Zologie Des Lois
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Transcript of Zologie Des Lois
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3
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X 2
gx (t)
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Ω
X Ω →Ω
ω →X (ω) = ω
Ω
P r (X (ω)∈E ) = P r (ω ∈X −1(E ))
Ω ⊂ R X (ω) ∈ R R nX (ω)
Ω =
{ω
∈R 2
|ω = ( i, j )
X (ω) = i + j Ω = {x∈N |2 ≤x ≤12}P r (X −1(k)) pr (X = k).P r (X = 4) = P r (X −1(4)) = P r ((1 , 3)) + P r ((2 , 2)) + P r ((3, 1))
P r (X = 4) = 336 = 112
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R
a∈E
R I ∩E = a
Z N {1k |k ∈N∗} R
F (x) = P r (X ≤x)
F (xn ) = p1 + p2 + . . . + pn
pn = P r (X = xn ) X = xi
x < x 1
x ≥xn card (Ω) = n
limx−→+ ∞
F (x) = 1
Ω
X (Ω) R
P r (X = x) = 0
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F (x) = P r (X
≤x)
a ≤ b (X ≤ a) ⊂ (X ≤ b) ≤
F (x) = P r (X ≤x) = x−∞f (t)dt
+ ∞−∞ f (t)dt = 1
P r (a < X < b ) = F (b) −F (a) = ba f (t)dt
µX
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(Ω, C, P r ) {α i}(i∈I )
pi = P r (X = α i)
E (X ) =i∈I
pix i
(Ω, C, P r )
E (X ) = Ω XdP r = Ω xP r x dx
E (X ) = Ω xf (x)dx
E (a) = aE (aX + b) = aE (X ) + bE (X + Y ) = E (X ) + E (Y )
E (X ) = 0
P r x
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V (X ) =n i (x i −x)2
N =
n i x2iN −x
2
V (X ) = E [X −E (X )]2 = E (X 2) −[E (X )]2 σX
σX = V (X )
E (X 2) = +
∞−∞ x2f (x)dx
V (X ) = + ∞−∞ x2f (x)dx −[E (X )]2
{x i}i∈I {y j } j∈J I, J ⊂N
pij = P (X = xi , Y = y j )
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P (X = xi) = j∈J
P (X = xi , Y = y j ) = j∈J
pij = pi.
P (Y = y j ) =i∈I
P (X = xi , Y = y j ) =i∈I
pij = p.j
Y = y j
P (X = xi |Y = y j ) = P (X = xi , Y = y j )
P (Y = y j ) =
pij p.j = p
j
i
i∈I j ∈J
P (X = xi , Y = y j ) = P (X = xi )P (Y = y j )
F (x, y) = P (X < x, Y < y )
f (x, y) = ∂ 2F (x, y )
∂x∂y
F X (x) = P (X < x ) = F (x, + ∞) et F Y (y) = P (Y < y) = F (+ ∞, y)
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f X (x) = + ∞−∞ f (x, y)dy et f Y (y) = + ∞−∞
f (x, y)dx
f X (x|Y = y) = f (x, y)
f Y (y) et f Y (y|X = x) =
f (x, y)f X (x)
f (x, y) = f X (x)f Y (y)
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X (Ω) = Ω P r (X = x i )
x i Ω
Ω card (Ω)
x i P r (X = xi ) = pi
x i
pi 1362
363
364
365
366
365
364
363
362
361
36
Ω
P r (X = xn )
Ω = 1
ω1; 2; . . . ; n; . . . ; X (ωn ) = 1n = xn
Ω = xn = 1n |n∈N∗
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P r X = xn = k.a n
a∈R δ a
R
δ a (x) = 1 si x = a
δ a (x) = 0 si x = a
δ a
E (X ) = a V (X ) = 0
( p∈[0, 1])
P (X = 1) = p
P (X = 0) = 1 − p = q.
X ∼> B ( p).
1A(ω) =1 si ω ∈A0 si ω /∈A
=1 si
0 si
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X = 1 A A = ω ∈Ω, X (ω) = 1 .
