Vecteurs aléatoires

(Ω, P )XΩX(Ω)
X X2X
X(Ω)
X(Ω)
X Y Ω
(X, Y ) : Ω→R2
ω7→ X(ω), Y (ω)
X Y
PX,Y (X, Y )X Y
(x, y)∈X(Ω) ×Y(Ω)
PX,Y {(x, y)}=P(X, Y )=(x, y)=P(X=x)∩(Y=y)
X Y (X, Y )
X(Ω) = {x1, . . . , xn}Y(Ω) = {y1, . . . , ym}xiyj
X Y
X\Y y1y2· · · ym
x1PX,Y {(x1, y1)}PX,Y {(x1, y2)}· · · PX,Y {(x1, ym)}
x2
xnPX,Y {(xn, y1)}· · · PX,Y {(xn, ym)}
6X Y

(k, n)∈[[1,6]] ×[[2,12]] P(X=k)∩(Y=n)n k
(X, Y )
PX,Y X(Ω) ×Y(Ω)
(xk, yk)∈X(Ω) ×Y(Ω) PX,Y ({(x, y)} ∈ [0,1]
X
(x,y)∈X(Ω)×Y(Ω
PX,Y (x, y)=
n
X
i=1 
m
X
j=1
P(X=xi∩Y=yj)
=
n
X
i=1
P(X=xi)=1
{Y=yj}j∈[[1, m]] Ω
(X, Y )PXPY
∀i∈[[1, n]] , P (X=xi) =
m
X
j=1
P(X=xi)∩(Y=yj)
∀j∈[[1, m]] , P (Y=yj) =
n
X
i=1
P(X=xi)∩(Y=yj)
i{X=xi}
{(X=xi)∩(Y=yj)}j∈[[1, m]]
PX,Y PXPY
(X, Y )
PX(xi)PY(yj)
X Y
Z= 7 −X X, Y Z {1,2,3,4,5,6}
(X, Y ) (X, Z)
123456
11
36
1
36
1
36
1
36
1
36
1
36
21
36
1
36
1
36
1
36
1
36
1
36
31
36
1
36
1
36
1
36
1
36
1
36
41
36
1
36
1
36
1
36
1
36
1
36
51
36
1
36
1
36
1
36
1
36
1
36
61
36
1
36
1
36
1
36
1
36
1
36
(X, Y )
123456
1000001
6
200001
60
30001
60 0
4001
6000
5 0 1
60000
61
600000
(X, Z)
X Y X
Z

x∈X(Ω) P(X=x)>0Y X =x
∀y∈Y(Ω), P (Y=y|X=x)
X λ > 0
p α
Y
Y X =n
Y λp
X Y (x, y)∈X(Ω) ×Y(Ω)
P(X=x∩Y=y) = P(X=x)P(Y=y)
(x, y)∈X(Ω) ×Y(Ω) P(X=x)>0
P(Y=y|X=x) = P(Y=y)
i∈[[1,6]]
P(Y=j|X=i) = 1
6=P(Y=j)∀j∈[[1,6]]
P(Z=j|X=i) = 1j= 7 −i
0j6= 7 −i
X Y
X Z X
Z
X Y
(x, y)∈X(Ω) ×Y(Ω)
PX,Y {(x, y)}=P(X=x)∩(Y=y)=P(X=x)P(Y=y)
X Y
Z
Z(ω) = 
X(ω)X(ω)63.
Y(ω)X(ω)>3Y(ω)>3.
Y(ω)+3 X(ω)>3Y(ω)63.
(X, Y ) (X, Z)
X Z
X Y (A, B)⊂X(Ω) ×Y(Ω)
P(X, Y )∈A×B=P(X∈A)P(Y∈B)

X Y f g
X(Ω) Y(Ω) f(X)g(Y)
X Y N
X+Y
∀n∈N, P (X+Y=n) = X
i+j=n
P(X=i)P(Y=j)
=
n
X
i=0
P(X=i)P(Y=n−i)
{X+Y=n}=
n
[
i=0
{X=i∩Y=n−i}
P(X+Y=n) = P n
[
i=0
{X=i∩Y=n−i}!=
n
X
i=0
P(X=i∩Y=n−i)
=
n
X
i=0
P(X=i)P(Y=n−i)
X Y
S=X+Y7
P(S= 7) = P(X+Y= 7) =
7
X
i=0
P(X=i)P(Y= 7 −i)
=
6
X
i=1 1
6×1 6=6 36 =1 6
X Y B(p)B(n, p)
S=X+Y
X Y α β
S=X+Y
X1, X2, . . . , XnΩ
(α1, α2, . . . , αn)∈X1(Ω) ×X2(Ω) × · · · × Xn(Ω)
P n
\
i=1
(Xi=αi)!=
n
Y
i=1
P(Xi=αi)
X1, X2, . . . , Xn
(A1, . . . , An)∈ PX1(Ω)× · · · × PXn(Ω)(Xi∈Ai)

X1, . . . , Xnp∈[[1, n]] f g
X1(Ω) ×. . . ×Xp(Ω) Xp+1(Ω) ×. . . ×Xn(Ω)
f(X1, . . . , Xp)g(Xp+1,· · · , Xn)
X1, . . . , Xn
(X1+· · · +Xn−1)Xn
X1, . . . , XnB(p)p∈]0,1[ X1+· · · +Xn
B(n, p)
n∈N∗
p Xi1 0
X1+· · · +Xn
B(n, p)
X Y
E(X+Y) = E(X) + E(Y)
X Y
E(XY ) = X
16i6n
16j6m
xiyjP(X=xi)∩(Y=yj)
f(X, Y )=
XY
E(XY ) = X
(xi,yj)∈X(Ω)×Y(Ω)
xiyjP(X, Y )=(xi, yj)
X Y
E(XY ) =
6
X
i=1
6
X
j=1
ijP (X=i)∩(Y=j)
6
X
i=1
6
X
j=1
ij
36
=1
36
6
X
i=1
i
6
X
j=1
j=1
36 ×21 ×21 = 49 4