Principe des méthodes de Monte-Carlo

X M xkk=
0, ..M −1xkpk
M−1
X
k=0
pk= 1
E(X) =
M−1
X
k=0
pkxk
var(X) =E(X−E(X))2
=EX2−2XE(X) + E(X)2
=E(X2)−E(X)2
=
M−1
X
k=0
pkx2
k− M−1
X
k=0
pkxk!2
∆X=pvar(X)
xiX

xk
N
yN=moy(X, N) = 1
N
N−1
X
i=0
xi
xi
vN=var(X, N) = 1
N
N−1
X
k=0
x2
i−(moy(X, N))2
yMN
yNN
var(x1+x2) = var(x1) + var(x2)
var(ax) = a2var(x)
var(yN) = var(X)
N
∆(yN) = pvar(X)
√N
vN
yN−1,96√vN
√N, yN+1,96√vN √N0,95
E(X) = 1
6(1+2+3+4+5+6)= 7 2= 3,5

var(X) = 1
6(1 −7/2)2+ (2 −7/2)2+···'2,9167
N
1000
E(X)=3.48 ±0.03

X
[a, b]p(x)
[x1, x2]
P(x1≤x≤x2) = Zx2
x1
p(x)dx
Zb
a
p(x)dx = 1
F(x)
F(x) = Zx
a
p(x0)dx0
p(x) = dF
dx
x[a, b]
x x +δx
δP 'p(x)δx
p
x
a b
xx+δx
Probabilité δP
f(X)
E(f(X)) = Zb
a
p(x)f(x)dx

[a, b]
p(x) = 1
b−a
x
E(x) = Zb
a
x dx
b−a=a+b 2
[−∞,∞]
p(x) = 1
√2πσ exp −(x−µ)2
2σ2
E(x) = µ
var(x) = σ2
[a, b]
δx [a, b]
M δx
x p(x)δx
[a, b]
[0,1[