E (X ) =2
i=1
x i pi = (0 ×q ) + (1 × p) = p
E (X ) =2
i=1
x2i pi
−(E (X )) 2 = [(0
×q ) + (1
× p)]
− p2 = p
0, 1, . . . , n ,
P (X = k) = C k
n pk
q n
−k
n
k=0
P (X = k) =n
k=0
C kn pkq n−k = ( p + q )n = 1
X →B (n, p ) X →B (1, p)
E (X ) = np et V (X ) = np(1 − p) = npq
X = X 1 + X 2 + . . . + X n X i
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P (X i = x)
E (X ) =n
i=1
E (X i ) = np
E (X i ) = p
V (X ) =n
i=1
V (X i )
V (X i ) = pq
p1, p2 . . . pk
k
i=1 pi = 1
X i
X = ( X 1, . . . , X k)
(n; p1, . . . , p k ) M (n; p1, . . . , p k ).
P r [X = ( x1, . . . , x k )] = C x1n C x2n−x1 . . . C
xkn−(x1 + ... + xk − 1 ) p
x 11 ...p
x kk
= n!
x1! . . . x k ! px11 . . . p
xkk
k
i=1 pi = 1
k
i=1x i = n
E (X i) = np i V (X i ) = np i (1 − pi )
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∀i, P (X = xi ) = 1n
P (X = xi ) 1
616
16
16
16
16
E (X ) = 16
6
i=1
i = 3 , 5
V (X ) = 16
6
i=1
i2 −(E (X ))2 = 2 , 92
x i
x i = i (∀i∈[1, n ])
E (X ) = n + 1
2 et V (X ) =
n2 −112
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N p ∈N
P (X = k) =C kNp C
n−kNqC nN
n
k=0
P (X = k) =n
k=0
C kNp C n−kNq
C nN = 1
m
p=0C pn C
m− pN −n = C mn
m
p=0C pn C
m− p0 = C mn = C mN
m
p=0C pn C
m− pN +1 −n =m
p=0C pn (C
m− p−1N −n + C m− pN −n )
=
m
p=0C pn C m− p−1N −n +
m
p=0C pn C m− pN −n
=m−1
p=0C pn C
m− p−1N −n +m
p=0C pn C
m− pN −n
= C m−1N + C mN
= C mN +1
X →H (N,n,p ).
E (X ) = np et V (X ) = npq N −nN −1
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N −nN −1
E (X ) =n
k=0
kP (X = k)
=n
k=0
kC kNp C
n−kNqC nN
= NpC nN
n
k=1
kNp
(Np)!k!(Np −k)!
C n−kNq
= NpC nN
n
k=1
(Np
−1)!
(k −1)!(Np −k)!C n
−k
Nq
= NpC nN
n−1
m =0C mNp−1C
n−m−1Nq (avec m = k −1)
= NpC nN
C n−1N −1 (carn−1
m =0
C mNp−1C n−m−1Nq
C n−1N −1= 1)
= N p nN
= np
E (X (X −1)) = np(Np −1) n−1N −1
V (X ) = E (X 2) −(E (X ))2
= E (X (X −1)) + E (X ) −(E (X )) 2
= np(Np −1) n
−1
N −1 + np + n2
p2
= np(1 − p)N −nn −1
V (X ) = npq N −nn−1
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P (X = k) = q k−1 p
k≥1P (X = k) =
k≥1q k−1 p = p
1 −q = 1
X →G( p).
E (X ) = 1 p et V (X ) =
q p2
E (X ) =k≥1
kq k−1 p = pk≥1
kq k−1 = p(1 −q )2
= 1 p
E (X (X −1)) =k≥1
k(k −1)P (X = k)
=k≥1
k(k −1)q k−1 p
= 2 pq
(1 −q )3 = 2
q p2
V (X ) = E (X 2) −(E (X ))2
= E (X (X −1)) + E (X ) −(E (X )) 2
= 2 q p2
+ 1 p −
1 p2
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V (X ) = q p2
λ
P (X = k) = λke−λ
k!
k≥0 p(X = k) =
k≥0λke−λ
k! = 1
λ X →P (λ).
