# Correlation between two intra correlated variables

I am looking into computing a correlation between two variables, x and y, each having a general normal distribution and not i.i.d. normal distribution. In particular these two variables have a particular variance-covariance structure, i.e., Var(x)=A and Var(y)=B. Is it still OK to compute correlation between x and y as shown bellow or should I use any other approach?

Cor(x, y) = Cov(x, y) / sqrt(Var(x)*Vary)

In detail I have a set of simulated/true values stored in the variable x that has a particular structure due to the nature of values - there is some underlying data generation process giving rise to variance-covariance structure Var(x)=A. I then try to estimate these values based on some data which gives me variable y. Due to the nature of x, there is also an underlying structure in y that can be described by Var(y)=B. A and B are not the same due to conditioning on the observed data. I want to evaluate Cor(x, y) to validate the quality of estimation and could use a simple standard approach, but I wonder if I am making a mistake here by treating x and y as iid observations.

Indeed, if they were independent, the correlation between the two popopulations would be $0$ and you would expect the calculated sample correlation to be close to $0$. So getting a result significantly different from $0$ gives you some potentially useful information.  