Chapitre 6 Couples et suites de variables aléatoires discrètes

X Y
(Ω,A, P )
(X, Y ) (X, Y )
•X(Ω) Y(Ω)
•(i, j)∈X(Ω) ×Y(Ω) pi,j =P[X=i]∩[Y=j]
X(Ω) Y(Ω) (X, Y )
X1X2
S
(X1, S)
HHHH
H
X1
SX1
S
HHHH
H
([X=i]∩[Y=j])(i,j)∈X(Ω)×Y(Ω)
(X, Y )
X
(i,j)∈X(Ω)×Y(Ω)
P([X=i]∩[Y=j]) = 1

{pi,j }i∈I,j∈J
• ∀(i, j)∈I×J, pi,j ≥0
•P
(i,j)∈I×J
pi,j = 1
{pi,j }i∈I,j∈J
(X, Y )
•X(Ω) = I
•Y(Ω) = J
• ∀(i, j)∈I×J P ([X=i]∩[Y=j]) = pi,j
a
X Y N∀(i, j)∈N2P([X=i]∩[Y=j]) = a
2i+1j!a
(X, Y )X Y
i∈X(Ω) P(X=i) (X=i)
(Y=j)j∈Y(Ω)
(X=i) = [
j∈Y(Ω)
(X=i)∩(Y=j)
P(X=i) = P
j∈Y(Ω)
P[X=i]∩[Y=j]
j∈Y(Ω) P(Y=j) (Y=j)
(X=i)i∈X(Ω)
P(Y=j) = P
i∈X(Ω)
P[X=i]∩[Y=j]

(X1, S)
S
i
P([S=i])
(X, Y )
(X=i)X
Y(X=i)Y(X=i)
j P(X=i)(Y=j)
S[X1= 2]
i
P[X1=2](S=i)

i∈X(Ω) j∈Y(Ω)
(X=i) (Y=j)
∀i∈X(Ω),∀j∈Y(Ω), P [X=i]∩[Y=j]=P(X=i)P(Y=j)
•
P((X=i)∩(Y=j))
P((X=i)∩(Y=j)) = 0 6=P(X=i)P(Y=j)
•
S X1
X Y
g(X, Y )
Z
X Y (X, Y )
g(X, Y )
(X, Y )
Z=g(X, Y )X Y Z =X+Y Z =XY
Z
•Z(Ω) X(Ω) Y(Ω)
•P([Z=k]) k∈Z(Ω) (X, Y )
(Z=k)X Y
Z=XY (Z=k) (X=i)i∈X(Ω)
(XY =k) = [
i∈X(Ω)
(X=i)∩(XY =k)

P(XY =k) = X
i∈X(Ω)
P[(X=i)∩(XY =k)] = X
i∈X(Ω)
P[(X=i)∩(iY =k)]
Z=X+Y
Z=XY
X → P(λ)Y → P(λ0)
X Y X +Y
(X+Y)(Ω)
k∈(X+Y)(Ω) P(X+Y=k) =
k
P
i=0
P(Y=k−i)P(X=i)
X+Y → P(λ+λ0)