Cours de Proba1

8/16/2019 Cours de Proba1

1/58


2/58

P (E )

⊂

E p E n E p E n 1 ≤ p ≤ n p

Ω


3/58


4/58

x ∈ E a E x ∈ x

Ω Ω ⊂ Ω E = {1, a, }

E = {x/x P }

N = {x/x } = {0, 1, 2,...}Z = {x/x } = {..., −2, −1, 0, 1, 2,...}

P (E )

E E E P (E )

•P (∅) ∅•P ({a}) {{a}, ∅}•P (a, b) {{a, b}, {a}, {b}, ∅}


5/58

⊂

E A E E A E A

⊂E

A ⊂ E ⇐⇒ (∀x, x ∈ A =⇒ x ∈ E )A ⊂ E ⇐⇒ (A ∈ P (E ))

(x ∈ E ) ⇐⇒ ({x} ⊂ E ) ⇐⇒ ({x} ∈ P (E ))

E A E A E AE E A

AE = {x ∈ E/x /∈ A}

∅E = E E ∅ = ∅

A B

A × B = {(x, y)/x ∈ A et y ∈ B} A B A ∪ B = {x/x ∈ A oux ∈ B} A ∪ B = {x/x ∈ A etx ∈ B}

A ∩ B = ∅ A B

I i ∈ I E i

i∈I

E i = {x/∃i ∈ I , tel que x ∈ E i} (E i)i∈I

i∈I E i = {x/∀i ∈ I, x ∈ E i} (E i)i∈I

A B P (E )

C A∩BE = C AE ∪ C BE

C A∪BE = C AE ∩ C BE


6/58

(∀

x∈

E )

non(x ∈ A e t x ∈ B) ⇐⇒ (x /∈ A) ou (x /∈ B)non(x ∈ A o u x ∈ B) ⇐⇒ (x /∈ A) et (x /∈ B)

C A∩BE = C AE ∪ C BE

C A∪BE = C AE ∩ C BE

A B E A B A

\B A B

A \ B = {x ∈ E/x ∈ A et x /∈ B}

(Ai)1≤i≤n E

i=ni=1

Ai

C =

i=ni=1

AC i

i=ni=1

Ai

C =

i=ni=1

AC i

A B E

AB = (A ∪ B) \ (A ∩ B). AB = (A \ B) ∪ (B \ A) = (A ∩ B̄) ∪ (B ∩ Ā)

E F f E source F but•

f surjective surjection

(∀ ∈ E ), (∃x ∈ E ) tel que y = f (x)•

f injective injection

(∀x, y ∈ E ), [f (x) = f (y) =⇒ x = y]• f bijective bijection

(∀y ∈ E ), (∃!x ∈ E ) tel que y = f (x)


7/58

• f : x → y = f (x) E F

f −1,

F

E

(∀x, y ∈ E × F ), [y = f (x) ⇐⇒ x = f −1(y)]

n p p ≤ n, [ p, n] = { p, p + 1,...,n} E F E F E F

P I P I

φ : P −→ I n −→ n + 1

ϕ

ϕ : R \ {1} −→ R \ {2}x −→ 2x+1x−1

R \ {1} R \ {1} ψ

χ : ] − 1, 1[ −→ Rx −→ x1−x2

] − 1, 1[ R E

n

E

[1, n]

n

cardinal

E

E = n

• ∅ = 0

• non f ini infini N,Z,Q,R,C

• {♠, ♣,,,, ♦,,, ℵ, ♥} = 10• [1, n] = n.• card[ p, n] = n − p + 1.


8/58

E F f E F f E = F

E F ·

E ⊂ F E ≤ cardF ·

(E ⊂ F E = F ) E < cardF f E F

· f E ≥ cardf (E ) = cardF · f E = cardf (E ) ≤ cardF · f E = cardf (E ) = cardF

E F

card(E ∪ F ) = cardE + cardF − card(E ∩ F )

E ,F,G

(E ∪ F ∪G) = cardE + cardF + cardG − card(E ∩ F ) − card(E ∩ G) − card(F ∩G) + card(E ∩ F ∩ G)

A B E

E = cardA + card Ā ( Ā = AE = E

\A)

E = card(A ∩ B) + card( Ā ∩ B) + card(A ∩ B̄) + card( Ā ∩ B̄)

E F

card(E × F ) = cardE × cardF

a n b m (a, b), nm

E 1, E 2,...,E p

card(E 1 × E 2 × ... × E p) = cardE 1 × cardE 2 × ... × cardE p

cardE n = (cardE )n


9/58

E n (n ≥ 1) n

E p E n

E p E n n p.

F (E p, E n) E p E n

ψ : F (E p, E n) −→ E n × E n × ... × E n = E pnf −→ (f (a1), f (a2),...,f (a p))

E p = {a1, a2,...,a p}ψ

cardF (E p, F n) = cardE pn = n p

E p (x1, x2,...,x p) E p = E × E × ... × E E

23 = 8. 3 − liste {P, F } r 1, 2,...,r

n 1, 2,...,n r

{(x1, x2,...,xr) xi i}

nr

E p E n 1

≤ p

≤n

E p E n n(n − 1)...(n − p + 1)

E p {1, 2,...,p}. E p E n 1 n

E n 1 2


10/58

1 n − 1 3 1 2

n(n − 1)...(n − p + 1) = n!

(n − p)!

A pn = n(n − 1)...(n − p + 1) = n!

(n − p)!

E E = n, un p E p E

p n (1 ≤ p ≤ n) A pn

· p > n, A pn = 0 : E p E n

· p = n, Ann = n! E p E n

· p = 0 A0n = 1

A58 = 336

A2323 = 23!

