Variables aléatoires : loi et espérance.

n m
m
k∈ {0, . . . , n}m k
k
Cm
nm
m k
k m Ck
mm−k n −m
Cm−k
n−mm−k≤n−m k ≥2m−n
Ck
mCm−k n−m=m!(n−m)!
k!(m−k)!2(n−2m+k)!.
P(n, m, k) = Ck
mCm−k
n−m
Cm
n
=m!2(n−m)!2
n!k!(m−k)!2(n−2m+k)!.
kmax(0,2m−n)≤k < m
k
P(n, m, k + 1)
P(n, m, k)=(m−k)2 (k+ 1)(n−2m+k+ 1) =(m−k)2 (n/2+1−(m−k))2−(n/2−m)2.
k k =m−1 1
1P(n, m, k)k
max(0,2m−n)≤k0≤m−2k≥k0
k0= max(0,2m−n)k0
P(n, m, k + 1)
P(n, m, k)>1⇔(m−k)2<(n/2+1−(m−k))2−(n/2−m)2.
P(n, m, k + 1) > P (n, m, k)⇔(n+ 2)k < m2+ 2m−n−1.

P(n, m, k)
k=k0:= 1 + m2+ 2m−n−1
n+ 2 =(m+ 1)2 n+ 2 .
(m+1)2
n+2 k0−1
k0k
P(n, m, k)k/m
m/n
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 )
∀(i, j)∈N2pi,j ≥0P(i,j)∈N2pi,j = 1
pi,j Pi∈Nxi= 1/(1 −x)x= 1 −α
x= 1 −β
X
(i,j)∈N2
pi,j =X
i∈NX j∈N
αβ(1 −α)i(1 −β)j=X
i∈N
α(1 −α)i= 1.
k∈NPj∈Nβ(1 −β)j= 1
P(X=k) = X
(i,j)∈N2
pi,j {k}(i) = X
j∈N
pk,j =α(1 −α)k.
X α P(Y=k) = β(1 −β)k
Y β
P(X < Y ) = X
(i,j)∈N2
i<j
pi,j =
∞
X
i=0
∞
X
j=i+1
pi,j .
i∈NP∞
j=i+1 β(1 −β)j= (1 −β)i+1
P(X < Y ) =
∞
X
i=0
α(1−α)i(1−β)i+1 =
∞
X
i=0
α(1−β) ((1 −α)(1 −β))i=α(1 −β)
1−(1 −α)(1 −β)=α−αβ α+β−αβ .

X Y α β
P(X > Y ) = β−αβ
α+β−αβ .
P(X=Y)pi,i
{X=Y} {X < Y } {X > Y }N2
P(X=Y)=1−P(X > Y )−P(X < Y ) = α+β−αβ −(β−αβ)−(α−αβ)
α+β−αβ =αβ α+β−αβ .
XNn≥0pn=
P(X=n)pn>0λ > 0
X λ
n≥1pn
pn−1
=λ
n
∀n∈Npn= exp(−λ)λn
n!
n
pn=p0
λn
n!.
p0Pn∈Npn= 1
p0Pn∈N
λn
n!= 1 p0exp(λ) = 1
XN
E[X] = X
n≥1
P(X≥n).
(am,n)m,n≥1
X
n≥1X m≥1
am,n =X
m≥1X n≥1
am,n.

X
n≥1
P(X≥n) = X
n≥1X m≥n
P(X=m)
=X
n≥1X m≥1
{m≥n}P(X=m)
=X
m≥1X n≥1
{m≥n}P(X=m)
=X
m≥1
mP(X=m)
=E[X],
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).
E(X)+E(Y)
E(X+Y)
X
n≥0
nP(X+Y=n) = X
n≥0
n
X
k=0
nP(X=k Y =n−k)
=X
n≥0X
k≥0
{k≤n}nP(X=k Y =n−k)
=X
k≥0X
n≥0
{k≤n}nP(X=k Y =n−k)
=X
k≥0X n≥k
nP(X=k Y =n−k)
=X
k≥0X l≥0
(k+l)P(X=k Y =l)
=X
k≥0
kX
l≥0
P(X=k Y =l) + X
l≥0
lX
k≥0
P(X=k Y =l).
l=n−k
k≥0l

{X=k}
l≥0k
{Y=l}
X
n≥0
nP(X+Y=n) = X
k≥0
kP(X=k) + X
l≥0
lP(Y=l),
(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
s∈(0,1] X(Ω) ⊂N
X(Ω) ⊂R+x∈R7→ sx∈R
X sX
1
s= 0 sX={X=0}
s t 0≤s≤t≤1ω∈Ω 0 ≤
sX(ω)≤tX(ω)≤1
0≤GX(s)≤GX(t)≤1GXGX(0) = P(X= 0)
GX(1) = 1
X