Variables aléatoires

TSI2
p= 0,15
XnYn=Xn
n
Yn1 + 1
n−(0,85)n
f(x) = ax + ln x a = ln(0,85) n f(n)>0
f(n)>0E(Yn)<1n
1/2
X P
P X
X
E(P)
2/3
1/3X
n>1pn
P(X=n)
(X= 2),(X= 3),(X= 4) p2, p3, p4
pn=2
9pn−2+1 3pn−1n>4
pnn
q∈]−1,1[
+∞
X
n=0
nqnE(X)
n∈N∗q
f(q) = (1 −q)−n, q ∈[0,1[
p= 1 −q
∀k∈N, P (X=k) = n+k−1
kpnqk,
Nn p

n1n2p=n1
n1+n2
Y
Y
T
λ(T)λ(1) = 1,7
X1X2
X1+X2
λ(2)
{X1= 0} {X2= 0}
X λ > 0Y
X Y
X Y X/2
Y
N α
P(N=k) = e−ααk
k!, k ∈N.
p
S pα
n∈N∗Yn
k
Rk:
Bk:
Y1
Ynn2
n P (Yn= 2)
n un=P(Yn= 1)

TSI2
u1u2=2
3
Ynn>2
un+1 =2
3un+2 3n+1 n= 1
n vn=un+2
3n
(vn)n∈N∗vnn v1
∀n∈N∗, un= 2 2
3n
−2
3n.
P(Yn)=0 n∈N∗
Yn
Z
Z(Ω)
k>2 (Z=k)YkYk−1
Z
n X X(Ω)
{0,1, . . . , n}X
E(X) =
n
X
k=1
kP (X=k)
E(X) =
n
X
k=1
P(X>k)
X23
4
P(X>1) + P(X>2)
E(X)
X X(Ω) {0,1, . . . , n}
k∈ {1, . . . , n −1}P(X=k)P(X>k)P(X>k+ 1)
n
2
3X
X(Ω)
Njj
k[[1, n]] (X>k)N1, N2, . . . , Nk
P(X>k)
E(X)

O
0k n
(n+ 1) (k+ 1) k+1
k+2 0
1
k+2
nNXnn X0= 0
nNXn(Ω, P )
un=P(Xn= 0)
Xn
X1(Ω) = {0,1}X1
n Xn(Ω) = {0,1, .., n}
∀n∈N∗,∀k∈ {1, .., n}, P (Xn=k) = k
k+1 P(Xn−1=k−1)
∀n∈N,∀k∈ {0,1, .., n}, P (Xn=k) = 1
k+1 un−k
n
X
k=0
P(Xn=k) = 1 ∀n∈N,
n
X
j=0
uj
n−j+ 1 = 1
u0u1u2u3
(k+ 1)P(Xn=k) = kP (Xn=k−1)
∀n∈N∗, E(Xn)−E(Xn−1) = un
n E(Xn)
(un)
n
n−1
X
j=0
uj
n−j
un+
n−1
X
j=0
uj
n−j+ 1 = 1
un=
n−1
X
j=0
uj
(n−j)(n−j+ 1)
nN∗un>1
n+1 lim
n→+∞E(Xn)
T
T(Ω, P )
T0O
0,0,1,2,0,0,1T= 1
1,2,3,0,0,1T= 4
kN∗(T=k)
Xi
∀k∈N∗, P (T=k) = 1
k(k+ 1)
a b ∀k∈N×,1
k(k+ 1) =a k+b k+ 1
P(T= 0) = 0
T