Variables aléatoires discrètes Espérances et variances Variables aléatoires

(Xn)n∈N
E N
(Ω,T)Y
∀ω∈Ω, Y (ω) = XN(ω)(ω)
Y
T
∀n∈N,P(T > n)>0
T(θn)n∈N
θn= P(T=n|T≥n)
T
θn
∀n∈N, θn∈[0 ; 1[
(θn)n∈NP(T≥n)
Pθn
(θn)n∈N
∀n∈N, θn∈[0 ; 1[ Xθn
(θn)n∈N
T
X[a;b]
X m [a;b]
X σ2
Y=X−m
t=X
y≥0
yP (Y=y), s =X
y≥0
y2P(Y=y)u= P(Y≥0)
t2≤su
Y
t2≤(σ2−s)(1 −u)
t2≤σ2/4
σ2≤(b−a)2/4
X
n p
X(Ω) = {n, n + 1, . . .}P(X=k) = k−1
n−1pn(1 −p)k−n
X1, . . . , Xn
p
X1+··· +Xnn
p
n p

1/2
2n−1
n
N
X1
X2
X1, X2, . . .
Sn=X1+X2+··· +Xn
n
Sn
n Sn=N
T= min {n≥1|Sn=N}∪{+∞}
T
T
N= 1 N= 2
N N N ≥2
k
k≥2
T
P(T=k) P(T=k+ 1)
n≥1
P(T=n+k) = N−1
NkP(T > n)
T
(Xn)n≥1
p∈]0 ; 1[
n≥2
Xn=Xn−1= 1
T
T= min {n≥2|Xn=Xn−1= 1}∪{+∞}
P(T= 2) P(T= 3) n≥4 P(T=n)
P(T=n−1) P(T=n−2)
T
X
X
(Ω,T,P) (En)n∈N
+∞
X
n=0
P(En)<+∞
X1XX

Z=P+∞
n=0 1EnZ= +∞
Z
F=ω∈Ωω En
FP(F)=1
Z
Ai
i
i−1i
qi= P(Ai)T
P[
i≥2
Ai= 1
Ann≥2qn
T
T
qn= P(T=n)n≥2
q2= 1/4q3= 1/8
∀n≥4, qn=qn−1
2+qn−2 4
T
n≥2M∈ Mn(R)mi,j
µ λ ∈RχM(λ)
X Y
V(X)>0a, b ∈R
E(Y−(aX +b))2
S
mSσ2
S
B S σ2
B>0
X=S+B
a, b ∈RY=aX +b S
E(Y−S)2
X1, . . . , Xn
(X1, . . . , Xn)
Σ = (Cov(Xi, Xj))1≤i,j≤n
X=a1X1+··· +anXnai∈R
XΣ
Σ
(U, V )
B(2,1/2)
n
p∈]0 ; 1[
S= (U−1)2+ (V−1)2S
T= (U−1)(V−1) + 1 ES(T−1)
T(S, T )S T

X Y
p
Z=X+Y
X λ > 0
n∈N(X=n)
n λ
(X=n)
X p
E1
X
X λ > 0
X
X Y
p, q ∈]0 ; 1[
P(X < Y )
T
P(T≤n)n∈N∗
T
X Y
p q > 0
A=X1
0Y
T
n∈Ns, t ∈R0≤s≤t A(n, s, t)
n[s;t[
m, n ∈N0≤r≤s≤t A(m, r, s)
A(n, s, t)
A(n, s, t)n
t−s
pn(t) = P (A(n, 0, t))
p0p0(0) = 1
t∈R+
+∞
X
n=0
pn(t) = 1
1−p0(t)−p1(t)=
t→0+o (p1(t))
p0
∀s, t ∈R+, p0(s+t) = p0(s)p0(t)

p0λ≥0
∀t∈R+, p0(t)=e−λt
p1(t)=
t→0+λt + o(t)∀n≥2, pn(t)= t→0+o(t)
n∈N∗
∀s, t ≥0, pn(s+t) =
n
X
k=0
pk(s)pn−k(t)
pn
∀t≥0, p0
n(t) = λ(pn−1(t)−pn(t))
pn(t)qn(t) = eλtpn(t)
X
T > 0X
λ
X Y N
X λ > 0Y
X=n n p ∈]0 ; 1[
(X, Y )
Y
X Y N
X Y
P (X=j, Y =k) = a
j!k!a∈R
a
X Y
X Y
X Y N
X Y
∀(j, k)∈N2,P(X=j, Y =k) = aj+k
2j+ka∈R
a
X Y
X Y
P(X=Y)
X Y Np∈]0 ; 1[
X Y
P (X=k, Y =n) = (n
kanp(1 −p)nk≤n
0a∈R
a
Y
∀x∈]−1 ; 1[,
+∞
X
n=kn
kxn−k=1 (1 −x)k+1
X
X Y
p > 0
1−p
m
Tmm
T1
Tmm∈N∗