DM bis no 3

\
n
n(n+ 2)
j j j
(j+ 2) n
j j
•
(j−1)
•
(j+ 1)
n= 1
X1
X2
X1X2(Ω,A,P)
X1
X2
X1X2
n
jNEjj
ej= P(Ej)e0= 1

\
n
∀n∈N∗, en=n
2n+ 2en−1+n+ 2 2n+ 2en+1.
∀n∈N, en>en+1
(en)
lim
n→+∞
en= 0
n un= (n+ 1)en
nN∗un+1 unun−1
unn e1
∀n∈N, en= (2e1−1) n
n+1 +1 n+1
e1n en
n
B= (e1, e2, e3)R3fR3
B
A=
212
−1−1−1
−1 0 −1
A26= 0 A3
(a) Ker f(b, c) Im f
Im f2= Ker f
gR3g26= 0 g3= 0 g◦g
g◦g◦g
M g BR3
M26= 0 M3= 0
Im g2= Ker g
g
g
g
uR3g2(u)6= 0
(u, g(u), g2(u) ) R3B0
N g B0
Im gKer g
Im g2

\
1
2Z
Z k k ∈N∗k
· · · , k,
X
  1 
0.
k k
1 0 k−1.
Z
X
1
k1 2kk∈N∗
Z
(i, k)N∗×N∗,P[Z=k](X=i)
∀i∈N∗P (X=i) =
+∞
P
k=i
1
k1 2k.
+∞
P
i=1
+∞
P
k=i
=
+∞
P
k=1
k
P
i=1
+∞
P
i=1
P (X=i)=1
i iP (X=i)61
2i−1
X
P
E(X) = 3
2
X
P
EX2=1
6
+∞
X
k=1
(k+ 1) (2k+ 1) 1
2k
a, b c ∀k∈N∗(k+ 1) (2k+ 1) = ak (k−1) + bk +c
EX2V(X) = 11
12
X
P (X>3) 611
27
P (X= 1) ,P (X= 2) P (X>3) .
x n
n
P
k=1
xk−1n
x

\
∀n∈N∗:
n
P
k=1
1
k1 2k= ln (2) −R1/2
0
xn
1−xdx
∀n∈N∗: 0 6R1/2
0
xn
1−xdx 61 2n
lim
n→+∞R1/2
0
xn
1−xdx
P (X= 1) = ln (2) P (X= 2)
P (X>3)
P (X>3) ln 2 '0,7
O.
k n
(n+ 1) (k+ 1) p0<p<1 0
1−p
nNXnn X0= 0
nNXn(Ω,A, P )
T
0012001
T= 1 1 2 3 0 0 1 T= 4
T(Ω,A,P)
kN×(T=k)
Xi
X1
P(T=k)kN×T
n Xn(Ω) = [[0, n]]
nN×(Xn−1=k)06k6n−1
P(Xn= 0) = 1 −p
∀n∈N,∀k∈ {1,2,...n+ 1}, P (Xn+1 =k) = pP (Xn=k−1)
∀n∈N×,∀k∈ {0,1,2...,n−1}, P (Xn=k) = pk(1 −p).
P(Xn=n)
n
P
k=0
P(Xn=k)=1
p=1
3
{0,1,2}
Xnn
∀n>2,
n−1
P
k=1
kpk−1=(n−1) pn−n pn−1+ 1
(1 −p)2

\
E(Xn) = p(1 −pn)
1−p
∀n∈N, E X2
n+1=pEX2
n+ 2E(Xn)+1
n un=EX2
n+ (2n−1) pn+1
1−p
un+1 =pun+p(1 + p)
1−p
unEX2
np n
V(Xn) = p
(1 −p)21−(2n+ 1) pn(1 −p)−p2n+1