Calcul d`intégrales : méthode de Monte-Carlo

f[a, b]xi
[a, b]
h= ∆x=b−a
N
[xi, xi+1]
f
[xi, xi+1]
Zb
a
f(x)dx 'b−a
N
N−1
X
i=0
f(xi)
Zπ
0
sin(x)dx = 2

h1h
h
Zb
aZb a
f(x, y)dxdy 'b−a
N2
X
i,j
f(xi, yj)
Zπ
0Zπ 0
sin(x) cos(y)dxdy = 0
d
10d

[a, b]
N xi[a, b]
p(x) = 1
b−a
SN= (b−a)1
N
N−1
X
i=0
f(xi)
f(X) (b−a)
X
[a, b]p(x)
g g(X)
E(g(X)) = Zb
a
p(x)g(x)dx
f(x) = p(x)g(x)
E(f/p) = Zb
a
f(x)dx
f/p
N xi[a, b]p
SN=moy(f/p, N) = 1
N
N−1
X
i=0
f(xi)
p(xi)
σ2f(X)/p(X)
σ2=E(f/p)2−E(f/p)2
var(SN) = σ2
N
d1/√N

1/N
d
σ2
vN=var(f/p, N) = 1
N
N−1
X
i=0 f(xi)
p(xi)2
−S2
N
N
SN−1,96√vN
√N, SN+1,96√vN √N
N1/√N
σ2f/p
p(x)
f(x)>= 0
po(x) = f(x)
Rb
af(x)dx
σ2=var Zb
a
f(x)dx= 0
po(x)
f(x)
f
p(x)

π/2
x
p
ab
1/2
1/4
c1c2
p
1,2,3,4