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Thread: Monte Carlo's Simulation

  1. #1
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    Monte Carlo's Simulation

    How do I use Monte Carlo's simulation to approximate part of the ellipsoid X^2/2+y^2/4+z^2/8=16, that lies in the first octant x>0,y>0,z>0?
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  2. #2
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    Quote Originally Posted by herbalist View Post
    How do I use Monte Carlo's simulation to approximate part of the ellipsoid X^2/2+y^2/4+z^2/8=16, that lies in the first octant x>0,y>0,z>0?
    Rejection sampling:

    Generate a point $\displaystyle (x,y,z)$ uniformly distributed over $\displaystyle [0,\sqrt{32}]\times[0,8]\times[0,\sqrt{128}]$ accept this point if:

    $\displaystyle \frac{x^2}{2}+\frac{y^2}{4}+\frac{z^2}{8}\le 16$

    If not accepted repeat until an acceped point is found

    The resultant $\displaystyle (x,y,z)$ is uniformly distributed over the required ellipsoid.

    The efficiency of this process is $\displaystyle \approx 0.507$ which means the volume of the ellipsoid in the first octant is $\displaystyle \approx 0.507 $ of the volume of $\displaystyle [0,\sqrt{32}]\times[0,8]\times[0,\sqrt{128}]$

    CB
    Last edited by CaptainBlack; May 19th 2010 at 01:35 AM.
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