1 Variables aléatoires finies, lois de probabilité et fonctions de

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•
•
•
•
•n
Ω
X: Ω →
RΩX(Ω)
X
ΩX(Ω)
X
X(Ω) = N R
(x1, x2)x1, x2∈[[1,6]]
Ω = [[1,6]]2
X: Ω →R
(x1, x2)7→ x1+x2.
X X(Ω) = {2,3,...,12}

3 32
3
C32 Ω = {P⊂ C : Card P= 3}
ω X(ω)ω X
X(Ω) = {0,1,2,3}
•a∈R
X: Ω →R
ω7→ a.
X(Ω) = {a}
•A⊂Ω
1A: Ω →R
ω7→ (1ω∈A
0ω /∈A
A
Ω = [[1,6]] A
A A ={1,3,5}A
1A: Ω →R
1A(1) = 1A(3) = 1A(5) = 1,1A(2) = 1A(4) = 1A(6) = 0.
X I ⊂Rω
X(ω)∈I
{X∈I}={ω∈Ω : X(ω)∈I}
{X∈I}Ω
[X∈I] (X∈I)
{X=x} ∩ {Y=y}X y
{X=x, Y =y}
x∈R
{X≤x}={ω∈Ω : X(ω)≤x}=X−1(] − ∞, x])
{X < x}={ω∈Ω : X(ω)< x}=X−1(] − ∞, x[)
{X=x}={ω∈Ω : X(ω) = x}=X−1({x})
X
{X≤x}={X < x}∪{X=x},
X X(Ω) ⊂Zk∈Z
{X≤k}={X≤k−1}∪{X=k}.

A⊂Ω
{1A= 1}=A, {1A= 0}=A
X
Ω = [[1,6]]2X(Ω) = {2,3,4,...,12}
{X= 2}={(1,1)}
{X= 3}={(1,2),(2,1)}
{X= 4}={(1,3),(2,2),(3,1)}
{X= 12}={(6,6)}
XΩX(Ω) = {x1, x2, . . . , xn}
X
({X=x1},{X=x2},...,{X=xn})
Ω
Ω = {X= 2}∪{X= 3}∪···∪{X= 12}
X Y Ω
X+Y
(X+Y)(ω) = X(ω) + Y(ω).
XY X/Y X2eX−Y. . .
X: Ω →R
fX:X(Ω) →[0,1]
x7→ P({X=x})
X
X
X X
XP({X=x})
x∈X(Ω)

P(X=x)P({X=x})
fX
X(Ω) = {x1, x2, . . . , xn}xi
fX(xi)xi
1Px∈X(Ω) P(X=x) = 1
X
PΩ
1/36 k∈[[2,12]]
{X=k}={(i, j)∈[[1,6]]2:i+j=k}.
P(X= 2) = P({(1,1)}) = 1
36
P(X= 3) = P({(1,2),(2,1)}) = 2
36
P(X= 4) = P({(1,3),(2,2),(3,1)}) = 3
36
xi2 3 4 5 6 7 8 9 10 11 12
P(X=xi)1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
pi
xipi
P(X=xi) = pi
pi1

(x1, x2, . . . , xn)∈Rn(p1, p2, . . . , pn)∈Rn
∀i∈[[1, n]] pi≥0
Pn
i=1 pi= 1.
X
∀i∈[[1, m]],P(X=xi) = pi
Ω = {x1, x2, . . . , xm}P
P({xi}) = piX
X: Ω →R
FX:R→[0,1]
x7→ P({X≤x})
X
X
xi12468
P(X=xi) 3/10 1/10 1/10 2/10 3/10
X
fxFX
FFXfx
FXX(Ω) = {x1, x2, . . . , xn}
FX(x) = 
0x<x1,
Pk
i=1 fX(xi)xk≤x<xk+1,
1xn≤x.
FXfX(xi)xi
FfXFx