[PDF] [PDF] Méthodes de Monte-Carlo (Cours et exercices) M1 IM 2018-2019

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X ?????IX1+X2++XNN I=Z [0;1] df(u1;:::;ud)du1:::dud;

I=E(f(U1;:::;Ud)) =E(X):

1;:::;U(1)

d;U(2)

1;:::;U(2)

d;::: (U(1)

1;:::;U(1)

d)?X2= (U(2)

1;:::;U(2)

I=Z R ng(x)f(x)dx R nf(x)dx= 1?? ????? ???? ??????? ??????? ? ??????? ????Rd?d2N?? ?? ??????? ???E(jX1j)<+1? ????? X

1++Xnn

????!n!+1E(X): ?????pn

X1++Xnn

E(X1) ???!n!+1N(0;1):

E(f(Zn))!n!+1E(f(Z)) =Z

R f(x)(x)dx: P apn

X1++Xnn

E(X1) b =E [a;b] pn

X1++Xnn

E(X1) n!+1Z b a

1ex2=2p2dx:

X1++Xnn

E(X)??? ???????pn

X1++Xnn

E(X)??? ? ??? ???? ?? ???N(0;2=n)?? ???? ??

E(X1)? ?? ???????

P(jnj 0;01)0;05;

P(jnj 0;01)0;95:

P(jnj 0;01) =P

0;01X1++Xnn

E(X1)0;01

=P

0;01pn

pn

X1++Xnn

E(X1)

0;01pn

?????n?????? ???????Z +0;01pn

0;01pn

e t2=2p2dt +0;01pn 1e t2=2p2dt1: ?? ???? ????n??? ???R+0;01pn

1et2=2p2dt0;975? ?? ??????? ???? ??? ????? ?? ?? ??? ???????

n(1;96)20;012:

X1++Xnn

2=V(X)? ?????

(X1X2)2+ (X3X4)2++ (X2n1X2n)22n????!n!+12: E(X21+X222X1X2) = 2E(X21)2E(X1)2<+1????E(X21)>E(X1)2??E(X21)<+1?? ???? ??? ???X? p n=X1++Xnn p: n(1;96)20;012: ???? ????p2[0;1]?p(1p)1=4? ???? ?? ???? ?? ??????? n1;9620;01214

E(eZ) =Z

+1 1 exex2=2p2dx Z +1

11p2exp

12 (x)2+22 dx =e2=2: ?? ???? ??????Z1;Z2;:::?????? ?? ???? ??? ???Z? ???? ?????? ?? ???? ??n??????? ? e

Z1++eZnn

E(eZ) +pn

Y

1E(eZ)

eZ1++eZnn

E(eZ)E(eZ)pn

E(jYj) =E(eZ)pn

r2

E(eZ)=pe

??1? ?? ???? ???? ???????n??? ???pnpe 21q2
1;96; [0;1] df(x)dx= limn!+11n dX

1i1;:::;idnfi1n

;:::;idn [0;1] df(x)dx1n dX

1i1;:::;idnfi1n

;:::;idn Cn ????x;y2[0;1]d?jf(x)f(y)j C0kxyk? ???? ????? ???? Z [0;1] df(x)dx1n dX

1i1;:::;idnfi1n

;:::;idn X

1i1;:::;idnZ

u

12[i11n

;i1n ]:::Z u d2hid1n ;idn if(x)fi1n ;:::;idn du

1:::dud

X

1i1;:::;idnZ

u

12[i11n

;i1n ]:::Z u d2hid1n ;idn if(x)fi1n ;:::;idn du1:::dud X

1i1;:::;idnZ

u

12[i11n

;i1n ]:::Z u d2hid1n ;idn iC0pd n du1:::dud=C0pd n P n I 1=Z 1 0 cos(x3)exp(x)dx; I 2=Z +1 0 sin(x4)exp(2x)expx22 dx; I 3=Z 1 0 ln(1 +x2)exp(x2)dx: ?????? ????D?? ?????? ?????R2? ?? ??????(0;0)?? ?? ?????1? I=Z +1 0pxe xdx:

0sin(px)dx?

n 1nX i=11 (U1i)2+(U2i)2+(U3i)2<1: S n=nX k=1X k:

ES4nCn2:

????? ???? ???? ????n1?

PjSnj n5=6cn4=3:

P n(x) =nX k=0

Cknxk(1x)nkfkn

8" >0;PSn(x)n

x"14n"2: lim n!+1sup x2[0;1] f(x) Pn(x)= 0: y

2x)2= (x2+y2)?

= 1 + cos()? ?? ??????? ??? ?? ?????? ????? ????? ???8=3? P apn Snn 1b; ????a??b???? ?????? x7!1[0;+1)(x)xn1(n)exp(x); n!n!+1p2nne n: ????0t1? F

1(t) = inffx; F(x)tg:

?????? ??U U([0;1])????? ???? ? ?U???? ?? ???U([0;1])??? ?????F1(U)? ???? ??? ???X? uF(t)? [uF(t)],[F1(u)t]: ???? ????? ?????? ???? ????t2R?

P(F1(U)t) =P(UF(t)) =F(t):

F(x) =8

>>:x3 ??x2[0;1[; x3 +13 ??x2[1;2];

1??x1;

0??x0;

F

1(t) =8

:3t??t20;13

1??t213

;23

3t1??t223

;1:

F:t2R7!nX

i=1p i?txi; F

1:u2[0;1]7!nX

j=1x j?p1++pj1uP(X > t) = exp(t):

1et?????t0? ??

F

1(x) =log(1x)

???1U? ???? ??? ???U?? ???? log(U) E(): (p2log(U)cos(2V);p2log(U)sin(2V)) 0 1 ????f2 C+ b(R2)?C+

E(f(X;Y)) =Z

R

2f(x;y)e(x2+y2)=22dxdy :

J

1(x;y) =

@x@r @y@r @x@ @y@ =cos() sin() rsin()rcos()quotesdbs_dbs13.pdfusesText_19

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