(c1, c2, c3)(ci ∈[1, 12]). A312 = 1320 l

ordre

AHMED

A D


11/58

p

E n (n ≥ 1) P p(E ) E p ( p

≤n).

p n (0 ≤ p ≤ n) n! p!(n− p)!

pn =

n

p

= n! p!(n− p)!

M p E n (M, E ) M E P p(E ) E p ϕ

(M, E )

−→ P p(E )

i −→ A = i(M ) = {i1, i2,...,i p}

(M, E ) = ∪A∈P p(E )ϕ−1(A) (M, E ) = p!.cardP p(E ) P p(E ) =

n! p!(n− p)! =

pn

E p p E

p > n pn = 0, E E

n

0n =

nn = 1

58 = 56

A312

3! = 220

552 = 2598 960

(x1, x2,...,x p) ∈ N p x1 + x2 + ... + x p = n p n

n+ p−1n

pn

n p (0 ≤ p ≤ n), pn = n− pn pn =

pn−1 +

p−1n−1


12/58

· E n A

⊂E p Ā = A

E

E n−

p

ϕ : P p(E ) −→ P n− p(E )A −→ Ā

P p(E ) P n− p(E ) pn = n− pn n − p p

· a ∈ E.

p a a.

a p−1 E \{a}

p−1

n−1. E \{a}. pn−1 pn =

p−1n−1 +

pn−1

a b n

(a + b)n =

p=n p=0

pnan− pb p

n ∈ N∗, • 0n + 1n + 2n + ... + nn =

p=n p=0

pn = (1 + 1)n = 2n

• 0n − 1n + 2n − 3n + ... + (−1) p pn + ... + (−1)nnn = p=n p=0

pn(−1) p = (−1 + 1)n = 0

• 0n + 21n + 222n + ... + 2 p pn + ... + 2nnn = p=n p=0

pn2 p = 3n

N k (k

≤N ),

N i(i ∈ [1, k]) i N 1 + N 2 + ... + N k = N n

n n− N n


13/58

n1, n2,...,nk n1 + n2 + ... + nk = n. n n1 1 n2 2,...,nk k

n!n1!n2!...nk!

N n11 N n22 ...N

nkk .

n n N AnN

n− n1 n2 2,...,nk k

n!n1!n2!...nk!

An1N 1An2N 2

...AnkN k .

n n nN .

(n1, n2,...,nk) n1 N 1, n2 N 2,...,nk k N k

n1N 1

n2N 2

...nkN k .

•A

•B • C A = 2!1!×0!×1!4

1 × 20 × 31 = 24 B = B1 ∪ B2 ∪ B3 B1 B2 B3


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cardB = cardB1 + cardB2 + cardB3

= 2!2! × 0! × 0! 42 × 20 × 30 + 2!0! × 2! × 0! 40 × 22 × 30 + 2!0! × 0! × 2! 40 × 20 × 32

= 29

C = C 1 ∪ C 2 C 1 C 2 C 3 = B3.

cardC = cardC 1 + cardC 2 + cardB3

= 2!

1! × 1! × 0!42

×21

×30 +

2!

0! × 1! × 1!40

×21

×31 +

2!

0! × 0! × 2!40

×20

×32

= 85

• • • • • • • • • •


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prvisions

f : I −→ Rt −→ x = f (t)

e


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Ω

Ω

erlancer P F

elancer P F P F

elancer P F P F P F P F

Ω = {P P P , P P F , P F P , P F F , F P P , F P F , F F P , F F F }

Ω = [1, 6]2

Ω = {n ∈ N/0 ≤ n ≤ 36}

Ω =

{P, F

} ×[1, 6]

Ω P (Ω)

P (Ω) Ω

{P P P , P P F , P F P , P F F }. {2, 4, 6}.


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· ∅

· Ω

· A B A B A ∩ B = ∅.

· Ω

p P (Ω) [0, 1]

p(∅) = 0, p(Ω) = 1

A ∩ B = ∅

p(A ∪ B) = p(A) + p(B) (Ω, P (Ω), p) Ω (Ω, P (Ω), p)

Ω = {P, F }

p : P (Ω) −→ [0, 1]P −→ 12F −→ 12

(Ω, P (Ω), p)

(Ω, P (Ω), p)

∀A, B ∈ P (Ω), p(A ∪ B) = p(A) + p(B) − p(A ∩ B).

∀A, B ∈ P (Ω) A ⊂ B, p(A) ≤ p(B).

∀A ∈ P (Ω), p( Ā) = 1 − p(A), Ā = AΩ


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∀A, B ∈ P (Ω)

A ∪ B) = (A − B) ∪ (B − A) ∪ (A ∩ B) A − B, B − A, A ∩ B

p(A ∪ B) = p(A − B) + p(B − A) + p(A ∩ B)

A = (A − B) ∪ (A ∩ B) (A − B) ∩ (A ∩ B) = ∅

B = (B − A) ∪ (A ∩ B) (B − A) ∩ (A ∩ B) = ∅

p(A) = p(A − B) + p(A ∩ B)

p(B) = p(B − A) + p(A ∩ B) p(A − B) p(B − A)

p(A

∪B) = p(A)

− p(A

∩B) + p(B)

− p(A

∩B) + p(A

∩B)

p(A ∪ B) = p(A) + p(B) − p(A ∩ B) A, B ∈ P (Ω) A ⊂ B.

