwWT\ZRMLùT\KVjCM'QD[0]JH0SzzH!SbONMPXLH

îKNMCKYORE4H0j TdKOYH!`0I!M'Q ^bjCQDZRS H!^T_QDI0Q<jH`']~Q 4 KNSijlH

T (C) =

⋃

A∈D

A∣

∣

∣ D ⊂ C

.

m b=%ni= $ >=?; B BC@6Zrq ZNQ<jΩ]~S ]~SiQ 4 H0ML` OWX!SbZNEc[!MKY[0T\Hw K σ KNT [!MPHH0SzzH!SbONMPXîKNMcT H0S:`0H0E4[0T\HOYH!`X 4 X0SH0E4H0SijC`X!T\X0EcH!SjKRQ<MPH'` ω, ω ∈ Ω

H!`'jP(Ω)

we 'bf?g B B 2\ b^>yQ7EF[BBm2U[7/>&8U;GzEF2U=<B<Zo[7>

Ω>&Bm;¦:876%2/[d@%n&872/457=<9/57E0>/zEO >&B?Q:9US\>yQ7=?2U5:9R5:6FE060B<9R5:EF>yZU[:>

E O 2U8 SD2/8:B?60@7nD=<>D=9u>GB`;(Ω,P(Ω))

.748:9 <; >=?; B B @ bc9jCM'QD[0][LZNMX!T_QDH0SiSbHYB?[7=

R>&Bm;¥E09J;<=?6057[d>D8:g/>D8@%=?n&>Q:9R=;<2/[:BE0>&B¥6F8o;?>&=?+9/EFE0>&B#)>&=?4dn&B

5'2/=<87n&B&.&bc>&BP>&8:Bm>&4u57E0>&B @%>SD>\;m;<>;?=<6057[uB?2/8o;c9RQ7Q'>DE0n&BH!Si`0H!Ec[!TdH'`[LZRMPX!T QDH!Si`z2U[V;<2/[%;Bm604dQ7E0>D4d>D8o;[LZNMX!T_QDH0S:`l.

4 ,C"N(, &F"F!/!!, ,,I (*# &-B(*# >-(?)B E@) @ P748:9 <; >=?; B BD@ r%2/6F;

(Ω, T )[78 >GBmQ9/S\>VQ7=<2/5:9/576FE06TB?9/57EF>U. Y87>J4>GBm[:=?>J@%>AQ7=?2U5:9R5:6FE06;<nVB?[7=

(Ω, T )>GB`;

[787>e9/Q7Q7E060S&9+;<6F2U8P : T → R

;<>DE0EF>eZo[7>d~-U.

∀A ∈ Tz0 6 P (A) 6 1

]L7.P (Ω) = 1h:. )Q7=<2/Q7=<60n\;?n¤@%>

σ _ 9/@:@%6;<6Fs6F;?nd2/[7= ;<2/[%; !9/4d6FE0EF> @7nD872U4u57=9R57E0> (An)n∈N

@Ô n&/nD8:>D4d>D8o;D[% (@%>D[%6F8:SD2/4dQ:9R;?6057EF>GBDz%2U8¤9~

P

(

⋃

n∈N

An

)

=+∞∑

n=0P (Ai)

.

m b=%n 8 8 B B rq ZNQ<jP]~SHFEcH!`L]~MHOYH^MZY[LKY[0Q<T_Q<jlX'``']~M

(Ω, T )wTdZNML`

w !XN@7@%6F;?60s6;<n ZN]~MjlZN]~j)ILZN] ^TdH(A,B)

O? X 4 X!SbH!EcH0SijC`.Q<SILZNE^:KRjCQD[0T\H!` s P (A∪B) = P (A)+P (B) Yw∀A ∈ T s P (A) = 1 − P (A) Rw∀(A,B) ∈ T s A ⊂ B =⇒ P (B \A) = P (B) − P (A)

w∀(A,B) ∈ T s A ⊂ B =⇒ P (A) 6 P (B)

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

Ww !ª72U=?4[7E0>A@7[fSD=?6057E0>I@%>A#2U6F8SD9R=<n T_]a`WX0SX0MPKRT\H!EcH0Sij s `!ZRQ<j (A1, . . . , An) ∈ T n wT\ZRML`

P

(

n⋃

i=1

Ai

)

=∑

I⊂[[1,n]]

(−1)|I|+1P

(

⋂

i∈I

Ai

)

=

n∑

k=1

(−1)k+1∑

16i1<···<ik6n

P (Ai1 ∩ · · · ∩ Aik).

B V 0n nb= 8 @

&<;?=a= B B rs2U6;(An)n∈N

[787>ABm[:6;<>V@PO nDUnD87>&4>&8o;<B&. 1 8@%6;NZo[7>(An)n∈N

>GB`;NS\=<2/6TB<B?9/8U;<>CBm6Q'2/[7=;?2/[7;

n ∈ NzAn ⊂ An+1

.sV>C443D4d>/z(An)n∈N

>GB`;y@%n&SD=?2U60B<B?9/8o;?>NBm6Q2U[7=¥;<2/[%;n ∈ N

zAn+1 ⊂ An

.F 8 =& B B )¥=<2/Q7=<60n\;?nI@%>IE06046F;?>I4d2U872R;<2/87>xYwqbZNQ<j

(An)n∈N

]~SH$`']~Q<jlHI0MPZNQv`L`0KNSjH4O? X 4 X!SbH!EcH0SijC`!wTdZNML` P(

⋃

n∈N

An

)

= limn→+∞

P (An)w

yzwqbZNQ<j(An)n∈N

]~SH$`']~Q<jlHOWXI0MPZRQv``0KRSijlHO? X 4 X0SH0E4H0SijC`0w?T\ZNML` P(

⋂

n∈N

An

)

= limn→+∞

P (An)w

748:9 <; >=?; B B\p £¤2%@%DE0>¥Q7=<2/59R5760EF6TBm;?>@%>V2/E042Ug/2U=?2+ Y8H'`kîKYILH?^MZY[LKY[0Q<T_Qv`0Xy>&Bm;^[78A;?=<60Q7EF>D;(Ω, T , P )265

(Ω, T )>&Bm;N[78f>&B?Q:9/SD>AQ7=?2U5:9R5:6FE060B<9R5:EF>Uzo>D;

P[78:>I4>GBm[:=?>I@%>IQ7=<2/59R5760EF6F;?nIB?[7=

(Ω, T ).

F 8 =& B B\ !Vn\¦:876F;?602/8 @Ô [78:>IQ7=?2U5:9R5:6FE06;<nIBm[7=N[:8 B?¬%B`;<D4d>eS\2U4Q:EF>D; qbZRQ<j

(Ω, T )]~SrH'`kîKYILH ^MPZW[LKY[0Q<T_Qv`0KY[0T\H s H0j C ⊂ T

]~S `6`'j !EcH ILZRE.^bT\H!jcOYX0SZRE4[0MPKW[!T\H O? X 4 X!SbH!EcH0SijC`H0SzzH!SbONMPKRSijTQDw<H w

T = T (C)Yw?qbZNQ<j

P : C → [0, 1]jH0T<T\H!]iH ∑

A∈C

P (A) = 1w T\ZRML`

P`0H^bMPZNT\ZRSzzHJOWH

EcKNSiQ 0MPH]~SiQD0]iHH0S]~SbH^MZY[LKY[0Q<T_Q<jlXP : T → [0, 1]

wV== B BHS qbZRQ<j

Ω]~S H0S:`0H!Ec[0T\HtuSQZR]OWX!SbZNEc[!MPKW[!TdHw T\ZRML` s ]~SHVEcH!`L]~MHFOYH ^bMPZY[LKY[0Q<T_Q<jlXF`']~M

(Ω,P(Ω))H!`'jOYXCtuSQDH OYH E4KRSiQ 0MPH4]~SiQD0]iHîKNMT\K OYZRSiSbXHO ]~SbH K^W^T_QDILKNjkQDZNS

P : Ω → [0, 1]jlH0T<T\H 0]iH

∑

ω∈Ω

P (ω) = 1w

,I BE@@ 2P*# # &-B &)! &!) +(CB # !)rs2U6;

Ω[78f[78760/>D=By¦:876 . 1 8fS\2U8:Bm6T@%&=?>JEO >&B?Q:9US\>AQ7=<2/5:9/576FE06TB?9/57EF>

(Ω,P(Ω)).

O 9/Q7=?GBEF>fS\2U=?2UEFET9R60=?> L%. h7. wsz#Q2U[7= @%n\¦:8:6F=d[787> 4d>&B?[7=<> @%> Q7=<2/5:9/5760EF6F;?nPB?[7=

(Ω,P(Ω))z60ENBm[d;p@%>

@%nD¦:8760=PBm[:=E0>&B B?6087g/E0>\;<2/8:Bd@%>fBm2U=m;<> Zo[7>fET9B?2/4d4d>fBm2U6; nDgo9RE0> (

1. 2/6TS\6N[:8 S&9/BQ:9R=?;?6TS\[:EF60>D=;?=<&B

604Q'2/=?;<9/8o;A~748:9 <; >=?; B B[X b 9 ^bMPZY[LKW[!Q<T_Q<jlX]~SiQ !ZNM'EcH !2/[AXL0]~Q<MPXl^:KRM'jCQ<jkQDZNSdBm[:=

(Ω,P(Ω))>GB`;dET9HQ7=<2/59R5760EF6F;?n

@%nD¦:8760>IQ:9R=I~∀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'`kîKYILH^bMPZY[LKW[!Q<T_Qv`0X]~SiQ !ZRM'E4HF`L]~MVT ]~SQ<4 H!MLùtuSQ Ω w?T\ZNML` s îZN]~MFjlZN]~j A ∈ P(Ω)

P (A) =|A|

|Ω|=

SZNEc[0MPHJOWHJILKN`!K 4 ZRMPKY[0T\H!`SbZNEc[!MHJOWHJILK6`îZ6`L`'QD[!TdH'` .,II !#1(R; @(*# !-P*D"DE(C @(*#748:9 <; >=?; B BD@]\ rs2/6F;

(Ω, T , P )[78f>DQ9/S\>JQ7=<2/5:9/5760EF6TBmnUz%>\;YB?2/6F;

A ∈ T.

-U. 1 8 @%6F;YZo[7>A>&Bm;SXYT_Q WHLKW[!T\HDz:2U[^MPH!`0!]iH0lQ<E.îZ6`L`'QD[0T\Hz:Bm6

P (A) = 0.

L7. 1 8 @%6F;YZo[7>A>&Bm;^MPH!`0!]iH0ILH!M'jlKNQ<SBm6

P (A) = 1.

h:. C87> Q7=<2/Q7=<6FnD;?n>&Bm;ds=9R60>c^bMPH'`00]iH$`~MPH0E4H0SijNB?6EO nDUnD87>&4>&8o;2AET9HQ:=?2UQ7=?60n\;<n >GB`;pB<9+;?6TBm!9R6F;?>5 >&Bm;Q7=?>GB?Zo[7> _ SD>D=?;<9/6F8^.

@ ?n B V n nb= 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[78fnDUnD87>&4d>D8o;NQ7=?>GB?Zo[7> _ 604dQ2oB?B?6057EF>Uzs4d9/60BQ9/B604Q'2UB<B?6F57E0>/.

4 *L#! (. ( J# (# ! * , ( * " (,Q_) BE@@ 2P*# "!)> +!)!-(R (R#1 8HS©7>&=<S©:>e4 9R608U;<>D8:9/8o; (pU2/60=VSD2/4d4d>D8o;VET9pS\2U878:9/60B<B?9/8:S\>@%>SD>D=?;<9R6087>GBC608%)2/=<4 9+;?602/8BVB?[7=YE0>=?nGBm[7EF;<9R;@PO [787>>\%Q'nD=<6F>&8:S\>42%@%6F¦:>EF>GBYQ:=?2U5:9R5760E06;<n&B&.2/[:=JSD>DET97z'2/8H@%nD¦:876F;JET9pQ:=?2U5:9R5760E06;<nS\2U8:@%6F;?602/878:>DE0EF>d@PO [78nDUnD87>&4d>D8o;

AB?9US©:9R8o;NZU[:>IEO nDUnD87>&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>UzA>D;B@%>&[%n&/n&87>D4d>D8o;B¥@%>

T;<>DETBZo[7>

B87>IB?2/6F;CQ:9/BQ7=<>&B<Zo[7> _ 604Q'2UB<B?6F57E0>/.%XYEF2U=<B&z%E09^bMPZY[LKY[0Q<T_Q<jlX4IZRSORQ<jCQDZRSiSH0T<T\HcOWH A H0S

B!2/[fE09^bMPZY[LKY[0Q<T_Q<jlX

OYHA`0KWIiKNSij

By>GB`;I~

PB(A) =P (A ∩ B)

P (B).

1 8;?=<2/[7U>Jn&gU9/EF>&4>&8o;B?2/[7U>D8o;NE09872/;<9R;?602/8P (A | B)

.m b=%ni= $ >=?; B B rqbZRQ<j

B]~S X 4 X0SH0E4H0SijVSZRS^MPH!`0!]iH0lQ<E^:ZN``'QD[0T\H w T\ZRML` PB

OWXDt SiQ<jV]~SHcEcH!`L]~MHOYH^MZY[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>&BQ7=<2/5:9/576FE06F;?n&B3S\2U8:@%6F;?602/8787>&EFE0>&BBm2U[:BET9)2/=<4d>IBm[760+9R8o;?>d~

P (A ∩ B) = P (A | B)P (B).

¥8¤> '>D;&z'2/8¤Bm>eB?>D=?;VB?2/[7U>D8o;C@%>&BYQ7=<2/5:9/576FE06F;?n&BNSD2/8:@76;<6F2U8787>DE0E0>&BCQ'2/[:=YSD9/E0SD[7EF>&=NEF>GBCQ7=<2/59R5760EF6F;?nGBC@PO 6F8o;<>D= _B?>&Sl;<6F2U8:B&z%>\;C872U8E O 608s/>&=<B?>/z%SD9/=EF>GBNQ7=<2/5:9/576FE06F;?n&BSD2/8:@%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[:BE02/608 [787>Ag/n&87nD=9RE060B<9+;<6F2U8p@%>IS\>D;m;?>J)2U=?4[7EF> ([787>A608U;<>D=Bm>GSl;<6F2U8@%>

nn&/n&87>D4d>D8o;B

)2U=?4[7E0>A@7>&BNQ7=<2/59R5760EF6F;?nGBS\2/4dQ'2UB?nD>GBDzs;<©7nD2U=?&4>eL7. K7. h

,Q_) JK!)>)PC(*!)>) !)(ij8o;?[76F;?60/>&4d>D8o;&zP@%>&[% nDUnD87>&4>&8o;D;BBm2U8U;A608:@%nDQ'>D8@79R8o;<BAZo6#E09 S\2U878:9R6TB<B?9/8:S\>u@7>E O [:8H87>4d2s@76¦:>

Q:9UBET9 Q7=<2/5:9/576FE06F;?nf@%>fEO 9R[%;<=?>Uz¥@%2/8SB?6P (A | B) = P (A)

>D;P (B | A) = P (B)

.bO 608:SD2/8s/n&876F>&8o;@%> S\>D;m;<>d@7n\¦:876F;?602/8 =<n&B?60@%>d@79/8:BAEF>!9R6F;uZo[Ô >DE0EF>8Ô >&Bm;A+9/E09/57EF> Zo[7>dBm2U[:BJ[787>d©s¬sQ2/;?©7GBm> @%>872/8 Zo[:9/B?6 _604Q'2UB<B?6F5760E06;<n/. 1 8fS\2U8:B`;9+;<>ASD>DQ'>D8:@:9R8o;YZo[7>I@79R8:BE0>eSD9UB@%>I872/8fZo[:9/B?6 _ 604dQ2oB?B?6F5:6FE06;<n/z%E0>&BC@7>D[% n&gU9/EF6F;?nGBB?2/8o;Yn&Zo[760+9RE0>D8o;?>GB( E O n&gU9/EF6F;?np~

P (A ∩ B) = P (A)P (B).P13>D;m;?>nDgo9RE06;<neQ2U[7+9R8o;3D;?=<>uS\2U8:Bm6T@%n&=?n&>eB?9/8:B

©s¬sQ2/;?©7GBm>Y@7>YZo[:9/B?6 _ 604Q'2UB<B?6F5760E06;<n/z/S/O >GB`;>&EFE0>VZo[7>Y872/[:BQ7=<>D872U8:B¥S\2U4d4>V@%n\¦:8:6;<6F2U8p@%>CE O 608:@%n&Q>&8:@79R8S\>u~748:9 <; >=?; B B r%2/6F;

(Ω, T , P )[78>&B?Q:9/SD>uQ7=<2/5:9/5760EF6TBmnUz>D;

(A,B) ∈ T 2 .Pb^>&BJn&/nD8:>D4d>D8o;\;BB?2/8o;FQ<SOYXlîH!SbOYKNSjDèB?6

P (A ∩B) = P (A)P (B).

748:9 <; >=?; B B\p r%2/6F;(Ω, T , P )

[78 >&B?Q:9/SD> Q7=<2/59R5760EF6TB?n/zc>D;(A1, . . . , An) ∈ T n .b^>GBn&/n&87>D4d>D8o;B

A1, . . . , An

Bm2U8o;E]~jk]iH!T<T\H0EcH!SjuQ<SOWXîH!SbOYKRSijCùBm6~

∀J ⊂ [[1, n]], P

⋂

j∈J

Aj

=∏

j∈J

P (Aj).

¥8fQ:9/=m;<60SD[7EF60>D=Gz%>D8fQ7=<>D89R8o;J = [[1, n]]

~

Bdp F = 8 =' 9 =?; ':; g W @Ypm b=%ni= $ >=?; B BdrqbZRQ<j

A1, . . . , An

OYH!`JX 4 X!SbH!EcH0SijC`VE]~jC]iH0T<T\H0E4H0Sij Q<SOYXlîH!SbOYKNSjD`0w?TdZNML`

P

(

n⋂

i=1

Ai

)

=

n∏

i=1

P (Ai).

1 8 />&=?=9f@79R8:BeEF> Q:9/=<9/g/=9RQ7©:>B?[7609/8o;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;B87>CB?2/8o;Q:9UB4u[%;<[7>DE0E0>D4d>D8o;608:@%n&Q>&8:@79R8o;B' )2U=?4[7E0>C@%>&BQ7=?2U5:9R5:6FE06;<n&B¥SD2/4dQ'2UB?nD>&B&z+;<©7nD2 _=<D4d>eL%.MK%. h l.748:9 <; >=?; B BTS r%2/6F;

(Ω, T , P )[:8 >&B?Q:9/SD>¤Q7=<2/5:9/576FE06TBmnUz¥>\;

(Ai)i∈I

[787>H!9R4d6FE0E0> 6F87¦:8760>@PO nDUnD87> _4d>D8o;<B&. bc>&BunD/n&87>D4d>&8U;BAi, i ∈ I

B?2/8o; E]~jC]iH0T<T\H0E4H0SijQ<SOWXîH!SbOYKRSijC`Bm63Q2U[7=e;<2/[%;n ∈ N

z EF>GBen&/n _87>&4>&8o;<BA1, . . . , An

B?2/8o;V4[%;?[:>DE0EF>&4>&8o;A608:@%n&Q>&8:@79/8U;BDzP@%2U8:SuB?6Q2U[7=J;?2U[%;AB?2/[B _ >D8:B?>D457E0>tuSQ J @7>Iz

P

⋂

j∈J

Aj

=∏

j∈J

P (Aj).

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

C1>\;

C2@%>&[%B?2/[:B _ >&8:B?>D4u5:EF>GBV@%> T . 1 8 @%6F;eZo[7>dEF>GBIS\ET9/B<Bm>GB C1

>\;C2B?2/8o;

608:@%nDQ'>D8@79R8o;?>GBCB?6PQ2U[7=;?2U[%;A1 ∈ C1

>\;N;<2/[%;A2 ∈ C2

zA1>\;A2Bm2U8o;608:@%n&Q>&8:@79/8U;BD.