E (X ) = λ et V (X ) = λ
E (X ) =k≥0
kλke−λ
k!
= e−λ
k≥0k
λk
(k −1)!
= e−λ λe λ
= λ
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E (X (X −1)) =k≥0
k(k −1)P (X = k)
E (X (X −1)) =k≥0
k(k −1) λke−λk!
= e−λk≥0
k(k −1) λk
(k)!
= e−λ λ2eλ
= λ2
V (X ) = E (X 2) −(E (X ))2
= E (X (X −1)) + E (X ) −(E (X )) 2
= λ2 + λ−λ2
V (X ) = λ
P (X = k) = C r−1k−1q k−r pr
k≥rP (X = k) =
k≥rC r −1k−1q k−r pr = 1
g(q ) = 11 −q
=l≥1
q l−1
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g(r −1) (q ) = (r −1)!(1
−q )r
=
l≥1(l −1)( l −2) . . . (l −r + 1) q l−r
=l≥r
(l −1)( l −2) . . . (l −r + 1) q l−r
=l≥r
(l −1)!(l −r )!
q l−r
1 pr
= 1(1 −q )r
=l≥r
C r−1l−1 q l−r
X →BN ( p, r ). X →BN ( p, 1)⇐⇒X →G( p).
E (X ) = r p
et V (X ) = rq p2
X = X 1 + X 2 + . . . + X r X i → G( p) X i E (X i ) = 1 p V (X i) = q p2
E (X ) = E (X 1) + E (X 2) + . . . + E (X r ) = r p V (X ) = V (X 1) + V (X 2) + . . . + V (X r ) = rq p2
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0 ≤a ≤b
F (x) =
1b−a si x∈[a, b]
0 sinon
b
a1
b−a dt = 1
F (x) =
x−ab−a si x∈[a, b]0 si x < a
1 si x > b
E (X ) = b
a1b−a tdt = a + b2
V (X ) = ba 1b−a (t −E (X ))2dt=
a2 + ab + b2
3 − (a + b)2
4
= (b−a)2
12
f (x) = 1σ√ 2π e−
( x − m ) 2
2σ 2
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m∈R σ ∈R∗+
+ ∞−∞
1
σ√ 2πe−
( x − m ) 2
2σ 2 dx =
+ ∞−∞
1
σ√ 2πe−t2 σ√ 2dt
( t = x −m
σ2 )
= 1√ π + ∞−∞ e−t 2 dt
= 1
∀
x
∈
R
σ X →N (m, σ ).
E (X ) = m et V (X ) = σ2
E (X ) = + ∞−∞ xσ√ 2π e−( x − m ) 2
2σ 2 dx
= + ∞−∞ tσ√ 2 + mσ√ 2π e−
t2 σ√ 2dt
( t = x −m
σ√ 2 )
= m 1√ π
+ ∞
−∞
e−t2 dt + σ√ 2√ π
+ ∞
−∞
te−t2 dt
= m + σ√ 2√ π [−
12
e−t 2 ]+ ∞−∞
=0= m
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V (X ) = + ∞−∞ (x −m)2
σ√ 2π e−( x − m )2
2σ 2 dx
= +
∞−∞
2σ2
t2
σ√ 2π e−t2
σ√ 2dt
( t = x −m
σ√ 2 )
= 2σ2
√ π + ∞−∞ t2e−t2 dt=
2σ2
√ π [−12
te−t2 ]+ ∞−∞
=0
− + ∞−∞ −12e−t2 dt= σ2
σ = 1 .
