Feuille 4 - Variables aléatoires : loi et espérance

n m
m
k∈ {0, . . . , n}m k
k
α, β ∈]0,1[ (i, j)∈N2pi,j =αβ(1−α)i(1−β)j
P({(i, j)}) = pij (i, j)∈N2
N2
(i, j)∈N2X((i, j)) = i Y ((i, j)) = j
X Y
P(X < Y )P(X=Y)P(X > Y )
XNn≥0pn=
P(X=n)pn>0λ > 0
X λ
n≥1pn
pn−1
=λ
n
XN
E[X] = X
n≥1
P(X≥n).
X, Y : (Ω,F,P)→N N
X
n≥0
nP(X=n) + X
n≥0
nP(Y=n) = X
n≥0
nP(X+Y=n).
(Ω,F,P)X: Ω →R
X(Ω) ⊂N

s∈[0,1] sX
00= 1
X
GX: [0,1] −→ R
s7−→ E[sX].
GX0
1
GXX
p∈[0,1]
n≥0p∈[0,1]
p∈]0,1[
λ > 0
X: (Ω,F,P)→Ra > 0
P(|X| ≥ a)≤E(|X|)
a
Ωf: Ω →R
(g, h) Ω R+f=g−h g ≤ |f|
h≤ |f|(˜g, ˜ h)
f= ˜g−˜ h g ≤˜g h ≤˜ h
p1−p
T1
T1
i≥1Tii
Tii≥1