B = A ∪ AΩ = A ∪ (B − A) A B − A

p(B) = p(A) + p(B − A) ≥ p(A) (carp(B − A) ≥ 0) Ω = A ∪ A,

1 = p(Ω) = p(A) + p(A)

(Ω, P (Ω), p) Ω = {a1, a2,...,an}. p({ai}) (Ω, P (Ω), p)

· ∀a ∈ Ω, p({a}) = 12· A = {ai1 , ai2 ,...,aip} ⊂ Ω, p(A) = pn = cardAcardΩ = nombre de cas f avorablesnombre de cas possibles


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· Ω = {a1, a2,...,an},

1 = p({a1}) + p({a2}) + ... + p({an}), ∀i, p({ai}) = 1n

· p(A) = p({ai1 , ai2 ,...,aip}) = k. 1n = kn

16

6, 1, 2, 3, 4, 5, 6 Ω ={1, 2, 3, 4, 5}

p Ω ={

ω1, ω2, ω3, ω4, ω5, ω6}

.

ω1 ω2 ω3 ω4 ω5 ω6 pi

415 x

14

112

112 y

x y

p({ω1, ω2, ω3}) = 2 p({ω4, ω5, ω6}).

415

+ x + 14

+ 112

+ 112

+ y = 1 x + y = 1960

4

15 + x +

1

4 = 2(

1

12 +

1

12 + y)

31

60 + x =

1

3 + 2y

x = 320 y = 16


20/58

p Ω. A B

p(A) = 1

3 , p(B) = 2

5 p(A ∩ B̄) =

1

6

p(A ∩ B), p(A ∪ B), p(B ∩ Ā), p( Ā ∩ B̄)

· A ∩ B A ∩ B̄ A

p(A) = p(A ∩ B) + p(A ∩ B̄),

p(A ∩ B) = 13 − 1

6 =

1

6

· p(A ∪ B) = p(A) + p(B) − p(A ∩ B)

p(A ∪ B) = 1730

· p(B ∩ Ā) + p(B ∩ A) = p(B) p(B ∩ Ā) = 730· Ā ∩ B̄ = A ∪ B p( Ā ∩ B̄) = 1 − p(AB) = 1330

Ω P (Ω) Ω

Ω = {P, F } × {P, F } = {(P, P ), (P, F ), (F, P ), (F, F )}

P (Ω) = A ∪ B ∪ C ∪ D

A = {∅, Ω}

B = {{(P, P )}, {(P, F )}, {(F, P )}, {(F, F )}}C = {{(P, P ), (P, F )}, {(P, P ), (F, P )}, {(P, P ), (F, F )}, {(P, F ), (F, P )}, {(P, F ), (F, F )}, {(F, P ), (F, F )}}D = {{(P, P ), (P, F ), (F, P )}, {(P, F ), (F, P ), (F, F )}, {(F, F ), (P, P ), (P, F )}, {(F, F ), (P, P ), (F, P )}}

∀E ∈ P (Ω), p(E ) = cardE 24

p(∅) = 0 p(Ω) = 1

∀E ∈ B, p(E ) = 116


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∀E ∈ C, p(E ) = 216

∀E

∈D, p(E ) =

3

16

A : ” ” B : ” ” C : ” ”

N 1, R2, R3, V 4, V 5, V 6.

Ω Ω

Ω

A2

6 = 6 × 5 = 30.· A A 3 × 2 = 6

p(A) = 6

30 =

1

5

· p(B) = 1 − p( B̄) B̄ : ” ” B̄

N 1, V 4, V 5, V 6. B̄ = 4 × 3 = 12

p(B) = 1

− p( B̄) = 1

−

12

30

= 3

5· p(C ) = 1 − p( C̄ ) C̄ : ” ”C̄ = C 1 ∪ C 2 C 1 : ” ” C 2 : ” ” C̄ 2 × 1 + 3 × 2 = 8

p(C ) = 1 − 830

= 11

15


22/58

Ω Ω 62 = 36.

A 3

×3 = 9 B̄ 4

×4

C̄, C̄ = C 1 ∪ C 2 ∪ C 3 C 1 : ” ”

C 2 : ” ”

C 3

cardC̄ = cardC 1 + cardC 2 + cardC 3 = 3 × 3 + 2 × 2 + 1 × 1 = 14

p(A) = 9

36

= 1

4

, p(B) = 1

−

16

36

= 5

9

, p(C ) = 1

− p( C̄ ) = 1

−

14

36

= 11

18

.

Ω = 26 = 15, 15 3 A 6 B̄ 4 C̄.

Ω

{1, 2, 3, 4, 5, 6}4, Ω = 64. A Ā = {1, 2, 3, 4, 5}4 Ā = 54 p(A) = 1 − 5464 = 0. 5177

(b, n) b n 62 = 36 Ω = 3624 B : ” 6”; B̄ = 3524 p(B) = 1 − 35243624 = 0. 4914


23/58

(Ω, P (Ω), p) B A B p(A/B)

pB(A), p(A/B) = p(A∩B)

p(B) .

pB

: P (Ω) −→ [0, 1] Ω : B

• pB

(Ω) = p(Ω∩B)

p(B) = p(B) p(B) = 1

• ∀A, A ⊂ Ω, A ∩ A = ∅ : pB

(A

∩A) = p[(A∩A

)∩B] p(B) =

p[(A∩B)∩(A∩B)] p(B) =

p(A∩B) p(B) +

p(A∩B) p(B) = pB(A) + pB(A

)

♠ 1 R D V 10 9 8 7♥ 1 R D V 10 9 8 7♦ 1 R D V 10 9 8 7♣ 1 R D V 10 9 8 7

R : ” tirer un roi ”

T : ” tirer un trèfle ”R ∩ T : ”tirer un roi de trèfle ” Ω = 32 p(T ) = 832 =

14 , p(R ∩ T ) = 132 p(R∩T ) p(T ) = 18

Ω = {1,R,D,V, 10, 9, 8, 7} Ω.

R/T p(R/T ) = 18 p(R/T ) =

p(R∩T ) p(T ) .

·

p(A/B)

P (A ∩ B)

P (A ∩ B) = P (A/B) × P (B).· P (A) = 0, P (A ∩ B) = P (B/A) × P (A).

