# probability involves bivariate gaussian

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!

• 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. – whuber Mar 17 at 20:51
• 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. – dynamic89 Mar 17 at 23:38
• 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. – whuber Mar 17 at 23:50