problemset4

Problemset4

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Unformatted text preview: 360 0.1680 0.1064 0.0507 r=3 0.4544 0.3788 0.3120 0.2505 0.1936 0.1401 0.0902 0.0435 r=4 0.3334 0.2824 0.2360 0.1936 0.1531 0.1135 0.0746 0.0367 r=5 0.2309 0.1979 0.1680 0.1401 0.1135 0.0870 0.0591 0.0298 r=6 0.1429 0.1238 0.1064 0.0902 0.0746 0.0591 0.0426 0.0227 r=7 0.0667 0.0583 0.0507 0.0435 0.0367 0.0298 0.0227 0.0145 Table 4: Computed probabilities for the three-person ballot problem for p = 8. A C++ codes for simulating the Cauchy first passage problem This listing provides a simple C++ code for generating the distribution of first passage times for a Cauchy walk. The code accepts two command line argumerts: the drift parameter d, and a random seed. Once all walks have been simulated, the first passage probabilities for each number of walk step are printed to the standard output. Walks which did not achieve first passage in n steps are listed in the final line of the output. #include <string> #include <iostream> #include <cstdio> #include <cmath> using namespace std; const double p=3.1415926535897932384626433832795; const long n=10000; //Cutoff number of steps const long w=10000000; //Number of walkers double d=0; //Drift inline int cauchy() { static double x;x=1; for(int i=0;i<n;i++) { x+=d+tan(((double(rand())+0.5)/RAND MAX −0.5)∗p); if (x<0) return i; } return n; } int main(int argc,char ∗ argv) { srand(atoi(argv[1])); M. Z. Bazant – 18.366 Random Walks and Diffusion – Problem Set 4 Solutions q q q q q q q q q =0 =1 =2 =3 =4 =5 =6 =7 =8 r=0 1.0000 0.8000 0.6362 0.5000 0.3849 0.2860 0.2000 0.1250 0.0588 r=1 0.8000 0.6544 0.5303 0.4236 0.3301 0.2477 0.1750 0.1103 0.0523 r=2 0.6362 0.5303 0.4379 0.3550 0.2801 0.2129 0.1521 0.0968 0.0462 r=3 0.5000 0.4236 0.3550 0.2921 0.2343 0.1807 0.1307 0.0841 0.0406 r=4 0.3849 0.3301 0.2801 0.2343 0.1914 0.1501 0.1103 0.0719 0.0351 r=5 0.2860 0.2477 0.2129 0.1807 0.1501 0.1203 0.0902 0.0600 0.0298 r=6 0.2000 0.1750...
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This note was uploaded on 01/23/2014 for the course MATH 18.366 taught by Professor Martinbazant during the Fall '06 term at MIT.

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