# Use a single Gaussian to represent a mixture of Gaussians

I want to merge a Gaussian mixture, $$\sum_{i=1}^{K} w_i \exp(x; \mu_{i}, \Sigma_{i})$$ into one single Gaussian, under the constraints that $$w_1 >> w_2 \geq \dots \geq w_K$$, i.e. we have a dominant Gaussian. Intuitively, the resulting Gaussian will be near the first Gaussian. I want to use a MLE estimate, seeking the optimal Gaussian $$\exp(x; \mu, \Sigma)$$. Obviously, formula for the expression of the mean is $$\mu = \sum_{i} w_i \mu_{i}$$ What about the covariance? Can any one help me with this problem?