1 8f@%n\¦:8:6;V@7>JET9443D4d>I4 9R8760D=<>AEO 6F8@%nDQ'>D8:@:9R8:SD>I4u[%;<[7>DE0EF>e@Ô [78:>A!9R4d6FE0E0>e@%>IS\ET9/B<Bm>GB(Ci)i∈I

.F 8 =& B B[ZAqbZNQ<j

C1H!jC2

OWH!]YcH0S:`0H0E4[0T\H!`VOYX0SZRE4[0MPKW[!T\H!`VO? X 4 X!SbH!EcH0SijC`!wE3S`'] ^YîZ6`0H$0]iHC1

H!jC2`0ZNSj)I :KWI!]~S$!ZNM'EcX!Ò X 4 X0SH0E4H0SijCÒYH0]Y JOWH!]YJQ<SbIZRE.îKNjkQD[0T\H'` s H!j0]iH C1

H0jC2

`0ZNSj%Q<SbOYXl^:H0SOWKNSijC`0wT\ZRML`$TdH'`FjkM'QD[!]a`H!SaWH0SORMPXLH'`

T (C1)H!j

T (C2)`0ZNSjXWKRT\H0E4H0SijQ<SOWXîH!SbOYKRSijlH'`0w

F 8 =& B BD@6\AqbZRQ<jC1, . . . , Cn

OYH!`VH0S:`0H0E4[0T\H!`VOYX0SZRE4[0MPKY[0T\H!`FO? X 4 X0SH0E4H0SijC`0wE3S`L] ^W^:ZN`0H!]iHVTdH'`Ci

`0ZNSijI :KWI!]~S !ZRM'E4X!Ò? X 4 X!SbH!EcH0SijCÒYH0]Y OWH!]Y Q<SbILZNE^:KRjCQD[0T\H!` s H0j!]iH C1, . . . , Cn

`0ZNSij$E]~jC]iH0T_T\H0E4H0SijQ<SOWXîH!SbOYKRSijC`!wuTdZNML`4T\H'`4jkM'QD[!]a` H!SaWH0SORMPXLH'`

T (C1), . . . , T (Cn)`0ZNSjXWKRT\H!EcH0SijE]~jk]iH!T<T\H0EcH!Sj

Q<SOWXîH0SOYKRSijlH'`0wV== B BC@:@ qbZRQ<j

(Ci)i∈I

]~SbHb!KNEQ<T<T\HO? H!Si`0H!Ec[!TdH'`.OYX0SZRE4[0MPKY[0T\H!Ò X 4 X0SH0E4H0SijC`0wR3S`'] ^YîZ6`0H0]iHT\H!`

Ci

`0ZNSjJI :KYI0]~S!ZRM'E4X!` O? X 4 X0SH0E4H0SijC` OWH!]Y OYH0]Y Q<SILZNE^:KRjCQD[0T\H!` s H!jJ0]iH (Ci)i∈I

H!`'j]~SbH!KNEQ<T<T\H OWH I0T\KN``0H!` EJ]~jk]iH0T<T\H!EcH0SijQ<SOYXlîH!SbOYKNSjH!`0wT\ZRML` T\H'` jkM'QD[0]a` H0SzzH0SONMXH!`

T (Ci) s i ∈ I s `!ZRSijXWKRT\H0E4H0SijEJ]~jk]iH0T<T\H!EcH0SijuQ<SbOYXl^:H0SOWKNSijlH!`!wV== B BC@ rqbZRQ<j

A1, . . . , An

OYH!`JX 4 X0SH0E4H0SijC`FEJ]~jk]iH!T<TdH!EcH!Sj Q<SbOYXl^:H0SOWKNSijC`0w%T\ZRML`FQ<TH0S H!`LjOYHEcG!EcHJOWHjlZN]~jlHu!KREJQ<T<TdH(B1, . . . , Bn) s ZîZN]~MFjlZN]~j i ∈ [[1, n]] s Bi = Ai

ZN]Bi = Ai

w

4 (,#?&)!,# F& . (,N *" ( J#%" , -EB ; & ( >)(*# BE@@ 2P*#2 2 (*#1 8fBm>IQ7ET9/SD>I@79R8:B[:8 >GBmQ9/S\>JQ7=<2/5:9/5760EF6TBmn

(Ω, T , P ).

748:9 <; >=?; Bdp BD@ r%2/6F;C ⊂ T

. 1 8 @%6;dZo[7>C>&Bm;u[78¨B?¬%B`;<D4d> Zo[:9/B?6 _ SD2/4dQ7E0>\;u@PO nD/n&87>D4d>&8U;BeB?6E0>&BnDUnD87>&4d>D8o;<BN@%>

CBm2U8U;N872U8604dQ2oB?B?6F5:EF>GBDz7@%>&[% (d@%>&[%608:SD2/4dQ:9+;<6F5:EF>GBDz%>D;YBm6 ∑

A∈C

P (A) = 1.

@Y ?n B V n nb= 8

XC608:B?6zPET9 @%6 nD=<>D8S\>d9/>&Su[78 Bm¬%Bm;?D4d>dS\2U4Q:EF>D;A=<n&B?6T@%>[:8760Zo[7>&4d>D8o;A@:9R8:BJE0>!9/6;eZo[7>EO [787602/8 @%>GBVn&/n _87>&4>&8o;<Be8cO >GB`;nDgU9/EF> (ΩZU[cO (¤[:8 >D8:B?>D457EF> 87nDgUEF60g/>G9R57E0> Q7=<&B !EO [787602/8¨>&Bm;uQ7=<>&B<ZU[:> _ S\>D=?;<9/6F8:> >\;u872U8S\>&=m;9R6087> \.

F 8 =& Bdp B ª72/=<4u[:EF>I@%>GBNQ7=<2/5:9/576FE06F;?n&B3;<2R;<9/EF>GBqbZRQ<j

(Ai)i∈I

]~S` 6`'j !EcHF0]iK6`'Q<ILZNE^T\H0j:tuSQ%ZN]OYX0SZNEc[0MPKY[0T\H s jlH!T_`V!]iHT\H!`VX 4 X0SH0EcH!SjD` Ai

SbH`!ZRSijîK6`^MH'`00]iH!lQ<E^:ZN`L`LQD[0T\H!` s H0ju`0ZNQ<j B ∈ T

wT\ZNML`P (B) =

∑

i∈I

P (B | Ai)P (Ai).

$GU(o$ N>\;<=?2U[7/>&=[787>IQ7=<2/5:9/5760EF6F;?nP (B)

S\2/8:8:9R6TB?B<9R8o;EF>GBQ7=<2/5:9/576FE06F;?n&BNSD2/8:@76;<6F2U8787>DE0E0>&BP (B | Ai)

.XC608:B?6zs2U8 Q'>D[%;CSD9/E0SD[7E0>D=

P (B)>D8 2C@%6TBm;?6087g/[:9/8o;B?[7609/8o;y[78 S\>&=m;9R608p872/457=<>J@%>ASD9UB5!B?[7609/8o;Zo[7>

Ai>&Bm;C=<n&9RE06TBmnlz%Bm6 S\>&BNS&9/BQ:9/=m;9RgU>D8o;yE O >D8:B?>D457EF>I@%>GBNSD9UBQ2oB?B?6F5:EF>GBD., -EB ; & ( >)(*# BE@@ 2P*#%";="#KPC(*#F 8 =& Bdp B !ª72U=?4[7E0>A@7>&BNQ7=<2/59R5760EF6F;?nGBS\2/4dQ'2UB?nD>GB qbZRQ<j

(A1, . . . , An) ∈ T n jlH!T!]iHP (A1 ∩ · · · ∩ An−1) 6= 0

lOWZNSbIA1 ∩ · · · ∩ An−1

S H!`'j îK6`^MPH!`0!]iH0Q<E^:ZN``'QD[0T\H RwT\ZNML`

P

(

n⋂

i=1

Ai

)

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

= P (A1)

n∏

i=2

P

Ai

∣

∣

∣

∣

∣

i−1⋂

j=1

Aj

.

$GU(o$ 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:BE0>SD9UB@Ô [78:>B?[:SDSD>&B<Bm602/8f@PO >\%Qn&=?60>D8S\>/.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&=?nGB87>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[:BZo[7>AE09dQ:=?2UQ2oBm6F;?602/8fL%. q:. W7.,I -EB ; & ( >)( "(*#F 8 =& Bdp B !ª72U=?4[7E0>A@7>exy9¬U>&B qbZRQ<j

(Ai)i∈I

]~S `6`'j !EcH0]iK6`'Q<ILZNE^T\H0jtuSQZR] OYX0SZNEc[0MPKY[0T\H s jH0T 0]iHVîZN]~M4jlZN]~j i ∈ I s P (Ai) 6= 0w

qbZRQ<jj ∈ I

wT\ZNML`P (Aj | B) =

P (B | Aj)P (Aj)∑

i∈I

P (B | Ai)P (Ai).

$GU(o$ E0>3B?¬%B`;<D4d>(Ai)

=<>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;BEF2U=n\;<9/Q>Bm[760+9R8o;?>@%>eE O >DsQ'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;<nZo[Ô [787>CS\>&=m;9R6087>S\2/8BmnGZU[:>D8:SD>

B9/6;¥E06F>&[^z/B<9/S©:9/8o;Zo[7>ET9JS&9R[:B?>

Ai

9A>&[E06F>&[^z/S/O >GB`; _ ( _ @%60=<>3ET9JQ7=?2U5:9R5:6FE06;<nS\2U8:@%6F;?602/878:>DE0EF>P (B | Ai)

!2/8=?>GBmQ'>&S\;?>N@79R8B#SD>yS&9/B#E O 2U=<@%=<>¥;<>D4dQ'2/=<>DEl.U13>&B@%2U8787n&>&BQ'>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=<4d6F89R8o;CE09Q7=<2/59R5760EF6F;?neZo[Ô [787>eSD>D=?;<9/6F87>IS&9R[:B?>Ai

9R6F;>&[pEF60>D[B?9US©:9R8o;ET9eS\2U8:B?n&Zo[7>D8S\>B.U2/[:=3S\>D;m;<>Y=9R6TBm2U8^zUSD>\;m;<>C)2U=?4[7EF>Y>&Bm;y9R[B?B?6B?2/[7U>D8o;9RQ:Q>&EFn&>

!ZNM'E]~T\HOYH.^MZY[LKY[0Q<T_Q<jlXOWH'`JILKR]a`!H!`l.

N N N

J9R8:BNSD>eS©:9RQ76F;?=<>/z7B<9R[7 4d>D8o;?602/8fS\2U8o;?=9R60=?>Uz(Ω, T , P )

@%n&B?6FgU87>A[78f>&B?Q:9/SD>AQ7=?2U5:9R5:6FE060B?nAZU[:>DETS\2/8ZU[:>/.

/ ", & (, (# V"F&0"F /0(, "/ "$#'&)(,I,E P ! "!#748:9 <; >=?; BC@:BD@ Y8 KNT\XLKSi]~EcX!M'QD0]iHA>&Bm;C[787>e9/Q7Q7E060S&9+;<6F2U8

X@%>

Ω′ @79/8:B Rz%;?>&EcZo[7>d~

%Ω′ >&Bm;C[78fBm2U[:B _ >D8Bm>&4u57E0>J@7> Ω% Q2U[7=;<2/[%;

a ∈ RzX−1(]−∞, a]) ∈ T

@%2/8SCE O n&/nD8:>D4d>D8o;X−1(]−∞, a])

9/@%4d>D;y[78:>YQ7=<2/5:9/5760EF6F;?nl.C8¤9/EFnG98s[74dn&=?6TZo[7>

X9UB?B?2%S\60>I@%2/8:S (dSD>D=?;<9R608:Bn&EFn&4>&8o;<BC@%>

Ω[787>eSD>D=?;<9/6F87>I+9/EF>&[7=N8s[74dnD=<60Zo[7>Uz74 9R6TB

;?2U[:BE0>&BNn&EFn&4>&8o;<BN@%>Ω8Ô 2U8o;CQ:9/By)2U=<SDnD4d>D8o;C@PO 6F4 9RgU>AQ:9R=

X.

748:9 <; >=?; BC@:B Y87> 4 KNM'QDKW[!TdHJKRT\XLKNjlZNQ<MPHMPXLH!T<T\H ) . 97. =&.o>D8 9/57=<nDg/n>&Bm;3[:8 9RE0n&9I8s[74dn&=?6TZo[7>C;<>DE ZU[:>Ω′ = Ω

]79/[%;?=<>D4d>&8U;Y@76;Gz%;?2U[:BEF>GBnDE0nD4d>D8o;BC@%>Ω2U8o;[:87>I6F4 9/g/>AQ:9R=

X.

m b=%ni= $ >=?; BD@:B 87%SbH!ZRSijkQDZNSX : Ω → R

H!`'j]~SbH 4 KNM'QDKW[!T\H KRT\XLKNjlZNQ<MPHMPXLH0T<T\H`'QH!j`!H0]~T\H0E4H0Sij`'QîZN]~MFjlZN]~j Q<SjH0M 4 KNT<T\H I s X−1(I)H'`'j ]~S X!TdX!EcH!SjOWHTdKcjkM'QD[!]

Tw

= >=%; BC@:B 1 8I@%2U8787>ET9872R;9+;<6F2U8J>&4dQ7EF2+¬UnD>[:B?[7>DE0E0>D4d>D8o;cQ'2/[7= S\>D=?;<9/6F8B nD/n&87>D4d>&8U;G.&rs2U6;(a, b) ∈

R2 >\; x ∈ 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) &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. =&.XSD2/4d4d>¤[78 9RE0n&9 8s[74dn&=?6TZo[7>

@%nD¦:8763B?[7=Q7=?>GB?Zo[7>d;<2/[%;<>&BuEF>GBe+9RE0>D[7=Be@%>Ωz SRO >&Bm; _ ( _ @%60=?>pB?[7= Ω′ ;?>&E3Zo[7> P (Ω′) = 1

.cXC608:B?6zBm>&[7E[:8B?2/[:B _ >&8:B?>D4u5:EF>87nDgUEF60g/>G9R57E0>d8Ô 9/@%4d>\;eQ9/BI@%> +9RE0>D[:=eQ:9R= X . 13>\;m;<>pS\2U8oU>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\2U44d>A[787>I . 97. =&.0.V-W n BD@:Bd 1 8 E09/8:S\> [:87>p608%¦:876F;?n @%> )2/6TBe[:87>pQ760&SD>/z^>D;

XS\2U=?=<>&B?Q2U8:@ 9R[ =<9/87g @%>pET9R8S\>D=@%[

Q7=<>D4d60>D=C¥60E0>u2/57;?>D8s[^.¥8H9U@%2/Q%;9R8o;YET9pS\2U8s/>D8o;<6F2U8Q7=<n&SDn&@%>&8o;?>/zX>&Bm;V[787>u'. 97. =G.Fz7Q:=?>&8:9R8o;YE0>&BY9/EF>&[7=B

@6X %n B 8 = 8

X(Ω) = N∗ . N>D4 9/=<Zo[7>yZo[7>E O n&/nD8:>D4d>D8o;238Ô 2U5%;?>&8760=¥ZU[:>@7>&B#!9/SD>&B5 zRZo[76%>GB`;Zo[:9UBm6 _ 6F4dQ'2UB<Bm6057E0>/z4 9R6TBQ:9UB6F4dQ'2UB<Bm6057E0>/z%8Ô 9/@%4d>D;CQ:9/BN@PO 604d9/g/>AQ:9/=X.

m b=%ni= $ >=?; BD@:BHS qbZRQ<jX

]~SH 4 w<KYwdM0w<wuTdZNML` [X = x], x ∈ X(Ω)H'`'j.]~S `6`'j 0EcH ILZNE^T\H0j ZN]

0]iK6`'Q<ILZNE^T\H0j.H!S KWOYZL^jlKNSijT\KILZNS 4 H0SijkQDZNS OWHTdK MPH0EcKNMP0]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).7Y2R;9+;?602/8)=<n&Zo[7>D8o;<>I4d9/60BQ9/B[78760/>&=<B?>DE0E0>~

AX

2U[TX

.I,E -"!)M +! >)( BPC) B2 2"!748:9 <; >=?; BC@:B[Z r%2/6F;

X : Ω → R[:87> . 97. =&.0.cbc9 )2U8:Sl;<6F2U8 @%>p=<nDQ:9/=m;<6;<6F2U8 @7>

X>&Bm;uET9 )2/8:S\;?602/8^z

872/;?nD>AgUnD87n&=<9/EF>&4>&8o;FX

z:@%n\¦876F>IQ'2/[:=;?2/[7;x ∈ R

Q:9R=FX(x) = P (X 6 x)

.F 8 =& BC@:BD@6\AqbZRQ<j

X]~SbH 4 w<KYwdM0w s H0j FX

`0K.!ZNSbI!jkQDZNS OYHJMPXl^:KRM'jCQ<jkQDZNSbw?T\ZRML`FxYwFX

H'`'j ]~SHu!ZRSI0jCQDZRS I0MPZRQv``0KRSijlHF`']~MR yzw

FX

H'`'jIZRSijkQ<Si]iH ONMZNQ<jlHH0S jlZN]~j^:ZRQ<SijOYHR |Ww

FX

KYORE4H0jOWH'`FT_Q<EQ<jlH'`JH0S±∞ s H0j lim

x→−∞FX (x) = 0 s lim

x→+∞FX(x) = 1

wF 8 =& BC@:BD@i@ KYONEQv`XLI!Q ^bMPZ !]iH0E4H0Sij s jZR]~jH!ZRSI0jCQDZRS F : R → R 4 X!M'Q tKNSj T\H!` ^bMPZ^bM'QDX!jlX!ÒR]j :XLZNM 0EcH|zw<xYw<xH!`'j T\K!ZNSbI!jkQDZNS OYHMPXl^:KRM'jCQ<jkQDZNS O ]~SHILH!M'jlKNQ<SbH 4 w<KYwdM0w<w

", * (, " &* !,* & #%(,I, P ! "!#b^>GBA@%n\¦:8:6;<6F2U8:BI@%6 '&=?>&8U;A[78 Q>&[ B?[760+9R8o;IEF>GBJSD2/8o;?>Ds;?>&BJ>D;JE0>&BA5'>&B?2/608:B&. b 9@7n\¦:876F;?602/8 B?[76F+9/8U;<>>GB`;AET94d2/608:BNS\2U8o;?=9R60g/8:9/8U;<>/zs>\;N>GB`;Y9/4Q: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 X ) . 97. =&. @P.%>D8¤9/57=?n&g/n>GB`;N[78:>I'. 97. =G.s;?>DE0E0>Zo[7>

X(Ω)B?2/6F;N[78f>D8:B?>D457EF>J¦:8:6^2/[¤@%n&872/457=<9/57E0>/.

O 9/Q7=?GBAEF>GBA=?&g/E0>&BA@%>pS&9R=@%6F89RE06;<nSD2/8:SD>D=<8:9R8o;AEF>GBA>D8:B?>D457EF>GBJ¦:876TBA2/[ @%n&872/457=<9/57E0>&B&z'2U82U5%;?60>D8o;I604 _4dn&@%6T9+;<>D4d>D8o;e~m b=%ni= $ >=?; B B rqbZRQ<j

XH0jYOYH0]Y 4 w<KYwdM0w<OWw s λ ]~S MPXLH0T!]iH0T\ILZNSb!]iH s H0j f ]~SHKL^Y^bT_QDILKNjkQDZNS OYH X(Ω)OYKRS:`

RwT\ZRML` s T\H'` 4 w<KYwdM0wuOWXDt SiQDH!ÒYHJT\KcEcKNSQ !MHF`']~Q 4 KNSijlH`!ZRSijH!SbIZRMPHONQv`0I!M !jlH'`xYw

X + Y : ω 7→ X(ω) + Y (ω) yzwλX : ω 7→ λX(ω) |WwXY : ω 7→ X(ω)Y (ω) wf(X) : ω 7→ f(X(ω))

wI, -"!)M +! >)( BPC) B2 2"!r%9/[%@79R8BV@%>GBVS&9/B2YQ9+;?©:2/E02/g/6TZo[7>&B5f=<>D8@79R8o;V[:87>e;<>DE0EF>@%>&B<S\=<6FQ7;?602/8 604dQ2oB?B?6F5:EF> )Q7=<n&B?>D8:SD>u@%>Q2U6F8o;<B@PO 9/S&S\[74[7E09R;?602/8 @79R8:BdE0>&Bp9/EF>&[7=BQ'2UB<B?6F57E0>&BdQ:=?6TBm>GBQ:9R=

XzQ:9/= >\%>D4dQ7E0>¤Bm6

X(Ω) = QlzE0>fg/=9RQ7©7>

B h=&:; \g; B B B B @6Z

@%>E09f)2U8:Sl;<6F2U8 @%> =<nDQ9R=?;?6F;?602/8 @PO [787> . 97. =&. @P.X>&Bm;u[787> )2/8:S\;?602/8 >D8¨Q7E09R;?>G9R[% Q7=<>D8:9/8U;@%>&Bu9/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;

X[787>C . 97. =&. @P.0./bc9AE02/6:@7>CQ7=<2/5:9/576FE06F;?nC@%>

X>GB`;¥ET9A)2/8Sl;?602/8

f : X(Ω) → [0, 1]@%nD¦:8760>IQ2U[7=;?2U[%;x ∈ X(Ω)

Q:9R=f(x) = P (X = x)

.V>J4 9/876F&=?>Jn&Zo[760+9RE0>D8o;?>Uz%Bm6P2U8@%6TBmQ'2UB?>A@PO [:87>AnD8s[74dnD=9+;<6F2U8

(xn)n∈N

@%>GBynDE0nD4d>D8o;<BN@7>X(Ω)

zsET9E02/6P@%>Q7=<2/5:9/5760EF6F;?nI@%>

X>&Bm;CE09d@%2U8787nD>e@7>JET9dBm[:6;<>

(pn)n∈N

;?>&EFE0>eZo[7>IQ2U[7=;?2U[%;n ∈ N

zpn = P (X = xn)

.e 'bf?g B B 2U[7=IQ2U[7/2U6F=IQ9R=<EF>&=A@7>pE09 E02/6@7>

Xz^60E!9R[%;u9+9/8U;A;<2/[%;9U2/60=J@7n&S\=<6F;eEO >D8:B?>D457E0>

X(Ω)@%>GBN+9RE0>D[7=BQ7=<60B?>&ByQ:9/=

X.

F 8 =& B B\pAqbZNQ<jX]~SH 4 w<KWwdM0w<OYw)H0j f : X(Ω) → R

`0KTdZNQDwbT\ZNML` ∑x∈X(Ω)

f(x)H!`'ju[!QDH0S OYXCtuSiQDH$H0j

∑

x∈X(Ω)

f(x) = 1.

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[Ô >DE0EF>>&Bm;dSD2/8s/>&=?gU>D8o;?> Q<SbOYXl^:H0SOWKNEE4H0SijJO? ]~SH X0Si]~EcX!MKNjkQDZNS OYH

ΩzSRO >&Bm; _ ( _ @%60=?> 608:@%n&Q>&8:@79/44d>&8U;@PO [78¨2U=<@7=?> @%> B?2/4d4 9+;<6F2U8^z#>\; ZU[:>ET9 +9RE0>D[:=@%> ET9B?2/4d4d> 87>f@%n&Q>&8:@¨Q9/B872/8Q7EF[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¥Q2oBm6F;?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[:4n&=<9R;?602/8 @72/8787n&>I@%>

X(Ω).

