Quantiles et Simulation : Cours Complet

X FXp
p X x X p
x est un p−quantile de X si P[X≤x] = p.
0,5
0,25 0,75
0,8 0,99
α
1−α X

I
I]0,1[
p0 1 x I
FX(x) = p P [X≤x] = p
x p X X
FX
1
XU[a, b]FXI=]a, b[ ]0,1[ p∈]0,1[
p X
(FX)−1(p) = a+ (b−a)p .
2
XE(λ)λ > 0FX:x7−→ 1−exp(λx)I=]0,+∞[
]0,1[ p∈]0,1[ p X
(FX)−1(p) = −ln(1 −p)
λ.
3
XN(µ, σ2)σ > 0FXI=] − ∞,+∞[ ]0,1[
p∈]0,1[ p X
(FX)−1(p) = µ+σΦ−1(p)
Φ−1Φ
Φ(x) = Zx
−∞
1
√2πexp(−u2
2) du .
[a, b]X
1 4 0,95
a b 0,025 0,975
0,025 1 2 a' −2,919922
0,975 1 2 b'4,919922
1
a+b
2= 1 .

X FX
p]0,1[ x P [X≤x] = p
X
X
U[a, b]U[c, d]a < b < c < d p 1−p x [b, c]
P[X≤x] = p
X FX
x P [X≤x] = p
p
X x1, x2, . . . , xn
p1, p2, . . . , pn
1FX
n+ 1 0, p1, p1+p2, p1+p2+p3,...,1
X
p0 1 p X X
x
(P[X < x ]≤p
P[X≤x]≥p.
x P [X < x ]≤p]− ∞, M ]
M M p X
x > M P [X≤x]≥p P [X≤M]≥p
FX
x P [X≤x]≥p[m, +∞[
m p X
x < m P [X < x ]≥p P [X < m ]≤p
x7−→ P[X < x ]
p X [m, M]
[Q−
X(p), Q+ X(p)] Q− X(p)p X
Q+
X(p)
Q−
X(p) = Inf{x∈;P[X≤x]≥p}
Q+
X(p) = Sup{x∈;P[X < x ]≤p}.

4
XU[a, b]U[c, d]a<b<c<d
p1−p X [b, c]Q−
X(0,5) = b Q+ X(0,5) = c
Q−
X(p) = Q+ X(p) = (a+ 2p(b−a) si 0 <p< 1
2
c+ (2p−1)(d−c) si 1
2<p<1.
5
X0 1 2
1
4
1
4
1
2
1
4X[0,1]
X[1,2]
Q−
X(p) = Q+ X(p) = 
0 si 0 <p< 1
4
1 si 1
4<p< 1 2
2 si 1
2<p<1
.
X]0,1[
p p X p
QX(p) [Q−
X(p), Q+ X(p)]
QX(p) = Q−
X(p)QX(p) = Q+ X(p)] QX(p) = 1
2Q−
X(p) + Q+ X(p)]
Q−
X(p)
QX
X X QX
]0,1[
QX(p) = Inf{x∈;P[X≤x)] ≥p}.

QX
QX]0,1[ 0
X1
QXFX
I]0,1[
FX
•x QXFX(x)≤x y x
FX(y) = FX(x)
•p0 1 FXQX(p)≥p p
FX
•QX◦FX◦QX=QX
•FX◦QX◦FX=FX
QXFX
(FX)−1
4
XU[a, b]U[c, d]a<b<c<d
p1−p
QX(p) = (a+ 2p(b−a) si 0 < p ≤1
2
c+ (2p−1)(d−c) si 1
2<p<1.
5
X0 1 2
1
4
1
4
1
2
QX(p) = 
0 si 0 < p ≤1
4
1 si 1
4< p ≤1 2
2 si 1
2<p<1
.