用蒙特卡洛计算积分.doc

用蒙特卡洛计算积分1用蒙特卡洛方法估计积分 ，和的值，并将估计值与真值进行比较。1. 的 真值为1clearclctrue=1;for t=1:1:50n=1000; a=(pi)/2*rand(1,n);for i=1:1:n b(i)=(pi)/2*a(i)*sin(a(i);endc(t)=mean(b);endcmean(c-true).*(c-true)结论 当N为200时 计算的值分别为c = Columns 1 through 8 1.0777 0.8619 1.0677 1.0151 1.0311 1.0291 0.9675 0.9751 Columns 9 through 16 0.9502 0.9672 0.9271 0.9416 0.9899 1.0854 1.0175 0.9739 Columns 17 through 24 0.9612 1.0555 0.9526 1.0375 1.0233 1.0361 0.9694 0.9995 Columns 25 through 32 1.0538 0.9941 1.0114 1.0576 1.1029 0.9244 1.0831 0.9029 Columns 33 through 40 1.0329 0.8919 1.0362 0.9808 0.9724 1.0233 1.0386 0.9858 Columns 41 through 48 0.9704 0.9825 0.9741 1.1060 1.0521 0.8751 1.0642 0.9985 Columns 49 through 50 0.9915 0.9792ans = 0.0032当N为500时 计算的值分别为c = Columns 1 through 8 0.9614 0.9748 0.9687 0.9646 1.0441 0.9816 0.8864 1.0066 Columns 9 through 16 0.9673 0.9897 1.0502 0.9713 1.0277 1.0071 1.0252 1.0193 Columns 17 through 24 1.0236 0.9899 0.9883 0.9983 1.0300 0.9259 1.0385 0.9855 Columns 25 through 32 1.0057 1.0056 1.0918 1.0327 0.9601 1.0232 0.9806 1.0470 Columns 33 through 40 1.0505 0.9945 1.0287 1.0317 1.0248 1.0316 1.0248 1.0013 Columns 41 through 48 0.9863 0.9696 0.9993 1.0207 1.0333 1.0618 1.0705 1.0144 Columns 49 through 50 0.9454 1.0030ans = 0.0014当N为1000时 计算的值分别为c = Columns 1 through 8 1.0036 0.9988 0.9848 0.9598 0.9653 0.9595 1.0015 1.0034 Columns 9 through 16 1.0378 1.0291 1.0006 1.0548 1.0178 1.0227 1.0087 0.9922 Columns 17 through 24 1.0389 0.9956 0.9669 1.0135 1.0392 1.0228 1.0088 1.0390 Columns 25 through 32 0.9871 1.0236 1.0245 1.0038 1.0065 0.9701 0.9918 1.0066 Columns 33 through 40 0.9790 0.9985 1.0474 0.9783 0.9997 1.0450 0.9524 1.0147 Columns 41 through 48 1.0001 0.9929 1.0384 0.9813 0.9696 1.0088 1.0085 0.9837 Columns 49 through 50 0.9938 1.0066ans = 6.1578e-004再用莱姆大为1的指数分布估计clearclctrue=1;for t=1:1:50n=1000;b=exprnd(1,1,n);for i=1:1:n if(b(i)0&b(i)0&b(i)1) c(i)=exp(b(i)*b(i); else c(i)=0; endenda(t)=mean(c);endavar(a)a = Columns 1 through 8 1.4675 1.4531 1.4591 1.4754 1.4712 1.4765 1.4461 1.4575 Columns 9 through 16 1.4629 1.4569 1.4541 1.4663 1.4605 1.4498 1.4505 1.4620 Columns 17 through 24 1.4675 1.4727 1.4772 1.4662 1.4650 1.4500 1.4546 1.4426 Columns 25 through 32 1.4635 1.4834 1.4513 1.4525 1.4911 1.4488 1.4798 1.4437 Columns 33 through 40 1.4525 1.4724 1.4457 1.4622 1.4588 1.4498 1.4504 1.4606 Columns 41 through 48 1.4631 1.4772 1.4689 1.4612 1.4699 1.4574 1.4715 1.4678 Columns 49 through 50 1.4600 1.4531ans = 1.2042e-004ans =1.4629for t=1:1:50n=1000;r=sqrt(rand(1,n); seta=2.*pi.*rand(1,n); x=r.*cos(seta); y=r.*sin(seta); for i=1:1:n c(i)=pi/sqrt(1+x(i)*x(i)*x(i)*x(i)+y(i)*y(i)*y(i)*y(i);enda(t)=mean(c);endavar(a)a = Columns 1 through 8 2.8452 2.8304 2.8501 2.8400 2.8424 2.8435 2.8325 2.8475 Columns 9 through 16 2.8390 2.8417 2.8350 2.8444 2.8417 2.8495 2.8256 2.8448 Columns 17 through 24 2.8589 2.8415 2.8364 2.8588 2.8487 2.8518 2.8426 2.8409 Columns 25 through 32 2.8440 2.8295 2.8550 2.8470 2.8377 2.8476 2.8554 2.8342 Columns 33 through 40 2.8381 2.8545 2.8395 2.8437 2.8498 2.84