¥8¤9/@72/Q%;9R8o;NE09dS\2U8s/>D8o;<6F2U8@%>IET9=?>&4d9/=<Zo[7>Ah7.0-/.MK%z%2U82U5%;?60>D8o;I~m b=%ni= $ >=?; B BHS 7%S KNT\XLK

X : Ω′ → RH'`'j%]~SH 4 w<KYwdM0w<OWw `'Q?H0j`0H0]~T\H!EcH0Sij`'Q ∑

x∈X(Ω′)

P (X = x) = 1w

m b=%ni= $ >=?; B BYXrqbZRQ<jX]~SbH

v.a.r.d.H0jFX

`0K.!ZRSI0jCQDZRS OWHMPXlîKNM'jkQ<jCQDZRSw?ZR]~MFjZR]~jx ∈ R s

P (X = x) = limy→x+

FX (y) − limy→x−

FX(y) = FX (x) − limy→x−

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

P (X = x)>&Bm;N4 9+%6F4 9/EF>U. C87>A;<>DE0EF>I+9RE0>D[:=N8Ô >GB`;NQ:9UBy)2/=S\n&4>&8o;[:8760Zo[7>U.

( J#%,-* ( " &* I,I #KPCB !( (*&)B ;= "(*!)!)(748:9 <; >=?; B BD@ r%2/6F;

X[787>N . 97. =&. @P.0.bOvH!`k^:X0MPKRSILHC@%>

X>&Bm;&zRB?2/[:B=<n&B?>D=</>y@%>IZRS 4 H!MkWH0SILHVKY[!`!ZRT_]iHI~

E(X) =∑

x∈X(Ω)

xP (X = x).

e 'bf?g B B bO >GBmQ'nD=9R8S\>JQ>&[%;C87>IQ:9UB>\%60Bm;?>&=I~% Bmn&=?60>I@%6FU>D=<g/>&8U;<>~

X(Ω) = N∗ >\; ∀n ∈ N∗ z P (X = n) =6

π2n2

.% Bmn&=?60>IBm>&46 _ S\2/8sU>D=<g/>D8o;<>e~ X(Ω) = (−1)nn, n ∈ N∗

z%>\;e~∀n ∈ N∗, P (X = (−1)nn) =

6

π2n2.

2\ %n B 8 = 8 I,I B !)( >)#K(CB #K"! 748:9 <; >=?; B B r%2/6F;

X[787>'. 97. =G. @P.y9/@%4d>D;m;<9/8o; [787>>&B?Qn&=<9/8:S\>

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

S\2U8s/>D=<g/>&8:S\>J@%>IET9B?nD=<60>/z%ET9+9/=?6T9R8:SD>J@7>X>GB`;I~

V (X) =∑

x∈X(Ω)

(x−E(X))2P (X = x).

b¥O n&S&9R=?; _ ;`¬sQ>J>&Bm;Y9RE02/=B@%n\¦:8:6^Q:9R=I~ σ(X) =√

V (X).

e 'bf?g B B b 99/=?6T9R8S\>JQ>&[%;C87>IQ:9UB>\%60Bm;?>&=N443D4d>eBm6PE O >&B?Qn&=<9/8:S\>A>\%6TB`;<>/.I,II 9: Q,QB* >N

f(X)J9R8:BNSD>\;m;<>eBm>GSl;<6F2U8^z

X>GB`;C[:87>I . 9:. =G. @^.%>\;

f[:87>e9RQ7Q7E06TSD9+;<6F2U8@%>

X(Ω)@79/8:B

R.

m b=%ni= $ >=?; B Bdp !bc2/6#@%>f(X)

H0S:`0H0E4[0T\HcOWH'` 4 KNT\H0]~ML`îZ6`L`'QD[!T\H!ÒWH f(X)H!`'j

Im(f) s H0j `0KT\ZNQH!`LjOYZRSiSXLH^:KRMJ∀y ∈ Im(f), P (f(X) = y) =

∑

x∈f−1(x)

P (X = x).

F 8 =& B B\ !@%>;?=9R8:Bm)>D=?; E(X)

HWQv`'jH`'Q$H0jV`0H!]~T\H0EcH!Sj`'Q ∑

x∈X(Ω)

g(x)P (X = x)ILZNS 4 H0MCzH

KY[!`0ZNT_]~EcH0Sij s H!jOYKRS:ÌLHJILK6` E(g(X)) =∑

x∈X(Ω)

g(x)P (X = x).

V== B BHS ZN]~MFjlZN]~j(a, b) ∈ R2 s E(aX + b) = aE(X) + b

wV== B BYXrqbZRQ<j

ϕH0jψOWH!]YV!ZRSI0jCQDZRS:ÒYH

X(Ω)OYKNSi`

RwTdZNML`

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!jkjKRSij]~SH 4 KRM'QDKNSILH w?T\ZRML`V (X) = E(X2) −E(X)2

H!j∀a ∈ R, V (X) = E((X − a)2) − (E(X) − a)2.

V== B BC@]\∀(a, b) ∈ R2 s V (X + b) = V (X)

H0jV (aX) = a2V (X)

w748:9 <; >=?; B BD@:@¨rs6

E(X) = 0z+2U8@%6F;Zo[7>

X>GB`;¥S\>&8o;?=<nD>¥]+B?6

V (X) = 1z+2U8@%6F;Zo[7>

X>GB`;=<n&@%[76F;?>U.

V>Y4 9R8760D=<>CgUnD87n&=<9/EF>YB?6E(X)

>D;V (X)

>\%60Bm;?>&8o;&zoE09I . 97. =&. @P.X∗ =

X −E(X)

σ(X)

>GB`;S\>D8o;<=?n&>C>\;y=?nG@%[76F;?>/.1 8 EO 9RQ7Q'>DE0E0> 4 w<KWwdM0w<OYwIH0SijkMPXLHFMXOR]~Q<jHJK6`L`0ZI0QDXLH X .I,IQ_ ";=(*!+2# ( ;=";=(*!+#%(*!+2BP*#748:9 <; >=?; 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~

mk(X) =∑

x∈X(Ω)

xkP (X = x) = E(Xk).

b^>V4d2/4d>D8o;NS\>D8o;<=?n&>V@PO 2/=@%=<>k>&Bm;&z%B?2/[:B3=<n&B?>D=</>C@PO >\%6TB`;<>D8:SD>J@%>

E(X)>\;NB?2/[:B=?nGBm>&=?U>Y@%>AS\2/8sU>D=<g/>D8S\>

9R5Bm2UEF[7>d~µk(X) =

∑

x∈X(Ω)

(x−E(X))kP (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>GBB?6;<[:9+;<6F2U8:B&.bc9dQ:EF[7Q9R=?;V@%>eS\>GBCE02/6TBC9RQ:Q:9R=9R6TB?B?>D8o;EF2U=<BY@PO >D%Qn&=?60>D8:SD>&BC@7>A;?60=<9/g/>&BY@%>I52U[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;

n5'2/[:EF>GB8s[74dnD=<2R;<nD>GB3@%>

1(n.%b 9

+9R=<609/57E0>X>&Bm;CEF>I8s[74dn&=?2d@%>IET95'2/[:EF>A;<6F=<nD>U.7XC608:Bm6 z72U8 @%nD¦:876F;u~

748:9 <; >=?; BC@:BD@ r%2/6F;n ∈ N∗ .C87>J . 9:. =G. X Bm[76F;3ET9eE02/6'[7876F)2/=<4d>J@7>YQ:9/=<9/4D;?=<>

nB?6X(Ω) = [[1, n]]>\;YB?6~

∀k ∈ [[1, n]], P (X = k) =1

n.

1 8 872R;<>X → U(n)

.

m b=%ni= $ >=?; BD@:B rqbZRQ<jX → U(n)

w?T\ZRML`E(X) =

n+ 1

2

H0jV (X) =

n2 − 1

12

w2/[7=A;<2/[%;I>&8:Bm>&4u57E0>¦:876

Ezc2/8H8:2R;?>dQ7E0[:BIgUnD87n&=<9/EF>&4d>D8o;

X → U(E)Q'2/[7=I[:87>d'. 97. =G.^Bm[760+9R8o;IE09E02/6

[7876F)2/=<4d>IBm[7=Xz7SRO >&Bm; _ ( _ @%6F=<>J[787>I . 9:. =G.s;?>&EFE0>eZo[7>d~

∀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:=?2UQ2U=m;<6F2U8

p@7> 52U[7EF>GBe57ET9R8S©7>&B&. 1 8

872/;?>XET9d+9R=<609/57E0>J9/EFnG9+;?2U6F=<>JnDgU9/EF>)(

1Bm6P2U8;<6F=<>A[787>I5'2/[7E0>A57ET9R8:S©7>Uz%>\;

0B?6F872U8^.

1 8 Q'>D[%; 9R[B?B?63@%nD¦:8760=X(HEO 9R6T@%> @%>p;<6F=9RgU>&B (HQ76FE0> 2U[ !9/SD>9U>&Sd[787>Q760&S\> @%n&B?n&Zo[760E06F57=<nD>Uz#@72/8o;uET9

Q7=<2/5:9/5760EF6F;?nI@%>IQ760E0>J>GB`;p]XQ7=<>D8:@f9/EF2U=<ByET9+9/EF>&[7=

1Bm6P2U8;<6F=<>AQ760EF>Uz7>\;

0B?6P2/8 ;?60=?>J!9US\>/.

XC608:B?6z[X = 1]

=?>&Q7=?nGBm>&8o;?>JEF>eB?[:SDSD&BN@79/8:B[787>I>D%Qn&=?60>D8:SD>I9¬U9R8o;[787>IQ7=<2/59R5760EF6F;?np@%>eB?[:SDSD&B&.

1 8f@%n\¦:8:6;V@72/8:S~

?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>

XBm[:6;Y[:87>eE02/6c@%>exy>&=?872U[7E0EF6c@7>eQ:9R=9 _4d\;<=?>

p)>D;C2/8 872R;<>

X → B(1, p)3B?6¥~

X(Ω) = 0, 1>\;

P (X = 0) = 1 − p

P (X = 1) = p

Vn&B?2/=<4d9/60B&zs2/8¤B?>A@72/8787>A[78f=<nD>&Ep ∈]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) = pH!jV (X) = pq

w

!, !,%, (, * (#'&" (, , ' (,%,? N, " (&)(. !,%(_- 9:" @!-";< (8!-"; @B( >-( # &) *#rs2U6;

n ∈ N∗ . 1 8 =<nDQ'\;?> n )2/6TB[787>>\%Qn&=?60>D8S\>@7> xy>D=<872U[7EFE06 z >D;2U8 872R;<> X EF>872/457=?> @%>fBm[:S&S\GB2/57;?>D8s[:B&.¥9R= >D%>D4dQ7E0> 2/8 S\2U4dQ%;?>¤E0>¤872/457=<>f@7> ¥60EF> 2U5%;?>&8s[:B @79R8B [787>HB?[:S&S\>&B<B?6F2U8 @%>

nET9R8:SD>D=B

)608:@%n&Q>&8:@79/8U;B 9U>&SV[787>IQ760&SD>e@%n&B?n&Zo[760E06F57=<nD>A@%2U878:9R8o;C¥60E0>e9/>&SV[787>IQ:=?2U5:9R5760E06;<nJ@%>p.

Vn\;?>&=?4d60872/8BYET9pE02/6@%>X.'| 2/[7;A@PO 9R5'2/=@Pz2/8HQ'>D[7;A9U2/60=Y@%>

0(nB?[:SDSD&B&z'9/6F8Bm6

X(Ω) = [[0, n]].Prs2/6F;

@%2U8:Sk ∈ [[0, n]]

.b¥O nD/n&87>D4d>&8U;[X = k]

>&Bm;N2/5%;<>D8s[fBm6^2U8 9>&[kBm[:S&S\GBD.:rs2U6;Gz%Q2U[7=;?2U[%;

i ∈ [[1, n]]zSiE O n&/nD8:>D4d>D8o;I~72/57;?>D8:6F=N[78¤B?[:SDSD&B*(ET9

i _ 6F&4d>J>D%Qn&=?60>D8:SD>/.7XYEF2U=<BA~

[X = k] =⋃

I⊂[[1,n]]|I|=k

⋂

i∈I

Si ∩⋂

i∈[[1,n]]\I

Si

b^>GBNnDUnD87>&4d>D8o;<BSi

zi ∈ [[1, n]]

z:B?2/8o;N4u[%;<[7>DE0EF>&4d>D8o;Y608:@%nDQ'>D8@79R8o;<B&z7>D;CEF>GBNnDUnD87>&4>&8o;<BN@%>eE O [787602/8¤Bm2U8o;@%>&[% (d@%>&[% 608:S\2U4Q9+;?6057E0>&B&.:XC608:B?6z

P (X = k) =∑

I⊂[[1,n]]|I|=k

∏

i∈I

p ·∏

i∈[[1,n]]\I

(1 − p)

=∑

I⊂[[1,n]]|I|=k

pk(1 − p)n−k =

(

n

k

)

pk(1 − p)n−k.

b^>S\2s> pS\60>D8o;C5760872U46T9RE#S\2U=?=<>&B?Q2U8:@ (dE09 Q'2UB?6F;?602/8¤@%>&BkBm[:S&S\GBDz

pr >&Bm;YET9dQ7=<2/5:9/576FE06F;?ne@PO 2/5%;<>D8o;?602/8 @7>S\>GBrB?[:S&S\&B>D;

(1 − p)r >GB`;CET9Q7=<2/5:9/576FE06F;?nI@PO 2/5%;<>D8o;?602/8f@%>&BCn&S©7>GSDB9R[% 9R[%;<=?>GBy;?60=<9/g/>&B&.1 8f@%n\¦:8:6;V@72/8:S~748:9 <; >=?; B BD@ r%2/6F;

X[787>e . 97. =&.0. 1 8 @%6F;VZo[7>

XB?[76F;YE09dE02/6 5760872/4d6T9RE0>e@%>eQ:9/=<9/4D;?=<>&B

(n, p))>\;

2/8 872/;?>X → B(n, p)

yB?6¥~

X(Ω) = [[0, n]],>D;

∀k ∈ [[0, n]], P (X = k) =

(

n

k

)

pkqn−k,265

q = 1 − p.

13>&SD6^@%n\¦876;V576F>&8f[787>AEF2U6c@%>AQ7=<2/5:9/5760EF6F;?nGBD.%¥8f> '>D;e~ n∑

k=0

(

n

k

)

pk(1 − p)n−k = (p+ (1 − p))n = 1.

m b=%ni= $ >=?; B B rqbZRQ<jX → B(n, p)

w T\ZNML`E(X)

H0jV (X)

HWQv`'jH0Sij s H!jE(X) = np,

H!jV (X) = npq.

1 8¨Q'>D[7;2U5%;?>&876F= SD> =<n&B?[7E;9+; Bm2U6;(HE O 9R6T@%>@PO [78 SD9RETS\[:E@76F=<>&S\;&zBm2U6;>D8=?>&4d9/=<Zo[:9/8o;Zo[7>X>&Bm;[78:>

B?2/4d4d>V@%>n9/=?6T9R5:EF>GB¥@%>Ax3>&=?8:2/[7E0EF66F8@%nDQ'>D8:@:9R8o;?>GBDzo>\;y>D8 [7;?60EF6TB?9/8o;3E0>&B¥=<n&B?[7EF;<9R;<By@%[ S©:9/Q76;<=?>VBm[760+9R8o;&.

B F = 0g g $ $ 9 ' $

_- 9:" DEPC";=P 2B $,&-( 2(*;=#%>:OQ (R!"2( >)&AB(*;< (*B # &) *#1 8SD2/8:B?6T@%D=<>u4 9R608o;?>&8:9R8o;A[787> Bm[SDS\>GB?B?602/8 608%¦:8760>d@PO >D%Qn&=?60>D8:SD>&BA@%>dxy>D=<872/[7E0E06#608:@%n&Q>&8:@79/8U;<>&BA@%>dQ:9 _=9R4d\;<=?>

p. 1 8fS\2U8:B?60@%&=?>

XE09+9R=<609/57E0>J9/EFnG9+;?2U6F=<>AS\2/=<=<>&B?Q2U8:@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@%>VET9R8S\>D=By9/>&SY[787>AQ7=<2/5:9 _5760EF6F;?ne@^O 2U5%;?>&8U;<6F2U8 @7>IQ76FE0>InDgo9RE0> (

pQ'2/[7=CS©9/Zo[7>J;?60=<9/g/>U.

Vn\;?>&=?4d60872/8BAE09EF2U6¥@%>X.^b^>GBJ9/EF>&[7=BVQ'2UB<B?6F57E0>&BI@7>

XBm2U8o;V;<2/[%;<>&BAE0>&BA+9RE0>D[7=BJ>D8o;?60D=<>&BABm;?=<60S\;?>&4>&8o;

Q'2UB?6;<6FU>&BA~X(Ω) = N∗ .rs2U6; k ∈ N∗ .:XYEF2U=<B&zs>D8f[%;<6FE060B<9R8o;CE0>&B443D4d>GBN872R;9+;<6F2U8:BZU[:>IQ7=?nGS\nG@%>D4d4d>D8o;&zP (X = k) = P (S1 ∩ · · · ∩ Sk−1 ∩ Sk) = P (S1) · · ·P (Sk−1)P (Sk) = pqr−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>

XBm[:6;CET9dEF2U6^g/n&2/4dn\;<=?6TZo[7>A@%>eQ:9/=<9/4D;?=<>

pz7>\;C2U8

872/;?>X → G(p)

zBm6¥~

X(Ω) = N∗,>D;

∀k ∈ N∗, P (X = k) = pqk−1.

13>DET9d@%n\¦876;C5:6F>&8 [:87>IEF2U6^@%>IQ7=<2/5:9/5760EF6F;?nd~ +∞∑

k=1

P (X = k) =

+∞∑

k=1

pqk−1 =p

1 − q= 1.

m b=%ni= $ >=?; B B qbZRQ<jX → G(p)

w?T\ZRML`X

KWONEcH!j]~SHJH'`k^iX!MKNSILHH!j ]~SH 4 KRM'QDKNSILH s H0j

E(X) =1

p

H!jV (X) =

q

p2.

_-I 9:" >-( #K rs2U6;

r ∈ N. 1 8S\2U8:B?60@%&=?>Y4d9/6F8o;<>D8:9/8U;yE09+9R=<609/57E0>V9RE0n&9R;?2U6F=<>YS\2U=?=<>&B?Q'2/8:@79/8o;9R[ =9R8:gu@%[

r _ 6F&4d>AB?[:S&S\&B@79/8:B[787>eB?[:SDSD>&B<Bm602/8 @PO nDQ7=<>D[7U>&BN@%>Ixy>D=<872U[7EFE06^@%>IQ:9/=<9/4d\;?=<>p608:@%n&Q>&8:@79/8U;<>&B&.

1 8f9X(Ω) = [[r,+∞[[

.rs2U6;k ∈ [[r,+∞[[

.72/[:=CZU[:>[X = k]

B?2/6F;N/n&=?6F¦:n/z%60EP!9/[%;e~% 9/2/60=y2/5%;<>D8s[f[78¤Bm[:S&S\GBN9R[

k _ 60D4d>J;?60=9Rg/>Uz79/>GSY[78:>JQ:=?2U5:9R5760E06;<n p z r − 1% 9/2/60=2/5%;<>D8s[r − 1

9R[%;<=?>IB?[:SDSD&BN9R[:Q:9R=9+9R8o;&zo6F8@%nDQ'>D8:@:9R8o;<BNE0>&B[:8:BC@%>GBC9R[%;<=?>GBDz7SD>IZU[:6c9R=<=?60/>A9/>GS[787>IQ7=<2/5:9/5760EF6F;?n

pr z% 9/2/60=y2/5%;<>D8s[

k − rnGS©7>&S&BBm[7=NE0>&BY9R[%;<=?>GB;<6F=9RgU>&B&z7S\>IZo[76c9/=?=<6FU>V9U>&SV[787>IQ7=<2/5:9/5760EF6F;?nI@%>

(1 − p)k z% >\;VSD>IQ2U[7=N;?2U[%;YS©72U6@7>eE09dQ'2UB?6F;?602/8¤@%>&B

r − 1Q7=<>D4d6F>&=<BNB?[:S&S\&BCQ:9R=<4d6PEF>GB

k − 1Q7=<>D4d60>D=By;?60=<9/g/>&B&z

S\>IZo[76^Q'>D[%;Y9/=?=<6FU>D=N@%> (k−1r−1

) !9 \2/8BD.XC608:B?6z:2/8 2/5%;<6F>&8U;e~

P (X = k) =

(

k − 1

r − 1

)

prqk . 1 8f@%n\¦876;Y@72/8:S~

748:9 <; >=?; B B\p r%2/6F;X[787>A . 9:. =G.0. 1 8f@%6F;CZU[:>

XB?[76;N[78:>JE02/6P@%>A9/B<SD9/EP@%>AQ:9R=9R4d\;<=?>GB

(r, p)z7>\;

2/8 872/;?>X → P(r, p)

B?6~

X(Ω) = [[r,+∞[[>\;

∀k ∈ [[r,+∞[[, P (X = k) =

(

k − 1

r − 1

)

prqk.

13>DET9d@%n\¦876;C5:6F>&8 [:87>IEF2U6^@%>IQ7=<2/5:9/5760EF6F;?nUz%Q7[76TB?Zo[7>I@PO 9RQ7=<&BET9)2/=<4u[7E0>I@%[f57608/4d>A87nDgo9+;<6~+∞∑

k=r

P (X = k) =+∞∑

k=r

(

k − 1

r − 1

)

prqk =pr

(1 − q)r= 1.