f (x) = 1√ 2π e
− x 22
F (x) = 1√ 2π x−∞e
− t 22 dt
12
−∞ ∞
12 X → N (m, σ )
X −
mσ →N (0, 1)
f (x) = 1√ 2π e− x 2
2
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−x−∞f (t)dt = + ∞x f (t)dt
∀t∈R
x
−x f (t)dt = 2
x
−∞f (t)dt −1
x−x f (t)dt
= x−∞f (t)dt − −x
−∞f (t)dt
= x−∞f (t)dt −(1 − + ∞−x
f (t)dt)
= x−∞f (t)dt −(1 − x
−∞f (t)dt )
= 2 x
−∞f (t)dt −1
γ
λ λX γ 1
γ r
f (x) = 1Γ(r )
e−x xr −1
∞0 f (x)dx = 1 Γ(r ) γ r
E (X ) = r
E (X ) = 1Γ(r ) + ∞0 x r e−xdx = Γ(r + 1)Γ(r ) = r
V (X ) = r
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V (X ) = E (X 2) −(E (X ))2 = 1Γ(r ) + ∞x xr +1 e−x dx −r 2
V (X ) = Γ(r + 2)
Γ(r ) −r2 = ( r + 1)
Γ(r + 1)Γ(r ) −r
2 = r (r + 1) −r 2 = r
β (n; p)
f (x) =1
B (n,p ) xn−1(1 −x) p−1 si 0 ≤x ≤1
0 sinon
B (n, p ) = 10 xn−1(1 −x) p−1dx = B( p, n)B (n, p ) =
Γ(n)Γ( p)Γ(n + p)
β (n, p )
Y = X 1−
X
f (y) =1
B (n,p )yn − 1
(1+ y)n + p si y ≥00 sinon
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E (X ) = nn + p
V (X ) = np
(n + p + 1)( n + p)2
E (X ) = n
p −1 V (X ) =
n(n + p −1)( p −1)2( p −2)2
λ
f (x) =λe−λx si x ≥0
0 sinon
f (a) = P r (X < a ) = a0 λe−λx dx = 1 −e−λa
E (X ) = 1λ
V ar(X ) = 1λ2
E (X ) = + ∞0 xf (x)dx=
+ ∞0 λxe −
λx
dx
= [xe−λx ]+ ∞0 + + ∞0 e−λx=
1λ
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X = eY
x →ey
f (x) =1
xσ √ 2π e− ( ln ( x ) m ) 2
2σ 2 si x ≥00 sinon
E (X ) = e−(m + σ2
2 ) et V (X ) = [e(σ2 ) −1]e(2m + σ
2 )
f (x) = 1
π(1 + x2)
F (x) = 1π
arctan x + 12
R xπ (1+ x2 ) dx
α
θ
α > 0
θ > 0
f (x) = αθx α−1e−θx α 1R (x)
F x (x) = 1 −e−θxα
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α = 1
γ (1, θ) 1−1 −e−θx P (X ≥x) = e−θx {X ≥ x}
e−θx
e−θx α
X α γ (1, θ)
x →x1/α
E (X ) = Γ(1 + 1α )
θ1/α
V (X ) = Γ(1 + 2α )
−Γ2(1 + 1α )
θ2/α
f x (x) = e(x−ex ) , x∈
R
F x (x) = 1 −e(
−ex )
E (X ) = −0.57722 V (X ) = π2
6
X 2
U 1, U 2, . . . , U p
X 2 pi=1 U 2i X 2
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X 2 X 2 γ T = U 2
g(t) = 1√ 2π t−
12 exp −
t2
E (X ) = p et V (X ) = 2 p
X 2n T n = U √ X/n
T n = U √ X/n Y = U √ n Z = √ X T n = Y Z
f T n (t) = + ∞−∞ |z|f Z (z)f Y (tz )dz (1)f Y (y) = 1√ n f U y√ n f U (u) = 1√ 2π e−u
22
f Y (y) = 1√ 2πn e−
y 22n ; f Y (tx ) =
1√ 2πn e−
t 2 x 22n
f X (x) =x
n2
− 1 e− x
2
2n2 Γ n2
si z ≥00 sinon
f Z (z) = 2 xf X (z2)
f Z (z) =1
2n2
− 1 Γ n2 zn−1e−z
22 si z ≥0
0 sinon
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f T n (t) = 1
2n2 −1√ nπ Γ n2
+ ∞0
zn e−t2 + n2n zdz
v = t2 + n2n z
2
zn = 2nt2 + n
n2