A B (Ω, P (Ω), p) p(A) = 0 p(B) = 0,

P (A/B) = P (A)

P (A ∩ B) = P (A) × P (B) P (B/A) = P (B).


24/58

A1, A2,...,An Ω. A1, A2,...,An Ω

∀i ∈ {1, 2,...,n}, Ai = ∅.

∀i, j ∈ {1, 2,...,n}, i = j Ai ∩ A j = ∅ A1 ∪ A2 ∪ ... ∪ An = Ω

7 4 2 i = 1 2 Ri ” ième ” N i ” ième ” V i ” i

ème ”

R1 = ∅, N 1 = ∅, V 1 = ∅

R1 ∩ N 1 = ∅, V 1 ∩ R1 = ∅, N 1 ∩ V 1 = ∅ R1 ∪ N 1 ∪ V 1 = ΩN 1, V 1, R1 Ω. N 2, V 2, R2 Ω.

A (Ω, P (Ω), p) A1, A2,...,An Ω

p(A) =i=ni=1

p(A/Ai) × p(Ai)

p(A) = p(A ∩ Ω) A ⊂ Ω= p(A ∩ (A1 ∪ A2 ∪ ... ∪ An)= p((A ∩ A1) ∪ (A ∩ A2) ∪ ... ∪ (A ∩ An))= p(A ∩ A1) + p(A ∩ A2) + ... + p(A ∩ An)) A1, A2,...,An = p(A/A1) × p(A1) + ... + p(A/An) × p(An) p(A ∩ B) = p(A) × p(B).

p ∈]0, 1[

T ” ” A ” ”


25/58

:Ω = T ∪ T̄ ,

p(A) = p(T ) p(A/T ) + p(T̄ ) p(A/T̄ )

= p.1 + (1 − p). 113

= 1 + 12 p

13

( p(A/T̄ ) = 113

p(B) > 0, k

∈ {0, 1,...,n

},

p(Ak/B) = p(B/Ak) × p(Ak)i=ni=1

p(B/Ai) × p(Ai)

∀k ∈ {0, 1,...,n},

p(Ak/B) = p(B ∩ Ak)

p(B)

=

p(B/Ak) p(Ak)i=ni=1

p(B/Ai) × p(Ai)

0.8 0.5 0.7

A B p(A/B).

p(A/B) = p(B/A) p(A)

p(B/A) p(A) + p(B/ Ā) p( Ā)

p(A/B) = 0.7×0.2

0.7×0.2+0.5×0.8 = 0.2592


26/58

A B

B ( p(B) = 0) A ( p(A) = 0), p(A/B) = p(A)

p(A ∩ B) = p(A) × p(B)

A B (Ω, P (Ω), p) p(A ∩ B) = p(A) × p(B)

·

· p(A) = 0 p(B) = 0 ( p(A ∩ B ) = p(A) × p(B) = 0)

R : ” ”

T : ” ”

R ∩ T : ” ” R : ” ” T : ” ”

p(R) =

1

8 , p(T ) =

1

4 et p(R ∩ T ) = 1

32 ,

p(R ∩ T ) = p(R) × p(T ). T R

n

M : F :

n n = 2n

n = 2

2 = {(G, G), (G, F ), (F, G), (F, F )}

M = {(G, F ), (F, G)}, F = {(G, G), (G, F ), (F, G)} M ∩ F = M


27/58

p(M ) = 2

4 =

1

2, p(F ) =

3

4, p(M ∩ F ) = 1

2

p(M ∩ F ) = p(M ).p(F ), M F n = 3

3 = {(G,G,G), (G,G,F ), (G,F,F ), (F,G,F ), (F,G,G), (F,F,G), (G,F,G), (F,F,F )}

M = {(G,G,F ), (G,F,F ), (F,G,F ), (F,G,G), (F,F,G), (G,F,G)}F = {(G,G,G), (G,G,F ), (F,G,G), (G,F,G)}

M ∩ F = {(G,G,F ), (F,G,G), (G,F,G)}

p(M ) = 6

8 =

3

4, p(F ) =

4

8 =

1

2, p(M ∩ F ) = 3

8

p(M ∩ F ) = p(M ).p(F ). M F

n > 3 M F

A

B

(Ω, P (Ω), p)

A B A B̄ Ā B Ā B̄.

A = (A ∩ B̄) ∪ (A ∩ B)

p((A ∩ B̄) = p(A) − p(A ∩ B)= p(A) − p(A).p(B)= p(A)(1 − p(B))

= p(A).p( B̄)

A B̄ Ā B B̄ A B̄ Ā (Ak)1≤k≤n (Ω, P (Ω), p) A1, A2,...,An Ai1 , Ai2 ,...,Aik

p(Ai1 ∩ Ai2 ∩ ... ∩ Aik) = p(Ai1).p(Ai2).....p(A)ik


28/58

A1, A2,...,An

a b A = {a } B = {b } C = {a b } p(A) = p(B) = p(C ) = 12 p(A ∩ B) = p(A ∩ C ) = p(B ∩ C ) = 14 A, B C p(A ∩ B ∩ C ) = 0 = 18 A, B C

• • • • • • • • •


29/58

Ω

R

X Ω R

X −1(x) = {ω ∈ Ω/X (ω) = x} X = x X −1(]a, +∞[) = {ω ∈ Ω/X (ω) > a} X > a X −1([a, b[) = {ω ∈ Ω/a ≤ X (ω) < b} a ≤ X ≤ b X −1(] − ∞, a]) ∩ X −1(] − ∞, b]) (X ≤ a, X ≤ b)

Ω = {P, F } X X ({P }) = 1 X ({F }) = 0

p(X = 0) = p(X = 1) = 1

2

p 1 − p n ω n X (ω) X (Ω) = [0, n]


30/58

X

X (Ω) = {0, 1, 2, 3, 4, 5, 6}.

