Poly de cours

{P, F }
1 6
1/6
1N
Ω
0 42
26
n[[−n, n]]
Ω
(Ω,T,P)
E
X(Ω)
x∈X(Ω) X−1({x}) = {ω∈Ω; X(ω) = x}y
T
ΩX
ΩXE
X−1(x1)
X−1(xn)
•x1
•x2
•xn

{X=x0}=X−1(x0) = X−1({x0}) = {ω∈Ω; X(ω) = x0}
P(X=x0)P(X>x0)P(X∈A)
ΩX(ω)
P(X=xi) = PX−1({xi})P(X∈A) = PX−1(A)A⊂E
TX
x∈X(Ω) X−1({x})∈ T A⊂E X−1(A)∈ T
TX(Ω)
X−1({x})E
X(Ω) E E =R
X(Ω) = R+
X(ω) Ω
X(ω)
E
X: Ω →EPXEPX(A) = P(X∈A)
PXP(E)
PXP(X=x)x∈E
A E
X(Ω) σ
PX(A) = X
x∈X(Ω)∩A
P(X=x).
R
P(X6t){X6t}
{X=x}x X(Ω)∩]− ∞, t]

X FX
∀t∈R, FX(t) = P(X6t}.
k X1, ..., XkY
{Y6n}=
k
\
i=1
{Xi6n}
FY=FX1FX2...FXkY
X
•FX0 1 −∞ +∞
•FXR\X(Ω)
•FX
•FXtP(X=t)=0
X FX
P(a<X6b) = FX(b)−FX(a)P(a6X6b) = FX(b)−lim
t→a−FX(t).
X
X=x0
X{xi|i∈N}xi(pi)i∈N
Ppi1
P(Ω,T)i∈N P(X=xi) = pi
xi
L {xi|i∈N}P(X=xi)
X → L

B(1/3)
p∈]0,1[ X
p X → B(p)X(Ω) = {0,1}P(X= 1) = pP(X= 0) = 1 −p
1
AA:x∈Ω7→ (1 si x∈A
0 sinon
P(A)
n > 0
B(p)X n
[[0, n]]
k∈[[0, n]] P(X=k) = n
kpk(1 −p)n−kX → B(n, p)
B(8,1/3)
X → B(n, p)kP(X=k)
k>1k−1
k
X(Ω) = N∗∀k∈N∗,P(X=k) = p(1 −p)k−1.
p X → G(p)
G(1/3)