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I'm working on a spatial project. I need to calculate the probability of a point being the closest to another. Say I'm given four points $y$, $x_1$,$x_2$ and $x_3$ in 2D plane, and let $Y'=y+Z$, where $Z$ is a bivariate normal with known mean and covariance. I want to know the probability that $x_1$ out of the three $x$'s is closest to $Y'$. How should I proceed? Thank you very much!

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    $\begingroup$ Are you familiar with Thiessen polygons (aka Voronoi tessellations, polygons of influence, and Dirichlet cells)? That construction reduces your question to an integration of the density over a (possibly infinite) polygonal region, which ordinarily is done numerically. $\endgroup$ – whuber Mar 17 at 20:51
  • $\begingroup$ Thanks. Yup I know of Thiessen polygons. That's an excellent suggestion!! But this operation needs to be repeated a lot of times, I'll have to look into the computation time. I am still trying to work out an analytic solution. $\endgroup$ – dynamic89 Mar 17 at 23:38
  • $\begingroup$ Perhaps there are some special features of your problem. Are there any constraints at all on the possibly configurations of the $x_i$ and $y$? Note that you can choose coordinates in which $y$ is the origin and $Z$ is bivariate standard Normal, thereby offering a little simplification. $\endgroup$ – whuber Mar 17 at 23:50

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