?n B F = g g m b=%ni= $ >=?; B BdrqbZRQ<j

X → P(r, p)wTdZNML`

XKYORE4H0j ]~SbHH!`k^:X0MPKNSbILHJH0j ]~SbH 4 KNM'QDKRSILH s H!j

E(X) =r

p

H!jV (X) =

rq

p2.

e 'bf?g B BTS 1 8Q>&[%;;?=<2/[7U>D=S\>GB3=<n&B?[7EF;<9R;<ByBm2U6;Q:9R=[78fSD9/E0SD[7EP@%60=?>GSl;&z%B?2/6F;y>&8=<>D4 9/=<Zo[:9/8U;yZU[:>X>&Bm;3[:87>VB?2/4d4d>Y@%>

r9/=?6T9R5:EF>GB6F8@%nDQ'>D8:@:9R8o;?>GB3B?[7609/8o;y@%>GBE02/6TB¥gUnD2/4dnD;?=<60Zo[7>GB¥@%>YQ9R=9R4d\;<=?>

p>D;3>&8

[%;<6FE060B<9R8o;CE0>&B=<DgUEF>GBy@Ô 9U@7@%6F;?60o6F;?nGBC@%>E>\;VnD;?[:@%60nD>GBN@79R8BEF>eS©:9/Q76F;?=<>AB?[760+9R8o;&.

_-Q_ 9:" @!-";< (!-PCD"R "(rs2U6;

r ∈ N∗ 13>D;m;<>VE02/6>&Bm;;?=<&B3Q7=<2%S©7>Y@%>VE09IE02/6 @%>V9UB?S&9RE z/4 9R6TB9/[pE06F>&[ @%>VSD2/4dQ%;<>D=yEF>V=<9/87ge@%[ r _ 6F&4d>B?[:SDSD&B&z+2/8 S\2U4Q7;?>NE0>N872/457=?>N@PO n&S©:>&S9+9R8o;EF>r _ 6F&4>VBm[:S&S\GBD./rs2U6; X ET9A'. 97. =G. @P.RSD2/4dQ%;9R8o;S\>&E09:.RXCE02/=B

X(Ω) = N.^Y>dQ7E0[:B&zPEF2U=<B<Zo[Ô 2U89/=?=<6FU>9/[

r _ 60D4d> B?[:SDSD&B&zP2/89 >&[ r Bm[:S&S\GBV>D; X n&S©:>&SDB&zP@%2/8S2U89!9R6F;

X + r>\%Q'nD=<6F>&8:S\>GBD.7XY6F8:B?6 z

X + r → P(r, p). 1 8f>D8f@%nG@%[76F;YEO >\%Q7=<>&B<B?6F2U8@%>IET9dEF2U6^@%>

Xz79/Q7Q>&EFn&>

E02/6P5760872/4d6T9RE0>A87nDgo9+;?60/>Uz%>\;C2U8 9/52U[%;?6F;'(E09d@%nD¦:876F;?602/8¤Bm[:6F+9R8o;<>~748:9 <; >=?; B B[X r%2/6F;

X[:87>V . 97. =&. @P.0. 1 8@%6;NZo[7>

XBm[:6;E09uEF2U65:6F872U4d609/EF>J87nDgo9+;<6FU>Y@%>JQ:9R=9R4d\;<=?>GB

(r, p)z:>\;C2/8 872/;?>

X → J (r, p)z:B?6¥~

X(Ω) = N>\;

∀k ∈ N, P (X = k) =

(

k + r − 1

k

)

prqk.

13>&SD63@%nD¦:876F;u5760>D8 [78:>pEF2U63@%> Q7=?2U5:9R5:6FE06;<n/z^Q:[760B<Zo[7>pET9¤Bm2U4d4>@%> S\>GBIQ7=<2/5:9/5760EF6F;?nGBI+9R[%;1@PO 9/Q7=<&BAET9

)2/=<4u[:EF>I@%[f57608/4d>A87n&gU9+;<6`.N>&4d9/=<Zo[7> Zo[7>¤EF>S\2s> pS\60>D8o; (r−1

k

) S\2U=?=<>&B?Q2U8:@ 9R[ S©72/6F @7> E09 Q'2UB?6F;?602/8 @%>&Bkn&S©7>GSDB Q:9/=?4d6CE0>&B

r − 1Q7=?>&4d6F>&=<B;<6F=9RgU>&B&z7Zo[7>

qk SD2/=<=?>GBmQ'2/8:@ (dET9 Q7=<2/5:9/5760EF6F;?ne@PO 9/2U6F=NE0>&B k n&S©7>GSDBC9/[%fQ2oBm6F;?602/8:BV9R608:Bm6@%nD;?>D=<4d6F8:nD>&B&z:>D;VZo[7>pr S\2U=?=<>&B?Q2U8:@ (dE09pQ7=?2U5:9R5:6FE06;<nu@PO 9/2U6F=C@%>GBYB?[:SDSD&BV9R[% r 9/[%;?=<>&BY>\%Q'nD=<6F>&8:S\>GB )¬S\2U4dQ7=?6TBET9@7>D=<876F&=?>l.

m b=%ni= $ >=?; B BYZrqbZRQ<jX → J (r, p)

w?T\ZNML`X

KWONEcH!j]~SHJH'`kîX!MPKRSILHH!j]~SH 4 KRM'QDKNSILH s H0jE(X) =

rq

p

H!jV (X) =

rq

p2.

!, !,%, (, * (#'&" (, , ' (,%,? N, ,%" ,-&(. 0,?(rs2U6;

N ∈ NzP>\;

p ∈]0, 1[zq = 1 − p

. 1 8 ;<6F=<>d4d9/6F8o;<>D8:9/8U;IB?[:SDSD>&B<Bm60/>&4>&8o;V>\;eB<9R8:BJ=<>D4d6TBm>d@79R8BV[:87>[7=<87>eS\2U8o;?>D89R8o;C6F8:6;<609/EF>&4>&8o;

N52U[7EF>GBDz@%2/8o;

Np5'2/[:EF>GBN57ET9R8:S©7>GBN>\;

Nq5'2/[7E0>&BC872/60=?>GBD.XC608:Bm6 z

p>GB`;

ET9Q:=?2UQ2U=m;<6F2U8 60876;<609/EF>e@7>I52U[7EF>GB57ET9R8:S©7>GBD. 1 8f9RQ:Q>&EFE0>eBm[SDS\GBEF>J!9/6;Y@%>A;<6F=<>D=N[:87>I52U[7E0>J5:E09/8:S©7>/._-I 9:" ,(*BDEPC";=P 2B $,&-(rs2U6;

n ∈ [[0, N ]]. 1 8f> >&S\;?[7>

n;?60=<9/g/>&B&zs>\;C2U8 8:2R;?>

XEF>I8:2/4u5:=?>A@%>eB?[:SDSD&B2/57;?>D8s[:B&.

XCE02/=BXQ>&[%;YQ7=<>D8:@7=?>I@%>GB9/EF>&[7=B>D8o;?=<>

0>D;nz(©:9R[7;?>D[:=C@%[f872/457=?>A4 9+%604 9RE^@%>eB?[:SDSD&BQ'2UB<Bm6057E0>/z

(pB<9/2U6F=Np.V>uQ:EF[:B&z Bm6

n > Nqz6FE#8Ô ¬¤9pQ:9UBYB?[pB?9/44d>&8U;J@%>u5'2/[7E0>&BV872U6F=<>&BNQ'2/[:=V87>Q:9/BC;?60=?>&=V@%>

5'2/[7E0>5:E09/8:S©7>¥]'Q7E0[:BYQ:=?nGS\6TBmn&4>&8o;&z'2/8@72/6F;Y;?60=<>D=A9R[ 4d2/608:Bn −Nq

5'2/[7E0>&BJ57ET9R8:S©:>&B&.'XY6F8:B?6 zX(Ω) =

[[max(0, n−Nq),min(Np, n)]].

rs2U6;k ∈ [[max(0, n − Nq),min(Np, n)]]

.cVn\;?>&=?4d60872/8BAE09 Q7=<2/5:9/5760EF6F;?n @%>dE O n&/nD8:>D4d>D8o;[X = k]

. 1 89RQ:Q>&EFE0> =<n&B?[7EF;<9+;@%>pE O >DsQ'nD=<60>D8:SD>pEF> Bm2U[:B _ >&8:Bm>&4u57E0>p@%>GB n 5'2/[7E0>&Be;?60=?n&>&Bu@7> E O [7=<87>Uzc>D8¨Q7=?>&8:9R8o;eQ'2/[7=©s¬sQ2/;?©7GBm>ZU[:>uE0>&BJ52U[7E0>&BJ@%>u443D4d>dS\2U[7EF>&[7=ABm2U8U;A@%6TB?SD>D=<8:9R57E0>&B&. 13©9/Zo[7>B?2/[B _ >D8:B?>D457E0> ( n n&EFn&4>&8o;

B F = 0g g $ $ 9 ?; ' $ :p

@%>IE O >&8:B?>D4u5:EF>I@%>A;<2/[%;<>&BE0>&B5'2/[7E0>&BC9ET943&4d>IQ7=?2U5:9R5:6FE06;<nI@PO 3D;?=<>J;?60=?nUz7Q7[76TB?Zo[7>J;<2/[%;<>&BNE0>&B5'2/[7E0>&B2U8o;@%>GBC= /E0>&BCB?¬s4nD;?=<60Zo[7>GBD.'XC608:B?6z'2/8¤>GB`;V@:9R8:BY[787>uS&9/BY@Ô nGZU[:6FQ7=<2/59R5760EF6F;?nUz:>D;Y2/8¤Q'>D[%;A[%;<6FE060B?>D=VE09 )2/=<4u[7E0>@%>eb 9RQ7ET9/SD>/.b^> 872U4u57=<>p;<2R;<9/Ey@7>=<n&B?[7EF;<9+;Bu>GB`;dE0>872U4u57=<> @%>fB?2/[:B _ >&8:B?>D4u5:EF>GB ( n nDE0nD4d>D8o; @PO [:8¨>D8Bm>&4u57E0> ( NnDE0nD4d>&8U;BDz( B<9/2U6F= (N

n

)

C8=?nGBm[:E;9+;39U@%4d60B<Bm6057E0>Y>GB`;y[78B?2/[:B _ >D8:B?>D457EF>CSD2/8o;?>&8:9R8o; k 52U[7EF>GB57ET9R8S©7>&B3>\; n− k5'2/[7E0>&B3872/60=<>&B&./ijE

B?[d;y@%2U8:SC@%>JS©72/6TBm60=¥E0>&Bk5'2/[7E0>&B57E09/8:S©7>GB;?60=<nD>&BQ:9R=<4d6:E0>&B

Np5'2/[7E0>&B57E09/8:S©7>GB@%2U8U;y2/8p@%6TB?Q2oBm>UzUS\>

Zo[76 Bm>Y!9R6F;y@%> (Npk

) !9 \2U8:B¥Q'2UB<B?6F57E0>&B&z/>D;@7>VS©72U60B?6F=3EF>GBn− k

52U[7EF>GB8:2/60=?>GB;?60=?n&>&B3Q:9R=<4d6:E0>&BNq

5'2/[7E0>&B872U6F=<>&B&z7S\>IZo[76cB?>A!9R6F;Y@%> ( Nq

n−k

) !9 D2/8:B&.7XC608:B?6ÊF>I8:2/4u5:=?>I@%>I=<n&B?[7EF;<9+;BN9/@%4d6TB?B?6057EF>GB>&Bm; (Npk

)(

Nqn−k

) . 1 8>D8 @%nG@%[76F;eE09 E02/6¥@%>

X>\;I2U8 @7n\¦:876F;eET9E02/6©s¬sQ'>D=<g/nD2U4dn\;?=<6TZU[:>@%>dQ:9/=<9/4d\;?=<>

(N,n, p)SD2/4d4d>nD;<9/8U;

S\>D;m;<>IEF2U6~748:9 <; >=?; B BD@ r%2/6F;

X[:87>e . 9:. =G.0. 1 8f@76;VZo[7>

XB?[76F;Y[787>IE02/6^©s¬sQ'>D=<g/nD2U4dn\;?=<6TZU[:>J@%>uQ:9R=9R4d\;<=?>GB

(N,n, p)z7>D;C2/8 872R;<>

X → H(N,n, p)z:B?6~

X(Ω) ⊂ [[0, n]]>\;

P (X = k) =

(

Npk

)(

Nqn−k

)

(

Nn

)

13>\;?;?>e>D%Q7=?>GB?B?602/8f@%n\¦:8:6;J576F>&8¤[787>eE02/6 @%>eQ7=<2/5:9/5760EF6F;?nUz:SD9/=CE09 B?2/4d4d>u@%>GBNQ7=<2/5:9/5760EF6F;?nGBCS\6 _ @7>&B<Bm[:BC>&Bm; 1@PO 9RQ7=<&BET9)2/=<4u[7E0>I@%>)¥9R8:@7>D=<42U8:@%>U.e 'bf?g B B 1 8HQ>&[%;AB?>SD2/8o;?>&8o;?>D=A@%>Q7=?nGS\6TBm>&=YE O 608:SDEF[:B?602/8

X(Ω) ⊂ [[0, n]]zPS&9R=YQ'2/[:=Y;<2/[%;

k ∈

[[0, n]]8cO nD;<9R8o;JQ:9/BV@79R8:B

X(Ω)zE O >D%Q7=?>GB?B?602/8¤@%n\¦8760B<B<9R8o;

P (X = k)>&Bm;V8s[7E0EF>UzQ:9/=VSD2/8s/>&8U;<6F2U8¤Bm[7=VEF>GB

S\2s> SD6F>&8o;<B5760872/4d6T9R[%P.132/4d4d>

X(Ω)>&Bm;¦:876 z%EO >\%6TB`;<>D8:SD>A@7>VE O >GBmQ'nD=9R8S\>J>\;C@%>AE09+9R=<609/8:S\>J>&Bm;C9R[%;<2/4 9+;<60Zo[7>Uzs>\;N2/82/5%;<6F>&8U;I~

m b=%ni= $ >=?; B B rqbZRQ<jX → H(N,n, p)

wT\ZRMLÈ(X) = np

H0jV (X) = npq

N − n

N − 1

w_-I (*;= # >NOQ 2(*!+2( >& )B(*;<(CB # &) R#1 8@7n\¦:876F; 4 9R608U;<>D8:9/8o;

XE09+9R=<609/57EF> 9RE0n&9R;?2/60=<> nDgo9RE0>9/[¨=9R87g @PO 9RQ7Q:9/=?6F;?602/8@%>ET9 Q7=<>D4d6F&=?>52U[7E0>

57ET9R8:S©:>/.7XC[fQ760=<>/z%2/8 ;<6F=<>I@PO 9/52U=<@ ;?2U[%;?>GBEF>GBN5'2/[7E0>&B872U6F=<>&B&.7XC608:B?6zX(Ω) = [[1, Nq + 1]]

.rs2U6;

k ∈ [[1, Nq+1]].s1y9/E0SD[7EF2U8:B

P (X = k). 1 8d;?60=?>

k)2/6TBD. 1 8p9RQ7Q'>DE0E0>Y=<n&B?[7E;9+;3@7>CEO >\%Q'nD=<6F>&8:S\>J@%>&B

k;?60=9Rg/>GB#E09A@%2U8787nD>N@%[ SD2/[7Q7E0>(E, b)

zR215E>&Bm;¥EF>NB?2/[:B _ >D8:B?>D457EF> ( k nDE0nD4d>D8o;<B@7>&B5'2/[7E0>&B#;<6F=<nD>GBDz+>D; b >GB`;ET9e5'2/[7E0>Y;<6F=<nD>VEF2U=<B3@%[

k _ 60D4d>C;<6F=9RgU>/zo@%2/8:S b >&Bm;y[:8pnDE0nD4d>D8o;N@%6TB`;<6F87gU[7nV@7> E .7132/4d4d>YQ:=?nGS\n&@7>D4d4d>D8o;&zE0>&B @76 nD=<>D8o;<B=<n&B?[7E;9+;BpQ'2UB<Bm6057E0>&BpB?2/8o; nGZU[:6FQ7=<2/59R57E0>&B&zQ7[:60B<Zo[7> E0>&Bp5'2/[:EF>GBek`2U[7>D8o;p;<2/[%;<>&B @%>&B =UEF>GB6T@%>D8o;?6TZo[7>&B&. N;<6FE06TBm2U8:BC@%2U8:SJE09)2/=<4u[:EF>I@%>eb 9RQ7ET9/SD>/.b^>Y872/457=?>C;?2R;9RE'@%>Y=<n&B?[7EF;<9+;B>&Bm;y@72/8787nYQ:9R=3EF>YS©:2/6Fd@PO [78 B?2/[B _ >D8:B?>D457E0>( k nDE0nD4d>D8o;<By@%>CE O >D8:B?>D457EF>@%>Y;?2/[7;?>&B3E0>&B¥5'2/[:EF>GB&!6FE'¬ 9 (N

k

) S©72/6FdQ2oB?B?6F5:EF>GB ¥>\;y@Ô [78 nDE0nD4d>D8o;3Q9R=<46'S\>GBk!6FE ¬d9

kS©72/6Fl.sXC608:B?6z

E0>I872/457=?>V;?2/;<9/Ec@%>A=?nGBm[:E;9+;GB`;k(

Nk

) .C8=?nGBm[:E;9+;9U@%4d60B<B?6F57E0> >GB`;d[78 S\2/[:Q7EF>

(E, b)2U[

E>GB`;[78B?2/[:B _ >D8:B?>D457EF>S\2/8o;<>D8:9/8U; k − 1

52U[7EF>GB872U6F=<>&B3>\;N[787>J5'2/[7E0>V5:E09/8:S©7>/zs>D;

b>&Bm;SD>\;?;?>J52U[7EF>A57ET9R8S©7>/.sXC608:B?6z%[78:>Y)2U60ByE0>AS©:2/6F @7>

E> >&Sl;<[7n/zs2U8

8Ô 9eQ7E0[:ByE0>VS©72U6dQ2U[7=b. 1 =GzoQ2U[7=yS©:2/6TBm60=

Ezo6FE'!9/[%;S©:2/6TBm60=

k− 15'2/[7E0>&B3872/60=<>&B¥Q9R=<46'E0>&B

Nq52U[7E0>&B

872U6F=<>&B&zR>\;15'2/[7E0>C57ET9R8:S©:>CQ:9R=<4d6:E0>&B

Np52U[7E0>&B357E09/8:S©7>GBDz/SD>YZo[76 Bm>N!9/6;@%> (Np

1

)(

Nqk−1

) !9 D2/8:B&./XC608:B?6 ~m b=%ni= $ >=?; B B KFT\ZRQ%ON]4jlH!E^Ò KRjCjlH0SijlH

XON]^bMPH!EQDH0M`']iII !` T\ZRMLÒ ]~S jkQ<MPKWH.`0KNS:`MPH0EQv`!HOYH

îKNMPKRE 0jCMH'`(N, p)

H!`'jX(Ω) = [[1, Nq+1]]

H0j∀k ∈ [[1, Nq+1]], P (X = k) =

Np

k·

(

Nqk−1

)

(

Nk

) .

: ?n B F = g g _-II (*;= # >NOQ 2(*!+2( >&

r 0 *;=(?# & *#V>I43&4d>/z7>&8fS©:2/6TBm6TB?B<9R8o;S\>\;?;?>A)2U60B

k − r5'2/[7E0>&BN8:2/60=?>GBDz

r5'2/[:EF>GB57E09/8:S©7>GBDz%>D;C[787>I5'2/[7E0>I@%60Bm;?6087gU[7nD>

Q:9/=?4d6^S\>GBr5'2/[:EF>GB57E09/8:S©7>GBDz%2U82U5%;?60>D8o;CET9E02/6^@7[;<>D4dQ:BC@^O 9R;m;?>&8o;?>e@%[

r _ 60D4d>eB?[:SDSD&B r ∈ N∗ J~

X(Ω) = [[r,Nq + r]]>\;

∀k ∈ [[r,Nq + r]], P (X = k) =r

k·

(

Npr

)(

Nqk−r

)

(

Nk

) .

* ( !,?,% 1 8@7n\¦:876F;d[787>f@%>D=<8760D=<> E02/6 zZo[7687>fS\2U=?=<>&B?Q'2/8:@ Q:9UB( [787> >\%Q'nD=<6F>&8:S\> Q7=<n&SD60B?>/z#4 9R6TBZo[76N9RQ:Q:9R=9T;B?2/[7U>D8o;SD2/4d4d>I4d2s@7nDE060B<9+;<6F2U8^z%2/[fS\2U44d>AE02/6^E06F4d6F;?>U.748:9 <; >=?; B BD@ r%2/6F;

λ ∈ R+>\;X[787>A . 97. =&.0. 1 8f@%6F;CZU[:>

XBm[76F;E09EF2U6P@%>A#2U60B<B?2/8 @%>AQ:9/=<9/4d\;?=<>

λz7>\;C2U8872/;?>

X → P(λ)zB?6~

X(Ω) = N>\;

∀n ∈ N, P (X = n) = e−λ ·λn

n!.

O 9/Q7=?GBE0>&B3=<n&B?[7E;9+;B3B?[7=yEF>GByB?nD=<60>&B¥>D%Q2U87>D8o;?60>DE0E0>&B&zoSD>DET9u@%nD¦:876F;5:6F>&8p[787>JEF2U6P@%>YQ7=<2/59R5760EF6F;?nU.oV>JQ7EF[BV~m b=%ni= $ >=?; B B rqbZRQ<j

X → P(λ)w%T\ZNML`

XKYORE4H0j]~SHH!`k^:X0MPKRSILHJH0j ]~SbH 4 KNM'QDKRSILH s H!j

E(X) = λH0j

V (X) = λ.

