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If we generate points from the distribution f (x), then the Monte Carlo estimate of the integral is n g(xi ) 1 (17.38) I n i=1 f (xi ) It turns out (see Rubinstein (1981), Press et al. (1992) or Fishman (1995)) that the optimal choice of f (x) is g(x) f (x) = (17.39) I This states essentially that, if we know the value of the integral, we can then nd the best importance function, but the whole purpose of the exercise is to nd the value of that integral. Nevertheless, Equation (17.39) does indicate that a good choice of f (x) is one that closely approximates the function to be integrated in the signi cant region. The trick is to nd f (x) without expending more effort than is saved by the increased ef ciency of the simulation. Two examples demonstrate how this is done. Example 17.2 Importance Sampling for Integration of a Function Section 17.1.1 showed how Monte Carlo simulation could be used to estimate the integral of cos2 between 0 and . We have already observed that the effort could be cut in half by integrating between 0 and /2 and doubling the result. Figure 17.7 shows two possible choices for f (x) compared with cos2 . The solid line is cos2 , the dotted line is a linear

The function that actually contains the overflow bug is CVariableSet:: AddExtensionControlBlock, which is implemented in idq.dll. Listing 7.2 contains a partial listing (I have eliminated some irrelevant portions of it) of that function. Notice that we have the exact name of this function and of other internal, nonexported functions inside this module. idq.dll is considered part of the operating system and so symbols are available. The printed code was taken from a Windows Server 2000 system with no service packs, but there are quite a few versions of the operating system that contained the vulnerable code, including Service Packs 1, 2, and 3 for Windows 2000 Server.

cossq( x ) linear( x ) 1 cubic( x )

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idq!CVariableSet::AddExtensionControlBlock: 6e90065c mov eax,0x6e906af8 6e900661 call idq!_EH_prolog (6e905c30) 6e900666 sub esp,0x1d0 6e90066c push ebx 6e90066d xor eax,eax 6e90066f push esi 6e900670 push edi 6e900671 mov [ebp-0x24],ecx 6e900674 mov [ebp-0x2c],eax 6e900677 mov [ebp-0x28],eax 6e90067a mov [ebp-0x4],eax 6e90067d mov eax,[ebp+0x8] . . . 6e9006b7 mov esi,[eax+0x64] 6e9006ba or ecx,0xffffffff 6e9006bd mov edi,esi . . . 6e9007b7 push 0x3d 6e9007b9 push edi 6e9007ba mov [ebp-0x18],edi 6e9007bd call dword ptr [idq!_imp__strchr (6e8f111c)]

Listing 7.2 Disassembled listing of CVariableSet::AddExtensionControlBlock from idq.dll. (continued)

How do you refocus Well, this depends largely upon the individual If the issue is acute then there are lots of techniques to help you get your head back in the race Some of these are listed below However, if the problem is chronic, taking some time away from the sport may be required This is more often a problem for those who are full-time sailing, (who should have some other interests in their lives to keep the motivation high), but it can happen to young sailors who have done little sailing during school time and then nd themselves on the water every day over the summer Remember, both training and racing need to be fun If you are not enjoying yourself it is harder to push yourself to do well The sooner you can refocus the better.

0.8 x

7

6e9007c3 6e9007c5 6e9007c6 6e9007c8 6e9007c9 6e9007cf 6e9007d1 6e9007d3 6e9007d4 6e9007d7 6e9007d8 6e9007de 6e9007e0 6e9007e1 6e9007e3 6e9007e4 6e9007e6 6e9007e8 6e9007ea 6e9007eb 6e9007f0 6e9007f2 6e9007f4 6e9007f5 6e9007f8 6e9007fa 6e9007fd 6e9007ff 6e900801 6e900804 6e90080b 6e90080d 6e900812 6e900814 6e900817 6e90081a 6e90081f 6e900820 6e900823 6e900828 6e90082b 6e90082e 6e900834 6e900835 6e900838 6e900839 6e90083c 6e900841

Figure 17.7 Function (cos2 ) to be integrated (solid line), linear importance function (dotted line), and cubic importance function (dashed line).

mov pop test pop je sub push push mov inc call mov pop test pop jz cmp jnb inc jmp mov sub inc mov jmp mov sub add mov cmp jb mov xor mov lea push push mov call mov push lea push lea push push call xor

function, and the dashed line is a cubic function. The linear and cubic functions are scaled to enclose unit areas. The functional forms are linearf ( ) = cubicf ( ) = 4 4 1 2 2

esi,eax ecx esi,esi ecx 6e9008d2 eax,edi 0x26 edi [ebp-0x20],eax esi dword ptr [idq!_imp__strchr (6e8f111c)] edi,eax ecx edi,edi ecx 6e9007fa edi,esi 6e9007f0 edi 6e9008e4 eax,edi eax,esi edi [ebp-0x14],eax 6e900804 eax,[ebp-0x10] eax,esi eax,ebx [ebp-0x14],eax dword ptr [ebp-0x20],0x190 6e900828 eax,0x80040e14 ecx,ecx [ebp-0x3c],eax eax,[ebp-0x3c] 0x6e9071b8 eax [ebp-0x38],ecx idq!_CxxThrowException (6e905c36) eax,[ebp+0x8] dword ptr [eax+0x8] eax,[ebp-0x1dc] eax eax,[ebp-0x20] eax dword ptr [ebp-0x18] idq!DecodeURLEscapes (6e9060be) ecx,ecx

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