vn2 etdz =
12
2nt2 + n
12
v−n2 dv
+ ∞0 zn e−t 2 + n2n zdz = 12 2nt2 + nn +1
2 + ∞0 v n +12 −1e−vdv= 2
n − 12
1 + t2n
n +12
Γ n + 12
f T n (t) = 1√ nπ
Γ n +12
Γ n2
1
1 + t2n
n +12
pour t ∈R
E (T n ) = 0
V (T n ) = nn −2
(n > 2)
X 2(n)
X 2(m)
F = X/nY/m F n,m
nm F =
X/ 2Y/ 2 β (
n2 ,
m2 )
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f F (x) = 1
β ( n2 , m2 )
nn/ 2mm/ 2 xn/ 2−1
(m + nx )n + m
21R + (x)
E (F n,m ) = 1m −2
(m > 2)
V (F n,m ) = 2m2(n + m −2)n(m −4)(m −2)2
(m > 4)
ϕx
ϕx (t) = E [eitX ] = R eitx dP X (x) P x [eitX ] = L
ϕX (t) = peit + q
ϕX (t) = ( peit + q )n
ϕX (t) = exp( λ(exp( it ) −1))
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E [exp(itX )] =x
x=0exp( itx ) exp(−λ)
λx
x! = exp( −λ)
x
x=0
λ exp( it )x
x!
= exp( −λ)exp( λ exp( it ))
ϕX (t) = eit −1
it
γ ( p, O)
ϕX (t) = (1 −Oit )− p
U →N (0, 1)ϕU (t) = e
− t 22
X →N (m, σ ) X = m + σU
ϕX (t) = E (eiu (m + σU ) ) = eium ϕU (uσ ) = eium e− u 2
2
gX (t) = E [etX ]
gx (t )
gX (t) =i
etx i pi
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pi = P r (X = xi)
gX (t) =
+ ∞
−∞
etx f (x)dx
(1 − p − pt)n
pt1−(1− p)t
n
eλ(t−1)
pt1−(1− p)t
eibt −eiati(b−a)t eimt
σ 2 t 22
1 − itλ −α
11−itλ
(1 −2it )−n2
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E (x) ∀λ > 1
P (x ≥λE (x)) ≤ 1λ
E (X ) λ
X (Ω)⊂R +
X (Ω)
A = {x∈X (Ω), x < λ E (X )}
B = {x∈X (Ω), x ≥λE (X )}
E (X ) =x∈X (Ω)
xP (X = x) =x∈A
xP (X = x) +x∈B
xP (X = x)
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x∈R + xP (X = x) ≥0
x
∈
A
xP (X = x) ≥0
∀x∈B x ≥λE (X )
x∈B
xP (X = x) ≥x∈B
λE (X )P (X = x)
E (X ) ≥λE (X )x∈B
P (X = x)
λE (X ) > 0
1λ ≥
x∈B
P (X = x)
x∈B
P (X = x) = P (X ≥λE (X ))
P (X ≥λE (X )) ≤ 1λ
X → G( 13 ) E (X ) = 3
P (X ≥60) = P (X ≥20 ×3) ≤ 120
5%
P (X < 60) =59
k=1
23
k−1(13
) ≥0.99999
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P (X ≥60) ≤0.00001
R − λ
E (X ) = + ∞−∞ xf (x)dx
E (X ) = + ∞0 xf (x)dx = λE (X )
0xf (x)dx + + ∞λ E (X ) xf (x)dx
x −→xf (x) R + 0 ≤λE (X )
λE (X )
0xf (x)dx ≥0
E (X ) ≥ + ∞λ E (X ) xf (x)dx [λE (X ), + ∞[ x ≥λE (X )
xf (x) ≥λE (X )f (x)
+ ∞λ E (X ) xf (x)dx ≥ + ∞λ E (X ) λE (X )f (x)dx
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+
∞λ E (X ) xf (x)dx ≥λE (X ) +
∞λ E (X ) f (x)dx
E (X ) ≥λE (X ) + ∞λ E (X ) f (x)dx λE (X )
1λ ≥ + ∞λ E (X ) f (x)dx
+ ∞λ E (X ) f (x)dx = P (X ≥λE (X ))
P (X ≥λE (X )) ≤ 1λ
E (X ) = 1
P (X ≥10) = P (X ≥10 ×1) ≤ 110
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10%
