Introduction aux probabilités et à la simulation aléatoire. Borel

0< p < 1An
pP(lim supnAn) = 0 p=1
2
m!∼√2πmm+1
2e−m
E F X Y
E F (X, Y )
E×F
X Y (Ω,A, P )
X Y E(XY ) = E(X)E(Y)
X, Y
3
n
X∈N
E(X) = X
n≥1
P(X≥n)

n n + 1
X1, X2
θ1>0θ2>0
X1+X2
X1X1+X2
E(X1|X1+X2)
N
N λ > 0
Gii N, G1, G2, . . .
p
1−p X
X1, . . . , Xn
p∈]0,1[ S=X1+. . . +Xn
s∈ {0, . . . , n}X1S=sE(X1|S)
k
Tn
n
(Xn)n∈N
p Sn=X1+··· +Xnm∈N∗ν= inf{n≥1, Sn=m}
ν
X{1, . . . , k}P(X=j) = pj,1≤j≤k. Zj=1(X=j)
s= (s1, . . . , sk)∈Rk
+
EhsZ1
1sZ2 2···sZk
ki.
X1, . . . , Xnn n
Nj=Pn
i=1 1(Xi=j).
EhsN1
1sN2 2···sNk
kia1, . . . , akn
P(N1=a1, . . . , Nk=ak) = n!
a1!. . . ak!pa1
1. . . pak
k.
(p1, . . . , pk)n