"F/!(" & "L# /!"$#%

C2U4 9R=9R4dD;?=<>&B C2R;9+;<6F2U8 ¥9/EF>&[7=B bc2/6E(X) V (X)

C876F)2/=<4d>n ∈ N∗ U(n) [[1, n]] P (X = k) =

1

n

n+ 1

2

n2 − 1

12xy>D=<872U[7EFE06p ∈]0, 1[ B(1, p) 0, 1 P (X = 1) = p p pq

xy6F8:2/4d609/EF>(n, p) B(n, p) [[0, n]] P (X = k) =

(

n

k

)

pkqn−k np npq

nD2U4nD;?=<60Zo[7>p G(p) N∗ P (X = k) = pqk−1 1

p

q

p2

9/B<SD9/E(r, p) P(r, p) [[r,+∞[[ P (X = k) =

(

k − 1

r − 1

)

prqk−r r

p

rq

p2

xy60872/4d6T9RE0>J8:nDgU9R;?60/>(r, p) J (r, p) N P (X = k) =

(

k + r − 1

k

)

prqk rq

p

rq

p

¬sQ'>D=<g/nD2U4dn\;?=<6TZU[:> (N,n, p) H(N,n, p) ⊂ [[0, n]] P (X = k) =

(

Npk

)(

Nqn−k

)

(

Nn

) np npqN − n

N − 1

Xy;m;<>D8o;?>e@7[1

B?[:SDSD&B ;<6F=9RgU>IB?9/8:B=?>&4d60B?> (N, p) [[1, Nq + 1]] P (X = k) =

Np

k·

(

Nqk−1

)

(

Nk

)

Xy;m;<>D8o;?>e@%[r B?[:S&S\&B

;<6F=9RgU>IB?9/8:B=?>&4d60B?> (r,N, p) [[r,Nq + r]] P (X = k) =r

k

(

Npr

)(

Nqk−r

)

(

Nk

)

2/6TB?B?2/8λ ∈ R+ P(λ) N P (X = k) = e−λ ·

λk

k!λ λ

§ v

1 8dBDO 6F8o;<nD=<>&B<Bm> (JE09AS\2U=?=<nDET9+;<6F2U8e>\%6TB`;9R8o;>D8o;?=<>Q7E0[:Bm60>D[:=<B#+9R=<609/57E0>&B9RE0n&9R;?2/60=<>&B&zG>&88:2/4u5:=?>3¦:876!/>GSl;?>&[7=B9RE0n&9R;?2U6F=<>&B V2/[ 608%¦:876) !9/460E0EF>GBu@%>p . 97. =&. @P. l.#V9/8:BI[78 Q:=?>&460>D=e;?>D4dQ:B&zc2U8 n\;<[:@%60> E0> SD9UBe@PO [78 S\2U[7Q7E0>(X,Y )

@%>'. 97. =G. @P.0z\@%2U8:S¥@PO [78A/>GSl;?>&[7=c9/EFnG9+;?2U6F=<>@%>¥@%604d>D8:B?6F2U8eL%.&b 9yQ7EF[:Q:9R=?; @%>GB^@%n\¦876;<6F2U8:BP>D;c=?nGBm[:E;9+;<BB?>JgUnD87n&=<9/EF6TB?>D8o;3!9US\60EF>&4d>D8o;Y9R[%/>&S\;?>&[7=<B9/EFnG9+;?2U6F=<>&B@%>e@76F4d>D8Bm602/8

nZo[7>&E0SD2/8:Zo[7>U.

4 !, * /!(J9R8:By;<2/[%;<>eS\>\;?;?>eB?>&S\;?602/8^z

X>\;Y@%nGBm60g/87>&8o;N@7>D[% '. 97. =G. @P.0.

,E B @& # #1 PC(=&)! "&)(?>)( Q,QB* >Nm b=%ni= $ >=?;hp BD@:BC@ K!KREJQ<T<T\HO X 4 X0SH0E4H0SijC` ([X = x] ∩ [Y = y]

)

(x,y)∈X(Ω)×Y (Ω)

H'`'j]~SH !KNEQ<T<T\HO X 4 X!SbH!EcH!SjD`OWH!]Y OWH!]YQ<SILZRE.^iKNjkQD[!TdH'` s OWZNSj T ]~SQDZNS H'`'jXzKNT\H Ω

we 'bf?g p BD@iB ijE8:>pBDO 9RgU6;A>D8 g/n&87nD=9REQ:9UBI@PO [78 Bm¬%Bm;?D4d> S\2U4dQ7EF>D;&zcS&9R=IS\>&=m;9R6087>&BJQ:9/=m;BAQ>&[7/>&8o;3\;<=?>As6T@%>&B&.748:9 <; >=?; p BC@:B bc>c`6`'j 0E4H4ILZNE^TdH!jKN``0Z I!QDXKN] ILZN] ^bT\H

(X,Y )>GB`;VEF>uB?¬%B`;<D4d>uSD2/4dQ7E0>\;Y2U5%;?>&8o[ (

Q:9/=m;<6F=C@%>AET9!9R4d60EFE0> ([X = x] ∩ [Y = y]

)

(x,y)∈X(Ω)×Y (Ω)

>D8fBm[7Q:Q7=?604 9R8o;NEF>GBNQ:9/=m;Bo6T@%>GBD.748:9 <; >=?; p BC@:B b 9 jCM'QD[0] K6`L`0ZI0QDXLH KR] TdHILZN] ^bT\H

(X,Y )>GB`;pET9;?=<6F5:[ >&87g/>&8:@%=<nD> Q:9R= E0>¤Bm¬%Bm;?&4>

S\2U4dQ7EF>D; 9/B<Bm2%SD6Fn ((X,Y )

. 1 8 E09 8:2R;?> B?2/[:/>D8o;A(X,Y )

2/[T(X,Y )

)872/;<9R;?602/8 ( =?>G@%n\¦876F=@79R8:Bp[78@%>&/2/60= l.m b=%ni= $ >=?;hp BD@:Bdp=3S KcTdH'`FQ<SI0T_]a`'QDZNSi`F`']~Q 4 KRSijlH'` AX ⊂ A(X,Y )

H0jAY ⊂ A(X,Y )

wbc9A@%nD¦:876F;?602/8 @%>NET9V;?=<6F57[ 9UB?B?2%S\60nD> (V[:8dSD2/[7Q:EF>NB?>Ng/nD8:nD=9RE060B?>3604d4dn&@%6T9+;<>D4d>D8o; (V;?2U[%;?>!9/460E0EF>

(Xk)k∈K@%>I+9R=<6T9R57E0>&B9RE0n&9R;?2U6F=<>&B&. 1 8872/;?>&=<9A(Xk)k∈K

.

,E 9:" "!*LK"!+ ( "#%;= BD"!- (*# = >=%; p BC@:B\ 2U[7=39/EFE0nDgU>D=E0>&B3872/;<9+;<6F2U8:B&z/2/8 872R;<>

P (X = x, Y = y)Q2U[7=

P ([X = x]∩ [Y = y]).

X ?n pB :g 8 = 748:9 <; >=?; p BC@:BTSb 9 T\ZNQ ILZRSG0ZNQ<SjH@%[fS\2U[7Q7E0>

(X,Y )>&Bm;CE09d@%2U8787nD>I@%>IET9)2/8:S\;?602/8 ~

p : X(Ω) × Y (Ω) −→ [0, 1]

(x, y) 7−→ P (X = x, Y = y).

;9R8o;@%2U8787np@%>GBenD8s[74dn&=<9R;?602/8:B(xi)i∈I

>D;(yj)j∈J

@%>X(Ω)

>\;Y (Ω)

I¦:87632/[

I = N] @%>p443D4d>

Q'2/[7=J\z/ET9J@72/8787n&>C@%>CET9AEF2U6:S\2U8+k`2/608o;?>N@%>

X>\;Y>&Bm;¥n&Zo[7609/EF>&8o;?>(AET9J@72/8787n&>C@%>CET9IBm[:6;<>C@%2/[75:EF>&4>&8o;

608:@%>\%n&>(pi,j)(i,j)∈I×J

z7215~∀(i, j) ∈ I × J, pi,j = P (X = xi, Y = yj).

748:9 <; >=?; p BC@:B[X -/.Cbc9F^MPH0EQ !MPHT\ZNQEcKNMkRQ<SbKNT\HI@7[S\2U[7Q7E0>(X,Y )

>GB`;E09E02/6P@%>JE09u9/=?6T9R5:EF>J9RE0n&9 _;?2/60=<>X.s;9R8o;C@%2/878:nV[78:>Vn&8o[:4n&=<9R;?602/8

(xi)i∈I

@%>X(Ω)

z%E09@%2/8:87nD>A@%>JE09Q7=<>D4d6F&=?>JE02/6 4 9R=<g/6 _8:9RE0>A>&Bm;nGZo[76F+9/EF>&8U;<> (E09d@%2U8787nD>I@PO [787>eB?[76F;?>Uz7g/n&87nD=9RE0>D4d>D8o;872R;<nD>(pi,•)i∈I

z7265 ~∀i ∈ I, pi,• = P (X = xi).

L7.Cbc9`!HLILZNSbOYHT\ZNQEcKNMCYQ<SKRT\H@%[ SD2/[7Q:EF>(X,Y )

>&Bm;NE09E02/6P@%>AE09+9R=<609/57E0>J9/EFnG9+;?2U6F=<>Y.%;<9/8o;N@72/8787n

[787>Yn&8s[74dnD=9+;?602/8(yj)j∈J

@%>Y (Ω)

zoET9u@72/8787n&>V@%>JET9uB?>&S\2U8:@%>CE02/6'4 9R=<g/608:9RE0>C>GB`;n&Zo[760+9RE0>D8o;?>'(uE09@%2/878:nD>I@PO [787>eB?[76;<>/z7gUnD87n&=<9/EF>&4d>D8o;y8:2R;?n&>

(p•,j)j∈J

z7215~∀j ∈ J, p•,j = P (Y = yj).

m b=%ni= $ >=?;hp BD@:BYZ1. ∀i ∈ I, pi,• =

∑

j∈J

pi,j 2. ∀j ∈ J, p•,j =∑

i∈I

pi,j

w

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[Ô 2U8u9YB?2/4d4dnB?[7= ;?2/[7;?>&BE0>&B+9RE0>D[7=BQ'2UB<Bm6057E0>&BN@%>IE O 608:@%6TS\>eZo[Ô 60EP=?>&4Q:E09US\>/.e 'bf?g p BD@iBD@:@¨b^>&BÊF2U60BP4 9/=?gU6F8:9/EF>GB^B?2/8o;c@%n\;<>D=<4d6F87n&>&BPQ9R=cET9E02/6USD2/8+k`2U6F8o;<>/. oU)#(+ # * "##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=<46087>&=C>D8:B?[76F;?>AE0>&BNE02/6TBN@%>X>D;Y.

m b=%ni= $ >=?;hp BD@:BC@ DMPZL^M'QDX0jX4ILKNMKYI0jX0M'Qv`'jCQD0]iH4O ]~SHT\ZRQ IZRSG0ZRQ<SijlHl MPXLI0Q ^MPZ !]iHKYOREJQv`0H 7%SbH$`']~Q<jlH

(pi,j)i∈I,j∈J

IH0jJtuSQv`ZR]OYX0SZRE4[0MPKY[0T\H!`H'`'j TdKcT\ZRQ O ]~S IZR] ^T\H

(X,Y )``'Q

xYw∀(i, j) ∈ I × J, pi,j > 0 yzw ∑

(i,j)∈I×J

pi,j = 1w

e 'bf?g p BD@iBD@ |c2U[%;?>GBÊ0>&B#Bm2U44d>GB SD2/8:B?60@7nD=<nD>&B @79R8BcS\>yQ:9/=<9/g/=9RQ7©7>B?2/8o; (C;?>&=?4d>&B Q'2UB?6;<6!B&.GV2/8SE0>D[7=SD2/8s/>&=?gU>D8:SD>n&Zo[7609/[%; (IE0>D[7=3S\2/8sU>D=<g/>D8S\>N9R5:B?2/E0[7>Uz+>\;E O 2U=<@7=?>C@%>VBm2U44 9R;?602/8d6F4dQ'2/=?;?>NQ'>D[^.sXC608:B?6z;?2U[%;?>GBEF>GBCB?2/4d4d>&BNZo[Ô 2/8f9SD2/8:B?6T@%nD=<nD>GByB?2/8o;N5760>D8¤@%nD¦:8760>&B&.,EI 9:"# "!)> "!!-(* (*#748:9 <; >=?; p BC@:BD@ rs2/6F;

y ∈ Y (Ω);?>&EoZo[7>

P (Y = y) 6= 0. 1 8e@%nD¦:876F; ET9 4 KRM'QDKY[0T\HKRT\XLKNjlZNQ<MPHIZRSORQ<jCQDZRSiSH0T<T\HV@%>

XB<9/S©9R8o;

Y = jzR>\;¥2U8872/;?>

(X | Y = j)E09J+9R=<609/57E0>9/EFnG9+;<2/60=?>yQ7=<>D8:9/8U;E0>&B+9RE0>D[:=<BQ'2UB<Bm6057E0>&B

@%>XB<9/S©:9/8o;yZo[7>

Y = yzs>\;C@%2/8o;E09E02/6 >&Bm;@72/8787n&>AQ2U[7=y;?2/[7;

x ∈ X(Ω)Q:9/=

P (X = x | Y = y).7bc9

T\ZRQ ILZNSORQ<jCQDZRSiSbH!T<T\H4OWHX`0KYI :KRSij

Y = j>&Bm;ET9dE02/6^@%>

(X | Y = j).

p B L; 8 n:; ?; 8 = Z

m b=%ni= $ >=?;hp BD@:BC@Yp jlKNSj OYZRSiSbXF]~SHFX0Si]~EcX0MPKNjkQDZNS(xi)i∈I

OYHX(Ω)

H0ju]~SHX!S]~E4X0MPKNjkQDZNS(yj)j∈JOYH

Y (Ω) s KRQ<S:`'Qu!]u ]~SQ<SbONQDILH j ∈ J s T\K4T\ZRQILZRSONQ<jkQDZNSSH0T<T\HOWH X `!KWIiKNSj !]iHY = yj

H!`'j OYZNSSXLH îKNM

∀i ∈ I, P (X = xi, Y = yj) =pi,j

p•,j

.

m b=%ni= $ >=?;hp BD@:BC@Y uH0jCjlHF`']~Q<jlH ( pi,j

p•,j

)

i∈I

OWXDt SiQ<j[!QDH0S]~SbHFT\ZRQ OYH^MPZW[LKY[0Q<T_Q<jX w

4 * , ( *" ( * ( "F&)!"F/!(, "/ "$#%!&(,, "&- (*# >-( ' BC!)>)PC(*!>- !+ (R#J9R8:BNSD>IQ:9R=9RgU=<9/Q7©7>/z

X>\;Y@%n&B?60g/87>&8U;C@%>&[% . 97. =&. @P.0.

748:9 <; >=?; p B BD@ 1 8 @%6F;Zo[7>X

H0jY`0ZNSjQ<SbOYXl^:H0SOWKNSijlH!` Bm63Q2U[7=I;?2U[%;

x ∈ X(Ω)>D;uQ'2/[7=I;<2/[%;

y ∈ Y (Ω)z7E0>&BnDUnD87>&4d>D8o;\;

[Y = y]Bm2U8o;608:@%n&Q>&8:@79/8U;BD.

m b=%ni= $ >=?;hp B B H'`^MZîZ6`'Q<jCQDZRS:`F`']~Q 4 KNSijlH!`V`0ZRSijXL0]~Q 4 KRT\H0SijlH'`xYwX

H!jY`0ZNSijQ<SOWXîH0SOYKRSijlH'` yzwT\H'`$`6`'j 0EcH'ÌLZNE^T\H0jD`

([X = x])x∈X(Ω)H0j

([Y = y])y∈Y (Ω)`0ZNSijQ<SOWXîH0SOYKRSijC` |Ww H!`

σKRT [0MPH!`

AX

H0jAY

`0ZNSj Q<SbOYXl^:H0SOWKNSijlH!`!wm b=%ni= $ >=?;hp B B rqQ

XH0jY`0ZNSij Q<SOWXîH!SbOYKRSijlH'` s KRT\ZNML`J

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

]~jkMPH!EcH0Sij ORQ<j s X0jKRSijOYZNSSXOYH!`JX!S]~E4X0MPKRjCQDZRS:` (xi)i∈I

H!j(yj)j∈J

∀(i, j) ∈ I × J, pi,j = pi,• · p•,j .

XC608:B?6z%B?6X>\;YB?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.

, - ;< (R#>)( ' BC!)>)PC(*!>- !+ (R#rs2U6;

(Xk)k∈K

[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

Xk

z Q'2/[7=k ∈ K

z B?2/8o;A4u[%;<[7>DE0E0>D4d>D8o;I6F8@%n _Q'>D8:@79/8o;?>&BJB?6 Q'2/[7=Y;?2U[%;?>u!9R4d6FE0E0>(xk)k∈K

;?>&EFE0>Zo[7>Q2U[7=Y;<2/[%;k ∈ K

zxk ∈ Xk(Ω)

z EF>GBYn&/nD8:>D4d>D8o;<B[Xk = xk ]

z%Q'2/[7=k ∈ K

z7B?2/8o;N4u[%;<[7>DE0EF>&4d>D8o;C6F8@%nDQ'>D8:@:9R8o;<B&.m b=%ni= $ >=?;hp B Bdp H'`^MZîZ6`'Q<jCQDZRS:`F`']~Q 4 KNSijlH!`V`0ZRSijXL0]~Q 4 KRT\H0SijlH'`xYwT\H'` 4 KNM'QDKW[!T\H!` Xk s îZN]~M k ∈ K s `!ZRSij EJ]~jk]iH!T<TdH!EcH!Sj Q<SbOYXl^:H0SOWKNSijlH!` yzwT\H'`$`6`'j 0EcH'ÌLZNE^T\H0jD`

([Xk = x])x∈Xk(Ω) s îZN]~M k ∈ K s `0ZNSijE]~jC]iH0T<T\H0E4H0SijuQ<SbOYXl^:H0SOWKNSjD` |WwT\H'`σKRT [0MPH'`

AXks ^:ZR]~M k ∈ K s `0ZNSj E]~jC]iH0T<T\H0E4H0Sij Q<SOYXlîH!SbOYKNSjH!`0w

2\ ?n pB :g 8 = ,I "PR!-(*;=(*!+# "!#62B & # OQ >)(?>)( Q,QBCN!)>)PC(*!)>) !"2(*#rs2U6;

(Xk)k∈K

[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@Ô [78 S\>&=m;9R608p872/457=<>Y@%>&B3+9R=<609/57E0>&B

Xk

~sB?2/6F;K ′ ⊂ K

]7@%60=?>eZo[7>E87>e@%n&Q>&8:@fZo[7>I@%>&BN+9/=?6T9R57E0>&B

Xk

z%Q'2/[:=k ∈ K ′ Bm60g/876F¦:>IZo[7> E ∈ A(Xk)k∈K′

.rs[7Q:Q2oBm2U8:BIZo[7>dEF>GBA9/=?6T9R5:EF>GB

Xk

zk ∈ K

z^B?2/8o;I4u[%;<[7>DE0E0>D4d>D8o;I6F8@%nDQ'>D8:@:9R8o;?>GBD.cbc>dEF>&44d> K7. L7. WQ>&= _4d>\;@7>d)2/=<4d9/EF6TB?>D=IE0> =<9/60B?2/878:>D4d>D8o;I6F8o;<[76;<6YB?[760+9R8o;p~cB?6E>\;FBm2U8o;u@%>D[% nDUnD87>&4>&8o;<Be@%n&Q>&8:@79/8U;

[7876TZo[7>D4d>&8U;Y@7>I9/=?6T9R5:EF>GBXk

B?9/8:B+9R=<609/57E0>JSD2/4d4u[78:> (E>\;Fz:9/EF2U=\;FBm2U8o;608:@%n&Q>&8:@79/8U;BD.

F '' p B B\rq ZNQ<j(Xk)k∈K

]~SH!KREJQ<T<T\HOYH 4 w<KWwdM0w<OYw%E]~jC]iH0T<T\H0E4H0SijQ<SOWXîH!SbOYKRSijlH'` s H0j`0ZNQ<j K ′ H!j K ′′OYH!`$`0ZN]a`'H!Si`0H!Ec[!TdH'ÒYHK

jlH!T `0]iHK ′ ∩K ′′ = ∅

w T\ZRML`VT\H!`σKNT [!MPH!`

A(Xk)k∈K′

H0jA(Xk)k∈K′′

`0ZNSijQ<SOWXîH0SOYKRSijlH'`0w13> E0>D4d4d>HB?>¤g/nD8:nD=9RE060B?>f576F>&8 B:= ( [78 872U4u57=<>¤Q7E0[:Bp604dQ2U=m;9R8o; @%>HB?2/[B _ >D8:B?>D457E0>&B @%>&[% ( @7>D[%@%6TB)k`2U6F8o;BN@%>

K. 1 8 2/5%;<6F>&8o;Y9RE02/=B3E O 6F8:@7nDQ'>D8:@79/8:S\>I4[%;?[:>DE0EF>e@%>GB

σ _ 9/EFgUD57=<>&BSD2/=<=?>GBmQ'2/8:@:9R8o;?>GBD.XC608:B?6zR@:9R8:B#ET9VB?6;<[:9+;<6F2U8@%nGS\=<6;<>3Q:EF[:B©9R[%;Q'2/[7=E0>&B#nDUnD87>&4>&8o;<BE>D;Fz+60Es>\%60Bm;?>@%>&[%B?2/[:B _ >&8:B?>D4u5:EF>GB@%6TB)k`2U6F8o;B

K ′ >D; K ′′ @%> K ;?>&E0B3ZU[:>E87>C@%nDQ'>D8@ ZU[:>Y@%>&B

Xk

Q'2/[7=k ∈ K ′ >\; F 87>Y@7nDQ'>D8:@ Zo[7>C@%>GB XkQ'2/[7=

k ∈ K ′′ .UXC[%;<=?>&4>&8o;@%6;C~ E ∈ A(Xk)k∈K′

>\;F ∈ A(Xk)k∈K′′

.£ 9R6TB@PO 9RQ7=<&B#E0>NEF>&44d>Uz

A(Xk)k∈K′>\;A(Xk)k∈K′′

Bm2U8o;608:@%n&Q>&8:@79/8U;<>&B&z:@%2U8:SE>D;FB?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>uQ9R=X>\;Y@7>D[% . 97. =&. @P.0z>D;JQ:9/=

g[787>)2/8Sl;?602/8@%>

X(Ω) × Y (Ω)@79/8:BR. 1 8f@%n\¦:8:6;CET9d+9R=<609/57EF>A9RE0n&9R;?2U6F=<>

g(X,Y )Q:9/=I~

∀ω ∈ Ω, g(X,Y )(ω) = g(X(ω), Y (ω)).1 8f9f~g(X,Y )(Ω) ⊂ Img

.,I B @& # #1 PC(

g(X, Y )m b=%ni= $ >=?;hp B BC@ qbZRQ<j

z ∈ ImgwTdZNML`

[g(X,Y ) = z] =⋃

(x,y)∈g−1(z)

[X = x] ∩ [Y = y].

m b=%ni= $ >=?;hp B B Ag(X,Y ) ⊂ A(X,Y )

w,I 9:" >-(

Z = g(X, Y )m b=%ni= $ >=?;hp B B KcT\ZRQ OYH

g(X,Y )H'`'jOYZRSiSXLH^:KRM

g(X,Y )(Ω) ⊂ ImgH0j

∀z ∈ Img, P (g(X,Y ) = z) =∑

(x,y)∈g−1(z)

P (X = x, Y = y).

