cours de proba1
TRANSCRIPT
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P (E )
⊂
E p E n E p E n 1 ≤ p ≤ n p
Ω
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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}, ∅}
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⊂
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
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(∀
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)
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• 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.
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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
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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
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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
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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
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· 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
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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
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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
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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
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(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).
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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 ” ”
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: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
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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
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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
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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
• • • • • • • • •
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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]
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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
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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
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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 )
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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
• • • • • • • • • •
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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 ?
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f (C ) 1.5 cm
x σ(G) 7 8.
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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.
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an. (an)n∈N
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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 :