Variables aléatoires

(Ω, P ) Ω
XΩR
X:Ω→R
ω7→ X(ω)
X(Ω) = {X(ω), ω ∈Ω}
X(Ω) = {x0, x1, . . . , xk, . . .}={xi, i ∈I}I= [[0, p]] N
ΩX(Ω) Ω X
Ω = {1,2,3,4,5,6}2.
ω= (i, j) Ω X(ω) = i+j
i\j123456
1234567
2345678
3456789
45678910
5 6 7 8 9 10 11
6 7 8 9 10 11 12
X(Ω) = {2,3,4,5,6,7,8,9,10,11,12}
X ω ∈Ω
X(ω)∈X(Ω)

X X
x∈X(Ω)
PX({x}) = PX−1(x)=P(X=x)
B⊂X(Ω)
PX(B) = X
xk∈B
P(X=xk)
x∈Rx∈/ X(Ω) PX({x})=0
PXX(Ω)
xk∈X(Ω) PX({x} ∈ [0,1] X−1({xk})xk∈X(Ω)
Ω
X
xk∈X(Ω)
P(X=xk) = X
xk∈X(Ω)
P(X−1({xk})) = P
[
xk∈X(Ω)
X−1({xk})
=P(Ω) = 1
Ω
X−1(3) = {(1,2),(2,1)}P(X= 3) = card (X−1(3))
card (Ω) =2 36
k2 3 4 5 6 7 8 9 10 11 12
P(X=k)1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
X1X2
X1X2Ω = {1,2,3,4,5,6}2
X1(Ω) = X2(Ω) = {1,2,3,4,5,6}
∀k∈[[1,6]] , P (X1=k) = 1
6, P (X2=k) = 1 6
X1X2PX1=PX2
X1{(1,2)}= 1, X2{(1,2)}= 2.
X1X2
1 6
X X
x∈R
FX({x}) = P(X6x) = X
xk∈X(Ω)
xk6x
P(X=xk)
FXR
x6y{X6x}⊂{X6y}

FX=FYPX=PY
X(Ω) = {x0, x1, . . . , xp}x0< x1<· · · < xpPX(x0) = FX(x0)
∀k∈[[1, p]] , PX(xk) = FX(xk)−FX(xk−1).
X(Ω) = {xn, n ∈N}(xn)n∈NPX(x0) = FX(x0)k>1
PX(xk) = FX(xk)−FX(xk−1)
X f R
f(X) : Ω→R
w7→ f(X)(ω) = fX(ω)
X f
X2:Ω→R
w7→ X2(ω) = X(ω)2X3:Ω→R w7→ X3(ω) = X(ω)3
X X(Ω) = {x0}
P(X=x0) = 1
X X(Ω) = {x1,· · · , xn}PX
∀k∈[[1, n]] , P (X=xk) = 1
n.
X
{1,2,3,4,5,6}
p∈[0,1] XB(p)p
X(Ω) = {0,1}
P(X= 1) = p P (X= 0) = 1 −p=q
X → B(p)

p q = 1 −p
X= 1 X= 0 X
p= 1/2
2 3 X= 1
0X p = 2/5
p∈]0,1[
Ri={ }
A={n}
B={n}
C={ }
RiA A
P(A)
I[[1, n]] k0< k < n J [[1, n]]
BI={i∈I}
RiBIP(BI)
B=[
I⊂[[1,n]]
card (I)=k
BIP(B)
k= 0 k=n
n>1 0 6P(C)6pnP(C)
p∈[0,1] n∈N∗XB(n, p)n p
X(Ω) = {0,1, . . . , n}
∀k∈[[0, n]] , P (X=k) = n
kpk(1 −p)n−k.
X → B(n, p)
n
p
10 2 3
X X n = 10 p= 2/5
[[0, n]]

p∈]0,1[ X
X= +∞
Ri={ }
Ri{X=k}
P(X=k)
P(X∈N∗)P(X= +∞)
p∈]0,1[ XG(p)p
X(Ω) = N∗
∀k∈N∗, P (X=k) = p(1 −p)k−1.
X → G(p)
p
X
X p = 1/6
N∗
X p
∀n∈N, P (X > n) = qnq= 1 −p
λ > 0XP(λ)λ X(Ω) = N
∀k∈N, P (X=k) = e−λλk
k!.
X → P(λ)
X
[0, T ]X
N