Suppose I have $n$ observations, which are all normally distributed with the same mean (which is unknown) but each has a different variance (the different variances could be called $v_1,...,v_n$ for example, which are all assumed to be known).
What is the best estimate of the mean? And further, how does one compute the variance of the mean?
Clearly, the sample average would be the best estimate if iid, but intuitively, an observation which has a relatively low variance would provide more information about the mean than the others, suggesting that the sample mean would not be the best estimate. My problem is that I don't know how to 'quantify' this intuition to generate an estimate of the mean and it's variance. Any ideas on how to do this? Thanks.