corrigé.

(Ω,F,P)A∈F
AP(A∩A) =
P(A)P(A)P(A) = P(A)2P(A) 0 1
X x ∈R
P(X≤x, X ≤x) = P(X≤x)P(X≤x)P(X≤x) = P(X≤x)2
X0 1
a= sup{x∈R:P(X≤x) = 0}.
x<a t x<t≤aP(X≤t)=0 x
a{x∈R:P(X≤x) = 0}
P(X≤x)≤P(X≤t)=0 x>a P(X≤x)>0a
{x∈R:P(X≤x)=0}P(X≤x)=1
X
P(X≤x) = [a,+∞[(x).
a X
X c A B
R P(X∈A, X ∈B)P(X∈A)P(X∈B) 1
c∈A∩B0X
A, B, C
P(A∩B∩C) = P(A)P(B)P(C)
A, B, C
A, B, C
P(A∩B) = P(A)P(B)P(A∩C) = P(A)P(C)
P(B∩C) = P(B)P(C)A, B, C

Ω = {1,2,3,4,5} F =P(Ω)
A:= {1,4} ⊂ B:= {1,2,4}C:= {3,4}A∩B=A A ∩C=B∩C=
A∩B∩CP(A∩B∩C) = P(A)P(B)P(C)P(A)P(B)
P(C) ]0,1[ P(A∩B) = P(A)6=P(A)P(B)P(A∩C)6=P(A)P(C)
P(B∩C)6=P(B)P(C)P(A)=1/3P(B)=1/2
P(C)=1/2P({4}) = P(A∩B∩C) =
P(A)P(B)P(C) = 1/12 P({1}) = P(A)−P({4}) = 1/4P({2}) = P(B)−P(A) = 1/6
P({3}) = P(C)−P({4})=5/12 P({5})=1−P({1,2,3,4}) = 1/12
Ω = {1,2,3,4} F =P(Ω) P
A={1,2}B={1,3}C={1,4}P(A) = P(B) = P(C)=1/2
P(A∩B) = P(A∩C) = P(B∩C)=1/4A B
A C B C P(A∩B∩C) = 1/46=
P(A)P(B)P(C) = 1/8A B C
n≥1 Ω {0,1}nP(Ω)
Pω= (ω1, . . . , ωn)∈Ωk∈ {1, . . . , n}
Xk(ω) = ωk
X1, . . . , Xn
p∈[0,1] Q(Ω,P(Ω))
(Ω,P(Ω),Q)X1, . . . , Xn
p
k∈ {1, . . . , n} {Xk= 1}={0,1}k−1× {1}×{0,1}n−k
2n−1
2n= 1/2X1, . . . , Xn
1/2B1, . . . , Bn⊂R
P(∀k∈ {1, . . . , n}, Xk∈Bk) = P(X1∈B1). . . P(Xn∈Bn).
Bk{0,1}
Bk0 1 i
k Bk0 1 P(Xk∈Bk) = 1/2
n−i k {0,1} ⊂ BkP(Xk∈Bk) = 1
2−i
2n−i
2n= 2−i
p∈[0,1] Q(Ω,P(Ω))
(Ω,P(Ω),Q)X1, . . . , Xn
p

E={x1, x2, x3}F={y1, y2, y3}R
P= (Pij )i,j=1,2,3(X, Y )
E×F i, j ∈ {1,2,3}P(X=xi, Y =
yj) = Pij X Y
P=
1
401 12
0 0 0
1
201 6
, P =
000
3
17
12
17
2
17
000
, P =
1
4
3
32
1
32
2
15
1
20
1
60
17
60
17
160
17
480
P=
1
4
3
32
1
32
2
15
1
20
1
20
1
4
1
10
3
80
.
X
Y
P(X=xi) = P(X=xi, Y =y1)+P(X=xi, Y =y2)+P(X=xi, Y =y3) = Pi1+Pi2+Pi3,
P(Y=yj) = P(X=x1, Y =yj)+P(X=x2, Y =yj)+P(X=x3, Y =yj) = P1j+P2j+P3j.
i, j Pij =P(X=xi)P(Y=yj)
n p m p p ∈[0,1] m, n
n p n
p
X1, . . . , Xn+m
p Y =X1+. . . +XnZ=Xn+1 +. . . +Xn+m
Y Z B(n, p)B(m, p)
Y+Z=X1+. . . +Xn+mB(n+m, p)
Y Z B(n, p)B(m, p)
Y Z GY(s) = (1 −p+sp)nGZ(s) = (1 −p+sp)m
GY+Z(s) = E[sY+Z] = E[sY]E[sZ] = (1 −p+sp)n+m.
n+m p
N1, . . . , Np
λ1, . . . , λpN1+. . . +Np
N1+. . . +Np
λ1+. . .+λpp= 2

