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I have to estimate the empirical mean value, covariance and correlation given a sample set $X_i$, $i=1, \dotsc , N$. I am trying to derive the mean value and I arrived at this point: $$ \sum_{i=1}^N {\boldsymbol \Sigma}^{-1} (x_i - \mu) = 0. $$ How can i continue?

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    $\begingroup$ I think this might qualify for a self-study tag since you seem to want some hints as to how to proceed, but not the final answer? Anyway, my suggestion would be to try distributing the multiplication with $\Sigma^{-1}$ over the $x_i$ and $\mu$, and then see if you can get it from there. $\endgroup$ – Ruben van Bergen Dec 14 '16 at 11:52
  • $\begingroup$ @RubenvanBergen Thanks. Yes, just some hints. Do I have to know any particular property of the covariance matrix in order to continue? Thanks. $\endgroup$ – wrong_path Dec 14 '16 at 11:58
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    $\begingroup$ Nope. But let me correct my previous comment as I just realized that wasn't really helpful as a first step. On second thought the first thing you really need to know at this point to proceed is that you can take the multiplication by $\Sigma^{-1}$ out of the sum over $i$. And then (without wanting to give away too much) you need to think about how to get rid of it ;). $\endgroup$ – Ruben van Bergen Dec 14 '16 at 13:38
  • $\begingroup$ @RubenvanBergen Thanks a lot, I was able to derive an estimate for both the mean value and the covariance. One last thing: the problem is now with the correlation: is it possible to estimate it with a similar process? Or it derives from the estimate for the covariance? Again, thank you. $\endgroup$ – wrong_path Dec 14 '16 at 17:43

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