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Suppose that X,Y and X,Z are bivariate normally distributed. We have

$E(X)=0, Var(X)=10$, $E(Y)=0, Var(Y)=6$ and $ρ_{xy}=0.87$

Moreover,

$E(X)=0, Var(X)=10$, $E(Z)=0, Var(Z)=4$ and $ρ_{xz}=0.87$

Will also Y and Z be bivariate normally distributed ? (I guess yes) If yes, which is their coefficient of correlation?

Posted also here: https://math.stackexchange.com/questions/3271570/correlation-coefficient-bivariate-normally-distributed

Added after comment of Whuber: Indicating as K the joint distribution of Y and Z,i know from the theory of the problem i'm dealing with that K is for sure a bivariate normally distributed. I expect this will pose some constraint on the value of $ρ_{yz}$

Andrea Mazzolari
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  • The correlation question is answered at https://stats.stackexchange.com/questions/72790, which shows there generally is a range of correlations consistent with this information. The bivariate normal question is answered (in the negative) at https://stats.stackexchange.com/questions/30159. – whuber Jun 23 '19 at 13:17
  • Comment after some thought on the problem i'm dealing with. Indicating as K the joint distribution of Y and Z, I know from the theory of the problem i'm dealing with that K is *for sure* a bivariate normally distributed. I expect this will pose some constraint on the value of $ρ_{yz}$ – Andrea Mazzolari Jun 23 '19 at 14:55
  • Also see [How to infer correlations from correlations](https://stats.stackexchange.com/q/122888/805) – Glen_b Jun 23 '19 at 16:08
  • @Glen_b thank you! that's exactly what i needed! Taking into account that K (my joint distribution) is for sure a bivariate normally distributed, is there a way to lower the interval of the inferred correlation ? – Andrea Mazzolari Jun 23 '19 at 16:52
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    The [second part of my answer](https://stats.stackexchange.com/a/72798/919) demonstrates that restricting the question about the possible range of correlations to bivariate Normal distributions doesn't change anything. The underlying idea is that for every possible covariance matrix, there is a (multivariate) Normal distribution whose covariance matrix is equal to it. The simplest proof proceeds from the expression of the characteristic function, which depends only on the mean and covariance. – whuber Jun 23 '19 at 18:34

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