Couples de variables aléatoires

(X, Y )
Ω
(X, Y ) : Ω →R2
X
Y
3 4 5 3
X Y
3
4
1
4X
Y
PPPPFFP X = 4 Y= 2
(X, Y )P((X=
i)∩(Y=j)) i∈X(Ω) j∈Y(Ω) (X, Y )
P((X=i)∩(Y=j)) P(X=i)
P(Y=j)
X Y
X(Ω) = {1; 2; 3; 4; 5; 6}Y(Ω) = {1; 2; 3; 4; 5; 6}
P((X=i)∩(Y=j)) = 0 i < j
P((X=i)∩(Y=yj)) = 1
36 i=j36
(i, i)P((X=i)∩(Y=yj)) = 1
18 i > j (i, j) (j, i)
X Y

123456
11
36
1
18
1
18
1
18
1
18
1
18
2 0 1
36
1
18
1
18
1
18
1
18
3 0 0 1
36
1
18
1
18
1
18
4 0 0 0 1
36
1
18
1
18
500001
36
1
18
6000001
36
X(Ω) = {0; 1; 2; 3}Y(Ω) = {0; 1; 2; 3}X+Y > 3
P((X=i)∩(Y=j)) = 3
i4 j 5 3−i−j
12
3
0123
01
22
3
22
3
44
1
220
14
22
6
22
3
55 0
23
22
9
110 0 0
31
55 000
(X=i)∩(Y=j) (X, Y ) = (i, j)
(X=i)∩(Y=j)i X(Ω) j Y (Ω)
X
i∈X(Ω);j∈Y(Ω)
P((X=i)∩(Y=j)) = 1
Ω
(X, Y )X Y
(X, Y )
∀i∈X(Ω) P(X=i) = X
j∈Y(Ω)
P((X=i)∩(Y=j))
Y
Y=j
Y
X
123456P(Y=j)
11
36
1
18
1
18
1
18
1
18
1
18
11
36
2 0 1
36
1
18
1
18
1
18
1
18
9
36
3 0 0 1
36
1
18
1
18
1
18
7
36
4 0 0 0 1
36
1
18
1
18
5
36
5 00001
36
1
18
3
36
6 000001
36
1
36
P(X=i)1
36
3
36
5
36
7
36
9
36
11
36
X
Y

0 1 2 3 P(Y=j)
01
22
3
22
3
44
1
220
14
55
14
22
6
22
3
55 028 55
23
22
9
110 0 0 12 55
31
55 0 0 0 1 55
P(X=i)21
55
27
55
27
220
1
220
X Y =j Z
∀i∈X(Ω) P(Z=i) = PY=j(X=i)Y
X=i
X X(Ω) = N∗X=k
k k P (X=k) =
3
4k
×1
4+1 4k
×3
4=3k+ 3
4k+1
Y(X, Y ) = (i, j)
i j i j
P((X=i)∩(Y=j)) = 3
4i1 4j
×3
4+1 4i3 4j
×1
4=3i+1 + 3j 4i+j+1
Y
(X=i)i>1P(Y=j) =
+∞
X
i=1
P(X=i)×PX=i(Y=j) =
+∞
X
i=1
P((X=i)∩(Y=j)) =
+∞
X
i=1
3i+1 + 3j
4i+j+1 =1 4j
+∞
X
i=1 3
4i+1
+3
4j+∞
X
i=1
1
4i+1 =1 4j3 421
1−3
4
+3
4j1 42
1
1−1
4
=32+ 3j−1
4j+1
Y X
Y=j PY=i(X=i) = P((X=i)∩(Y=j)
P(Y=j)=3i+1 + 3j 4i+j+1 ×4j+1 32+ 3j−1=3i+1 + 3j 4i(32+ 3j−1)
X=i Y =j∀i∈X(Ω) ∀j∈Y(Ω) P((X=i)∩(Y=j)) =
P(X=i)P(Y=j)
X
Y X Y
P((X=i)∩(Y=j)) = 1
36 16i66 1 6j66
X
Y
P(X= 8) = 5
36 P(Y= 15) = 1 18 P((X= 8) ∩(Y= 15)) = 1 18
X Y =j X

X Y
(X, Y )
X Y 0
X
X=i Y =j
P(X= 1) = 3+3
42=3 8P(Y= 1) = 32+ 1 42=5 8P((X= 1) ∩(Y= 1)) =
32+ 3
43=3 16 6=3 8×5 8
X Y
E(X) =
+∞
X
k=1
k×3k+ 3
4k+1 =3
42
+∞
X
k=1
k3
4k−1
+3
42
+∞
X
k=1
k1
4k−1
=3
42 1
(1 −3
4)2+1
(1 −1
4)2!=
3 + 1
3=10 3
Y E(Y) =
+∞
X
k=1
k×32+ 3k−1
4k+1 =32
42
+∞
X
k=1
k1
4k−1
+1
42
+∞
X
k=1
k3
4k−1
=32
42×1
(1 −1
4)2+
1
42×1
(1 −3
4)2= 1 + 1 = 2
2
X Y V (X+Y) = V(X) +
V(Y)
(n, p)n
p p(1 −p)
np(1−p)
X Y X +Y
P(X+Y=k) = XP(X=i)P(Y=k−i)i
P(X=i)P(Y=k−i)
P(X+Y=k) = XP(X=i)i∈X(Ω)PX=i(X+
Y=k)PX=i(X+Y=k) = PX=i(Y=k−i)P(X+Y=k) = PPi∈X(Ω)((X=i)∩(Y=
k−i))
X Y {1; 2; . . . ; 6}
{1; 2; 3; 4}
X+Y
i2 3 4 5 6 7 8 9 10
P(X+Y=i)1
24
1
12
1
8
1
6
1
6
1
6
1
8
1
12
1
24

X Y
Bm,p Bn,p X+YBm+n,p
P(X+Y=k) =
i=k
X
i=0
P(X=i)P(Y=k−i) =
i=k
X
i=0 n
ipi(1 −p)n−im k−ipk−i(1 −p)m−k+i=pk(1 −p)n+m−k
i=k
X
i=0 n
i m k−i=n+m kpk(1 −
p)n+m−k
(n+m, p)
X Y
λ µ X +Y λ +µ
P(X+Y=k) =
i=k
X
i=0
P(X=i)P(Y=k−i) =
i=k
X
i=0
e−λλi
i!×e−µµk−i (k−i)! =e−(λ+µ)
i=k
X
i=0
1
i!(k−i)!λiµk−i=e−(λ+µ) k!
i=k
X
i=0 k
iλiµk−i=e−(λ+µ)(λ+µ)k k!
λ+µ
X Y
FXFYZ= max(X, Y )
FZ=FXFY
FZ(x) = P(Z>x) max(X, Y )6x
X6x Y 6x FZ(x) = P((X6x)∩(Y6x)) = P(X6x)×P(Y6x) =
FX(x)FY(x)
X
Y FXFY
[−∞; 1[ FX(x) = 0 [1; 2[ FX(x) = 1
6[2; 3[ FX(x) = 2 6Z= max(X, Y )
Z[1; 2[ FZ(x) = 1
36 [2; 3[ FZ(x) = 4 36
P(Z=i) = FZ(i)−FZ(i−1)
i123456
P(Z=i)1
36
3
36
5
36
7
36
9
36
11
36
X+Y= min(X+Y)+max(X+Y)
GX(x) = 1 −FX(x) = P(X > x)U= min(X, Y )
GU(x) = GX(x)GY(x)