p∈[0,1]
X Y
X x1x2
P(X=x1)>0P(X=x2)>0y1y2
P(Y=y1)>0P(Y=y2)>0x1< x2y1< y2
x1+y1< x1+y2< x2+y2z1=x1+y1z2=x1+y2z3=x2+y2
P(X+Y=z1)≥P(X=x1, Y =y1)X Y
P(X+Y=z1)≥P(X=x1)P(Y=y1)>0P(X+Y=z2)>0
P(X+Y=z3)>0X+Y
X+Y
(Ω,F,P) (Xn)n≥1
(Ω,F,P)
p∈]0,1[
n≥1ω∈Ω
Sn(ω) = k∈ {1, . . . , n}Xk(ω) = 1.
Sn(Sn)n≥1
ω∈Ω
T1(ω) = min{n≥1 : Xn(ω)=1},
min ∅= +∞P(T1= +∞)T1
ω∈Ω
T2(ω) = min{n > T1(ω) : Xn(ω) = 1}.
T2T2−T1T1T2
T1T2−T1
Sn=X1+· · · +Xnn
p n p
(Sn)n≥1
Sn+1 −Sn=Xn+1 P(Sn+1 −Sn= 2) = 0 n≥1P(Sn+1 = 2)P(Sn= 0) >0

k≥1Xnω∈Ω
P(T1=k) = P(X1=· · · =Xk−1= 0, Xk= 1) = P(X1= 0) . . . P(Xk−1= 0)P(Xk= 1) = p(1−p)k−1.
P(T1<+∞) = Pk≥1P(T1=k) = 1 P(T1=
+∞)=0 T1p1−p
j≥1k≥1P(T1=j, T2=k) = 0 k≤j
k > j
P(T1=j, T2=k) = P(X1=· · · =Xj−1= 0, Xj= 1, Xj+1 =· · · =Xk−1= 0, Xk= 1) = P(X1= 0) . . . P(Xj−1= 0)P(Xj= 1)P(Xj+1 = 0) . . . P(Xk−1= 0)P(Xk= 1) = p2(1−p)k−2.
k j = 1, . . . , k −1
P({T2=k) = (k−1)p2(1 −p)k−2.
kP(T2= +∞) = 0 T1
{1,2,3, . . .}T2{2,3,4, . . .}
P(T1< T2) = 1 T1T2
P(T1= 2, T2= 2) = 0 <P(T1= 2)P(T2= 2)
k=i+j
P(T1=j, T2−T1=i) = P(T1=j, T2=k) = p2(1 −p)k−2=p(1 −p)j−1p(1 −p)i−1.
jP(T2−T1=i) = p(1 −p)i−1
P(T1=j, T2−T1=i) = P(T1=j) = P(T2−T1=i).
T2T1T1
X Y N(µ1, σ2
1)
N(µ2, σ2
2)a b c aX +bY +c
gR→R
E[g(X+Y)] = 1
2πσ1σ2Z+∞
−∞ Z+∞ −∞
e−(x−µ1)2
2σ2
1
−(y−µ2)2
2σ2
2g(x+y)dydx.
u=x+y y
E[g(X+Y)] = 1
2πσ1σ2Z+∞
−∞ Z+∞ −∞
e−(x−µ1)2
2σ2
1
−(u−x−µ2)2
2σ2
2g(u)dudx.
(x−µ1)2
2σ2
1
+(u−x−µ2)2
2σ2
2
=σ2
1+σ2 2
σ2
1σ2 2(x−λu)2+µ2
1
σ2
1
+(u−µ2)2
σ2
2
−σ2
1+σ2 2
σ2
1σ2 2
λ2
u,