There's a stochastic variable $X$ which I assume is normally distributed, and I'd like to know the parameters of that normal distribution. (Mean $\mu_X$ and variance $\sigma_X^2$.)
I can't actually measure this variable directly, but I have several ways to estimate it, with some error. Each estimate's error is also normally distributed, with a known standard deviation. Furthermore, these estimates are actually correlated.
So I have my different estimates, say $\mu_1 ... \mu_N$, with covariance matrix $\Sigma$, where the diagonals are the variances of the estimates' errors, $\sigma_1^2 ... \sigma_N^2$.
How do I now get an estimate for the mean $\mu_X$ and variance $\sigma_X^2$ of $X$?
(Should you want an example use case: I want to know (the mean and standard deviation of the distribution of) the temperature in a room. I have ten different thermometers, and they all give different results. Some thermometers are similar types to each other (and hence their results will be correlated), and each thermometer comes with a (nominal, but assumed true) uncertainty.)
I considered taking the weighted average of these estimates, $\hat{\mu}_X=\sum_{i=1}^{N}w_i\mu_i$, with weights inversely proportional to the standard deviation, $w_i = \frac{1}{\sigma_i}/\sum_{k=1}^{N}\frac{1}{\sigma_k}$ but that has to be incorrect, because if I have two estimates which are perfectly correlated, then the second adds no more information that the first hadn't, and so the weight of these two estimate shouldn't be more than the weight of one of the two.
The standard deviation, I'm even less certain how to calculate.