Variables aléatoires réelles discrètes Rappels et Compléments

n1n2r n =n1+n2
n1n2r
prr
p0p1
prpr−1
prr>1
3/4
1/4 3/4
1/4AnBn
Cnn anbncn
Xn=
an
bn
cn
M Xn+1 =MXn
Mn
a b c
m m >2
p p >2
m= 2
m= 3
m am,p m
m>2
am+2,p = (p−2)am+1,p + (p−1)am,p
m+ 1
am,p = (p−1)m+ (−1)m(p−1)
πm,p
(πm,p)p m
(πm,p)m p
(πm,m)m
p q = 1 −p0< p < 1
X

P(X= 2)
[X= 3]
k>2P(X=k)
k k
+∞
X
k=2
P(X=k)=1
X
n X1X2
[[1, n]]
a[[1, n]] Y
Y(ω) = X1(ω)X2(ω)6a
X2(ω)X2(ω)> a
Y
Y X1
a
a b [[1, n]] Z
Z(ω) = 
X1(ω)X2(ω)6a
X2(ω)a<X2(ω)6b
X1(ω)X2(ω)> b
Z
XN
n∈N∗
n
X
k=0
kP (X=k) =
n−1
X
k=0
P(X > k)−nP (X > n)
X n
06nP (X > n)6
+∞
X
k=n+1
kP (X=k)
P(X > n)
+∞
X
n=0
P(X > n) = E(X)
P(X > n)
nP (X=n)X
n
p
j n Xjj
Xj
X n
X XjP(X > k)k

X
(Xi)i∈N(Ω,A, P )N
N
Z=Sup (X0, X1, . . . , XN)Z
n F
x60F(x)=0
x∈k
2n;k+ 1 2n(k∈N, k 62n−1) F(x) = k 2n
x>1F(x)=1
F X
Pai,j ai,j =1
iji>2
j>2
an,p =1
n2−p2n6=p an,n = 0
+∞
X
n=0
+∞
X
p=0
an,p
+∞
X
p=0
+∞
X
n=0
an,p
Pai,j ai,j =(i+j)
i!j!2i+j
α n p
an,p =1
(n2+p2)αk∈N∗uk=X
16n6k
16p6k
an,p
k∈N∗
2k+ 1
2α(k+ 1)2α6uk+1 −uk62k+ 1 (k+ 1)2α
α61uk+1 −uk
u+∞Pan,p
α > 1uPan,p
(an)n > 0an=1
n(n+ 1)
PanPnan
XZ∗P(X=n) = 1
2a|n|
(An)An= [X=n]∪[X=−n]n > 0
X Ann∈N∗
E(X|An)P(An)
X
1,2,· · · , n, · · ·
XnYn
XnYna b
XnYnXi
Yj
Zn=XnYnn>1
ZnZn

Zn
Sn=
n
X
k=1
XkTn=
n
X
k=1
Yk
SnTn
SnTn
SnTn
R
R E(R)V(R)
n k 16k < n P[R=n](Xk= 1) P[R=n](Xk= 1)
k=n k > n
U
U[R=n]
E(U|[R=n])
PP(R=n)E(U|[R=n])
E(U)