Feuille de TD No1 : Rappels sur les probabilités discrètes

o1
n∈N∗Xn
B(n, pn) limn→+∞npn=λ
P(Xn=k)
limn→+∞P(Xn=k) = e−λλk
k!
NP(N=k) = e−λλk
k!
(Xn)n∈N∗λ
λ
j
P(” >”)+P(” 6”)+P(”=”)=1 P(” >”) = P(” 6”)
A B
P(A∩B)>P(A) + P(B)−1.
A1, . . . , An
P(A1∪A2∪ ··· ∪ An)6P(A1) + ··· +P(An).
A, B, C
P(A∪B∪C) = P(A) + P(B) + P(C)−P(A∩B)−P(A∩C)−P(B∩C) + P(A∩B∩C).

A1, . . . , An
P(A1∪A2∪ ··· ∪ An) = . . .
0.95
2n
n k k 6=n
n r
13
5+ 213 6+13 7=n r
r n
2n2n2n
n
n!∼√2πn(n/e)n
X λ
X
λ89 = 10
λ62 = 12
XNm, n ∈N
P(X≥m+n|X≥m) = P(X≥n).
P(X= 0) = a X
P3(N) 1 N
P3= lim
N→∞ P3(N) = 1/3.

A A(N)A
N A
P(A) = lim
N→∞
A(N)
N,
0
1
A
A
P(A)
b n k
B
b
1b
B=
b
X
i=1
1{ }.
B k
B=
k
X
j=1
1{ },
σ{1, . . . , n}σ:{1, . . . , n} →
{1, . . . , n}i∈ {1, . . . , n −1}σ(i)>
σ(i+ 1)
n−1
n n 2n/n n
2n/n
1−1
2+1 3−1 4+. . .