Can the Difficult Double Integral Be Simplified with Approximate Functions?

[Jeff.Nevington]

Here is the beast
$$\iint_{(ax+\mu _{1})^{2}+(bx+cy+\mu _{2})^{2}\leqslant z}\frac{1}{2\pi \sigma ^{2}}e^{-(\frac{1}{2\sigma ^{2}})(x^{2}+y^{2})}dxdy$$

The integral gives the C.D.F. of $(ax+\mu _{1})^{2}+(bx+cy+\mu _{2})^{2}\leqslant z$ where $x$ and $y$ 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 $x^{2}+y^{2}\leqslant z$ (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?

 

[maajdl]

Use coordinates matching the boundaries:
x'=ax+m1 and y'=bx+cy+m2
This will likely make your work easier.

 

[Jeff.Nevington]

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,
Jeff