X (Ω, P (Ω), p)

pX : X (Ω) −→ [0, 1]a

−→ p(X = a)

X (Ω), X X pX .

• X (Ω) =

a∈Ω

(X = a)

pX (X (Ω)) =a∈Ω

p(X = a) = 1

• B B X (Ω) A A Ω X (A) = B X (A) = B

pX (B ∪ B) = p(

a∈B∪B

X = a)

= p(A ∪ A)= p(A) + p(A)

= p(X −1(A)) + p(X −1(A))

= pX (B) + pX (B)

X


31/58

X Ω

cardΩ = 36 = 20

{1, 1, 1}, {1, 2, 2}, {1, 1, 3}, {1, 2, 3}, {2, 2, 2}, {2, 2, 3}

X 4, 5, 6, 7, X (Ω) = {4, 5, 6, 7}.

p(X = 4) = 22

13

20 =

3

20

p(X = 5) = 12

23 +

22

11

20

= 7

20 p(X = 6) =

1213

11 +

33

20 =

7

20

p(X = 7) = 23

11

20 =

3

20

X

xi ∈ X (Ω) 4 5 6 7 p(X = xi)

320

720

720

320

X (Ω, P (Ω), p) X F X R

∀x ∈ R, F X (x) = p(X ≤ x)

X

X X X


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Ω

cardΩ = 6!

X (Ω) = {0, 1, 2, 3} 1er X = 0 2me X = 1 3me X = 2 4me X = 3

p(X = 0) = p(E 0)

E 0 1er

cardE 0 = 135!

(

1

3

1◦

5!

5

p(X = 0) = 135!

6! =

1

2

p(X = 1) = p(E 1)

E 1 1er 2◦

cardE 1 = 13

134!

(13

1

◦ 1

3

2

◦

4!

p(X = 1) = 13

134!

6! =

3

10

p(X = 2) = p(E 2)

E 2 1er 2◦ ◦

cardE 2 = A23

133!

(A2

3

13 3

◦ 3!

p(X = 2) = A23

133!

6! =

3

20

p(X = 3) = p(E 3)

E 3 E 3 = 3!3!


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(3!

p(X = 2) = 3!3!

6!

= 1

20 X

k ∈ X (Ω) 0 1 2 3 p(X = k) 12

310

310

110

X x ∈ R x


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X Ω, P (Ω), p) X (Ω) =

{x1, x2,...,xn

}

X E (X )

E (X ) =i=ni=1

xi p(X = xi)

X U R R DX .

E (U (X )) =i=ni=1

U (xi) p(X = xi)

X (Ω) = {x1, x2,...,xn} Y = aX + b E (Y ) = aE (X ) + b.

X Y = X − E (X ) Y X E (X ) = 0 E (X ) = 0 X

X Ω, P (Ω), p) X (Ω) = {x1, x2,...,xn}

· v.a.r X V ar(X )

V ar(X ) =i=ni=1

[xi − E (X )]2 p(X = xi)

· X σ(X )

σ(X ) =

V ar(X )

V ar(X ) =i=n

i=1 [xi − E (X )]2 p(X = xi) : [xi − E (X )]2 p(X = xi).


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V ar(X ) =i=n

i=1[xi − E (X )]2 p(X = xi)

=i=ni=1

[x2i − 2xiE (X ) + E 2(X )] p(X = xi)

=i=ni=1

x2i p(X = xi) − 2E (X )i=ni=1

xi p(X = xi) + E 2(X )

i=ni=1

p(X = xi)

= E (X 2) − 2E 2(X ) + E 2(X ) (i=ni=1

p(X = xi) = 1)

= E (X 2) − E 2(X )

V ar(X ) = E (X 2) − E 2(X ) X

Ω, P (Ω), p) a, b

V ar(aX + b) = a2V ar(X ) et σ(aX + b) = |a| σ(X )

V ar(aX + b) = E [(aX + b)2] − E 2(aX + b)= a2E (X 2) + 2abE (X ) + b2 − a2E 2(X ) − 2abE (X ) − b2= a2[E (X 2) − E 2(X )]= a2V ar(X )

σ(aX + b) =

V ar(aX + b)

=

a2V ar(X )

= |a|

V ar(X )

= |a| σ(X )

Y = X −E (X )

σ(X )

X

Y E (Y 2) = 1


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· E (Y ) = E ( X −E (X )σ(X ) ) = 1σ(X ) [E (X ) − E (E (X ))] = 1σ(X ) [E (X ) − E (X )] = 0

· V ar(Y ) = ( 1

σ(X ))2

V ar(Y ) = 1 (

V ar(Y ) = σ2

(Y ))· V ar(Y ) = 1 =⇒ E (Y 2) − E 2(Y ) = 1 ⇒ E (Y 2) = 1

E (X ) =

k∈X (Ω)

kp(X = k)

= 0 × 12

+ 1 × 310

+ 2 × 320

+ 3 × 120

=

3

4

E (X 2) =

k∈X (Ω)

k2 p(X = k)

= 02 × 12

+ 12 × 310

+ 22 × 320

+ 32 × 120

= 27

20

V ar(X ) = E (X 2)

−(E (X ))2 =

27

20 −(

3

4

)2 = 63

80

σ(X ) =

V ar(X ) = 0.88741

Y = X −E (X )σ(X ) =

574X −3

3

X Ω σ ε

p(

|X

−m

| ≥ε)

≤ σ2

ε2

σ2 = V ar(X )

=

xi∈Ω

(X (xi) − m)2 p(X = xi) avec Ω = {x1, x2,...,xn}


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Ω = |X − m| ≥ ε = {xi ∈ Ω / |X (xi) − m| ≥ ε}

σ2 =

xi∈Ω

(X (xi) − m)2 p(X = xi)

≥ ε2 p(Ω) p(X = xi)≥ ε2 p(Ω) = ε2 p(|X − m| ≥ ε)

p(|X − m| ≥ ε) ≤ σ2

ε2

ΩC ,

p(|X − m| < ε) ≥ 1 − σ2

ε2

1 − p(Ω) ≥ 1 − σ2ε2 , p(|X − m| < ε) ≥ 1 − σ2

ε2 (ΩC = |X − m| < ε)

X Ω X (Ω) = {x1, x2,...,xn} X X (Ω) X → U n p(X = xi) = 1n .