^ S ^:KRM'jCQDI0]~T_QDH0M s `'Q X H!jY`0ZNSj Q<SOWXîH0SOYKRSijlH'`J

∀z ∈ Img, P(

g(X,Y ) = z)

=∑

(x,y)∈g−1(z)

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

V-W n 4p B B -/.∀z ∈ R, P (X + Y = z) =

∑

x+y=z

P (X = x, Y = y).

L7.∀z ∈ X(Ω) ∪ Y (Ω)

zP (min(X,Y ) = z) =

∑

y>z

P (X = z, Y = y) +∑

x>z

P (X = x, Y = z).

h:.Yrs6X → G(p)

>\;Y → G(p)

z9/EF2U=<BX + Y → P(2, p)

.

p B ' g Z = g(X,Y )

@,II #KPCB !( >)(

g(X, Y )F 8 =& p B B\pkj :XLZRM !EcH$OYHjkMPKNSi`<!H0M'j\ H!`CîX0MPKNSbIH$OWH

g(X,Y )HPWQv`'jH`'Q%H0j?`0H0]~T\H0E4H0Sij%`'QT\K`!X0M'QDH

I0Q<K^M '`JILZNS 4 H0MCzHJKW['`0ZRT_]~E4H0Sij s H0jOWKNS:ÌLHIKNÈ(g(X,Y )) =

∑

(x,y)∈X(Ω)×Y (Ω)

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

F 8 =& p B B\<3S `'] ^YîZ6`0H4QDI0Q0]iHΩH!`'jOYX0SZNEc[0MPKY[0T\H s H0j!]iHTP H'`kîKYILH^MZY[LKY[0Q<T_Qv`0KY[0T\H IZRS:`'QDOYX0MPXH!`Lj

(Ω,P(Ω))w?q ZN]a`$MPX'`0H0M 4 HJOYHILZNS 4 H0MCWH0SILHJKY[!`!ZRT_]iHc XL0]~Q 4 KRT\H!Sj cT HPWQv`'jlH!SbIHJOYHT H'`kîX!MKNSILH ZNS K

E(g(X,Y )) =∑

ω∈Ω

g(X(ω), Y (ω))P (ω).

,IQ_ (*;=(*# #1P*B !)( >-(X + Y

>-(XY

F 8 =& p B BTSoqbZNQ<jXH!jYOYH0]Y 4 w<KYwdM0w<OYwKWONEcH!jkjlKNSij)]~SbHVH!`k^:X0MPKRSILHwbT\ZRML`T H!`CîX0MPKNSbIHFOWH X+YHYQv`'jlH s H0j E(X + Y ) = E(X) +E(Y )

wV== pB BYX T_Q<SXLKNM'Q<jlXFOYHT H!`k^:X0MPKRSILHlqbZNQ<j

XH0jYOWH!]Y 4 w<KYwdM0w<OYw s H0j λ H0j µ OYH0]YMPXLH!T_`!wT\ZNML`

E(λX + µY ) = λE(X) + µE(Y )w

e 'bf?g p B B[Z Xy;?;?>&8U;<6F2U8^z:@%>A4 9R8:6F&=?>Ag/n&87nD=9RE0>/zE(XY ) 6= E(X)E(Y )

.F 8 =& p B BD@6\AqQ

XH!jY`0ZNSijQ<SOYXlîH!SbOYKNSjH!` H!j$KYONEcH0jCjlH!Sj.]~SHH'`kîX!MPKRSILH s KRT\ZNML` XY KWONEcH!j

]~SHJH'`kîX!MKNSILH 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 ILZ 4 KNM'QDKRSILH@%> X >D;

YQ:9R=I~

cov(X,Y ) = E(

(X −E(X))(Y −E(Y )))

.

e 'bf?g p B BD@ rs6cov(X,Y ) > 0

zX−E(X)

>\;Y −E(Y )

2U8U;;?>D8@79R8:SD>&(F3D;?=<>V@%>V443D4d>VB?6FgU87>/z@%2U8:S

X>D;Y2U8o;N;?>D8@79R8:SD> (B?>eB?6;<[7>D=C@%[f443D4d>IS /;?ne@%>AE0>D[7=Y>&B?Qn&=<9/8:S\>U.sXY6F8Bm6¥~

%cov(X,Y ) > 0

B?6FgU876¦>AZo[7>X>\;Y2/8o;N;<>D8:@79/8:S\> (3\;<=?>e@7>J+9/=?6T9+;<6F2U8Q:9/=<9/EFE0DE0>¥]

%cov(X,Y ) < 0

B?6FgU876¦>AZo[7>X>\;Y2/8o;N;<>D8:@79/8:S\> (3\;<=?>e@7>J+9/=?6T9+;<6F2U82UQ7Q2oBmn&>¥]

m b=%n 8 8 4p B BD@ xYwcov(X,Y ) = E(XY ) − E(X)E(Y ) yzw ^ S4îKNM'jkQDI!]~T_QDH0M s cov(X,Y )

HPWQv`'jlHF`'QX

H0jYKWONEcH!jkjlH!SjOYH!`H'`kîX!MKNSILH!`0w

|Wwcov

H!`'j `NEcX!jkM'QD0]iHJcov(X,Y ) = cov(Y,X)

ww

covH!`'j[0Q<T_Q<SbXKRQ<MPH w%]~jkMPH!EcH0Sij ONQ<j s îZN]~MFjlZN]~jlH'` 4 w<KYwdM0w<OYw X s X ′ s Y H0j

Y ′ lK'oT_Q<SXLKRM'Q<jX.îKNMFMPKL^YîZNM'j cT\K$^MPH0EQ !MPH 4 KNM'QDKY[0T\H

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

l[ T_Q<SXLKRM'Q<jX.îKNMFMPKL^YîZNM'j cT\Kc`0HILZRSOYH 4 KNM'QDKY[0T\H ∀(λ, µ) ∈ R2, cov(X,λY + µY ′) = λcov(X,Y ) + µcov(X,Y ′);

YwqQX

H!jY`0ZNSj Q<SbOYXl^:H0SOWKNSijlH!` s cov(X,Y ) = 0

w

?n pB :g 8 = m b=%ni= $ >=?;hp B BC@ qQ

XH!jYKYORE4H0jkjH0Sij ]~SH 4 KNM'QDKRSILH s KNTdZNML` X + Y

XzKNT\H0E4H0Sij s H0jV (X + Y ) = V (X) + V (Y ) + 2 · cov(X,Y ).

F 8 =& p B BD@WprqQXH0jY`0ZNSjQ<SOWXîH!SbOYKRSijlH'` s H0j KYONEcH0jCjlH!Sju]~SH 4 KNM'QDKRSILH s KNT\ZRML` X + Y

KN]a`L`'Q sH0jV (X + Y ) = V (X) + V (Y )

wm b=%ni= $ >=?;hp B BC@Yrq ZNQ<j

X1, . . . , Xn

OYH!` 4 w<KWwdM0w<OYwK YKNSjF]~S EcZNEcH0SijO ZRMPONMH 2 X1 + · · · + XnKYORE4H0j]~SH 4 KNM'QDKRSILH4H!jV (X1 + · · · +Xn) = V (X1) + · · · + V (Xn) + 2 ·

∑

16i<j6n

cov(Xi, Xj).

F 8 =& p B BD@ S qbZR]a`VT\H!`FE4G0EcH'` îZNj !`0H'` s `LQ X1, . . . , Xn

`0ZNSjC@%>D[7 (d@%>D[7 Q<SbOYXl^:H0SOWKNSjH!`V (X1 + · · · +Xn) = V (X1) + · · · + V (Xn).

748:9 <; >=?; p B BD@6X b^> ILZH I0QDH0Sij.OYHILZNM'MPX0T\KNjkQDZNS T_Q<SbXKRQ<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>\;YBm[:6;[:8pS\>D=?;<9/6F8d;`¬sQ'>Y@%>CE02/6 zU9/EF2U=-(R# +# @!-";< (*#m b=%ni= $ >=?;hp B BC@ qbZRQ<j

(m,n) ∈ (N∗)2H!jp ∈]0, 1[

w$qQX → B(n, p)

H0jY → B(m, p)

`0ZNSijQ<SOWXîH0SOYKRSijlH'` s KRT\ZNML` X + Y → B(n+m, p)

w^ ShîKNM'jkQDI!]~T_QDH0M `LQ X → B(n, p) s X H'`'jcT\K `0ZNEEcH OYH

n 4 KNM'QDKW[!TdH'` OYH H!M'SbZN]~T<T_Q OWH^:KRMPKNE !jkMPHpE]~jC]iH0T<T\H0E4H0Sij Q<SOYXlîH!SbOYKNSjH!`0w

,Q_) `'2@ 2P>-(R# +#%>-( # m b=%ni= $ >=?;hp B B rqbZRQ<j

(r, s) ∈ (N∗)2 s H!j p ∈]0, 1[w qQ

X → P(r, p)H!jY → P(s, p)

`0ZRSijQ<SbOYX0îH!SbOYKNSjH!` s KNT\ZRML` X + Y → P(r + s, p)

w^ S îKNM'jkQDI!]~T QDH!M `'Q X → P(r, p) s X H'`'jFT\K `0ZREJEcH OYH

r 4 KNM'QDKW[!T\H!``']~Q 4 KRSij]~SbH T\ZNQ.zXZRE4X0jkM'QD!]iH OWHîKNMPKRE 0jCMHp s EJ]~jk]iH!T<TdH!EcH!Sj Q<SbOYXl^:H0SOWKNSijlH!`!w

,Q_)I `'2@ 2P>-(R# +# @!-";< (*#%!-P*DE2 "(*#m b=%ni= $ >=?;hp B B rqbZRQ<j

(r, p) ∈ (N∗)2H0jp ∈]0, 1[

wbqQX → J (r, p)

H!jY → J (s, p)

`0ZNSjQ<SOWXîH!SOYKRSijlH'` s KRT\ZNML` X + Y → J (r + s, p)

w,Q_)Q_ `'2@ 2P>-(R# +#%>-( "#K#K"!m b=%ni= $ >=?;hp B B qbZRQ<j

(λ, µ) ∈ (R+)2w qQ

X → P(λ)H0jY → P(µ)

`!ZRSijQ<SbOYXl^:H0SOWKNSijlH!` s KRT\ZNML`X + Y → P(λ+ µ)

w

t N

(&( (+( & "F!/!'# ,E P ! "!748:9 <; >=?; BC@:BD@ r%2/6F;

(Xn)n∈N

[787> B?[76;<>@7> . 97. =&. @P.PB?[7=(Ω, T , P )

zP>D;X[787> 9R[7;?=<>

v.a.r.d.. 1 8

@%6F;YZo[7>Xn

ILZNS 4 H0MCzHJH0Sc^bMPZY[LKY[0Q<T_Q<jlX )2/[ ILZNS 4 H0MCzH`'jlZI :KN`'jCQD0]iH!EcH0Sij 3/>&= 0, lim

n→+∞P(

|Xn −X | 6 ε)

= 0.

,E JK!-PCD" 2P >)(% (*!)-; P 0 (C@ (F 8 =& BC@:B !SXzKNT_Q<jlXcOYHKNMWZ 4 q ZNQ<j X ]~SH 4 w<KYwdM0w<OYw `']~M (Ω, T , P )

jlH!T<T\Hc0]iHX(Ω) ⊂ R+ sKYORE4H0jkjKRSij ]~SbHH!`k^:X0MPKNSbILH872/8 8s[7EFE0>

E(X)wTdZNML`

P(

X > λE(X))

61

λ.

F 8 =& BC@:B !SXzKNT_Q<jlXOYH QDH0SK NE4X0 IiH[ YI :H 4 qbZRQ<j Y ]~SH 4 w<KWwdM0w<OYwVKYORE4H0jCjlKNSj]~SbHH!`k^:X0MPKRSILHmH!j]~SH 4 KRM'QDKNSbIH σ2 w?T\ZNML`

∀ε > 0, P(

|Y −m| > ε)

6σ2

ε2.

,EI 9:" @(?>)(*# DEB !>)# !-"; @B(*#F 8 =& BC@:B ZNQ!KRQD[!T\H OYH!` RMPKRSON`SZRE4[0MPH!` q ZNQ<j

(Xn)n∈N∗

]~SH `']~Q<jlH OYH 4 w<KWwdM0w<OYw$E]~jC]iH0T_T\H0E4H0SijQ<SOWXîH0SOYKRSijlH'``L]~Q 4 KNSjJ]~SbH EcG0E4H T\ZRQ s K YKRSij4]~SH H!`CîX0MPKNSbIH m H0jJ]~SbH 4 KNM'QDKRSILH σ2 w$qbZNQ<j(Zn)n∈N∗

OYXCtuSiQDHîKNM∀n ∈ N, Zn =

X1 + · · · +Xn

n.

T\ZRML`(Zn)n∈N∗

ILZNS 4 H0MCzHJH0S4^MZY[LKY[0Q<T_Q<jlX 4 H0ML`FT\K 4 KNM'QDKW[!T\HcILH0M'jKRQ<SHXWKRT\H m wT_]a`^MPXLI0Qv`0X!EcH!Sj

∀ε > 0, ∀n ∈ N∗, P(

|Zn −m| > ε)

6σ2

nε2.

F 8 =& BC@:B\p iXZRM 0E4H O ZRMOYH H0M'SZR]~T<T_Q qbZNQ<j(Xn)n∈N∗

]~SH`']~Q<jlHOYH 4 KRM'QDKY[0T\H!` EJ]~jk]iH!T<TdH!EcH!Sj Q<SOWXîH0SOYKRSijlH'` s `']~Q 4 KNSj jlZN]~jlH'`$]~SHT\ZNQ OWHH0M'SZR]~T<T_Q OYHîKNMPKRE 0jCMH p w%qbZNQ<j∀n ∈ N∗, Zn =

X1 + · · · +Xn

n.

?n BV=%; :; ?nin= W >=?;

T\ZRML`(Zn)n∈N∗

ILZNS 4 H0MCzHJH0S4^MZY[LKY[0Q<T_Q<jlX 4 H0ML`FT\K 4 w<KYwdM0w IH0M'jlKNQ<SHXWKRT\H p wT_]a`^MPXLI0Qv`0X!EcH!Sj∀ε > 0, ∀n ∈ N∗, P

(

|Zn − p| > ε)

61

4nε2.

132/8S\E0[:Bm602/8 ~b 9 )=?nGZo[7>D8:SD>HBm;<9+;<60Bm;?6TZo[7> @PO [78 nDUnD87>&4d>D8o; ;?>&8:@ />&=<BdET9 Q7=<2/5:9/5760EF6F;?n¤@PO 9RQ7Q9R=<6;<6F2U8 @7>E O n&/nD8:>D4d>D8o;&.,EQ_ !A((*;= ( # # $,&-(m b=' BD@iB\=3SjkQ<MPH4x $!ZRQv` ^Q<T\HcZR]!KYILHcK 4 HLI]~SH.^bQ ILHcOWX'`0XL!]~Q<T QD[!MPXLH OWZNSjTdKF^bMPZY[LKW[!Q<T_Q<jlXO ZW[!jlH0SijkQDZNS OWHJQ<TdHH!`'j

pw3S ZY[0jCQDH0Sij$ !ZNQv`Q<T\H wZNSSH0MJ]~S Q<SjH0M 4 KRT<T\H I jH0Tu!]iH T\KJ^bMPZY[LKY[0Q<T_Q<jlX

0]iHp ∈ I

`0ZNQ<j `'] ^:X0M'QDH0]~MPH Yw Wwrs2U6;Q2U[7=y;?2U[%;

i ∈ [[1, 1000]]zXi

E09+9R=<609/57E0>V@%>Axy>D=<872/[7E0E06 @7>VQ:9/=<9/4d\;?=<>pS\2U=?=<>&B?Q'2/8:@79/8o;39R[=?nGBm[7EF;<9R;

@%>IET9i _ D4d>A>\%Q'nD=<6F>&8:S\>Uz%>\;YB?2/6F; Z1000 = 1

1000 (X1 + · · · +X1000).

rs2U6;ε > 0

.¥uO 9RQ7=<&BdE0> ;?©7n&2/=<D4d>¤@PO 2/= @%> xy>D=<872/[:EFE06zP (|Zn − p| > ε 6 1

4nε2

.ijECB?[d; @%2U8:SfZU[:>1

4nε2 6 0.1z%Bm2U6;

ε 6 0.05.%XY6F8:B?6 z

P (0.52 6 p 6 0.62) > 1 − 0.1 = 0.9.

,E !-( "!)> +!8# & #K !"2( >)(?"!"(*BDE(*!(?(R!A)BE@ @ 2PF 8 =& BC@:BTSoqbZNQ<j

(Xn)n∈N

]~SH`']~Q<jlHOWH 4 w<KYwdM0w<OWw s H0j X ]~SH KR]~jCMH 4 w<KYwdM0w<OYw s K YKRSijjlZN]~jlH!`]~SbHH!`CîX0MPKNSbIHVH0j?]~SH 4 KRM'QDKNSbIH wqQ limn→+∞

E(Xn) = E(X)H0j`'Q

limn→+∞ V (Xn−X) = 0 s KNT\ZRML` (Xn)n∈NILZNS 4 H0MCzHJH0Sc^bMPZY[LKY[0Q<T_Q<jlX 4 H0ML` X w

(&( (+( /!, P ! "!748:9 <; >=?; B BD@ r%2/6F;

(Xn)n∈N

[787>YB?[76F;?>Y@%>Y'. 97. =G. @P./>\;X[78:>C . 9:. =G. @^.F. 1 8 B?[7Q7Q'2UB?>YZo[7>NQ'2/[7=;?2/[7;

n ∈ NzXn(Ω) ⊂ X(Ω)

. 1 8f@%6;YZo[7>(Xn)n∈N

ILZRS 4 H!MCzHH!S T\ZNQ3/>D=B X B?6~∀x ∈ X(Ω), lim

n→+∞P (Xn = x) = P (X = x).

, )B -;= "! >)( "@ !-+;= (%) B>-(*# "# (CBDEPC+; PC B $,&-(*#F 8 =& B B AqbZNQ<j s îZN]~M jlZN]~j N ∈ N∗ jlH!T0]iH

Np ∈ N s XN → H(N,n, p)

w.T\ZRML`(XN )ILZNS 4 H0MCzHJH0ST\ZRQ 4 H!ML` X → B(n, p)

wJ (+"%$G ##*¥ U #G!¯#°U(+* #0 3*# $ "%# (+^#*U( ## *!' P*o('sÜ$&(+ ##*

H(N,n, p)"%( ## *¤3! # "7!*B(n, p)

GN > 10n rs60g/876F¦S&9+;?602/8H~7b^2/=B<ZU[:>CEF>J872U4u57=<>V@%>J5'2/[7E0>&B@%>JEO [7=<87>J>&Bm;3;<=?GB3gU=<9/8:@PzoEF>V!9R6F;N@%>V=<>\;<6F=<>D=y[787>J5'2/[7E0>J@7>

E O [7=<87>A87>IS©:9/87g/>JQ:9/B5'>&9/[:S\2U[7Q ET9Q7=<2/Q'2/=?;?602/8@7>J5'2/[:EF>GB*(B?[:S&S\&B&.%XC608:B?6z72U8>GB`;NQ7=<>&B<Zo[7>A@79R8:BE0>ISD9UB@PO [78 ;?60=<9/g/>A9/>GSV=<>D4d6TBm>U.,I )B -;= "! >-( " >-( "# #K"!8) B >-(*# "# @!)";< (R#F 8 =& B B AqbZNQ<j

(Xn)n∈N

]~SbH$`']~Q<jHOYH 4 w<KYwdM0w<OWwujlH!T<TdH'`0]iHîZN]~MjlZN]~j n ∈ N s Xn → B(n, pn) sZ (pn)n∈N∗

H!`'j ]~SbHF`']~Q<jlHJjH0T<T\H4!]iHlim

n→+∞npn = λ ∈ R∗

+

w?T\ZRML`(Xn)n∈N∗

ILZNS 4 H0MCWH4H0S ^bMPZY[LKW[!Q<T_Q<jlX4 H!ML`F]~SH 4 w<KWwdM0w<OYw X → P(λ)

wuK6`^:KRM'jCQDI0]~T_QDH0MJ∀n ∈ N, Xn → B(n, λ

n)w

J (+"%$G ##*' /' G!¯#°o(* #F' *# $"%##(c#*o( ## *H!'¥! # "7!*B(n, p)

"%( ## #*¤!'¯#* '!G&'

P(np)Gn > 30

p 6 0, 1

*U$np 6 10

¨ N¨

#c$ Q9R=?;?60=C@%>u@%2U8787nD>GBN57=<[%;?>GBDzZU[9RE06;9+;?60/>GB )8:2/8¤S©76 '=<nD>GB y2U[fZo[:9/8o;?6F;<9+;<6FU>&B S©76 '=<nD>GB \z:@%n\¦876F=@%>GB^Q:9/=<9/4D;?=<>&B'Q'>D=<4>D;m;9R8o; @%>SD>D=<87>D= @%[IQ7=<>D4d6F>&=cS\2U[7QI@PO I6FEUEF>GB ;?>&8:@79R8S\>&B^g/nD8:nD=9RE0>&BP@%>¥ET9Q'2/Q7[7ET9+;<6F2U8n\;<[:@%60nD>~:=<>DQ7=<n&B?>D8o;<9R;?602/8:BgU=<9/Q7©76TZo[7>&B&z72/=@%=<>e@%>egU=<9/8:@%>D[:= )4d2+¬/>&8787>Uz74d2s@7>/z'4nG@%6T9R87>U.F.0. \z:>GB`;<6F4 9+;<6F2U8@%>IE O nD;<9/EF>&4>&8o; !n&S&9R=?; _ ;`¬oQ'>/z7Zo[:9/=m;<6FE0>&B&.0.F. l.

