Consider a mixture of two normal distributions: f(x) = p * N(x|u1, S1) + (1-p) N(x|u2, S2) N() is the normal pdf. p, S1, and S2 are known. The means are not.

You can get the MLE of u1 and u2 from the EM algorithm.

My question is, for a sample of size n, what are the variances of the EM estimates? My intuition tells me S1/(np) and S2/(n(1-p)) respectively, but I am not sure. Further, do the two estimates have a covariance? What would it be?