Définition d`une probabilité

P (N, ℘(N))
P ({n})−→
n→+∞0
(an)n∈N
λ∈RPN, ℘(N)
P{n, n + 1, . . .}=λan
(Ω,T,P) A, B ∈ T
d(A, B) = P(A∆B)
A∆B A B
A∆B= (A∪B)\(A∩B)
d(A, C)≤d(A, B) + d(B, C)
|P(A)−P(B)| ≤ P(A∆B)

P(N)=1 N=Sn∈N{n}σ
+∞
X
n=0
P ({n})=1
P({k}| k≥n)−→
n→+∞0
PP(N)=1 λa0= 1
λ= 1/a0
P ({n}) = P ({n, n + 1, . . .})−P ({n+ 1, n + 2, . . .}) = an−an+1
a0
P
pn=an−an+1
a0
pn(an)n∈N
+∞
X
n=0
pn=1
a0
+∞
X
n=0
(an−an+1)=1
(an)n∈NPN
P({n}) = pn
P ({n, n + 1, . . .}) =
+∞
X
k=n
pk=an
a0
A∆C⊂(A∆B)∪(B∆C)
P(A∆C)≤P(A∆B) + P(B∆C)
P(A) = P(A∆∅)≤P(A∆B) + P(B∆∅) = P(A∆B) + P(B)
P(A)−P(B)≤P(A∆B)
P(B)−P(A)≤P(A∆B)