Zologie Des Lois

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    3

    3

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    X 2

    gx (t)

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    X Ω →Ω

    ω →X (ω) = ω

    P r (X (ω)∈E ) = P r (ω ∈X −1(E ))

    Ω ⊂ R X (ω) ∈ R R nX (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

    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

    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=

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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

    {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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