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This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this.

Q1)

Are there any general results / relationships to get the Joint Density and Covariance between Two Random Variables with the same Mean and Variance?

Two Observations

1)

Is there a general result that says two random variables are the same if they have the same mean and variance? In which case the covariance between two random variables with the same mean and variance becomes the variance.

2)

Also, to get the joint density (ex:- bivariate normal) we need the correlation coefficient which is based on the covariance. And to get the covariance we need the joint density? Seems like a cyclical issue, which came first the chicken or the egg problem?

Any other suggestions / pointers / links to resources would be appreciated.

Steps Tried,

The example of the bivariate normal distribution at this link should make the above problems clear. Happy to elaborate if necessary.

http://mathworld.wolfram.com/BivariateNormalDistribution.html

Related Question Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables

Also, happy to delete the question, if this has been answered or if it is too basic.

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