E (X ) = 1

n

ni=1

xi et V ar(X ) = 1

n

ni=1

x2i − E 2(X )

X (Ω) = {1, 2,...,n}, X · P (X = xi) = 1n , i = 1, 2,...,n· E (X ) =

n+1

2

· V ar(X ) = n2−112

X v.a.r X p X → β (1, p)


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p ∈]0, 1[ X X = 1 X = 0 X v.a.r

X → β (1, p)

E (X ) = 0(1 − p) + 1.p = p et V (X ) = (0 − p)2(1 − p) + (1 − p)2 p = p(1 − p) = pq

p q = 1

− p.

p(B) p(B) = p p(N ) = q = 1 − p.

B N p q = 1 − p n

n p

n p n n

B N 2n

n p Ωn 2

n

X n

X (Ωn) = {0, 1,...,n}

n

p

k 0 n (X = k) kn p

k(1 − p)n−k

(X = k) kn B...B

k

N...N n−k

p(B...BN...N ) = p...p k

q...q n−k

= pk(1 − p)n−k


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kn (X = k) B P (X = k) = kn p

k(1 − p)n−k n p Ωn p(X = k) =

kn p

k(1− p)n−k

B(n, p) X n p X → B(n, p)

1%. X

100 (100, 10−2).

2

1 − p(X


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·

E (X ) =n

k=0

kp(X = k) =n

k=0

kkn pk(1 − p)n−k = n

k=1

kkn pk(1 − p)n−k

et

kkn = n!

(k − 1)!(n − k)!

E (X ) = n p

nk=1

(n − 1)!(k − 1)!(n − k)! p

k−1(1 − p)n−k

= np

n−1

i=1

(n−

1)!

i!(n − i − 1)! pi(1 − p)n−1−i

= np( p + 1 − p)n−1= np

·

V ar(X ) = E (X (X − 1)) + E (X ) − E 2(X )= E (X (X − 1)) + np − n2 p2

E (X (X − 1))

E (X (X − 1)) =n

k=0

k(k − 1)kn pk(1 − p)n−k

=

nk=2

k(k − 1)kn pk(1 − p)n−k( )

k(k − 1)kn = n(n − 1)k−2n−2

E (X (X −

1)) = n(n−

1)n

k=2 k−2

n−2 pk(1

− p)n−k

= n(n − 1) p2n

k=2

k−2n−2 pk−2(1 − p)n−k

= n(n − 1) p2n−2i=0

in−2 pi(1 − p)(n−−2)−i (avec i = k − 2)

= n(n − 1) p2( p + 1 − p)n−2= n(n − 1) p2


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V ar(X ) = E (X (X − 1)) + np − n2 p2 = n(n − 1) p2 + np − n2 p2 = npq

1, 2, 3, 4 pi i (i = 1, 2, 3, 4)

p1 = 1

12, p2 =

7

36, p3 =

11

36, 4 =

5

12

X A

X

N N p N q p 1 − p

n

X X N, n p X → (N,n ,p) X → (N,n ,p), X (Ω) ⊂ [0, n] ∀k ∈ [0, n] p(X = k) =kN p×

n−kN q

nN

X → (N,n ,p), E (X ) = np V (X ) = npq N −nN −1

kN p×

n−kN q

nN

k

n pk

(1 − p)n−k

N

+∞. X n p

N

(N,n ,p) B(n, p) n p N > 10n.


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(Ω, P (Ω), p) C = (X, Y )

X (Ω) = {x1, x2,...,xn} Y (Ω) = {y1, y2,...,ym} pij = p((X = xi) ∩ (Y = y j)), (i, j) ∈ [1, n] × [1, m] pi. = p((X = xi), i ∈ [1, n] p.j = p(Y = y j)), j ∈ [1, m]

f : I → Rt → x = f (t)

C = (X, Y )

p : X (Ω) × Y (Ω) → [0, 1](xi, y j) → pij = p((X = xi) ∩ (Y = y j))

C = (X, Y )

p · : X (Ω) → [0, 1]xi → pi. = p((X = xi)

C = (X, Y ) X

p· : Y (Ω) → [0, 1]y j → p.j = p(Y = y j ))

Y

−→C = (X, Y )

X \Y y1 y2 • • • ym X x1 p11 p12 p1m p1•x2 p21 p22 p2m p2•••xn pn1 pn2 pnm pn•

Y p•1 p•2 p•m

X


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Y

ercas : ecas :

X \Y

0 1

0 9491249

37

1 12491649

47

37

47 1

X \Y

0 1

0 1727

37

1 2727

47

37

47 1

C = (X, Y )

• ∀x ∈ X (Ω) p(X = x) = j=m j=1

p(X = x, Y = y j)

• ∀y ∈ Y (Ω) p(Y = y) =i=ni=1

p(X = xi, Y = y)

• i=ni=1

j=m j=1

p(X = xi, Y = y j) = 1

· ∀x ∈ X (Ω),

(X = x) = (X = x) ∩ ( j=m j=1

Y = y j)

