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Difficult double integral

  1. Nov 15, 2013 #1
    Here is the beast
    [tex]\iint_{(ax+\mu _{1})^{2}+(bx+cy+\mu _{2})^{2}\leqslant z}\frac{1}{2\pi \sigma ^{2}}e^{-(\frac{1}{2\sigma ^{2}})(x^{2}+y^{2})}dxdy[/tex]

    The integral gives the C.D.F. of [itex](ax+\mu _{1})^{2}+(bx+cy+\mu _{2})^{2}\leqslant z[/itex] where [itex]x[/itex] and [itex]y[/itex] are identically distributed gaussian random variables with zero mean and unit variance.

    The integrand can be easily evaluated with polar coordinates over the less complex domain [itex]x^{2}+y^{2}\leqslant z[/itex] (In this case it becomes chi-square with two degrees of freedom). I am quite certain however that over the ellipse-shaped domain that I require, there is no analytical solution. On the other hand it would greatly speed up the numerical solution if I could just get rid of one of the integrals and/or solve in terms of approximate functions like ERF and Bessel of the first kind.

    Any ideas? Anyone seen anything similar to this before?
  2. jcsd
  3. Jan 7, 2014 #2


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    Gold Member

    Use coordinates matching the boundaries:
    x'=ax+m1 and y'=bx+cy+m2
    This will likely make your work easier.
  4. Feb 4, 2014 #3
    Thanks for the reply maajdl,

    I altered the coordinates in this way as you suggested before and I didn't feel any closer to a solution, the integrand became a massive long mess and I couldn't simplify it. If I find my notes I will post it up (took me a while to re-arrange it).

    I think for now I will admit defeat on this. It doesn't take so long to evaluate numerically for its purpose.

    Thanks again,
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