L#! , (&) 0 /0!( / . ( J#%"'&(

E `'2R#6$'&)(*#

748:9 <; >=?; SBC@:BD@ Y87> `'jlKNjkQv`'jCQD0]iH>&Bm;V[78:>9RQ:Q7EF6TSD9R;?602/8X : Ω → C

z 265Ω>&Bm;V[:8H>D8Bm>&4u57E0>I¦:876 z >\;

C[78f>D8:B?>D457EF>IZo[7>&E0SD2/8:Zo[7>U.

&<;?=a= SBD@iB bc>&B¥2U5%k`>\;B>&8Ik`>D[ Q2U=m;<>D8o;y@%>GB¥872U4 B¥;<=?GB¥E06FnGB(IE O n\;?[@%>J@7>YET9IQ2UQ7[7ET9+;<6F2U8 @PO [78n\;9+;Gz:Zo[76P>&Bm;CEO 2/=<60g/6087>J@7>&BCBm;<9R;?6TB`;<60Zo[7>GBA~%

Ω>GB`;C9RQ:Q>&EFnVîZ^b]~T\KNjkQDZNSz

% [78fnDE0nD4d>D8o;x@%>

Ω>&Bm;Y9RQ7Q'>DE0n4Q<SONQ 4 QDON]%z%

C>GB`;Y9/Q7Q>&EFn H0S:`0H0E4[0T\HOYH!ÌLKRMPKYI0j 0MPHDz

% [78fnDE0nD4d>D8o;c@%>

C>GB`;N[78 ILKNMPKWI!j 0MPH\z

% [78oXLIiKNSijkQ<T<T\ZRS>GB`;e[78 B?2/[B _ >D8:B?>D457E0>d@7> Ω. 1 8 n\;<[:@%60> Bm2U[7/>&8o;A[:87>pB`;9+;<60Bm;?6TZo[7>p>D8 9/8:9RE0¬%B?9/8U;eB<9

=?>GB`;<=?6TSl;<6F2U8B?[7=C[78 n&S©:9/8o;?60EFE02/8 EF>IQ:EF[:B=<>DQ:=?nGBm>&8U;9+;<6#Q2oB?B?6F5:EF>I@%>IET9Q2UQ7[7ET9+;?602/8c.XC608:B?6z [787>pBm;<9R;?6TB`;<60Zo[7>p>GB`;e[:87> )2/8Sl;?602/8 Zo[76 (¤S©9/Zo[7> 608:@%60o6T@%[ 9UB?B?2%S\60>[:8 S&9R=9/S\;?D=<>/. 13>pS&9R=9/S\;?D=<>Q'>D[%;.3D;?=<>eS©76 =<n!E09;9R60EFE0>/z7E0>IQ'2/6T@7BDz%E O Rg/>U.F.0. 2U[8:2/8 )[78:>eS\2/[:EF>&[7=&z%E0>J!9R6F;Y@%>IQ'2/=?;?>D=C@7>&BE0[787>\;?;?>GBD.0.F.

&<;?=a= SBD@iB 1 8 @%6F;JZo[Ô [78 S&9R=9/Sl;<D=<>A>&Bm;40]iKNT_Q<jlKNjkQ B&O 6FE 87>eQ'>D[%;JQ:9/B3\;?=<>uS©76 =?nUz:>\;4!]iKRSijkQ<jKRjCQ B&O 6FEPQ'>D[%;VEO 3\;<=?>U.2/[7=E O n\;?[@%>N4 9+;?©:nD4 9+;<60Zo[7>C@%>GBB`;9+;<60Bm;?6TZo[7>&B&zR2/8dB&O 608o;?nD=<>&B<B?>D=9C>&B<B?>D8o;?60>DE0EF>&4d>D8o; (A@%>&BS&9R=9/Sl;<D=<>&BZo[:9R8 _;?6F;<9R;?6F!BD.7XY6F8Bm6 z

C>&Bm;nDgU9/E([78fBm2U[:B _ >&8:Bm>&4u57E0>Y@%> R

. V[76F;m;?>)(Q7=?2UEF2U87g/>&=XBm[7=B?2/8 >&8:B?>D4u5:EF>J6F4 9/g/>/z

2/8 Q'>D[%;VB?[7Q7Q'2UB?>D=NZo[7>C = R

.

: ?n SB ' 8 &:; >f%g E N(*M BP*$,&-(R!)(rs2U6;

X : Ω → R[787>yB`;9+;<60Bm;?6TZo[7>Zo[:9R8o;<6;9+;?60/>U.&b 9NQ2UQ7[7ET9+;<6F2U8

ΩnD;<9/8U; ¦:8760>')©s¬sQ'2R;?©:&B?>3@%>3E09C@7n\¦:876F;?602/8

@PO [787>B`;9+;?6TBm;?6TZU[:> lz B?2/8¤604d9/g/>X(Ω)

Q:9/=X>GB`;JnDgU9/EF>&4d>D8o;C¦:8760>/. 1 8¤872/;?>

x1, . . . , xpE0>&BYnDE0nD4d>D8o;X(Ω)

.oijE BDO 9RgU6;3@Ô [78Bm2U[:B _ >D8Bm>&4u57E0>N¦:876 @%> RzoZo[Ô 2/8 Q'>D[%;N@%2U8:SN2/=@%2U8787>D=G. 1 8pB?[7Q7Q'2UB?>D=9A;<2/[+k`2U[7=eZo[7>IE0>&B608:@%6TS\>&BNB?2/8o;CS©72U60B?60BN@7>eBm2U=m;<>JZo[7>x1 < · · · < xp

.748:9 <; >=?; SBC@:B r%2/6F;

xi ∈ X(Ω)z(a, b) ∈ R2 .

% b¥O<HHLI0jCQ ni

@7> E09+9RE0>D[7=xi

>&Bm;eEF>d872U4u57=<>@Ô 608:@%60s60@%[Be@%>Ω9¬o9R8o;JE0> SD9/=<9USl;?&=?>

xi

.cXC608:B?6zni =

|X−1(xi)|.

% b¥O<HHLI0jCQ I0]~E]~T\XH0Sxi

>GB`;NE0>A872/457=?>A@PO 6F8:60@%60s60@7[ @%2U8o;E0>ISD9/=<9USl;?&=?>V>&Bm;C9R[ Q7EF[Bxi

~%60E^BDO 9RgU6;N@%2/8S@%>

|X−1(] −∞, xi])|z:B?2/6F;N>D8:SD2/=<>I@%> i

∑

j=1

nj

.% b¥O<HHLI0jCQ OWHTP Q<SijlH!M 4 KNT<T\H ]a, b]

>GB`;|X−1(]a, b])|

~7S/O >GB`;CE0>A872/457=?>I@PO 608:@%60o6T@%[:BY@%2/8o;NEF>eS&9R=9/S\;?D=<>Y>GB`;@79R8:BE O 6F8o;?>&=?+9/EFE0>

]a, b].

% ¥EF[BNg/n&87nD=9RE0>D4d>D8o;GzoE O<HHLI0jCQ JO ]~SQ<SjH0M 4 KNT<T\H I >&Bm; |X−1(I)|.

% b¥O<HHLI0jCQ I!]~E]~T\X H!Sb)872U8 Q7E0[:BI)2U=<SDnD4d>D8o;e[787>+9RE0>D[7=u@7>

X(Ω)>GB`;EO > >&S\;?6FY@%>

] − ∞, b]z Bm2U6;

|X−1(] −∞, b])|.

748:9 <; >=?; SBC@:B\p r%2/6F;xi ∈ X(Ω)

z(a, b) ∈ R2 .rs2U6; n = |Ω|

.% bc9LMPXL!]iH0SILH

fi

@%>NET9J9/EF>&[7=xi

>&Bm;¥E09AQ7=<2/Q'2/=?;?602/8@PO 6F8:@76Fs6T@%[:B¥9¬o9R8o;EF>CS&9R=9/Sl;<D=<>xi

./XY6F8Bm6 zfi = ni

n

.% bc9$LMPXL!]iH0SILHI!]~E]~T\XLHH0S

xi

>&Bm; i∑

j=1

fj = 1n

i∑

j=1

ni

.% bc9$LMPXL!]iH0SILHO ]~S Q<SjH0M 4 KNT<T\H I >GB`; 1

n· |X−1(I)|

.

EI `,PCB (*# #12R#6$'&)(*#bc9Y@%2/878:nD>@%>E09)2U8:Sl;<6F2U8eB`;9+;<60Bm;?6TZo[7>

X>&Bm; B?2/[7U>D8o;^608o[7;?60EF>Uz&Q'>D[I4 9R8:609/57EF>UzDU2/60=?>872U8eBm2U[7©:9/6;9R57E0>)872U8

9R8:2/8s¬s4>l.s2/[7=NE O n\;?[@%>Ag/E02/5:9/EF>A@%>AE09Q'2/Q7[7ET9+;<6F2U8^z%60EP>&Bm;604dQ2U=m;9R8o;C@%>IB?9U2/60=3E0>A872/457=<>J@PO 608:@%60o6T@%[:B9¬U9/8o;I;?>DESD9/=<9USl;<D=<>/z^4 9R6TBeE0>D[:=u6T@%>D8o;?6F;?n6F4dQ'2/=?;?>Q>&[^. 1 8S\=<nD>p@%2U8:Sp[789R872U8o¬s4 9+;e>&8 87>=?>D;?>D89R8o;@PO [787> Bmn&=?60> Bm;<9+;<60Bm;?6TZo[7>Zo[7>EF>GBe)=<n&Zo[7>&8:S\>GBe2U[ EF>GBu> '>GSl;<6!B&. 1 8¨2/5%;<6F>&8U;d9R608:B?63S\> Zo[Ô 2U8¨9RQ:Q>&EFE0> [78:>B?nD=<6F>IBm;<9+;<60Bm;?6TZo[7>/. 1 8¤@%6TBm;?6087g/[7>IB?[7609/8o;NZo[7>IE O >D8:B?>D457EF>I@%>GBCSD9/=<9USl;<D=<>&B3Q2oB?B?6F5:EF>GB>&Bm;y¦876^2/[f872U8^.748:9 <; >=?; SBC@:B\ Y87>c`0X0M'QDHJ`LjlKNjkQv`'jCQD0]iHONQv`0I!M !jlH>&Bm;Y[787>I!9/4d6FE0EF>

(xi, ni), i ∈ Iz:265

I>GB`;N¦:8:6c2/[

@%n&872/457=<9/57E0>¥]sEF>GBxi

=?>&Q7=<n&B?>D8o;?>&8U;E0>&By+9RE0>D[7=ByQ2oB?B?6F5:EF>GBN@PO [:8 S&9R=9/Sl;<D=<>Y>D;ni

>GB`;E O > '>GSl;?6FS\2/=<=<>&B?Q2U8 _@79/8U;G. 1 8f=?>&Q7=?nGBm>&8o;?>JnDgU9/EF>&4d>D8o;NQ:9R=?)2/6TBy[787>eB?nD=<6F>IBm;<9+;<60Bm;?6TZo[7>AQ:9R=NE0>&By)=<n&Zo[7>&8:S\>GBA~(xi, fi), i ∈ I

J9R8:BE0>NSD9/B#215E O >&8:B?>D4u5:EF>@%>&B+9/EF>&[7=<B#Q'2UB<Bm6057E0>&B@%[dSD9/=<9USl;?&=?>38Ô >&Bm;Q9/BnD8s[74dn&=<9/57EF>Uz2/[u443&4>C@:9R8:B#E0>SD9UB2656FE:>&Bm;n&8s[74dnD=9R57E0>4d9/60BgU=<9/8:@ @%nD8:2/4u5:=<9/57EF>Uz2/[d443D4d>y¦:876 zR4 9R6TB@%>Ng/=9R8:@SD9/=<@%608:9/Ez+@%2U8:SQ>&[4 9R876T9R5:EF>lz%2/8 > >&S\;?[7>e@7>&B=<>Dg/=<2/[:Q>&4>&8o;<By@%>I+9RE0>D[7=BDzsQ'2/[7=CB?>A=<9/4d>D87>&= ([78 872U4u57=<>AQ7EF[:>J=<>&Bm;?=<>D608o;)gUnD87n&=<9/EF>&4d>D8o;3¦876 C@%>A9/EF>&[7=BD.748:9 <; >=?; SBC@:BTS Y8 I!T\KN`L`!Hu@%>eSD9/=<9USl;<D=<>Y>&Bm;N[78f=?>&g/=<2/[7Q'>D4d>&8U;N@%>A+9RE0>D[7=BQ'2UB<Bm6057E0>&B@%[¤SD9/=<9USl;?&=?>U.rs2U[7/>&8U;Gz7Bm6

C = Rz:2/8 > >&Sl;<[7>IE0>e@%n&SD2/[7Q:9/g/>ABm[:6F+9R8o;I~

] −∞, a1], ]a1, a2], . . . , ]ap−1, ap], ]ap,+∞],

265a1 < a2 < · · · < ap

. 1 8fQ:9R=<EF>&=<9@%>GBNS\ET9/B<Bm>GBy@7n\¦:8760>&BCQ:9R=NE0>&B5'2/=<87>&B(a1, . . . , ap)

.

SBC@ 748:9 <; >=%; '<;?=z= 8 8 &:; S

1 8 =?>&4 9R=ZU[:>D=9Zo[Ô [78 @%nGS\2/[:Q:9RgU>d9U@%4d60B<B?6F57E0>p@%>C@%2U6;uSD2/8:B?6TB`;<>D=I>D8 [:87>4îKNM'jkQ<jCQDZRS@%>

C9+¦8 Zo[7>

S©:9UZU[:>J608:@%60s60@7[fB?2/6F;CS\2U4Q7;?nI[787>A>D;C[787>eB?>D[7E0>J)2/6TB748:9 <; >=?; SBC@:B[X Y87>.`0X0M'QDH`'jlKNjkQv`LjkQD0]iHIZRSijkQ<Si]iHDzD2U[JRMZN] îXH3>GB`;#E09V@%2/878:nD>@PO [787>3Q:9R=?;?6F;?602/8

Ci, i ∈

I@7>

C>D8f6F8o;<>D=<9/EFE0>&B&z7>D;V@PO [787>A!9R4d60EFE0>

(Ci, ni), i ∈ Izni

S\2U=?=<>&B?Q'2/8:@79/8o;*(dE O > >&Sl;<6¥@%>uE09 SDE09UB?B?>Ci

.748:9 <; >=?; SBC@:B[Z r%2/6F;

(Ci, ni), i ∈ [[1, p]][787>B?nD=<60>Bm;<9+;<60Bm;?6TZo[7>SD2/8o;?608s[7>/zy265 Q2U[7= ;<2/[%;

i ∈

[[1, p]]zCi =]ai, ai+1]

z(ai)i∈[[1,p+1]]

nD;<9/8U;C[:87>eBm[:6;<>uS\=<2/6TB?B<9R8o;<>J@%>I=<nD>&E0B&.:r%9 ONQv`0I0MPX0jCQv`0KNjkQDZNS >&Bm;CET9 Bmn&=?60>Bm;<9+;<60Bm;?6TZo[7>p@%6TB?SD=?D;?> )¦:8760>

(xi, ni), i ∈ [[1, p]]z^215 Q'2/[:=I;?2U[%;

i ∈ [[1, p]]zxi = ai+ai+1

2

.#XC608:B?6z^2U8S\2U8:S\>&8o;?=<>JS©:9UZo[7>IS\ET9/B<Bm>A>D8fB?2/8 460E06F>&[^.e 'bf?g %SBD@iBD@]\rs6EF>GBA6F8o;<>D=<9/EFE0>&BJ>Do;<=?n&4 9R[%B?2/8o;A872/8 5'2/=<87n&B&zP2/[ Q>&[ =?>&Q7=?nGBm>&8o;<9+;<6!B&z 6FE¥9/=?=<60/>B?2/[7U>D8o;Zo[Ô 2U8 872/[B@%2U8787>CE0>&B9/EF>&[7=B¥4d2+¬/>&8787>&B9R[pB?>D608p@%>YS\>GBSDE09UB?B?>&B&. 1 8 Q'>D[%;9RE02/=B>D8:SD2/=<>C@%n\¦876F=[787>e@760B<S\=<n\;<60B<9+;<6F2U8 >D8¤SD2/8:SD>D8o;?=9R8o;EO > >&S\;?6F@7>ASD>&BNSDE09UB?B?>&B>&8E0>D[7=Y9/EF>&[7=4d2+¬/>D8:87>/. EQ_ (CBP*#K(*!+22"!)# DEB $,&-(*#± co(*s¯#!GR(+°o$&*s-U. " :(+"7¨ *se*s %$G 1 8 4d>\;C>&8 9/5:B?SD60B<B?>VE0>&B+9RE0>D[:=<B@%[fSD9/=<9USl;?&=?>Uzs>\;N>D8 2/=@%2/8:87nD>JEF>GB> >&S\;?6F!B )Q'2/[7=NE0>e@%6T9Rg/=9R4d4d>@%>&B> '>GSl;?6F!B y2/[fEF>GBy)=?nGZo[7>D8:SD>&B !Q2U[7=NEF>e@7609/g/=9R4d4d>J@%>GB)=<n&Zo[7>D8S\>&B l.b¥O > >&Sl;<6@PO [787>I+9RE0>D[:=

xi

>GB`;N=<>DQ7=<n&B?>D8o;?nJQ:9/=N[78f5 +;<2/8f@PO 9/5:B<S\6TB?B?>xi

@%>I©:9/[%;?>&[7=ni

.L7. ' #*¤¯ *sd**sR$G ijEP>&Bm;C2/5%;<>D8s[>&8f=?>&EF6T9R8o;NEF>IB?2/4d4d>\;Y@%>GB5 +;<2/8:BN@7[f@7609/g/=9R4d4d>V>&8 5 +;?2U8^.

h:. " :(+"7¨ **o$' #*¤¯ *sd**sR$G (+ ##*s U*s # ##!s9/=?>&6FE Zo[7>JQ2U[7=EF>GB3> '>GSl;<6!B&zs4d9/60By>&8S\2U8:B?60@%n&=<9/8o;3E0>&By> >&S\;?6F!BNS\[74[7EFnGB>&8 S©9/S\[:87>V@%>GBy9/EF>&[7=Bxi

.%¥8¤Q:9R=?;?6TS\[:EF60>D=GzsE09;9R60EFE0>I@%>&B5 R;?2U8:B>&Bm;YS\=<2/6TB?B<9R8o;<>/.q. ' /$&!' ¯ * (o"%($&!$&!' bc9I)2U8:Sl;<6F2U8@%>J=<nDQ:9/=m;<6;<6F2U8 @PO [787>ABmn&=?60>VBm;<9R;?6TB`;<60Zo[7>

X>&Bm;E09u)2/8:S\;?602/8

FX

@%>R@79/8:B

[0, 1]Zo[76(

;?2/[7;x ∈ R

9/B<Bm2%S\60>JE09)=<n&Zo[7>D8S\>eS\[:4u[7E0nD>FX (x) =

∑

i|xi6x ni

.b^>CgU=<9/Q7©7>Y@%>VE09e)2/8:S\;?602/8p@%>V=?n&Q:9R=?;?6F;?602/8 >GB`;3)2/=<4dnY@%>YQ:E09R;?>&9/[% =?>&EF6T9R8o;E0>&ByB?2/4d4d>\;<B3@%>&B35 R;?2/8B9R[5 +;<2/8dBm[760+9R8o;)608:S\E0[:B@%[ S R;?ngo9R[:S©7>Uz>\7S\E0[:B@%[ S /;?nN@%=<2/6F; ]+60E7>GB`;¥S\2U4dQ7EFnD;?nQ:9/=[78dQ7E09R;?>G9R[nDgU9/E(-e9/EFET9R8o;*(

+∞>\;C[78fQ7ET9+;<>&9R[ nDgo9RE(d 9RE0E09/8o; (

−∞.

