Probabilités discrètes

AjJ
([
j∈J
Aj)c=\
j∈J
Ac
j
I n {1, ..., n}
JN
R
Pun
lim
p→+∞X
n>p
un= 0
(un)φN N
+∞
X
i=1
ui=
+∞
X
i=1
uφ(i)
(un)P+∞
i=1 uiφ
N N
+∞
X
i=1
ui=
+∞
X
i=1
uφ(i)
Pn∈Nun
ΩP(Ω) Ω
A
Ω∈ A
A∈ A Ac∈ A
∀n∈N, An∈ A ⇒ Sn∈NAn∈ A
A
∅ ∈ A
A1, A2, ..., AnnASn
i=1 Ai∈ A
A A
AΩ (Ω,A)
A

(Ω,A)
pAR+
p(Ω) = 1
n∈NAnAAnp(Sn∈NAn) =
Pn∈Np(An)
(Ω,A, p)
(Ω,A, p)
p(∅) = 0
A∈ A p(Ac) = 1 −p(A)
A p(A)∈[0,1]
A⊂B p(A)≤p(B)
A, B ∈ A p(A∪B) = p(A) + p(B)−p(A∩B)
p(Ω,P(Ω)) Ω
ΩgΩR+
X
ω∈Ω
g(ω) = 1
p(Ω,P(Ω)) ω∈Ω
p(ω) = g(ω)
g p
(Ω,A, p)
Anp(Sn∈NAn) = lim p(An)
Anp(Tn∈NAn) = lim p(An)
(Ω,A) (E, E)XΩ
E A ∈ E X−1(A)∈ A
X−1(A) = {ω∈Ω/X(ω)∈A}
X∈A X−1(A)
XΩE X(Ω)
∀x∈E, x ∈ E X
∀x∈E, X−1(x)∈ A
(Ω,A) (E, E)XΩ
E
•X(Ω)

•x∈E X−1(x)∈ A
X(Ω,A, p)
(E, E)PX:A→p(X−1(A)) E[0,1]
E
U U1U2r r1
r2n
U k1U10≤k1≤n
Ω{A∈ P(U)||A|=n} |Ω|= ( r
n) Ω P(Ω)
p(Ak1)
Ak1={A∈Ω||A∩U1|=k1}
Ak1n−(r−r1)≤k1≤r1
p(Ak1) = (r1
k1)( r−r1 n−k1)
(r
n)
(n
p)p > n n < 0p < 0
X U1
p(X=k1) = (r1
k1)( r−r1 n−k1)
(r
n)
A B (Ω,A, p)
p(A∩B) = p(A)p(B)
(Ai)II
J⊂I
p(\
j∈J
Aj) = Y
j∈J
p(Aj)
X1(Ω,A, p) (E1,E1)X2
(Ω,A, p) (E2,E2)X1X2
A1∈ A1A2∈ A2X−1
1(A1)X−1 2(A2)
i∈I Xi(Ω,A, p) (Ei,Ei) (Xi)i∈I
(Ai)iinI
i∈I Ai∈ Ei(X−1
i(Ai))i∈I
(Xi)i∈{1,...,n}
(Xi) (x1, ..., xn)∈E1×.... ×E2
p(
n
\
i=1
Xi=xi) =
n
Y
i=1
p(Xi=xi)
(Ω,A, p) (Ai)i∈N
p
N ω inf{k∈N|ω∈Ak} {k∈N|ω∈Ak}
+∞
Sn=Pn
i=0 1Ai|{k∈N|ω∈Ak}|

(Ω,A, p)N∪ {∞}
N
p(N=k) = qkp
q= 1 −p
G(p)
Sn
p(Sn=k) = (n
k)pkqn−k
B(n, p)
p
k
k
p
(pn)N]0; 1[
lim npn=λ λ > 0SnB(n, pn)
lim
n→+∞(1 −λ
n+o(1 n))n−k=e−λ
lim
n→+∞p(Sn=k) = e−λλk
k!
P(λ) : k→e−λλk
k!N
λ
XPx∈Rxp(X=x)
X
Px∈Rxp(X=x)X
E(X)
L1
d(Ω,A, p)
X
E f E RY=f◦X
f(X)Y
X
x∈E
|f(x)|p(X=x)
E(X) = X
x∈E
f(x)p(X=x)
L1
d(Ω,A, p)E
L1
d(Ω,A, p)
•X X = 1AE(X) = p(A)
•SnB(n, p)E(Sn) = np

•NG(p)E(N) = q
p
•XP(λ)E(X) = λ
Lp
d(Ω,A, p)
X Xp∈ L1
d(Ω,A, p)
L2
d(Ω,A, p)⊂ L2 d(Ω,A, p)
E(|X|)≤(E(X2))1/2
X Y L2
d(Ω,A, p)XY ∈ L2 d(Ω,A, p)
E(|XY |)≤(E(X2))1/2(E(Y2))1/2
X∈ L2
d(Ω,A, p)X
var(X) = E((X−E(X))2)σXσ2
X=var(X)
σ2
X=X
x∈val(X)
(x−E(X))2p(X=x)
X Y
σ(X+Y)2=σ2
X+σ2 Y
σ2
1A=p(1 −p)
XB(n, p)σ2
X=np(1 −p)
XG(p)σ2
X=q
p2
XP(λ)σ2
X=λ
X∈ L1
d(Ω,A, p) > 0
p(|X|> )≤E(|X|)
X∈ L2
d(Ω,A, p) > 0
p(|X−E(X)|> )≤σ2
X
2
(Xn)
X  > 0
lim
np(|Xn−X|> ) = 0
(Xn)
(Ω,A, p)
lim
n→+∞
1
n
n
X
j=1
E(Xj) = m
lim
n→+∞
1
n2
n
X
j=1
σ2
Xj= 0
Xn=1
nPn
j=1 Xjm