Suppose we have $n$ normal random variables with variance $1$ and unknown mean.
Suppose we have $n$ samples of size 1 from those random variables.
Suppose also we know the correlations between the $n$ random variables.
Is it possible to estimate the mean of each of the random variables ?
Is there formula for this ?
Example would be say 2 variables with correlation 1 . Suppose samples are 2 and 3, then mean estimation is 2.5 for each variable. If correlation is 0 mean estimation for each is 2 and 3. These are two trivial cases.
First of all, two variables being perfectly correlated does not mean they come from the same distribution, just that they are linearly related.
To address your questions, you could use the Bayesian or maximum likelihood parameter estimates for a multivariate normal distribution (generalization of the bivariate).
In the MLE approach you set up the multivariate normal density using your known covariance matrix (based on your correlations and common standard deviation of 1), then you will have n realizations of a random vector of size n, each vector will have the same covariance matrix and unknown mean vector,$\overrightarrow{\mu}$, but with different values for the $x_i$. Now you need to maximize the likelihood of the product of n multivariate gaussian likelihoods wrt the mean vector $\overrightarrow{\mu}$