From the connotation of "Maximum likelihood estimator" I am inclined to think that the maximum likelihood estimator of the mean of a distribution should equal the mean of the sample values drawn from that distribution. What else could the "maximum likelihood estimate" of the mean be?
Also, by calculus, the least squares estimator of the mean is again equal to the mean of the sample values of a sample drawn from the distribution. So is the MLE of the mean of a distribution always equal to the least squares estimator of the mean? If it is not, can someone give a counter-example?