Can someone explain me what the relation is between bivariate normal distribution and dependence and independence of two variables? When i search for this topic i get answer in dependence and independence of two variables :(
If you know that $X$ and $Y$ are jointly a bivariate normal distribution then $X$ and $Y$ are independent if and only if they are uncorrelated.
However, the above statement is contingent on $X$ and $Y$ having a bivariate normal distribution. If all we know are the following facts:
The marginal distributions of $X$ and $Y$ are normally distributed.
The variables are uncorrelated.
Then it is possible to construct examples where the variables are not independent of each other. (Also see https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent for an example.)