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This is probably a stupid question but I will have to ask it.

If you had a set of N correlated random variables and knew the correlation matrix, can one compute the joint probability distribution of all variables?

Does it make a difference if the correlation matrix was built using Pearson's rho or Kendall's Tau for example?

Thanks, Bogdan.

nyz
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Bogdan
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    Knowing the correlation matrix is not enough to determine the joint distribution. This is like saying that the mean of a single random variable is enough to determine its distribution. – Stefan Hansen Mar 07 '13 at 10:10
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    @StefanHansen: In your statement, you might want to change 'mean' to 'variance'. – Aang Mar 07 '13 at 10:15
  • @Avatar: Or 'mean and variance'. – Stefan Hansen Mar 07 '13 at 10:21
  • I had this impression too. But do you know of any method at least to estimate the joint? What would be the minimum amount of information required to approximate the joint out of the marginals? – Bogdan Mar 07 '13 at 10:28
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    Like @Stefan said, there is no way to estimate the joint distribution from the correlation matrix. Even the exact marginals + the exact correlation matrix are not enough to estimate the joint distribution. – Did Mar 07 '13 at 10:37
  • I wanted to ask besides the correlation matrix what would be required to estimate the joint? – Bogdan Mar 07 '13 at 10:41
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    If you know or are given that the random variables are jointly normal, then knowledge of the mean vector and covariance matrix gives the joint distribution explicitly. Otherwise, as others have already told you, there is nothing that can be said. I recommend this answer on stats.SE (to a different question) for some very interesting insights. – Dilip Sarwate Mar 07 '13 at 12:05

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