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Thread: How to applying the monte carlo method : importance sampling

  1. #1
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    Question How to applying the monte carlo method : importance sampling

    Hello.I have a little problem with applying a Monte Carlo method : Importance Sampling.I need to calculate :
    integral(0 to infinity) integral(0 to infinity) 1/(2 * pi * sqrt((1 + x^2 + y^2)^3))dxdy
    Can somebody help me ? Thanks in advance.
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  2. #2
    Grand Panjandrum
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    Re: How to applying the monte carlo method : importance sampling

    Quote Originally Posted by cristinelm View Post
    Hello.I have a little problem with applying a Monte Carlo method : Importance Sampling.I need to calculate :
    integral(0 to infinity) integral(0 to infinity) 1/(2 * pi * sqrt((1 + x^2 + y^2)^3))dxdy
    Can somebody help me ? Thanks in advance.
    More information of why you mention importance sampling. I would first switch to polars, then you are effectivly left with a one dimensional integral, then you need to find a density $\displaystyle p(x)$ such that:

    $\displaystyle \frac{r}{2\pi \sqrt{(1 + r^2)^3}\; p(r)}\approx \text{ constant}$

    The main consideration for choosing $\displaystyle p(x)$ is that it's cumulative distribution be easily inverted, so that you can easily generate pseudo-random numbers with the required distribution. Which probably means that $\displaystyle p(r)$ should be piecewise polynomial.

    CB
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    Re: How to applying the monte carlo method : importance sampling

    Quote Originally Posted by CaptainBlack View Post
    More information of why you mention importance sampling. I would first switch to polars, then you are effectivly left with a one dimensional integral, then you need to find a density $\displaystyle p(x)$ such that:

    $\displaystyle \frac{r}{2\pi \sqrt{(1 + r^2)^3}\; p(r)}\approx \text{ constant}$

    The main consideration for choosing $\displaystyle p(x)$ is that it's cumulative distribution be easily inverted, so that you can easily generate pseudo-random numbers with the required distribution. Which probably means that $\displaystyle p(r)$ should be piecewise polynomial.

    CB
    Thanks very much,it was very usefull for me your answer.I have anoather problem now,how can i make a goodness fit test for sample from bivariate cauchy distribution ? Is there any book that i could find some example about goodness fit test for multivariate/bivariate distribution ?
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