I am trying to implement a Multivariate Statistics test for the bivariate Gaussian distribution but I am honestly very confused. I am confused because I am not sure how to test for bivariate normality or in fact why should I test for bivariate normality. I basically have a matrix Nx2 and I would like to see how well a mixture of gaussians fits to this experimental data. I have performed clustering (K-means-10 clusters)and then applied EM to estimate the means and covariance matrices for each cluster. My problem is that I dont know (how to prove)how good my fit of 10 mixtures of gaussians is to my data. By drawing a 3d plot I have seen that it is good but i would like to know how good. Each cluster is described by a bivariate gaussian distribution and not a univariate gaussian.

I was thinking of creating a cdf plot of the experimental data but the problem is that in Matlab you can only do that for one vector (so in my case for 1 column of the Nx2 matrix). This would be fine if I assume that the correlation coefficient of the bivariate gaussian model is zero because the marginal distributions will be independent and normal. But this cannot always be the case. Then I thought of performing the chi squares test but I am not sure how it works for multivariate cases.

I am honestly confused and I would appreciate someones help.