=

j=m

j=1(X = xi, Y = y j) ( )

p(X = x) = p(

j=m j=1

(X = xi, Y = y j) =

j=m j=1

p(X = x, Y = y j )

·

∀y ∈ Y (Ω), p(Y = y) =i=ni=1

p(X = xi, Y = y)

·

∀i = 1, 2,...,n

p(X = xi) =

j=m j=1

p(X = x, Y = y j ) =⇒ 1 =i=ni=1

p(X = xi) =i=ni=1

j=m j=1

p(X = x, Y = y j)


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· ∀i ∈ [1, n], pi· = j=m j=1

pij

· ∀ j ∈ [1, m], p· j =i=ni=1

pij

·i=ni=1

j=m j=1

pij = 1

C = (X, Y )

X (Ω) =

{x1, x2,...,xn

}et (Ω) =

{y1, y2,...,ym

} pij = p((X = xi) ∩ (Y = y j)), (i, j) ∈ [1, n] × [1, m] pi. = p(X = xi), i ∈ [1, n] p.j = p(Y = y j)), j ∈ [1, m]

p(Y = y j) = 0,

p(X = xi/Y =yj ) = p((X = xi) ∩ (Y = y j))

p(Y = y j) =

pij p· j

p((X = xi) = 0,

p(Y = y j /X =xi) = p((X = xi) ∩ (Y = y j))

p(X = xi) =

pij

p j·

C = (X, Y ) X Y = y j X (Ω) [0, 1]

xi → p(X = xi/Y =yj) = pij p· j

Y X = xi Y (Ω) [0, 1]

yi → p(Y = yi/X =xi) = pij pi·

X /Y = 0

p 3747

X /Y = 1

p 3747


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p(X = 0/Y =1) =9

493

7

= 37)

Y /X = 0

p 3747

Y /X = 1

p 3747

X Y Y X X Y X/Y = j = X Y/X =i = Y

X /Y = 0

p 1212

X /Y = 1

p 2313

p(X = 0/Y =1) =2

73

7

= 23)

Y /X = 0

p 1212

Y /X = 1

p 2323

X/Y =0 = X Y/X =1 = Y.

(Ω, P (Ω), p) C = (X, Y ) X Y

p((X = xi) ∩ (Y = y j )) = p(X = xi) × p(Y = y j )) pour (i, j) ∈ [1, n] × [1, m]

pij = pi· × p· j pour (i, j) ∈ [1, n] × [1, m]

X Y

n

n

A ” ”

B ” ”


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X n Y 0 n 1 n

(X, Y ).

X Y

Ω = 63.

· A A = 4 × 4 × 4 p(A) = (46)3 = 827· B B = 6 × 6 × 2 p(B) = 13 .

X (Ω) = {0, 1, 2, 3} et Y (Ω) = {0, 1} (X = 0, Y = 0) p00 = p(X = 0, Y = 0) = 0

(X = 0, Y = 1) A = (X = 0, Y = 1), p01 = p(A) =

827

(X = 1, Y = 0)

p10 = p(X = 1, Y = 0)

= 4 × 4 × 2

63

= 4

27

p11 = 8

27, p20 =

4

27, p21 =

2

27, p30 =

1

27, p31 = 0

X \Y 0 1 X 0 0 827

827

1 4

27827

1227

2 427227

627

3 127 0 1

27

Y 9271827 1

p(X = 0, Y = 0) = 0 p(X = 0) × p(Y = 0) = 827 . 927 , p(X = 0, Y = 0) = p(X = 0) × p(Y = 0) X Y


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A (A = ( pij )1≤i≤n,1≤ j≤m)

u(X, Y )

X Y Ω. Z = u(X, Y ) u

E (Z ) = (x,y)∈X (Ω)×Y (Ω) u(x, y) p(X = x, Y = y)

E (X + Y ) =

(x,y)∈X (Ω)×Y (Ω)

(x + y) p(X = x, Y = y)

=

(x,y)∈X (Ω)×Y (Ω)

(x,y)∈X (Ω)×Y (Ω) xp(X = x, Y = y) = x∈X (Ω)[ y∈Y (Ω) xp(X = x, Y = y)]=

x∈X (Ω)

x[

y∈Y (Ω)

p(X = x, Y = y]

y∈Y (Ω)

p(X = x, Y = y) = p(X = x)

(x,y)∈X (Ω)×Y (Ω)

xp(X = x, Y = y) =

x∈X (Ω)

xp(X = x) = E (X )

(x,y)∈X (Ω)×Y (Ω)

yp(X = x, Y = y) = E (Y )

E (X + Y ) = E (X ) + E (Y )

· E (XY ) = (x,y)∈X (Ω)×Y (Ω) xyp(X = x, Y = y)


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X Y

E (XY ) =

(x,y)∈X (Ω)×Y (Ω)

xyp(X = x, Y = y)

=

(x,y)∈X (Ω)×Y (Ω)

xyp(X = x) p(Y = y)

=

x∈X (Ω)

xp(X = x)

y∈X (Ω)

yp(X = y)

= E (X )E (Y )

X Y

E (XY ) = E (X )E (Y )

(Ω, P (Ω), p) C = (X, Y )

E [(X − E (X ))(Y − E (Y )] X Y Cov(X, Y ).

X Y Ω, E [(X − E (X ))(Y − E (Y )] =E (X.Y ) − E (X )E (Y )

E [(X − E (X ))(Y − E (Y )] = E [XY − E (Y )X − E (X )Y + E (X )E (Y )]

E [(X − E (X ))(Y − E (Y )] = E (XY ) − E (Y )E (X ) − E (X )E (Y ) + E (X )E (Y )= E (X.Y ) − E (X )E (Y )

X Y B(1, p) Cov(X, Y ) = p(X = 1, Y = 1) − p2.

