I wonder that what is the mu and sigma formula MLE(maximum likelihood estimates) for a 3 dimension guassian ? It is the same form as 1 and 2 dimension (+ 1 mu and sigma for the new vector) ?
As often happens with MLEs, it's just what you would expect.
The MLE for the matrix of covariances has the sample size $n$ in the denominator and the entries are the sample variances and sample covariances. This matrix has a Wishart distribution with $n-1$ degrees of freedom.
When I first saw the derivation of the MLE of the $3\times3$ or $p\times p$ variance, I was struck by the technique by which it turned out to be useful to view a scalar as the trace of a $1\times 1$ matrix. See this section: http://en.wikipedia.org/wiki/Estimation_of_covariance_matrices#Maximum-likelihood_estimation_for_the_multivariate_normal_distribution