± co(*sU $G! ##*s-U. !&$& :(+"7 ¨*1 8 =<>DQ:=?nGBm>&8U;<> >D8 9/5:B?SD60B<B?> EF>GBe5'2/=@7Be@%>GBe608U;<>D=<+9RE0EF>GBI=?>&Q7=<n&B?>D8o;<9/8U;IE0>&BS\ET9/B<Bm>GBe@%>pET9 B?nD=<6F>pBm;<9 _;?6TB`;<60Zo[7>U. C8:>uS\ET9/B<Bm>

]ai, ai+1]>&Bm;Y=<>DQ7=<n&B?>D8o;<nD>AQ:9R=Y[78f=?>GSl;<9/87g/E0>e@%>e59/B?>IEF>eB?>DgU4>&8o;Y@PO 9/5:B<S\6TB?B?>

[ai, ai+1]z%>\;Y@%2U8U;CE O 9R60=?>J>&Bm;CnDgo9RE0> (dE O > '>GSl;<6

ni

@%>IET9SDE09UB?B?> !@%2U8:SA@%>I©:9/[%;?>&[7= ni

ai+1−ai

. L7. ' #*¤¯ *sd**sR$G (+ ##*s U*sijEP>&Bm;C2/5%;<>D8s[>&8f=?>&EF6T9R8o;NEF>A4d60EF60>D[¤@%>GBCS\2/;?n&B©9R[%;BN@%>&B=<>&S\;<9R8:g/E0>&B@%>IE O ©76TBm;?2/gU=<9/4d4>U.

h:. " :(+"7¨ **o$' #*¤¯ *s (+ ##*s U*s ## ##!s1VO >GB`;p[78@%609/g/=9R4d4d>f>D8 5 R;?2/8B =?>&Q7=<n&B?>D8o;<9/8U; >&8 S©:9/SD[787>H@%>GBpS\2s2U=<@%2U8787n&>&B

ai

B?nDQ:9/=<9/8o;dE0>&BS\ET9/B<Bm>GB3ET9)=?nGZo[7>D8:SD>IS\[74[7EFn&>I>D8fS\>GB9/EF>&[7=BD.

X ?n SB ' 8 &:; >f%g

q. ' /$&!' ¯ * (o"%($&!$&!' bc9)2/8Sl;?602/8@%>e=<nDQ:9/=m;<6;<6F2U8 >GB`;JE09 )2/8:S\;?602/8 Zo[76(

x9/B<B?2sSD6F>IET9d)=<n&Zo[7>&8:S\>uSD[74u[:EFn&>

FX (x)>&8

xz

9/>&SyE O ©s¬sQ'2R;?©:&B?>CZU[:>CEO > >&S\;?6Fc@%>&B3S\ET9/B<Bm>GB>&Bm;[7876F)2/=<4dnD4d>D8o;=<nDQ9R=?;?6:@79/8:B¥S©:9UZo[7>CS\ET9/B<Bm>U./XC608:B?6z>DE0EF>A>&Bm;Y9 87>IB?[7=CS©:9UZU[:>J608o;?>&=?+9RE0E0>

]ai, ai+1].:r%2/8f>\%Q7=<>&B<Bm602/8>&Bm;C@%2/8:Sd~

∀i ∈ [[1, p]], ∀x ∈]ai, ai+1], f(x) =

i−1∑

j=1

fj +x− ai

ai+1 − ai

· fi.

XC608:Bm6 zPE0>gU=<9/Q7©7>d@%>ET9)2/8:S\;?602/8 @%>d=?n&Q:9R=?;?6F;?602/8 >GB`;IE0>dQ2UEF¬sg U87>u=<>DE06T9R8o;JE0> Bm2U4d4>D;A@7>&BA5 R;?2/8B@%[ @7609/g/=9R4d4d>@%>GBJ)=?nGZo[7>D8:SD>&BAS\[:4u[7E0nD>GBDz^SD2/4dQ7E0n\;?ndn&/>D8o;<[7>DE0EF>&4d>D8o;IQ:9R=

0>\;

1(go9R[:S©:>>D; (

@%=?2U6;<>/.

"F&)" #?&)(, * ( ,?L#! #c$ XcU2/60=[78f2/=@%=?>A@%>IgU=<9/8:@%>D[:=@7>&B+9RE0>D[7=BQ7=<60B?>&BQ9R=NE0>AS&9R=9/S\;?D=<>/.

,>)(*# N;= "(R!)!-(R#748:9 <; >=?; SB BD@ % C8 EcZ OYH@PO [:87>uB?nD=<6F>eBm;<9R;?6TB`;<60Zo[7>e@%6TB<S\=<\;?>I>GB`;V[787>I+9RE0>D[7=

x@%[ SD9/=<9USl;<D=<>J@%2U8o;

EO > >&S\;?6F>GB`;N4 9+%604 9RE .% C87>I0T\KN``0HEcZOWKNT\H@PO [787>Bmn&=?60>eBm;<9+;<60Bm;?6TZo[7>Ig/=<2/[:Qn&>I>&Bm;Y[787>uSDE09UB?B?>AQ2U[7=YET9/Zo[7>&EFE0>IET9pS\2U8:S\>&8U;<=<9R;?602/8@%>¤EO > >&S\;?6F )> >&S\;?6FG;9R60EFE0>H@%>¤ET9 S\ET9/B<Bm>>&Bm;p4d9R%6F4 9RE0>/z¥@72/8:S¤SD2/=<=?>GBmQ'2/8:@:9R8o;( [78 =<>&S\;<9R8:g/E0>f@7>©:9R[%;<>D[7=N4 9Rs604 9RE0>I@79R8:BE O ©760Bm;?2Ug/=9R4d4d>/.

e 'bf?g %SB B bc>A42%@%> !2/[ E09 SDE09UB?B?>J42%@79/EF>38^O >GB`;CQ9/By)2/=S\n&4d>D8o;N[7876TZU[:>/.748:9 <; >=?; SB B 2U60SD6c@%6 'n&=?>&8o;<By;`¬sQ>GBN@%>I4d2+¬/>&8787>&B@PO [787>eB?nD=<60>A@760B<S\=<\;<>

(xi, ni), i ∈ [[1, p]]~

% £¤2+¬/>&8787>A9R=<6;<©74dn\;?6TZo[7>x~x = 1

n

p∑

i=1

nixi =p∑

i=1

fixi

.% £¤2+¬/>&8787>Jg/nD2U4dn\;?=<6TZU[:>

g~g = n

√

p∏

i=1

xini =

p∏

i=1

xfi

i

.% £¤2+¬/>&8787>J©:9R=<4d2/876TZo[7>

h~ 1

h= 1

n

p∑

i=1

ni

xi=

p∑

i=1

fi

xi

.% £¤2+¬/>&8787>AZU[9/@%=9+;<60Zo[7>

q~q =

√

1n

p∑

i=1

nix2i =

√

p∑

i=1

fix2i

.

748:9 <; >=?; SB B r%2/6F;(xi, ni), i ∈ [[1, p]]

[787>eB?nD=<60>JBm;<9R;?6TB`;<60Zo[7>e@%6TB<S\=<\;?>Uz%>\;YB?2/6F;k ∈ N

.b^>A4d2/4d>&8U;Y@^O 2U=<@%=<>

k>GB`;

mk =p∑

i=1

fixki

.

V-W n SbB Bdpm0 = 1

zm1 = x

zm2 = q2

.

748:9 <; >=?; SB B\ 1y9/B@%>&BBmn&=?60>&BeBm;<9R;?6TB`;<60Zo[7>GBegU=?2U[7Qn&>&B l. 1 8 @7n\¦:876F;E0>&B@%6 'n&=?>&8U;<>&Bu42+¬U>D878:>&BI>D;E0>&B4d2/4d>D8o;BCS\2U44d>An\;9R8o;NEF>GBN4d2+¬/>&8787>&By>D;C42U4d>D8o;<BN@%>IE0>D[:=C@%60B<S\=<n\;<60B<9+;<6F2U8^.

SB m ' 0n >=%; Z

P*>) !)(bc94dn&@%6T9R8:>V>GB`;N608U;<[76F;?60/>D4d>&8U;NET9u+9/EF>&[7=N@%[ S&9R=9/S\;?D=<>VB?nDQ9R=9R8o;3E O > '>GSl;<6#>D8f@%>D[%Q:9R=?;<B3nDgo9RE0>&B&.713>D;m;<>872/;?602/8 9 @7nDQ'>D8:@%=<>I@%>IET9Q9R=<6;<nA@7>

|Ω|.

"7 ¯ ## *¤Go(*¤D$G"%$&D$G ##*¤¯ &/(+°U$G*rs2U6;

(xi, ni), i ∈ [[1, p]][787>eB?nD=<6F>AB`;9+;<60Bm;?6TZo[7>e@%6TB?SD=?D;?>/.

1 8p2U=<@%2U8787>NE0>&B36F8@%6Fs6T@%[:By@%>ΩQ9R=2U=<@7=?>CSD=?2U60B<B?9/8o;'!9R[Bm>&8:BET9R=<g/>@%>VS&9R=9/S\;?D=<>I~

Ω = ω1, . . . , ωnz

265X(ω1) 6 · · · 6 X(ωn)

.C2/;?> ¤Zo[7> Q7[76TB?Zo[7>¤@79/8:B[787>¤B?nD=<6F> B`;9+;?6TBm;?6TZU[:>/z2/8Q>&=<@¨EF>GB @%2/8:87nD>GBS\2U8:S\>&=?89R8o;

Ω(Q:9/=m;YB?2/8¤SD9/=<@76F8:9/Ez7SD>DET9=?>&o60>D8o;(d=<>&S\=<nD>&=y¦Sl;<6FU>D4d>D8o;

ΩzS/O >GB`; _ ( _ @%60=<> (d=<>D4d>\;?;?=<>I@%>&Bn\;<60Zo[7>D;m;<>&B&z%4d9/60BN9/872/8s¬s4d>&ByBm[:=NEF>GBy@76 nD=<>D8o;?>GBQ>&=<B?2/8:87>&B&zs[78Q'>D[¤SD2/4d4d>VE0>&BNSD2s@7>&B@PO 9R872U8o¬s4 9+;E02/=B

@PO >\79R4d>&8:B2/[fS\2U8:S\2U[7=<B&. 748:9 <; >=?; SB BTSb 94nG@%6T9R87>

m@%>IET9dBmn&=?60>

(xi, ni), i ∈ [[1, p]]>GB`;I~

%m = X(ωn+1

2

)B?6n = |Ω|

>&Bm;CQ:9R60=c]%m = X(ωn

2) +X(ωn

2+1)

Bm6n>&Bm;C604Q9R60=&.

N>&4d9/=<Zo[7>IZU[:>mQ>&[%;CQ7=<>D8:@%=<>A[787>I+9RE0>D[:=y©:2/=BN@%>

C.

"7 ¯ ## *¤Go(*¤D$G"%$&D$G ##*¤/' $& # *rs2U6;

(Ci, ni), i ∈ [[1, p]][787>pB?nD=<60>dBm;<9R;?6TB`;<60Zo[7> gU=?2U[7Q'nD> !@%2U8U;e2U8 95'2/=<87n E0>&BA>Do;<=P3&4d>&BIB?6¥5>GBm2U6F8lz

@%2U8U;EO > >&S\;?6FV@%>S©9/Zo[7> SDE09UB?B?> >&Bm;872U8 8s[7E.XCE02/=BeET9f)2U8:Sl;<6F2U8¨@%>p=<nDQ:9/=m;<6;<6F2U8FX

>&Bm;SD2/8o;?608s[7>p>D;Bm;?=<60S\;?>D4d>&8U;CSD=?2U60B<B?9/8o;?>/zs>D;Y@%n\¦:8:6;Y@%2U8:SJ[787>I576 k`>&S\;?602/8¤@%>

RBm[7=

[0, 1].7XC608:Bm6 z72U8Q'>D[%;J@%n\¦:8:6F=I~

748:9 <; >=?; SB B[X b 9u4dnG@%609/87>m@%>AE09Bmn&=?60>

(Ci, ni), i ∈ [[1, p]]>&Bm;yE O [7876TZU[:>A9/8o;?n&SDn&@79/8o;

F−1X

(

12

)

@%> 12

Q:9/=FX

.7XY6F8:B?6 zm>&Bm;NE09 B?>D[:EF>A+9RE0>D[7=;<>DE0EF>eZo[7>

FX (m) = 12

.e 'bf?g %SB B[Z r%6[787>dS\ET9/B<Bm>>&Bm;A@^O > '>GSl;?6F8o[:EzPET9p)2U8:Sl;<6F2U8 @7>=<nDQ:9/=m;<6;<6F2U8HQ7=<n&B?>D8o;<>[78Q7ET9+;?>G9R[^.rs6#[78 ;<>DE#Q7ET9+;?>G9R[ Q:=?>&8:@¤ET9p+9RE0>D[7= 1

2

z 2U8 87>uQ'>D[7;JQ:9UBY@%nD¦:8760=VET9p4dnG@%609/87>@7>uET9p4 9R8760D=<>uQ:=?nGS\n&@7>D8o;?>U.bc94dn&@7609/87>IBm>&=<9d9RE02/=ByEF>A4d60EF60>D[¤@%[fQ7ET9+;<>&9R[c.1y9RETS\[:EPQ7=<9R;?6TZo[7>d~1 8e=<>DQ'D=<>ET9YSDE09UB?B?>3Zo[76U!9R6F;#Q9/B<Bm>&=cE O > '>GSl;?6F S\[74[7E0n 9R[u@%>GB?B?[:B#@%>yE09Y42U6;<6Fn )2U[uET9N)=<n&Zo[7>D8S\>3SD[74u[:EFn&>9R[p@7>&B<Bm[:B@%> 1

2

l. 1 8 > >&Sl;<[7>Y>&8:B?[76;<>C[787>Y=?&g/E0>C@%>VhIB?[7=3SD>\;?;?>YSDE09UB?B?>/.srs2/6F;Ci =]ai, ai+1]

S\>D;m;?>VS\ET9/B<Bm>Uz@%>A)=<n&Zo[7>&8:S\>GBNS\[74[7EFn&>&B

gi

>&8ai

>D;gi+1

>&8ai+1

.7XCE02/=BA~m− ai

ai+1 − ai

=12 − gi

gi+1 − gi

.

1 8 Q>&[%;Y5760>D8¤B 7=NEF>J!9/6F=<>I9/>&SVET9)=?nGZU[:>D8:SD>ASD[74u[:EFn&>/z%>D8f=<>D4dQ7ET9 &9R8o; 12

Q:9R= n2

.

"F&)" #?&)(, * (* !,C (&,%! I B2 0 ,("7 ¯ ## *¤Go(*¤¯ &/(+°U$G*rs2U6;

(xi, ni)i∈[[1,p]][78:>eBmn&=?60>JBm;<9R;?6TB`;<60Zo[7>e@760B<S\=<\;<>/.

748:9 <; >=?; SB BD@ % b¥O<XLILKNM'jKNM'Q<j aEcX!jkQD!]iHEcZ YH0SH>&Bm;E = 1

n

p∑

i=1

ni|xi − x| =p∑

i=1

fi|xi − x|.

\ ?n SB ' 8 &:; >f%g

% b¥O<XILKRM'j0]iKYORMPKNjkQD!]iHEcZ YH0S )2U[ XLILKNM'jklj ^:Hy>&Bm;σ =

√

1n

p∑

i=1

ni(xi − x) =

√

p∑

i=1

fi(xi − x).

% bc9 4 KRM'QDKNSbIHu>&Bm; V = σ2 .748:9 <; >=?; SB B 1 8 9RQ7Q'>DE0E0>Q7E0[:BJg/n&87nD=9RE0>D4d>&8U;J4d2/4d>D8o;IS\>D8o;<=?n@PO 2U=<@7=?>

kk ∈ N

YE09 Zo[:9R8o;?6F;?n ~µk =

p∑

i=1

fi(xi − x)k .

V-W n SbB B µ0 = 1

zµ1 = 0

zµ2 = V

."7 ¯ ## *¤Go(*¤/' $& # *1 8@7n\¦:876F;e@%>43&4d>E O nGSD9/=m;A9R=<6;<©74dn\;?6TZo[7>42+¬U>D8^z'E O nGSD9R=?; _ ;`¬sQ>Uz E09 9/=?6T9R8S\>u>D;JE0>&BJ4d2/4d>&8U;BAS\>D8o;<=?nGB@PO [787>eB?nD=<6F>IBm;<9R;?6TB`;<60Zo[7>IS\2U8o;?608o[:>/z%>D8¤SD2/8:B?60@7nD=9R8o;B<9d@%6TB?SD=?nD;?6TB?9R;?602/8^.m b=%ni= $ >=?;SB B H0T\KNjkQDZNS OYH4ZH0SiQ YF] 0 :H0S:`

σ2 = q2 − x2 s `0ZNQ<j µ2 = m2 −m21

w I &) !+(*#C2U[:Be9U2/8:BA@%nmk (fo[ ET9 4nG@%6T9R87>UzcZo[76¥Q'>D[%;9/[:B<Bm63D;?=<>s[ S\2U4d4> [:8 Q9R=9R4d\;<=?> @%>p@%6TB?Q>&=<B?6F2U8^zPQ:9/=S\2U4dQ:9R=9R6TBm2U8(eE09I4d2+¬U>D8787>U.Rij8o;?[76F;?60/>&4d>D8o;&zoBm6

m < xz/E O > '>GSl;<6^>GB`;y>D8 4 9+k`2U=?6F;?nCB?2/[:BE09e42+¬U>D878:>/.R9R=

S\2U4dQ>&8:B?9R;?602/8^z'E O nGSD9R=?;YQ:9/=V=9RQ:Q2U=m;&( E09 4d2+¬/>D8:87>@%>GBY608:@%60s60@7[:BV9/[H@%>GB?B?[:BJ@%>ET9p4d2+¬U>D8787>@%2/6F;3D;?=<>Q7E0[:BI604dQ2U=m;9R8o;eZU[:>E O n&SD9/=m;IQ:9/=I=<9/Q7Q'2/=?; ( ET9 42+¬U>D878:>@%>GBI608:@%60o6T@%[:BuBm> B?6;<[:9R8o;I>D8 @%>GB?B?2/[:B&.^XC608:B?6zET9¤Bmn&=?60> >&Bm;uQ7E0[:BenD;<9/EFn&>pBm[7=EF>GBI+9RE0>D[7=BeB?[7Q'nD=<6F>&[7=?>GBDzc>\;uQ7E0[:BS\2/8S\>D8o;<=?n&>dB?[7=eE0>&Bu+9RE0>D[7=BI608%)nD=<6F>&[7=<>&B&.ij8s/>&=<B?>D4d>D8o;NBm6

m > x.

C2U[:B/2+¬/2U8:Bu@Ô 9/[%;?=<>&BuQ:9R=9R4d\;<=?>GBuB?6F4d60E09/6F=<>&B( ET9 4dnG@%609/87>/zZo[76yQ>&=?4d>D;m;?>&8o;d@%>fB?6;<[7>D=EF>fZo[:9R=?;&z#E0>@%6F%6F&4>J>\;SR.7@%>JE O > '>GSl;?6F`. # # ! )$G #I"# /"7d¯ ## *fGo(* (+# s*

(Ci, ni), i ∈ [[1, p]]@%2U8U;NE0>&BN> '>GSl;<6!BC@7>eS©:9/Zo[7>AS\ET9/B<Bm>IB?2/8o;N872/8 8s[7ETBDz7>D;CZU[cO 2U8f95'2/=<87nI9R[%>\s;?=3D4d>&B748:9 <; >=?; SB B\p % b^>^bMPH!EQDH0MJ!]iKRM'jCQ<TdHI>&Bm;CE O [78:60Zo[7>I+9RE0>D[:=

q1;?>&EFE0>eZo[7>

FX (q1) = 14

.% b^>4jCMZNQv`'Q !EcH4!]iKRM'jCQ<TdHI>&Bm;CE O [78:60Zo[7>I+9RE0>D[:=

q3;?>&EFE0>eZo[7>

FX (q3) = 34

.% b¥OdQ<SijlH!M 4 KNT<T\HJQ<SjH0MP0]iKNM'jkQ<T\HI>GB`;CE O 608o;?>&=?+9RE0E0> [q1, q3]

.% b¥O<H'`kîKYILHQ<SjH0MP0]iKNM'jkQ<T\Hu>&Bm;ET9EF2U87g/[:>D[7=C@%>eSD>\;N608U;<>D=<+9RE0EF>Uz(B<9/2U6F=

q3 − q1.

XC608:B?6zUEO 608U;<>D=<+9RE0EF>C608o;?>&=<Zo[:9/=m;<6FE0>>&Bm;EO 608U;<>D=<+9RE0EF>C=<>DgU=?2U[7Q:9/8U;ET9A4d2/6F;?60nYS\>&8U;<=<9/EF>C@7>&B¥608:@%60s60@7[:BD.ob¥O >&B?Q:9/SD>608U;<>D=Zo[:9R=?;?60EF>J>&Bm;Y@PO 9R[%;9R8o;NQ7E0[:BNQ'>\;?6F;VZo[7>AET9B?nD=<60>J>&8¤S\2/8S\>D8o;<=?n&>V>&8fB?2/8fS\>&8o;?=<>/.1 8f@%n\¦:8:6;V@7>J443&4> ~748:9 <; >=?; SB B\ % rs2U6;

k ∈ [[1, 9]].7bc>

k _ 60D4d> OYXLI0Q<T\Hu>&Bm; dk

;<>DEcZo[7>FX(dk) = k

10

.% rs2/6F;

k ∈ [[1, 99]].7bc>

k _ 60D4d> IH0SijkQ<T\H )2/[îH!MIH0SijkQ<T\Hy>&Bm; ck ;<>DEcZo[7> FX(ck) = k100

.% rs2/6F;

k ∈ [[1, 999]].%bc>

k _ 60D4d>4EQ<T<T_Q<T\H>&Bm; milk ;<>DEcZo[7> FX (milk) = k1000

.¥=<6F8S\60Q>e@%>eS&9RETS\[7E~1VO >&Bm;E0> 443D4d>fZo[7> Q'2/[7=ET9H4dn&@7609/87>H~>D8S\2/8Bm6T@%nD=9R8o;uE0>&B)=<n&Zo[7>&8:S\>GBe2/[EF>GBu> '>GSl;<6!BdS\[:4u[7E0n&B&z 2U8=<>DQ'D=<>JET9dS\ET9/B<Bm>A@79/8:BNET9/Zo[7>DE0E0>AB?>IBm6F;?[:>IEF>IZo[:9R8o;<6FE0> ( SD9RETS\[:EF>&=&zs>\;C2U8> '>GSl;<[7>I[787>I=<DgUEF>I@%>Ih7.

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