Cours 5 - IMJ-PRG

X X ϕX
R
ϕX(t) = E(eitX )∀t∈R.
X
ϕX(t) =
n
X
j=1
eitxjP(X=xj).
X
ϕX(t) =
+∞
X
j=1
eitxjP(X=xj).
X
ϕX(t) =
+∞
Z
−∞
eitx f(x)dx.
X
X
|ϕX(t)| ≤
+∞
X
j=1 |eitxjP(X=xj)|=
+∞
X
j=1
P(X=xj)=1
X
|ϕX(t)| ≤
+∞
Z
−∞ |eitx f(x)|dx =
+∞
Z
−∞
f(x)dx = 1
X a b
t∈R|ϕX(t)| ≤ 1
ϕX(0) = 1
t∈RϕX(−t) = ϕX(t)
t∈RϕaX+b(t) = eitb ϕX(at)
ϕXR
X
ϕX

t∈R
X∼ U(n)ϕX(t) = 
1
n
1−eitn
e−it −1si t6= 2kπ pour tout k∈Z
1 si t= 2kπ, k ∈Z
X∼b(n, p)ϕX(t)=(peit + 1 −p)n
X∼ G∗(p)ϕX(t) = peit
1−(1 −p)eit
X∼ G(p)ϕX(t) = p
1−(1 −p)eit
X∼ P(λ)ϕX(t) = eλ(eit−1)
X∼ U(]a, b[) ϕX(t) = eitb −eita
it(b−a)t6= 0 ϕX(0) = 1
X∼ E(λ)ϕX(t) = λ
λ−it
X ϕX(t) = e−|t|
X∼ N(m, σ)ϕX(t) = eimt−t2σ2/2
t∈R
XB(p)b(1, p)ϕX(t) = peit + 1 −p
XU(] −a, a[) ϕX(t) = e−ita −eita
−2ita =sin(at) at t6= 0 ϕX(0) = 1
XN(0,1) ϕX(t) = e−t2/2
X X
ϕ X ϕ
+∞
Z
−∞ |ϕ(t)|dt < +∞,
X f R
f(x) = 1
2π
+∞
Z
−∞
e−itx ϕ(t)dt ∀x∈R.
X X
ϕX(t) = e−|t|∀t∈R.

ϕ(t) = e−|t|R
X f x ∈R
f(x) = 1
2π
+∞
Z
−∞
e−itx e−|t|dt =1
2π
0
Z
−∞
e−itx+tdt +1
2π
+∞
Z
0
e−itx−tdt =e(1−ix)t
2π(1 −ix)0
−∞
+−e−(1+ix)t
2π(1 + ix)+∞
0
=1
2π(1 −ix)+1 2π(1 + ix)=1−ix +1+ix 2π(1 −ix)(1 + ix)=1 π(1 + x2).
X ϕ 
k(k∈N∗)X
E(Xk)
k X
X k
E(Xk) = 1
ikϕ(k)(0).
X f E(Xk)x7→ (ix)keitx f(x)
ϕXk ϕ(k)(t) =
+∞
Z
−∞
(ix)keitx f(x)dx
t∈R
ϕ(k)(t) = E(iX)keitX .
t= 0
ϕ(k)(0) = E(ikXke0) = ikE(Xk).
E(X)
E(X2)
E(X) = −i ϕ0
X(0) et E(X2) = −ϕ00 X(0).
X1Xnn S =
n
P
j=1
Xj
S Xj
ϕS(t) =
n
Y
j=1
ϕXj(t)∀t∈R.
ϕS(t) = ϕX1+...Xn(t) = E e
it
n
P
j=1
Xj!
=E e
n
P
j=1
itXj!=E
n
Y
j=1
eitXj
.
X1Xn
eitX1eitXn
ϕS(t) =
n
Y
j=1
E(eitXj) =
n
Y
j=0
ϕXj(t).
X∼b(n, p)Y∼b(m, p)X⊥⊥Y X +Y∼b(n+m, p)
X∼P(λ)Y∼P(µ)X⊥⊥Y X +Y∼P(λ+µ)
X∼ N(m1, σ1)Y∼ N(m2, σ2)X⊥⊥Y X +Y∼ N(m1+m2,pσ2
1+σ2 2)

X Y X +Y
fX+Y(u) =
+∞
Z
−∞
fX(u−v)fY(v)dv
=
+∞
Z
−∞
fX(v)fY(u−v)dv.
X∼ N(m1, σ1)ϕX(t) = eim1t−t2σ2
1/2Y∼ N(m2, σ2)ϕY(t) = eim2t−t2σ2 2/2
X Y
ϕX+Y(t) = ϕX(t)ϕY(t) = ei(m1+m2)t−t2(σ2
1+σ2 2)/2=ei(m1+m2)t−t2(√σ2
1+σ2 2)2/2.
m1+m2pσ2
1+σ2 2X+Y∼
N(m1+m2,pσ2
1+σ2 2)
X Y
ϕX+Y(t) = ϕX(t)ϕY(t)
=
+∞
Z
−∞
fX(x)eitx dx
+∞
Z
−∞
fY(v)eitv dv
=
+∞
Z
−∞
fY(v)
+∞
Z
−∞
fX(x)eit(x+v)dx
dv
u=x+v
ϕX+Y(t) =
+∞
Z
−∞
fY(v)
+∞
Z
−∞
fX(u−v)eitu du
dv
=
+∞
Z
−∞
eitu 
+∞
Z
−∞
fY(v)fX(u−v)dv
du.
X+Y
fX+Y(u) =
+∞
Z
−∞
fY(v)fX(u−v)dv.
(Xn)n∈N
(Ωn, Sn,Pn)X(Ω, S, P)FnXnF
XC(F)F
(Xn)n∈NX Xn
L
−→ X x ∈ C(F)
lim
n7→+∞Fn(x) = F(x).
FnF F
(Xn)n∈Nϕn
X ϕ
Xn
L
−→ X⇐⇒ ∀ t∈R,lim
n7→+∞ϕn(t) = ϕ(t).

R
ϕnϕ
(Xn)n∈NXn∼b(n, pn)
npnλ > 0n+∞XnX
P(λ)
t∈R
ϕn(t)=(pneit + 1 −pn)n=enln(pneit+1−pn).
lim
n7→+∞pn= 0 lim n7→+∞npn=λ n +∞
ln(pneit + 1 −pn)pneit −pn
lim
n7→+∞ϕn(t) = lim n7→+∞en(pneit−pn)=eλeit−λ.
X∼ P(λ)
Xn
L
−→ X
X= (X1, X2)T= (t1, t2)
R2
X= (X1, X2)
(X1, X2)ϕR2C
ϕ(T) = ϕX(T) = E(ei<T,X>)∀T∈R2
< T, X > R2
< T, X >=t1X1+t2X2.
X f T ∈R2
ϕX(T) = E(ei<T,X>) = ZZRR2 ei(t1x1+t2x2)f(x1, x2)dx1dx2.
ϕ
ϕR2
ϕ(0,0) = 1
ϕ(−T) = ϕ(T)T∈R2
|ϕ(T)| ≤ 1T∈R2
ϕ(X1, X2) (t1, t2)∈R2
ϕX1(t1) = ϕ(t1,0) et ϕX2(t2) = ϕ(0, t2).