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R
XR
(Ω,A, P r)
B∈BP r(B)
{X∈B}=X−1(B)A
(Ω,A, P r)
XΩR
B∈BX−1(B)∈ A
XΩR(Ω,A, P r)
X−1(] − ∞, x]) = {X≤x} ∈ A
F X
X
F(t) = P r(X≤t),∀t∈R.
U[0,2]
X X(u) = u2X
[0,2],B[0,2], P rB[0,2]
[0,2] P r
[0,2]
U
FU(t) = 
0t < 0
t
20≤t < 2
1t≥2.
X
ha, bi[0,4] Xh√a, √bi[0,2]
X FX(x)
0x < 01x≥4
0≤x≤4
x∈[0,4]
FX(x) = P r(X≤x) = P r(U≤√x) = FU(√x) = √x/2.
FX(x) = 
0x < 0
√x
20≤x < 4
1x≥4.

O
1
2
U
1
O4
Fonction de répartition de
t
t
F(t) F(t)
X U
[0,2]
X=U 2
Fonction de répartition de tiré uniformément dans
(Ω,A, P r)A∈ A
P r(./A)A
[0,1]
P r(B/A) = P r(A∩B)
P r(A),∀B∈ A.
A P r(B/A)
B A
(Ω,A, P r(./A))
A
P r(A∩B) = P r(A)P r(B/A).
A1, A2, ..., AnΩA
P r(Ai)6= 0, i = 1, ..., n B ∈ A
P r(B) =
n
X
i=1
P r(Ai)P r(B/Ai).
(Ω,A, P r)
Rk
ΩRkB∈Bk{X∈B}=X−1(B)
A
XRk(Ω,A, P r)
(Rk,Bk)B∈Bk
P r({X∈B})
A

XΩRk
x∈Rk{X≤x} ∈ A
X
F(x) = F(x1, ..., xk) = P r (X≤(x1, ..., xk)).
X= (X1, ..., Xk)
P r(X1≤x1, ..., Xk≤xk) = F(x1, ..., xk).
X= (X1, ..., Xk) (Ω,A, P r)
X1, ..., XkB1, ..., Bn∈B
P r (X1∈B1, ..., Xk∈Bk)=
k
Y
1
P r(Xi∈Bi).
F(x1, ..., xk)
Fi(xi),i= 1, ..., k limxj→∞,∀j6=iF(x1, ..., xk).
X1, ..., Xk
F(x1, ..., xk) =
k
Y
1
Fi(xi).
X1, ..., Xk
ϕ1, ..., ϕkϕ1(X1), ..., ϕk(Xk)
X= (X1, ..., Xk)∈Rk
f(t1, ..., tk), x1, ..., xk∈R
F(x1, ..., xk) = Zxk
−∞ ... Zx1
−∞ f(t1, ..., tk)dt1...dtk.
Z∞
−∞ ... Z∞
−∞ f(t1, ..., tk)dt1...dtk=1.
•XR+
f
EX=Z∞
−∞ xf(x)dx,
X
•X
X X =X+−X−
X+= max{X, 0}X−=−min{X, 0}. X+X−
X
EX=EX+−EX−.
X

X
[a, b]a < b
f(x) = 
0x < a
1
b−aa≤x < b
0x≥b.
EX=Zb
a
x
b−adx ="x2
2(b−a)#b
a
=b2−a2
2(b−a)=b+a
2.
ϕ(X)
X2
EX2=Zb
a
x2
b−adx ="x3
3(b−a)#b
a
=b3−a3
3(b−a)=b2+a2+ab
3.
X X
m σ
f(x) = 1
σ√2πexp −(x−m)2
2σ2!, x ∈R.
N(m, σ)
R∞
−∞ f(x)dx = 1
y=x−m
√2σ
Z∞
−∞ e−y2dy =√π
Gauss et sa courbe

f(x) = 1
π
1
1 + x2,x∈R.
F(t) = P r(X≤t) =1
2+Arctan(t)
π,t∈R.
R∞
−∞ =1
Comparaison de la densité de la loi de Cauchy et de la normale