Variables aléatoires discrètes

Ω,P(Ω), P 
XΩR
IR{ω∈ΩX(ω)∈I} ⊂ P(Ω)
X(Ω)
(X=x){ω∈ΩX(ω) = x}
(a<X){ω∈Ωa<X(ω)}
(X≤b){ω∈ΩX(ω)≤b}
(X= 3)
(X < 5)
((X≥3) ∩(X≤5)) = (3 ≤X≤5) = ((X= 3) ∪(X= 4) ∪(X= 5))
X, Y (Ω,A)λ
X+Y, λX, XY, max(X, Y ),min(X, Y ) (Ω,A).
P= 5 ×A+ 4 ×R+ 3 ×D+ 1 ×V

 PX
X(Ω,P(Ω), P )
PX:P((X(Ω)) →[0; 1], A 7→ P(X∈A)
(X(Ω),P((X(Ω)))
PXX
X
X
X X(Ω)
x X(Ω)
•(X=x)
•(X=x) =
•P(X=x) =
R R(Ω) = [[0; 4]]
(X=xk)xk∈X(Ω)
P(B) = P(B∩(X=x1)) + ··· +P(B∩(X=xn))
∀xk∈X(Ω) P(X=xk)6= 0
P(B) = PX=x1(B)P(X=x1) + ··· +PX=xn(B)P(X=xn)
X(Ω,P(Ω), P ).
X F
F:
R→R
x7→ P(X6x)
X Y
X
Y X
Y
X
Y
X Y

X(Ω,P(Ω)) F
∀x∈R, F (x)∈[0; 1]
FR
lim
x→−∞F(x) = 0 lim x→+∞F(x) = 1
∀a, b ∈R, a < b P (a < X 6b) = F(b)−F(a)
X
X
XN
∀k∈N
P(X=k) = F(k)−F(k−1)
n∈N∗{(xi, pi), i ∈[[1; n]]}
(Ω,P(Ω))
card (Ω) ≥n
∀i∈[[1; n]] pi>0
n
X
i=1
pi
X(Ω,P(Ω), P ){(xk, pk), k ∈ {1; ..;n}}.
X E(X)
E(X) = x1p1+x2p2+··· +xnpn
IE(X)X
1 2 2
X
X.
X a, b
E(aX +b) = aE(X) + b
X Y 2
E(X+Y) = E(X) + E(Y)

B
N
B E(B)
B N
N
X
Xi
X Xi
pi=P(Xi= 1) pi=−1
3pi−1+1 3
XiE(X)
X E(X)=0
X−E(X)X
r
X(Ω,P(Ω), P ).
r
r
mr(X) = E(Xr)
X(Ω,P(Ω), P ).
V(X) = E(X−E(X))2
X(Ω,P(Ω), P )
V(X) = E(X2)−E(X)2
a b V (aX +b) = a2V(X)
n p
q= 1 −p.
k2ki`eme
(k−1)i`eme
Xnn
X2
X3
E(X3) = 4pq V (X3) = 2pq(3 −8pq)

E(X)V(X)
X(Ω,P(Ω), P ).
σ(X) = pV(X)
X(Ω,P(Ω), P ).
X σ(X)=1
σ(X)6= 0 X∗=X−E(X)
σ(X)
X.
2 4
B
B
◦
B(Ω) = [[1,5]].
Bi
i
P(B= 1) = P(B1) = 2
6,
P(B= 2) = P(B1∩B2) = P(B1)PB1(B2) = 4
6×2 5=4 15
P(B= 3) = P(B1∩B2∩B3) = P(B1)PB1(B2)PB1∩B2(B3) = 4
6×3 5×2 4=1 5
P(B= 4) = P(B1∩B2∩B3∩B4) = P(B1)PB1(B2)PB1∩B2(B3)PB1∩B2∩B3(B4)
=4
6×3 5×2 4×2 3=2 15
P(B= 5) = P(B1∩B2∩B3∩B4∩B5)
=P(B1)PB1(B2)PB1∩B2(B3)PB1∩B2∩B3(B4)PB1∩B2∩B3∩B4(B5)
=4
6×3 5×2 4×1 3×2 2=1 15
E(B)=1×2
6+ 2 ×4 15 + 3 ×1 5+ 4 ×2 15 + 5 ×1 15 =7 3
E(B2)=12×2
6+ 22×4 15 + 32×1 5+ 42×2 15 + 52×1 15 = 7
V(B) = E(B2)−[E(B)]2=14
9
E(X)V(X)
X(Ω,P(Ω), P )
X
E(X)≥0
E(X)=0 P(X= 0) = 1
X Y (Ω,P(Ω), P )
X≤Y⇒E(X)≤E(Y)