I am doing a discriminant analysis problem, and I am running into the following subproblem. I have a sample $ (x_{1}, g_{1}), \ldots, (x_{m},g_{m})$ where $ g_{i} \in \{1,2\} $, taking the value $ 1 $ with probability $ p $, determines if the observation $ x_{i} $ comes from the group $ \mathcal{G}_{1} $ or $ \mathcal{G}_{2} $. I also know that $ x_{i} | g_{i}=g $ is distributed as a Normal $ p-$ varied with mean $ \mu_{g} $ and matrix of variances $ \Sigma $ (the same for all observations) . Here $ \mu_{g} $ is unknown for $ g = 1,2 $.
The first $ n $ observations $ x_{1}, x_{2}, \ldots, x_{n} $ have mean $ \mu_ {1} $ and the remaining ones up to the observation $ x_{m} $ have mean $ \mu_{2} $. I always have the same variance matrix $ \Sigma $. I need to estimate $ \Sigma $, in order to continue. Any help is welcome.
I know the usual forms but I need the identically distributed variable hypothesis, which is not the case in this problem.