ESPACES PROBABILISES FINIS Exercice 1

(Ω,P(Ω), P )
A={ }, B ={ },
C={ },
D={ }, E ={ },
F={ }, G ={ },
H={ }, I ={ }.
Ω
a b c
n
n
(Ωn,P(Ωn), Pn)
J={ }
Ωn
p
i∈ {1, ..., n}
Ci={ }.
CiJ
K={n}
L={n}.
J K L
J K L p

n≥2k∈ {2, . . . , n}
M
M
0 1
M
r≥k M r
`∈ {k−2, . . . , n −2}` M
k= 2
M
A B E n
A∪B
A∪B=E
A∩B
A⊂B
r n
µr,n(k)k r n
+∞r
nλ > 0
µr,n(k)→µ(k) = λk
k!e−λ,∀k∈N.
r1, . . . , rnr1+· · · +rn=r r1
1. . . rnn
r
r
n
10 4 6
3
k

10 8 20
1 10 i∈ {1, ..., 10}Xi1
i0
XiE(Xi) = 14
95.
1/12
4
R1, R2R3
20%
R150% R230% R3
0,05,0,15 0,30.
0,3
0,5
0,2.
An
pnAnp1= 0,3.
p2
(pn)pnn.
n≥3. Bnn
k
P(B1)P(B2)P(Bk)k∈ {2, ..., n −1}
n
p= 1/4
X
X E(X)V(X)
M
M E(M)

2 8 n
2 3. XnYn
Xn, E(Xn)V(Xn)
YnXn. Yn, E(Yn)V(Yn)
0,1,2, . . . , n
Xn
n Ynn
Yn, E(Yn)V(Yn).
XnYnn. E(Xn)V(Xn)Xn
XB(n, p). E(1
1+X).
n+ 1 U0, U1, . . . , Un{0,1, ..., n}Ukk
n−k
Z k k >1
k UkX
Z0X
X(Ω)
i= 0 1 6i6n PZ=0 (X=i)
i=n06i6n−1PZ=n(X=i)
k{1,2, ..., n −1}06i6k k < i 6n
PZ=k(X=i)
P(X= 0) =
n−1
X
k=1 n−k
2nk
+1
2n
P(X=n) = 1
2n
i{1,2, ..., n −1}P(X=i)
Pn
i=0 P(X=i)=1
n n 1n n
i∈ {1, ..., n}i n −i
1

n N 1N
X n
X
E(X) = n(N+ 1)
n+ 1
Y n Y.
n−2 2
X Y
X Y
(X, Y ).
X Y
cov(X, Y ).
i∈[[1,3]] Xii`eme
X1
(X1, X2). X2.
T T =X2−X1.
T
(T, X1)T.
X3.
U1U21 6
U11 2 U23,4,5 6
n∈NXnU1n
1,3,2,3,5
U1
X1E(X1)
(X1, X2)X2
(X1, X2)
nN×
P(Xn+1 = 0) = 1
6P(Xn= 1), P (Xn+1 = 6) = 1 6P(Xn= 5)
∀k∈ {1, .., 5}P(Xn+1 =k) = 7−k
6P(Xn=k−1) + k+ 1 6P(Xn=k+ 1)
nN×E(Xn+1) = 2
3E(Xn)+1
E(Xn)nlim
n→+∞E(Xn)