P (X ≥10) = + ∞10 e−x dx = e−10 ≈4.45.10−5
V (X ) = σ
P (|X −m| > λσ ) ≤ 1λ2
V (X ) = σ2 = E (X −m)2 =k
(xk −m)2P (X = xk)
∀ε > 0, σ2 =
k, |xk −m |>ε(xk −m)2P (X = xk ) +
k, |xk −m |≤ε(xk −m)2P (X = xk )
≥k, |xk −m |>ε
(xk −m)2P (X = xk )
≥ε2
k, |xk −m |>ε P (X = xk )
≥ε2P (|X −m| > ε )
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ε = λσ
P (|X −m| > λσ ) ≤ σ2
λ2σ2 =
1λ2
X n −→ p a ∀ε, η > 0 ∃N (ε, η n > N =⇒P r (|X n −a| > ε ) < η
(X n )
X n
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(X n )
α n
P r (X n = 0) = 1
− 1
n P r (X n = αn ) =
1
n (X n )
E (X n ) = αn
n
n → ∞ αn E (X n )
α n = √ n E (X n ) →0 αn = n E (X n ) = 1αn = (−1)n n E (X n ) = ( −1)n α n = n r E (X n ) → ∞
(X n )
P r [ω/X (ω) = Y (ω)] = 0
(X n )
X n → p.s. xP r [ω/limX n (ω) = X (ω)] = 0
(X n
)
F n
F n
X n −→x
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x0
F n (x0) F (x0)
X n N (0; 1n )
(X n ) n −→ ∞ F n X n
F n (x) = P r (X n < x )
n −→ ∞ F n (x) x ≤ 0 F n (x)
F (x) = 0 si x
≤0 et F (x) = 1 si x > 0
∀n F n (0) = 0 , 5 F (0) = 0 = F n (0) F n (x)
(X n )
E [(X n −X )q] −→ 0 n −→ ∞
X n B(n; p) X n −np√ npq → LG(0;1) X n ( pexp( it ) + 1 − p)n
X n −np√ npq
ϕ(t) = pexp it√ npq + 1 − p
nexp −
itnp√ npq
ln ϕ = n ln p exp it√ npq −1 −
itnp√ npq
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ln ϕ n ln 1 + p it√ npq −
t2
2npq − itnp√ npq
ln ϕ n pit√ npq −
pt2
2npq +
p2t2
2npq − itnp√ npq
Soit : lnϕ −t2
2q +
pt2
2q =
t2
2q ( p −1) = −
t2
2
ϕ(t) −→exp(−t2/ 2)
(X n ) (σ)2
1n2
n
i=1
σ2i −→0 quand n −→ ∞
X n = 1n
n
i=1
X i
(X n )
(X n ) (σ)2
k≥1σ2kk2
< ∞
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X n = 1n
n
i=1
X i
(X n )
σ n −→ ∞1
√ nX 1 + X 2 + . . . + X n −mn
σ =
n
i=1
X i −µσ√ n =
X −mσ/ √ n
(X i )
σi F i (x)
X i −m i
S 2n =n
i=1
σ2i
1S 2n
n
i=1 |x |>S n x2dF i(x)
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n −→ ∞1
S n
n
i=1
(X i −m i )
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P (λ)
λ
. . .
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A pn
1 ≤ p ≤n n, p∈N∗ A pn = 0
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A pn = n p, 1 ≤ p ≤n
AP n = n!
(n − p)!, 1 ≤ p ≤n
P n = n!
pn = n!k!
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ω Ω
Ω = ( l.1), (1.2), (1.3)... Ω
Ω
Ω
{x1, . . . , x k} {x1, x2, . . . , x n ; . . .}
{x1, x2, . . . , x k ; . . .}
Ω∈A
∀B ∈A =⇒B (Ai) i∈I
∪i∈I Ai ∈A, v = A
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(Ω,ϑ ,P )
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