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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 :(

Khan Saab
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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:

  1. The marginal distributions of $X$ and $Y$ are normally distributed.

  2. 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.)

Community
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  • +1 This was very well done and exactly what I would have said. – Michael R. Chernick Dec 15 '16 at 14:52
  • @whuber The link to cardinal's beautiful answer that you inserted via your edit might not be completely relevant here since, to the best of my belief, _none_ of the 6 examples in cardinal's answer has _uncorrelated but independent_ marginally normal random variables; they are all instances of marginally normal but not jointly normal random variables – Dilip Sarwate Dec 15 '16 at 16:58