# How to characterize the distribution of the intersection of 2 bivariate normals

I have two 2-dimensional Gaussian distributions:
$$D_1 := \mu_1=\pmatrix{x \cr y}, \quad \Sigma = \pmatrix{{\rm var}(x) &{\rm cov}(xy) \cr {\rm cov}(yx) &{\rm var}(y)} \\ D_2 := \mu_2=\pmatrix{x \cr y}, \quad \Sigma = \pmatrix{{\rm var}(x) &{\rm cov}(xy) \cr {\rm cov}(yx) &{\rm var}(y)}$$ and I want to characterize their 'intersection space' either analytically or with computationally generated samples. My ultimate goal is to be able to say something about the collisions of D_1 with D_2. Both are probabilistic representations of moving objects in 2-space.

One sampling idea was to sample each space (using a normal random number generator) and find the joint probability for each point. This seems to require me to pick a sensible threshold, which doesn't seem like the right direction.

I came across this CV thread, which yields a percentage. Can this be extended to 2 dimensions?

Any help would be appreciated by way of solutions, articles or avenues.

• Is your question to find the proportion of area in overlap (as with the link you give), or to find out something else? – Glen_b Feb 8 '14 at 23:44
• @Glen_b It looks like something like the Bhattacharyya coefficient gives the proportion of overlap with not too much difficulty in 1-d. I am wondering 1) how to compute this for bivariate gaussians and/or 2) some other characterization of the space other than a proportion. ie I would be willing to normalize the overlap space and I would like to know mean and variance of that new gaussian. – djh Feb 9 '14 at 1:10
• Now I am very confused. What new Gaussian? – Glen_b Feb 9 '14 at 2:25