Chapitre 21 : Variables aléatoires discrètes - Pierre

r∈N
2

(Ω,A, P )
Ω = {1,2,3,4,5,6}
X: Ω −→ R
ω7−→ (0ω= 1,2,3,4 5
1ω= 6
.
6
X
X
X X(Ω) X
X X X(Ω) = {0,1}
[X= 0] = {ω∈Ω|X(ω)=0}={1,2,3,4,5}
[X= 1] = {ω∈Ω|X(ω)=1}={6}
T
T
T
[T= 1] [T= 2]
Ω (uk)k∈N∗uk=P uk=F
T: Ω −→ R
ω7−→ n.

X(Ω) = N∗
[T= 1] = P . . . [T= 2] =
F−P . . .
X(Ω,A, P )X
ΩR
X: Ω −→ R
ω7−→ X(ω),
x∈R
{ω∈Ω|X(ω)≤x} ∈ A.
X
XΩR
X X(Ω) X
[X=x] = {ω∈Ω|X(w) = x}
X x
[X≤x] = {ω∈Ω|X(w)≤x},
X
x
[X∈I] = {ω∈Ω|X(ω)∈I}
X I

Ω =
{1,2,3,4,5,6}
X: Ω −→ R
ω7−→ (0ω= 1,2,3,4 5
1ω= 6
.
[X= 1] [X= 0] [X≤0] [X < 0] [X≤1] [X∈ {1}]
[X= 1] = {6}
[X= 0] = {1,2,3,4,5}
[X≤0] = {1,2,3,4,5}
[X < 0] = ∅
[X≤1] = Ω
[X∈ {1}] = {6}
X(Ω,A, P )X
P PXIR
R
PX(I) = P(X∈I).
PXP X L(X)X
(Ω,A, P )
(Ω,A, P )

{x1, x2, . . . , xk, . . .}xk∈Rk∈N∗
X T
X(Ω,A, P )
X(Ω,A, P )
X(Ω) = {x1, x2, . . . , xk, . . .}
P(X=xk)xk∈X(Ω)
xkx1x2. . . xk. . .
P(X=xk)P(X=x1)P(X=x2). . . P (X=xk). . .
P(X=x)
X FX
X(Ω) P(X=xk)xk∈X(Ω)
X
6S
SS(Ω) = {2,3, . . . 12}
P(S= 2) = P(∩) = 1
36