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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. (Also asked at: https://math.stackexchange.com/questions/1663910/joint-density-and-covariance-between-two-random-variables-with-the-same-mean-and)

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  • $\begingroup$ The covariance can take on any value in $[-\sigma^2,\sigma^2]$ even for the case of jointly normal random variables enjoying a bivariate normal density. The joint density can take on infinitely different shapes even for the case when the (marginal) density of the random variables is normal. For examples of marginally normal random variables that are not jointly normal, see this answer. So, yes, you are missing quite a bit. $\endgroup$ – Dilip Sarwate Feb 26 '16 at 14:12
  • $\begingroup$ @DilipSarwate Thanks for your comment. Are you saying that for the sub-question 1) above that even though two random variables have the same mean and variance, their co-variance can vary between the range you mention depending on the value of the correlation co-efficient between the two variable? $\endgroup$ – texmex Feb 27 '16 at 2:43
  • $\begingroup$ @DilipSarwate How about sub-question 2) above? Could you please add your thoughts on that? $\endgroup$ – texmex Feb 27 '16 at 2:45
  • $\begingroup$ The joint density is needed in order to determine the covariance or correlation coefficient. In general, the calculation of the covariance/correlation etc requires evaluating integrals. In special cases, the answer can be read off from the joint density without needing the evaluation of integrals. The bivariate normal density is one such joint density for which the covariance can be determined more easily than by grinding out integrations. $\endgroup$ – Dilip Sarwate Feb 27 '16 at 3:44
  • $\begingroup$ @DilipSarwate Thanks again for discussing this. Exactly my point and hence the sub question 2); we need the joint density to determine the covariance or correlation coefficient. But in the joint density, we use the correlation coefficient. That is, the joint density is defined using the correlation coefficient. (Seen easily from the bivariate normal and the math world link above; but perhaps true for other distributions as well). So which one comes first ... Is this not a cylical link? A chicken or egg came first problem? Could you please clarify this? $\endgroup$ – texmex Feb 27 '16 at 5:03

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