TD Modélisation Statistique

X f F
Y=αX +β α, β ∈RZ=eX
α6= 0
Y Z
Y1Y2f1f2X
p Y1Y2
Y=Y1X+Y2(1 −X).
(X, Y )>R2
f(x, y) = 1
2πexp −x2+y2 2, x, y ∈R.
X=Rcos(T)Y=Rsin(T)R≥0
T∈[0,2π[R T
(U, V ) [0,1]
(X, Y ) = p−2 ln(U) cos(2πV ),p−2 ln(U) sin(2πV ),
R2
X∼ N(0,1) X
ϕX(t) = exp(−t2/2) , t ∈R.
X= (X1, ..., Xk)>
Y∼ N(m,Σ)
Yloi =m+ Σ1/2X
X= (X1, X2, X3)>
∀1≤i<j≤3, E(Xi) = 0 ,E(X2
i) = 1 ,E(XiXj)=1/2.
X1−X2+ 2X3
a∈RX1+aX2X1−X2
X
X∼ N(0,1) Y X P(Y= 1) = P(Y=−1) = 1/2
Z=XY
Z
X+Z(X, Z)>

X∼ N(0,1) a > 0
Ya:= X1{|X|< a} − X1{|X| ≥ a}
Ya
b > 0Rb
0x2e−x2/2dx =√2π/4
cov(X, Yb) (X, Yb)>
−1<ρ<1
Σ = 1ρ
ρ1.
X= (X1, X2)>Σ
Y1= (X1+X2)Y2= (X1−X2)Y= (Y1, Y2)>
Y1Y2
YR2
X f F
1{X≤x}ps =1{F(X)≤F(x)}
F(X)F
X1, ..., XnX F
Kn:= sup
x∈R
1
n
n
X
i=1
1{Xi≤x} − F(x)
F
U]0,1[
Y=−log(U)Z= tan π(U−π/2)
F
X:= F−(U) = inf{x∈R:F(x)≥U}
F
X1, ..., XnF
Fn
x∈RFn(x)
x, y cov(Fn(x), Fn(y))
√nFn(x)−F(x)
Fn(y)−F(y).
n x1, ..., xn
f
fnf
K(s) = 1
√2πexp −s2
2, s ∈R,
h > 0

RRxfn(x)dx
Y{x1, ..., xn}
YN(0, h2)Z:= Y+
gn(x) = 1
2nh
n
X
i=1
1{xi∈]x−h, x +h]}.
X F
α∈]0,1[
F−(α)α
q0.5
q0.25
P(X≤q0.25)>0.25 P(X≥q0.25)>0.75.
X1, ..., XnF
f X(1) ≤... ≤X(n)
X(n)X(1)
f[a, b]X(1) X(n)a
b n → ∞
X(1) < ... < X(n)
a1, ..., an > 0
Pn
\
i=1{ai< X(i)≤ai+}= 0
ai
S {1, ..., n}a1< ... < an
Pn
\
i=1{ai< X(i)≤ai+}=X
σ∈S
Pn
\
i=1{ai< Xσ(i)≤ai+}.
fX(1),...,X(n)n(X(1), ..., X(n))
fX(1),...,X(n)(a1, ..., an) = lim
→0+
1
nPX(1) ∈]a1, a1+], ..., X(n)∈]an, an+].
U1, ..., Un]0,1[
α∈]0,1[ U(dnαe)
k U(k)
X1, ..., Xn+1 1
k= 1, ..., n + 1 Sk=Pk
i=1 XiΓ(k, 1)
γk(x) = xk−1
(k−1)!e−x, x > 0.
Sk/Sn+1 U(k)

Sdnαe− dnαe/pdnαe
lim
n→∞ dnαe
n=αlim
n→∞ √ndnαe
n−α= 0,
√nSdnαe
n−αloi
−−−−→
n→∞ N(0, α)√nSn+1 −Sdnαe
n−(1 −α)loi
−−−−→
n→∞ N(0,1−α).
√nαSn+1 −Sdnαe
n−(1 −α)Sdnαe nloi
−−−−→
n→∞ N0, α(1 −α),
Yn:= √nαSn+1 −Sdnαe
Sdnαe−(1 −α)loi
−−−−→
n→∞ N0,1−α
α.
U(dnαe)
loi
=α/(1+Yn/√n)ξn
0Yn/√n
U(dnαe)=α−α
(1 + ξn)2
Yn
√n.
Zn:= √nU(dnαe)−αloi −−−−→
n→∞ N(0, α(1 −α)).
F F −(U(dnαe))
X(dnαe)X1, ..., XnF
F α qαF
qα
√nX(dnαe)−qαloi
−−−−→
n→∞ N0,α(1 −α)
F0(qα)2.
X Y
cov(X, Y ) cor(X, Y )
α, β ∈Rcov(αX +β, Y ) = αcov(X, Y )
cor(αX +β, Y ) cor(X, Y )α6= 0
n(X1, . . . , Xn) (Y1, . . . , Yn) (R1, . . . , Rn)
(S1, . . . , Sn)XiYiXi
Yi1n
γn(X1, . . . , Xn) (Y1, . . . , Yn)
γn
(R1, . . . , Rn) (n+ 1)/2
(n2−1)/12
γn=12
n(n2−1)
n
X
i=1
RiSi−3n+ 1
n−1.
Di=Ri−Si12 Pn
i=1 RiSi=n(n+ 1)(2n+ 1) −6Pn i=1 D2
i
γn= 1 −6Pn
i=1 D2
i
n(n2−1) .

(X, Y )
L(Y|X) = a∗X+b∗
E(aX +b) = E(Y) cov(Y−(aX +b), X) = 0 (a, b)=(a∗, b∗)
(X1, Y1), ..., (Xn, Yn)
(X, Y )
y=anx+bnan
bn
anbn
(Xi, Yi)i= 1, ..., n n ≥2 (X, Y )
Y=a∗X+b∗+  X
Y=
Y1
Yn
∈RnW=
X11
Xn1
∈Rn×2.
∀i= 1, ..., n Yi=a∗Xi+b∗+iθ∗= (a∗, b∗)>
X fXW>W
ˆ
θ
θ7→ kY−Wθk2= (Y−Wθ)>(Y−Wθ), θ ∈R2.
ˆ
θ1ˆ θ2Xn, Y n,ˆσ(X, Y )
var(Y|X) = E(Y2|X)−E(Y|X)2.
var(Y) = var E(Y|X)+Evar(Y|X)
X 
Y=g(X) + 
g
E() = 0 E(Y|X) = g(X)
E(Y|X)E() = m6= 0
(X, Y )
fXY (x, y) = p1−ρ2
2πexp −1 2x2+y2+ 2ρxy, x, y ∈R,
ρ∈]−1,1[
fXY (u, v) = p1−ρ2x, y +ρx
E|X|k<∞k∈N E(X) var(X)
X Y
φ∗(x) = E(Y|X=x)