Exercices corrigés de SQ20

Printemps 

α P N
P({n}) = α
2nn∈N
(un)n∈Nn
un=1
2un+1 +1 2un−1
(un)n∈N
p n 06n6p
I n
A B p
unA
u0up
F
AnA
n16n6p−1
un=1
2un+1 +1 2un−1
unn p
1/2 1/2
n
p
p n

X1
2n
+∞
X
n=0 1
2n
=1
1−1
2
= 2
α P (N) =
+∞
X
n=0
α
2n=α
+∞
X
n=0 1
2n
= 2α= 1 α=1
2
r=u1−u0n∈Nun+1 −un=r
•n= 0 r •k∈Nuk+1 −uk=r
k+ 1
uk+1 =1
2uk+2 +1
2ukuk+2 +uk= 2 uk+1
uk+2 −uk+1 =uk+1 −uk
uk+1 −uk=r
uk+2 −uk+1 =r
•n
un+1 −un=r∀n∈N, un+1 =un+r
(un)n∈N
 u0= 1 A A  up= 0 B
A
P(F) = P(F) = 1
2
F
16n6p−1
n+ 1
PF(An) = un+1
n−1
PF(An) = un−1
F F Ω
P(An) = P(F)×PF(An) + P(F)×PF(An)
un=1
2un+1 +1
2un−1
un=1
2un+1 +1
2un−1n
p−1u0, u1, u2, . . . , up
r
n06n6p un=u0+n×r
u0= 1 up=u0+p r = 0
1 + p r = 0 r=−1
p
n06n6p un= 1 −n
p
BnB vn=P(Bn)
B A
p I m =p−n
vn= 1 −m
p

vn= 1 −(p−n)
p=n
p
Cn
An, BnCn
Ω
P(An) + P(Bn) + P(Cn)=1 P(Cn) = 1 −P(An)−P(Bn)
= 1 −un−vn= 1 −1−n
p−n
p. P (Cn)=0 A
un= 1 −p
n

3X Y Z
n n >3. n
Xn(Yn, Zn)X(
Y, Z)
XnYn, Zn
n−Xn
n−Xnn
Xn+Yn+ZnYn+Zn
P([Xn= 0] ∪[Yn= 0] ∪[Zn= 0]).
n
U
U
U
U U
XUA1
Ua
a=P(A1)
a
P(X= 1), P (X= 2) P(X= 3)
n>2P(X=n) = a(1 −a)n−2
Z=X−1
Z X