Probabilité discrète Fichier

ΩP(Ω)
P(Ω,P(Ω))
p: Ω →[0,1] ω→P({ω}.
X
ω∈Ω
p(ω) = X
ω∈Ω
P({ω}) = P([
ω∈Ω
({ω}) = P(Ω) = 1,
A⊂Ω
P(A) = P([
ω∈A
({ω}) = X
ω∈A
P({ω}) = X
ω∈A
p(ω).
pΩ [0,1] P
ω∈Ω
p(ω) = 1
P:P(Ω) →[0,1] ω→P(A) = P
ω∈A
p(ω)
P
P(P) = {ω∈Ωp(ω)>0}

Ων
∀A⊂Ων(A) = (A)A ν(A) = +∞A .
µ(Ω,P(Ω)) µ
ν h ∀ω∈Ωh(ω) = µ({ω})≤+∞
P P ν
µ=hν A Ω
µ(A) = µ(S
ω∈A
{ω}) = P
ω∈A
µ({ω}) = P
ω∈A
h(ω) = P
ω∈A
h(ω)ν({ω})
=P
ω∈A
h(ω)R1{ω}(x)dν(x) = P
ω∈AR1{ω}(x)h(ω)dν(x) = P ω∈AR1{ω}(x)h(x)dν(x)
=R[P
ω∈A
1{ω}]hdν =R1Ahdν =RAhdν =hν(A)
h0AΩµ(A) = RAh0dν
A={ω}µ({ω}) = R{ω}h0dν =h0(ω)ν({ω}) = h0(ω)
h0(ω) = µ({ω}) = h(ω)h=h0
(Ω,F, P )
P
Ω0Ωω∈Ω0{ω}∈F P(Ω0) = 1
PΩ0p: Ω0→[0,1] ω→P({ω})
P(P) = {ω∈Ω0:P({ω})>0}
P
p: (P)−→ [0,1] ω−→ P({ω}).
P=P
ω∈Ω0
p(ω)δω=P
ω∈(P)
p(ω)δω

ϕ(Ω,F, P ) [0,+∞]
ZϕdP =X
ω∈Ω0
p(ω)ϕ(ω).
ϕ ϕ (Ω,F, P )Rϕ
P
ω∈Ω0
p(ω)|ϕ(ω)|<+∞RϕdP =P
ω∈Ω0
p(ω)ϕ(ω)
Ω0Ω0={ωnn∈N}n→ωn
P(Ωc
0)=0
ZϕdP =ZΩ0
ϕdP +ZΩc
0
ϕdP =ZΩ0 ϕdP.
1Ω0=P
n∈N
1{ωn}1Ω0ϕ=P
n∈N
1{ωn}ϕ
ZϕdP =ZΩ0
ϕdP =Z1Ω0ϕdP =Z(X
n∈N
1{ωn}ϕ)dP =X
n∈NZ1{ωn}ϕdP.
ω∈Ω
[1{ωn}ϕ](ω) = 1{ωn}(ω)ϕ(ω) = 1{ωn}(ω)ϕ(ωn) = [1{ωn}ϕ(ωn)](ω),
1{ωn}ϕ= 1{ωn}ϕ(ωn)
Z1{ωn}ϕdP =Z1{ωn}ϕ(ωn)dP =ϕ(ωn)Z1{ωn}dP =ϕ(ωn)P({ωn}) = ϕ(ωn)p(ωn).
RϕdP =P
n∈NR1{ωn}ϕdP =P n∈N
ϕ(ωn)p(ωn) = P
ω∈Ω0
ϕ(ω)p(ω)
ϕR|ϕ|dP =P
ω∈Ω0
|ϕ(ω)|p(ω)<+∞
ZϕdP =ZΩ0
ϕdP +ZΩc
0
ϕdP =ZΩ0 ϕdP =Z1Ω0ϕdP.
n∈Nϕn=
n
P
i=0
ϕ(ωi)1{ωi}
(ϕn)n∈N
n|ϕn|≤|ϕ|
∀ω∈Ω lim
n→+∞ϕn(ω) = ϕ(ω)

Z1Ω0ϕdP = lim
n→+∞Z1Ω0ϕndP
= lim
n→+∞Z[
n
X
i=0
ϕ(ωi)1{ωi}]dP
= lim
n→+∞
n
X
i=0 Zϕ(ωi)1{ωi}dP
= lim
n→+∞
n
X
i=0
ϕ(ωi)Z1{ωi}dP
= lim
n→+∞
n
X
i=0
ϕ(ωi)P({ωi})
=X
i∈N
ϕ(ωi)p(ωi)
=X
ω∈Ω0
ϕ(ω)p(ω).
X(Ω,F, P )PX
DRPX(D) = 1 pXPX
D
ϕR[0,+∞]
Eϕ(X) = Zϕ(X)dP =ZϕdPX=X
d∈D
ϕ(d)pX(d).
X λ Pλ
Eϕ(X) = X
n∈N
ϕ(n)e−λλn
n!.
a∈RϕR R
a δaRϕd(δa) = ϕ(a)
X Y X Y
GpGq0< p, q < 1P(X≤Y)

Γ = {(x, y)∈R2x≤y}
P(X≤Y) = P((X, Y )∈Γ)
=Z1Γ(X, Y )dP
=Z1Γ(x, y)dP(X,Y )(x, y)
=Z Z 1Γ(x, y)dPX(x)dPY(y)
=X
n≥0X
k≥0
1Γ(n, k)p(1 −p)nq(1 −q)k
=pq X
n≥0X
k≥n
(1 −p)n(1 −q)k
=pq X
n≥0
(1 −p)nX
k≥n
(1 −q)k
=pq X
n≥0
(1 −p)n(1 −q)n
q
=pX
n≥0
[(1 −p)(1 −q)]n
=p
1−(1 −p)(1 −q)
=p
p+q−pq .
PR∀n∈NP({n}) = e−22n
n!ϕR
Rϕ(x) = 7x= exp{xln(7)}Eϕ
ϕ: (R,B1, P )→R, x →7x
Eϕ =ZϕdP =X
n≥0
ϕ(n)P({n}) = X
n≥0
e−22n
n!7n=e−2X
n≥0
1
n!2n7n
=e−2X
n≥0
1
n!(2 ×7)n=e−2X
n≥0
14n
n!=e−2e14 =e12 .