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[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?

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# Difficult double integral

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