# sampling from two multivariate gaussian distributions at the same time

The phenomena being modelized corresponds to the sampling from two multivariate gaussian distributions (with different mean vectors, but the same spherical variance matrix). So, given a point in this real n-dimensional space, what is the probability of this point belonging to the first distribution or the second one, taking into account that it may also belong to both of them. At what point would it be reasonable to consider the independency of the two gaussian variables? Is it reasonable to use some alternative based on some kind of Gaussian Mixture?

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Please clarify what you mean by spherical variance matrix. One interpretation of spherical would be that the covariance matrix is $\sigma^2I$, i.e. the individual Gaussian random variables are (conditionally) independent with common variance $\sigma^2$. In this case, the maximum-likelihood or Bayesian decision boundary is a hyperplane perpendicular to the straight line joining the mean points. Also, what is meant by a point possibly belonging to both distributions? Your observation can take on any value regardless of which distribution it comes from. How can it come from both? –  Dilip Sarwate Mar 4 '13 at 12:18
The spericity is the less important issue,yes, I meant that the covarince matrix is $σ^{2}I$ –  dynkin Mar 4 '13 at 12:46
The principal issue is: we have a point in a real n-dimensional space that belongs to the sample which is a mix of the samples of the two gaussian distribution. In other words: we have these two gaussian distrubution with different means/location factors and the same variance-covariance matrix. Then we withdraw samples from these two variables, then mix them at random. Now, if we take a point in this sample, it may come from the sample of the first variable, or from the smaple from teh second variable. Or it may also happen that this point (this vector) "occurs"/takes place in both samples –  dynkin Mar 4 '13 at 12:49
The complete question would be: given two gaussian multivariate distrubution with certain location (means vectors), what would be the probability of a point from the sample from these two distribution(configured in the way described before) to belong to a certain region of the real n-dimensional space( which is the image of both of the variables). My proposal would be to describe it as follows: $P(X)=P_{x_{1}} (X)+P_{x_{2}} (X)-P_{x_{1},x_{2}} (X)$ –  dynkin Mar 4 '13 at 12:57
I would like to find an alternative to a gaussian mixture ( which, i guess, might be justified if the sampling were made by mixing two samples with equal probability). But here the probability of a point in the major sample to proceed from a sample of one of these variables is not constant. –  dynkin Mar 4 '13 at 14:38