Variables aléatoires infinies

X X
X
(An) (Ω,T, P )
∀n∈NAn⊂An+1 ∀n∈NAn+1 ⊂An
(An)P +∞
[
n=0
An!= lim
n→+∞P(An)
(An)P +∞
\
n=0
An!= lim
n→+∞P(An)
100 000
50
100 000
An
n(An)
n+ 1 n p =1
501 000
P(An) = (1 −p)n
P +∞
\
n=0
An!= lim
n→+∞P(An) = 0 p > 0 1 −p < 1
1
1
A
Ω
A
A

XΩ
X: Ω →Z
X
6
X
X(Ω) = {2; 3; . . .}
X=i X >i
X Y ΩX+Y XY
max(X, Y ) min(X, Y )
XΩg:Z→Z
g(X) : ω7→ g(X(ω)) g(X))
X X
P(X=k)k X(Ω)
P(X=k) = 5
6k−1
×1
6
X=k k
1
3k−1
(k−1) 2
3k−2
×1
3
P(X=k) = (k−1) 1
322 3k−2
P
k∈X(Ω)
P(X=k) = 1
+∞
X
k=1 5
6k−1
×1
6=1 6
+∞
X
k=0 5
6k
=1
6×1
1−5
6
= 1
+∞
X
k=2
(k−
1) 1
322 3k−2
=1
9
+∞
X
k=1
k2
3k−1
=1
9×1
1−2
32= 1

X
FX:R→RFX(x) = P(X6x)
FX
X
kP (X=k)X E(X) =
X
k∈X(Ω)
kP (X=k)
6
E(X) =
+∞
X
k=2
k(k−1) 1
322 3k−2
=1
9×2
1−2
33=1
9×54 = 6
X Y
ΩX+Y E(X+Y) = E(X) + E(Y)
X E(X)>
0X Y X 6Y
E(X)6E(Y)
X g :R→R
g(X)E(g(X)) = X
k∈X(Ω)
g(k)P(X=k)
X r X
r Xrmr(X) = E(Xr)
X m
(X−m)2V(X) = E((X−m)2)
X X X2
X p ∈[0; 1]
X(Ω) = N∀k∈N∗P(X=k) = p(1 −p)k−1X∼ G(p)
n
X∼ G(p)X E(X) = 1
pV(X) = 1−p p2

E(X) =
+∞
X
k=1
pk(1 −p)k−1=p×1
(1 −(1 −p))2=p p2=1 p
E(X)2=1
p2X(X−1)
E(X(X−1)) =
+∞
X
k=1
pk(k−1)(1 −p)k−1=p(1 −p)2
(1 −(1 −p))3=2(1 −p) p2
V(X) = E(X(X−1)) + E(X)−E(X)2=2(1 −p)
p2+1 p−1 p2=2(1 −p) + p−1 p2=1−p p2
X∼ G(p)∀(n, m)∈N∗2PX>n(X >
n+m) = P(X > m)
P(X > m) = X
k>m
P(X=k) =
+∞
X
k=m+1
pqk−1=p
+∞
X
j=0
qj+m=
pqm
+∞
X
j=0
qj=pqm
1−q=qmP(X > n) = qnP(X > n +m) = qn+m
PX>n(X > n +m) = P(X > n +m)
P(X > n)=qm
X λ ∈R+
X(Ω) = N∀k∈NP(X=k) = e−λλk
k!X∼ P(λ)
1P(X=k) 0 k
+∞
X∼ P(λ)X E(X) = V(X) = λ
E(X) =
+∞
X
k=0
ke−λλk
k!=λe−λ
+∞
X
k=1
λk−1
(k−1)! =λe−λeλ=λ E(X(X−1)) =
+∞
X
k=0
k(k−1)e−λλk
k!=λ2
+∞
X
k=2
λk−2
(k−2)! =λ2V(X) = E(X(X−1)) + E(X)−E(X)2=
λ2+λ−λ2=λ
n p
λ∈R+(Xn)Xn∼ B n, λ
n
∀k∈Nlim
n→+∞P(Xn=k) = e−λλk
k!

k∈
NXn∼ B n, λ
nn>k P (Xn=
k) = n
kλ nk1−λ nn−k
λk
nk
n
k=n! k!(n−k)! =n(n−1)(n−2) . . . (n−k+ 1) k!
n k
n nknkn
k∼nk k!
1−λ
nn−k
=e(n−k) ln(1−λ
n)lim
n→+∞
λ
n= 0
ln 1−λ
n∼ −λ n(n−k) ln 1−λ n∼n−k n(−λ)→
n→+∞−λ
lim
n→+∞1−λ
n=e−λP(Xn=k)∼nk k!×λk
nk×e−λ∼e−λλk
k!