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  1. if two processes X and Y are marginally univariate gaussian, and given the fact we know all the parameters for those processes, then how could I find the joint probability function, p(X,Y)?

  2. if those two are instead marginally univariate gaussian mixture model, and given the fact that we know all the parameters of those processes, then how could I find the joint probability function, p(X,Y)?

Kevvy Kim
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  • What do you know about the association between X and Y – gammer Jan 31 '17 at 02:58
  • their correlation coefficients. The reason I'm asking is to find a way to calculate the mutual information for those X and Y – Kevvy Kim Jan 31 '17 at 03:49
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    In the case of (1), are linear combinations of $X$ and $Y$ normally distributed? If so, the information given means that their joint pdf is multivariate normal. – gammer Jan 31 '17 at 04:27
  • Yes, any linear combinations of X and Y are also normally distributed. Would that mean I could use bivariate gaussian for X and Y to calculate p(X,Y) ? – Kevvy Kim Feb 04 '17 at 01:26
  • but what if X and Y are gaussian mixtures then? what could I do for (2) case? – Kevvy Kim Feb 04 '17 at 01:27
  • it gets more complicated in that case. You'd probably have to characterize the joint distribution within each mixture component and go from there. Maybe ask the person who answered your question below – gammer Feb 05 '17 at 00:36

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The answer to both your questions is NO, you cannot find the joint probability from the marginal probabilities.

Ignore the stochastic processes part of the question and just concentrate on two Gaussian random variables $X$ and $Y$ with known means and variances. There is no way to determine what the joint distribution of $X$ and $Y$ is unless you have some other information such as $X$ and $Y$ are independent random variables or that $X$ and $Y$ are jointly Gaussian random variables with known correlation coefficient. Note: knowing only that $X$ and $Y$ are correlated Gaussian random variables (as you say in a comment) is not enough to determine their joint distribution. The same idea -- that marginally Gaussian random variables can arise from different joint distributions -- carries over to your second question about Gaussian mixture distributions.

To know more about the possible joint distributions of Gaussian random variables, see this great answer on this forum.

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Dilip Sarwate
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  • What if we know that any linear combination of those gaussian processes is also a gaussian? Couldn't I model p(x,y) via bivariate gaussian? – Kevvy Kim Feb 11 '17 at 01:27
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    If $X$ and $Y$ are random variables such that for _every_ choice of $a, b \in \mathbb R$, $aX+bY$ is a Gaussian random variable, then yes, $(X,Y)$ has a bivariate Gaussian distribution. The very _definition_ of bivariate Gaussian (also called jointly Gaussian) random variables $X$ and $Y$ is that for _every_ choice of $a, b \in \mathbb R$, $aX+bY$ is a Gaussian random variable! So yes, if you really truly know what you claim to know, then you already know more than you stated in your question: that $X$ and $Y$ are marginally Gaussian random variables. You know that they are jointly Gaussian. – Dilip Sarwate Feb 11 '17 at 14:37
  • I just wanted to confirm that I could model it via bivariate gaussian. But, the real question is I have no clue about is what if X and Y are gaussian mixtures? Given the values of mixture weights, ...etc, is there any way I could model them jointly? – Kevvy Kim Feb 13 '17 at 06:54