E (XY ) = (x,y)∈X (Ω)×Y (Ω) xyp(X = x, Y = y) = p(X = 1, Y = 1) X,Y,X , Y (Ω, P (Ω), p).

Cov(X, Y ) = C ov(Y, X ) Cov(X + X , Y ) = Cov(X, Y ) + C ov(X , Y ) Cov(X, Y + Y ) = Cov(X, Y ) +

Cov(X, Y ) Cov(λX,Y ) = C ov(X,λY ) = λCov(X, Y ) Cov(X, X ) = V ar(X ) ≥ 0 V ar(X + Y ) = V ar(X ) + 2Cov(X, Y ) + V ar(Y ) X Y Cov(X, Y ) = 0


49/58

X V ar(X ) = 0 X

V ar(X ) =i=ni=1

[xi − E (X )]2 p(X = xi) = 0 ⇐⇒ ∀i ∈ [1, n], [xi − E (X )]2 p(X = xi) = 0

⇐⇒ ∀i ∈ [1, n], xi − E (X ) = 0⇐⇒ ∀i ∈ [1, n]; xi = E (X )

X (Ω) = {E (X )} X E (X ) X, Y (Ω, P (Ω), p).

|Cov(X, Y )| ≤ σ(X )σ(Y )(∗) Y = aX + b a, b

λ ∈ RT (λ) = V ar(λX + Y ), T (λ) = V ar(λX + Y )

= V ar(λX ) + 2Cov(λX,Y ) + V ar(Y )

= λ2V ar(X ) + 2λCov(X, Y ) + V ar(Y )

· V ar(X ) = 0, T λ, ≥ 0 λ ∈ R

= [Cov(X, Y )]2 − V ar(X )V ar(Y ) ≤ 0

|Cov(X, Y )| ≤ σ(X )σ(Y ) (σ(X ) =

V ar(X ))

· V ar(X ) = 0 X Y Cov(X, Y ) = 0 (∗)

|Cov(X, Y )| = σ(X )σ(Y ), T (λ) c T (c) = V ar(cX + Y ) =

0, cX + Y b cX + Y = b =⇒ Y = −cX + b = aX + b (c = −a).

X Y (Ω, P (Ω), p) X Y Cov(X, Y ) = 0) Y (X, Y ) Cov(X,Y )σ(X )σ(Y )


50/58

X, Y (Ω, P (Ω), p). −1 ≤ (X, Y ) ≤ 1 X Y =⇒ (X, Y ) = 0

(X, Y ) = 1 −1 a > 0 a < 0 b

Y = aX + b.

X {−1, 0, 1}

X (Ω) = {−1, 0, 1} et p(X = −1) = p(X = 0) = p(X = 1) = 13

(X n, X m) (n, m) ∈ N2.

∀k ∈ N∗, k

E (X k) = (−1)k p(X = −1) + 0k.p(X = 0) + 1k.p(X = 1) = 23

V ar(X ) = E (X 2k) − E 2(X k) = 23 − ( 2

3)2 =

2

9

k

E (X k) = (−1)k p(X = −1) + 0k.p(X = 0) + 1k.p(X = 1)= − p(X = −1)) + p(X = 1) = 0

V ar(X ) = E (X 2k) − E 2(X k) = 23 − 0 = 2

3

(X n, X m) = Cov(X n,X m)

σ(X n)σ(X m)

(X n, X m) = E (X n+m) − E (X n)E (X m)

V ar(X n).V ar(X m)

· n m

(X n, X m) =23 − (23)2

29 × 29

= 1

· n m n m

(X n, X m) = 0 − 0 × (23)2

29 × 23

= 0


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· n m

(X n, X m) =23 − 0 × 0 23 × 23 = 1

• • • • • • • • • •


52/58

7 5 2

x (x ≥ 0) 10x

y 3

G

G

y G y E (G) = 0

y 3

y σ(G) G x

f x

f (x) =

110x2 + 60

21

α limx→+∞

[f (x) − αx] = 0 f (x) − αx x ≥ 0 ?


53/58

f (C ) 1.5 cm

x σ(G) 7 8.

• • • • • • • • •


54/58

L O P j −2; −1;0;1;2.

L P j , j ∈ {−2, −1, 0, 1, 2} t P k, k ∈ {−2, −1, 0, 1, 2} t + 1

t\ t+1 P −2 P −1 P 0 P 1 P 2P −2 0 1 0 0 0

P −1 0.5 0 0.5 0 0

P 0 0 0.5 0 0.5 0

P 1 0 0 0.5 0 0.5

P 2 0 0 0 0 1

X n t = n

X i, i = 0, 1, 2, 3, 4. X i, i = 0, 1, 2, 3, 4.

(X 3, X 4).

X n+1

X n

an ”X n = 0”

αan+2 + βan + γan−2 = 0, n ≥ 2. (un)n∈N

αun+1 + βun + γun−1 = 0, n ≥ 1.


55/58

an. (an)n∈N

• • • • • • • • • •


56/58

H 1 : 1, 2, 3.

H 2 :

H 3 :

1.

Ai(n), i ∈ {1, 2, 3} n ≥ 1, i n αi(n) = p[Ai(n)], i ∈ {1, 2, 3} n i

H 4 : α1(1) = 0.2; α2(1) = 0.45; α3(1) = 0.35H 5 : i i ∈ {1, 2, 3}

n + 1 n

aij = p[Ai(n + 1)/A j(n)] 1 ≤ i ≤ 3, 1 ≤ j ≤ 3, i n + 1, n j.

∀i, j ∈ {1, 2, 3}, aij aij M

M =

a11 a12 a13a21 a22 a23

a31 a31 a33

=

0.3 0.1 0.60.5 0.3 0.2

0.2 0.6 0.2

H 6 :

Cours de Proba1

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Transcript of Cours de Proba1