Corrigé D07M

p∈]0 ; 1[
q= 1 −p
n n ∈N∗n
(n+ 1)
kN∗PkFkk
L1L2
◦nN∗(L1=n)PkFkk∈J1 ; n+ 1K
n∈N∗
(L1=n) = P1∩P2∩. . . ∩Pn∩Fn+1∪F1∩F2∩... ∩Fn∩Pn+1
∀n∈N∗,P(L1=n) = pnq+qnp
P1∩P2∩. . . ∩Pn∩Fn+1 F1∩F2∩. . . ∩Fn∩Pn+1
P(L1=n) = PP1∩P2∩. . . ∩Pn∩Fn+1+PF1∩F2∩. . . ∩Fn∩Pn+1
P1, P2, . . . , Pn, Fn+1
F1, F2, . . . , Fn, Pn+1
P(L1=n) = P(P1)P(P2). . . P(Pn)P(Fn+1) + P(F1)P(F2). . . P(Fn)P(Pn+1) = pnq+qnp
+∞
X
n=1
P(L1=n)=1
∀N>1,
N
X
n=1
P(L1=n) =
N
X
n=1
(pnq+qnp) =
N
X
n=1
pnq+
N
X
n=1
qnp
p q |p|<1,|q|<1
N
X
n=1
P(L1=n)−→
N→+∞
pq ×1
1−p+pq ×1 1−q=p+q= 1
+∞
X
n=1
P(L1=n) = 1
L1

L1X
n>1
nP(L1=n)
X
n>1
nP(L1=n)
∀N>1,
N
X
n=1
nP(L1=n) =
N
X
n=1
n(pnq+qnp) = pqN
X
n=1
npn−1+
N
X
n=1
nqn−1
|p|<1,|q|<1
N
X
n=1
nP(L1=n)−→
N→+∞
pq1
(1 −p)2+1 (1 −q)2=p q+q p=p2+q2 pq
L1E(L1) = p2+q2
pq
◦n, k N∗(L1=n)∩(L2=k)PkFkk∈J1 ; n+k+ 1K
n, k ∈N∗
(L1=n)∩(L2=k) =
P1∩. . . ∩Pn∩Fn+1 ∩. . . Fn+k∩Pn+k+1∪F1∩. . . ∩Fn∩Pn+1 ∩. . . Pn+k∩Fn+k+1
(L1, L2)
L1L2N∗
∀n, k ∈N∗,P(L1=n)∩(L2=k)=pnqkp+qnpkq=pn+1qk+qn+1pk
◦∀k∈N∗,P(L2=k) = p2qk−1+q2pk−1
n(L1=n) ; n∈N∗ok∈N∗
P(L2=k) =
+∞
X
n=1
P(L1=n)∩(L2=k)
=
+∞
X
n=1
(pn+1qk+qn+1pk) = p2qk×1
1−p+q2pk×1 1−q
=p2qk−1+q2pk−1
∀K>1,
K
X
k=1
P(L2=k) =
K
X
k=1
p2qk−1+
K
X
k=1
q2pk−1−→
K→+∞
p21
1−q+q21 1−p=p+q= 1
+∞
X
k=1
P(L2=k) = 1

L2E(L2)=2
L2X
k>1
kP(L2=k)
X
k>1
kP(L2=k)
K>1
K
X
k=1
kP(L2=k) =
N
X
k=1
k(p2qk−1+q2pk−1)
=p2K
X
k=1
kqk−1+Xq= 1Kkpk−1
−→
K→+∞
p2×1
(1 −q)2+q2×1 (1 −p)2= 1 + 1 = 2
L2E(L2)=2
◦L1L2
âP(L1= 1) ∩(L2= 1)=p2q+q2p=pq(p+q) = pq P(L1= 1) P(L2= 1) =
(2pq)(p2+q2)
pq = (2pq)(p2+q2)⇐⇒ 1 = 2(p2+q2)pq 6= 0
⇐⇒ 2(p2+ (1 −p)2)−1=4p2−4p+ 1
⇐⇒ (2p−1)2= 0
⇐⇒ p=1
2
âp6=1
2P(L1= 1) ∩(L2= 1)6=P(L1= 1) P(L2= 1) L1L2
âp=1
2∀n, k ∈N∗,
P(L1=n)∩(L2=k)=1
2n+k+1 +1 2n+k+1 =1 2n+kP(L1=n)P(L2=k) =
1
2n×1 2k
∀n, k ∈N∗,P(L1=n)∩(L2=k)=P(L1=n)P(L2=k)L1L2
L1L2p=1
2

(L1, L2)
L1L2
L1L2X
n,k>1
n k P(L1=n)∩(L2=k)
ân∈N∗
K
X
k=1
n k P(L1=n)∩(L2=k)=
K
X
k=1
n k(pn+1qk+qn+1pk)
=npn+1q
K
X
k=1
kqk−1+nqn+1p
K
X
k=1
kpk−1
−→
K→+∞
npn+1q×1
(1 −q)2+nqn+1p×1 (1 −p)2=nqpn−1+npqn−1
+∞
X
k=1
n k P(L1=n)∪(L2=k)=nqpn−1+npqn−1
â
N
X
n=1
(nqpn−1+npqn−1)−→
N→+∞
q×1
(1 −p)2+p×1 (1 −q)2=1 q+1 p=p+q pq =1 pq
L1L2E(L1L2) = 1
pq
(L1, L2)
Cov(L1, L2) = E(L1L2)−E(L1)E(L2) = 1
pq −2(p2+q2) pq =1−2(p2+q2) pq =−(2p−1)2 pq
âp=1
2Cov(L1, L2) = 0
â
◦
p]0 ; 1[
L1L2

E(L1)E(L2) Cov(L1, L2)
∗
− ∗
− −
− ∗ −
∗∗ ∗ −
−
−
−
n
p=q=1
2
nN∗Nnn
N1=N2= 1, N3=··· =N6= 2, N7=N8= 3, N9=··· =N11 = 4
N12
◦N1N2N3