Couples de variables aléatoires - Page de M. Bailloeuil

Ω
(X, Y ) (Ω,A, P ) Ω
E×E0E E0
(Ω→E×E0
ω7→ (X(ω), Y (ω))
X Y (Ω,A).
E×E0=R2(X, Y )
(X, Y ) (Ω,A, P )
((X=x)∩(Y=y))(x,y)∈X(Ω)×Y(Ω)
Ω
(X, Y ) (Ω,A, P )
(X, Y ) (X, Y )X Y )
{((x, y), P [(X=x)∩(Y=y)]), x ∈X(Ω) y∈Y(Ω)}
X
Y
X Y
(X, Y )
X Y (X, Y ).
X Y )X Y ) (X, Y ).

(X, Y ) (ω, A, P )
∀x∈X(Ω), P (X=x) = X
y∈Y(Ω)
P[(X=x)∩(Y=y)]
∀y∈Y(Ω), P (Y=y) = X
x∈X(Ω)
P[(X=x)∩(Y=y)]
(Y=y)y∈Y(Ω) (X=x)x∈X(Ω)
X
Y
(X, Y )
X\Y0 1 2 3
0 0.25 0.20 0.05 0
1 0.20 0.10 0.05 0.05
2 0.05 0.02 0.02 0.01
X Y.
E(X)E(Y)σ(X)σ(Y)
(X, Y ) (Ω,A, P )y∈Y(Ω)
P(Y=y)6= 0
[Y=y]X
(xi, P[Y=y](X=xi))i∈I
x∈X(ω) [X=x]Y
[Y= 2]
X
[Y= 2] X

(X, Y ) (Ω,A, P )
x∈X(Ω) y∈Y(Ω) P(X=x)6= 0 P(Y=y)6= 0
•
P[X=x](Y=y) = P([X=x]∩[Y=y])
P(X=x)P[Y=y](X=x) = P([X=x]∩[Y=y]) P(Y=y)
•
P((X=x)∩(Y=y)) = P[X=x](Y=y)P(X=x)P((X=x)∩(Y=y)) = P[Y=y](X=x)P(Y=y)
•
P(X=x) = X
y∈Y(Ω)
P((X=x)∩(Y=y)) P(Y=y) = X
x∈X(Ω)
P((X=x)∩(Y=y))
•
P(X=x) = X
y∈Y(Ω)
P[Y=y](X=x)P(Y=y)P(Y=y) = X
x∈X(Ω)
P[X=x](Y=y)P(X=x)
i∈[[1,2]] Xi
i`eme
X1
(X1, X2). X2.
T T =X2−X1. T
X, Y (Ω,A, P )
X+Y, XY, max(X, Y ),min(X, Y ) (Ω,A).
g X (Ω) ×Y(Ω)
g(X, Y )
X, Y (Ω,A, P )g
X(Ω) ×Y(Ω)
Z=g(X, Y )X Y.
Z(Ω) = {g(x, y), x ∈X(Ω) y∈Y(Ω)} ∀z∈Z(Ω)
P(Z=z) = X
x y g(x,y)=z
P((X=x)∩(Y=y))
X, Y (Ω,A, P )g
X(Ω) ×Y(Ω)
Z=g(X, Y )X Y.
E(Z) = X
x∈X(Ω) X y∈Y(Ω)
g(x, y)P((X=x)∩(Y=y))

X1X2
P(Xi= 0) = 1
6, P (Xi= 1) = 5 6
S=X1+X2, P =X1X2.
(S, P )S P
S P
E(S)E(P)V(S)V(P)
X, Y 2 (Ω,A, P )E F
X Y ∀x∈X(Ω) ∀y∈Y(Ω)
P((X=x)∩(Y=y)) = P(X=x)×P(Y=y)
(X, Y )
X\Y0 1 2
0 1/20 1/4 0
1 17/60 1/4 1/6
X Y
E(X)E(Y)
X Y (Ω,A, P )
E F A ⊂E B ⊂F
P((X∈A)∩(Y∈B)) = P(X∈A)P(Y∈B)
X Y
X Y
X\Y0 1 2 3 loideX
0 0.04
1 0.02
2 0.06 0.09 0.06
3
loideY 0.2
(X, Y )
X\Y−1 0 1
0 1/4a1/8
1 1/5b1/10
X Y.
a b X Y

X Y (Ω,A, P )f, g
X(Ω) Y(Ω)
f(X)g(Y) (Ω,A, P )
X Y (Ω,A, P )
E(XY ) = E(X)E(Y)
V(X+Y) = V(X) + V(Y)V(X−Y) = V(X) + V(Y)
X Y E(XY ) = E(X)E(Y)
E(XY )6=E(X)E(Y)X Y
(Xi)i∈Np
Yi=XiXi+1
Yi
∀n∈N∗Sn=
n
X
i=1
YiE(Sn)
X Y X Y
cov(X, Y ) = E[(X−E(X))(Y−E(Y))]
X Y
cov(X, Y ) = E(XY )−E(X)E(Y)
Icov(Y, X) = cov(X, Y )cov(X, X) = V(X)
X Y cov(X, Y )=0
cov(X, Y )=0 X Y
cov(X, Y )6= 0 X Y
X
xi
P(X=xi)1
6
1
4
1
6
1
4
1
6
Y=X2
Y(X, Y )
X Y
(X, Y )