Variables aléatoires

X n p p ∈]0 ; 1[
b(k, n, p) = P(X=k)
m k b(k, n, p)
f:x7→ xm(1 −x)n−m[0 ; 1]
m∈[np ; (n+ 1)p]
bm, n, m
n+ 1≤b(m, n, p)≤bm, n, m n
m∈[(n+ 1)p−1 ; np]
n!∼√2πnnne−n
b(m, n, p)
X n p
Y=n−X
n
p
X
Y
Z=X+Y
Z
n p
X
Y
Z=X+Y
X Y
p q
Z= max(X, Y )
n
p q
X Y Ω
∀(x, y)∈X(Ω) ×Y(Ω),P(X=x, Y =y) = P(X=x)P(Y=y)
X Y
X Y
X+Y X −Y

X Y a1, . . . , an
P(X=ai) = P(Y=ai) = pi
X Y
P(X6=Y) =
n
X
i=1
pi(1 −pi)
X(Ω,P) f
X(Ω)
X f(X)
X
E (X)2≤EX2
X{0,1, . . . , N}
E(X) =
N−1
X
n=0
P(X > n)
Z
X
ϕX:R→C
ϕX(u) = E eiuX 
ϕX2πC∞
ϕX(0) ϕ0
X(0) ϕ00 X(0)
X
p
n p
X x0
P(X=x0) = 1
2πZ2π
0
ϕX(u)e−iux0du
ϕX=ϕY=⇒X=Y
X Y
ϕX+Y=ϕXϕY
p∈]0 ; 1[ (Xk)k∈N∗
P(Xk= 1) = pP(Xk=−1) = 1 −p
Xk
Yn=
n
Y
k=1
Xk
Ynpn= P(Yn= 1)
pnn→+∞
X Y J1 ; nK
E(X) P(X≥k)
X Y
min(X, Y ) max(X, Y )
|X−Y|

X n p ∈]0 ; 1[
Y=1
X+ 1
N
XN
XN
1 000
(Ω,T,P)
XN
k∈J1 ; NK
(2N−k−1)P(XN=k+ 1) = 2(N−k)P(XN=k)
XNE(XN)
X Y (Ω,P)
|(X, Y )| ≤ pV(X)V(Y)
n
X1, X2, X3
X1
X1X2X1+X2
X1X2
X g :R+→R+
∀a≥0,P (|X| ≥ a)≤E (g(|X|))
g(a)
X p n
ε > 0
P
X
n−p≥ε≤pp(1 −p)
ε√n
X n
p
λ, ε > 0
P (X−np > nε)≤E (exp(λ(X−np −nε))
X µ σ2
P (µ−ασ < X < µ +ασ)≥1−1
α2
X µ σ2
Y= [α(X−µ) + σ]2
α > 0
P (X≥µ+ασ)≤1
1 + α2

b(k, n, p) = n
kpk(1 −p)n−k
k∈J1 ; nK
b(k, n, p)
b(k−1, n, p)=n+ 1 −k k
p
1−p
b(k, n, p)
b(k−1, n, p)≥1⇐⇒ k≤(n+ 1)p
(b(k, n, p))0≤k≤n
m(n+ 1)p
m=b(n+ 1)pc
f
f0(x)=(m−nx)xm−1(1 −x)n−m−1
f[0 ; m/n] [m/n ; 1]
m∈[np ; (n+ 1)p]m/(n+ 1) ≤p≤m/n f
[0 ; m/n]
f(m/(n+ 1)) ≤f(p)≤f(m/n)
m∈[(n+ 1)p−1 ; np]m/n ≤p≤(m+ 1)/(n+ 1)
f[m/n ; 1]
bm, n, m+ 1
n+ 1 ≤b(m, n, p)≤bm, n, m n
n→+∞m=b(n+ 1)pc ∼ np →+∞
n−m∼n(1 −p)→+∞
n!∼√2πnnne−n, m!∼√2πmmme−m
(n−m)! ∼p2π(n−m)(n−m)n−men−m
bm, n, m
n∼rn 2πm(n−m)∼1
p2πnp(1 −p)
bm, n, m
n+ 1bm, n, m+ 1 n+ 1 
b(m, n, p)∼1
p2πnp(1 −p)
X(Ω) = Y(Ω) = J0 ; nKk∈J0 ; nK
P(X=k) = n
kpk(1 −p)n−k
P(Y=k) = P(X=n−k) = n
kpn−k(1 −p)k
Y n q = 1 −p
X
n p
P(X=k) = n
kpk(1 −p)n−k
k∈J0 ; nK(X=k)n−k
1/4
P(Y=j|X=k) = n−k
j1 4j3 4n−k−j
j∈J0 ; n−kK

ZJ0 ; nK
k∈J0 ; nK(Z=k)
(X=j, Y =k−j)j∈ {0,1, . . . , k}
P (Z=k) =
k
X
j=0
P(X=j, Y =k−j)
P(X=j, Y =k−j) = P(Y=k−j|X=j)P(X=j)
P(Z=k) =
k
X
j=0 n−j
k−j1 4k−j3 4n−kn jpj(1 −p)n−j
n−j
k−jn j=n! (k−j)!(n−k)!j!=k jn k
P(Z=k) = n
k(1 −p)n−k3 4n−kk
X
j=0 k
j1 4(1 −p)k−j
pj
P(Z=k) = n
k(1 −p)n−k3 4n−k1 4(1 −p) + pk
P(Z=k) = n
kqk(1 −q)n−kq=1 4+3 4p
E(Z) = (3p+ 1)n
4V(Z) = 3n(3p+ 1)(1 −p) 16
X
n p
P(X=k) = n
kpk(1 −p)n−kk∈J0 ; nK
Y
n−X
P(Y=`|X=k) = n−k
`p`(1 −p)n−k−``≤n−k
P(Z=m) =
m
X
k=0
P(X=k, Y =m−k)
P(Z=m) =
m
X
k=0
P(X=k)P(Y=m−k|X=k)
P(Z=m) =
m
X
k=0 n
kn−k m−kpm(1 −p)2n−k−m
n
kn−k m−k=n! k!(m−k)!(n−m)! =n mm k
P(Z=m) = n
mpm(1 −p)2n−m×
m
X
k=0 m
k 1 1−pk
P(Z=m) = n
mpm(1 −p)2n−m(2 −p)m (1 −p)m=n mqm(1 −q)n−m
q=p(2 −p)
Z