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I found that the polar coordinates of 2-dimensional Gaussian distribution with mean zero $$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({x^2+y^2})/{2\sigma^2}\big) \,\mathrm{d}x\,\mathrm{d}y$$ is $$\frac{1}{\sigma^2}\int_{0}^{\infty}\exp\big(-r^2/{2\sigma^2}\big) \,r\mathrm{d}r$$ What if we consider non-zero mean, that is what would exactly be the following equation in polar coordinate system? $$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({(x-\mu_x)^2+(y-\mu_y)^2})/{2\sigma^2}\big) \,\mathrm{d}x\,\mathrm{d}y$$

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up vote 3 down vote accepted

If you mean a polar coordinate system with respect to the origin, then the result is a complicated mess. However, expressed in a polar coordinate system with respect to the point $(\mu_x,\mu_y)$, your third expression is again equal to your second expression, where $r$ now stands for the distance from the point $(\mu_x,\mu_y)$.

P.S.: I suggest to take more care to use terms precisely. These expressions are neither distributions, nor coordinates, nor equations; they're normalization integrals over distributions expressed in certain coordinates.

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Thanks a lot for your reply. Actually I am looking for overlapping probability of two 2D Gaussians centred at $(\mu_x,\mu_y)$. For the amount of overlap I need to consider the centre point of both the Gaussian functions, since both normally distributed points are located at some distance from each other –  shaikh Jul 4 '11 at 11:36
    
Any suggestion, how can I find the overlapping probability of two 2D Gaussian functions??? –  shaikh Jul 4 '11 at 11:51
    
I don't know the term "overlapping probability" -- could you please define it? Also I'm not sure from what you wrote whether the two Gaussians are both centred at the same point $(\mu_x,\mu_y)$, or at different points. –  joriki Jul 4 '11 at 12:00
    
Assuming that there are two 2D Gaussians at $(\mu_{x1},\mu_{y1})$ and $(\mu_{x2},\mu_{y2})$ respectively. Since the Gaussian functions are infinite, I am interested in their overlapping area. –  shaikh Jul 4 '11 at 12:04
    
Sorry, I don't see what could be meant by their "overlapping area". Their support is the entire plane, so you can't be referring to the area of overlap of their support. Please define what you mean. –  joriki Jul 4 '11 at 12:12

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