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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- Dec 18th 2011, 09:09 AMcristinelmHow 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. - Dec 18th 2011, 12:47 PMCaptainBlackRe: How to applying the monte carlo method : importance sampling
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 - Dec 21st 2011, 12:33 AMcristinelmRe: How to applying the monte carlo method : importance sampling
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 ?