cours - Jean

Ω
F P(Ω) P:F → [0,1]
P(Ω) = 1
A B ΩP(A∪B) = P(A)+P(B)
(An)
Ω
P([
n∈N
An) = X
n∈N
P(An).
(Ω,F,P) Ω
F
Ω
Ω
R
P
[a, b]R
P([a, b]) = Zb
a
f(x)dx,
fR+∞
−∞ f(x)dx = 1 f
P
(Ω,F,P)A B
P(¯ A)=1−P(A)
P(A∪B) = P(A) + P(B)−P(A∩B)

0≤k≤n n
k
nΩn{P, F }
2n
P(Ω) Ω
Ωp
1−p
n
P P F P p2(1 −p)p n X = (x1, x2,...,xn)∈Ω
P(X) = pj(1 −p)n−jj P n X n −j
F
Ω
Akk
n
kk P n
Fn
kn k P n
pk(1 −p)n−kAkΩn
k
k n
P(Ak) = n
kpk(1 −p)n−k
nΩk k
nΩ = Fn
k=0 AkP(Ω) = Pn k=1 n
kpk(1 −p)n−k= (1 + (1 −p))n= 1
(Ω,F,P)A B
P(B)6= 0
A B
PB(A) = P(A|B) = P(A∩B)
P(B).
A7→ PB(A)
ΩB
P(B)PB
B
P(B|A) = P(A|B)P(B)
P(A).
P(A) = P(B)P(A|B) + P(¯ B)P(A|¯ B).
A B
P(A∩B) = P(A)P(B),PB(A) = P(A).
A B
B A
A B

(Ω,F,P)
X: Ω →R
PXR
∀A⊂R,PX(A) = P(X−1(A)) = P(X∈A).
ΩR
X(ω) = ω
X
X
FX:R→[0,1]
x7→ PX(] − ∞, x]) = P(X≤x).
FXlimx→−∞ FX(x)=0
limx→+∞FX(x)=1
X FXX
FX
X
X
fX:R→[0,1]
x7→ PX(x).
X
X fX∀a, b ∈R
PX([a, b]) = Zb
a
fX(x)dx.