Probabilités sur un univers dénombrable

Ω (ωn)n∈N
Ω Ω = {ωn|n∈N}
N,Z Q
R
Ω
Ω
ΩP:P(Ω) →[0,1]
P(Ω) = 1
(An)n∈N
P [
n∈N
An!=
+∞
X
n=0
P(An).
Ω
(Ω, P ) Ω
PΩ
P: Ω →[0,1] Ω
P(∅)=0
A1, . . . , An
P n
[
i=1
Ai!=
n
X
i=1
P(Ai),
∀A∈P(Ω) P(¯ A)=1−P(A)
∀(A, B)∈P(Ω)2A⊂B⇒P(A)6P(B)P(B\A) = P(B)−P(A)
∀(A, B)∈P(Ω)2P(A∪B) = P(A) + P(B)−P(A∩B)
AB
A∩B
Ω
p: Ω →[0,1] X
ω∈Ω
p(ω)=1
P:P(Ω) →[0,1]
∀ω∈Ω, P ({ω}) = p(ω).

PΩ
PΩ 1
P
∀A∈P(Ω), P (A) = X
ω∈A
P({ω}).
Ω = NΩ
∀k∈Ω, P ({k}) = 1
2k+1 .
A B P (B)>0
A B
P(A|B) = PB(A) = P(A∩B)
P(B).
A6
B
P(A|B) = P(A∩B)
P(B)=1/6 1/2=1 3.
B PB:P(Ω) →[0,1]
PB
A1, . . . , AmP(A1∩ · · · ∩ Am)6= 0
P m
\
i=1
Ai!=P(A1)P(A2|A1)P(A3|A2∩A1)· · · P(Am|Am−1∩ · · · ∩ A1).
(Ai)i∈I
(Ai)i∈I
(Ai)i∈IΩ Ω = [
i∈I
Ai
A1
A2
A3A4
Ω

(Ai)i∈I
(Ai)i∈I
X
i∈I
P(Ai)=1
B(An)n∈N
PP(B∩An)
P(B) =
+∞
X
n=0
P(B∩An) =
+∞
X
n=0
P(B|An)P(An).
A1
A2
A3A4
B∩A1
B∩A2
B∩A3B∩A4
B
Ω
A B
P(A|B) = P(B|A)P(A)
P(B).
A B P (A∩B) = P(A)P(B)
P(B)>0A B P (A|B) = P(A)
B
A
A1, . . . , An
∀I⊂J1, nK, P \
i∈I
Ai!=Y
i∈I
P(Ai).
n
Ω = {1,2,3,4}A=
{1,2}B={1,3}C={1,4}
P(A∩B∩C) = 1
46=1 8=P(A)P(B)P(C).

X: Ω →R
XΩ
U⊂R
(X∈U) = X−1(U) = {ω∈Ω|X(ω)∈U}
x∈R
(X=x) = {ω∈Ω|X(ω) = x},(X6x) = {ω∈Ω|X(ω)6x}
f:R→Rf(X)
f◦X
∀ω∈Ω, f(X)(ω) = f(X(ω)).
PX:P(X(Ω)) →[0,1] PX(U) = P(X∈U)
X(Ω)
X(Ω) X
X(n, p)X(Ω) = J0, nK
PXX
PXP(X=x)x∈X(Ω)
P(X=x)x∈X(Ω)
FX:R→RFX(x) = P(X6x)
X
FX
R
±∞
lim
x→−∞ FX(x) = 0,lim x→+∞FX(x) = 1.
X →B(5,1/2) X
x
y
••
•
•••
01 2 3 4 5
1

PXFX
x∈R
P(X=x) = P(X6x)−P(X < x) = FX(x)−lim
t→x−
FX(t).
XNX(Ω) = N
∀n∈N, P (X=n) = P(X6n)−P(X6n−1) = FX(n)−FX(n−1).
X
X
E(X) = X
x∈X(Ω)
xP (X=x).
X
X
X(Ω) = N∗∀k∈N∗, P (X=k) = 6
π2k2,
X
+∞
X
k=1
kP (X=k) =
+∞
X
k=1
6
π2k
X
X(Ω) = N∀k∈N, P (X=k) = 1
2k+1 ,
X
E(X) =
+∞
X
k=0
kP (X=k) =
+∞
X
k=0
k
2k+1 = 1.
YΩa, b ∈RX Y
aX +bY
E(aX +bY ) = aE(X) + bE(Y).
f:R→Rf(X)
E(f(X)) = X
x∈X(Ω)
f(x)P(X=x).
X2
E(X2) =
+∞
X
k=0
k2P(X=k) =
+∞
X
k=0
k2
2k+1 = 3.