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As the question says, is it possible to compute the joint PDF of two random variables using their marginal PDFs?

For example, if we let $X$ and $Y$ be Gaussian random variables with known mean, standard deviation and correlation coeffient, we could write their joint PDF by using the bivariate Gaussian PDF, right?

Can we do something like this in general? Or is the example that I provided not always true?

Thank you for your time

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2 Answers 2

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No. Consider the two different joint distributions on $X$, $Y$, both with values in ${0,1}$: $$ P_1(0,0) = \tfrac12, P_1(0,1)=0, P_1(1, 0)=0, P_1(1, 1)=\tfrac 12$$ and $$ P_2(0,0) = P_2(0, 1) = P_2(1, 0) = P_2(1, 1)=\tfrac 14$$

The two different joint distributions have identical marginal distributions (namely, both $X$ and $Y$ are uniformly distributed on $\{0,1\}$).

In your Gaussian example, $X$ and $Y$ could either be independently distributed Gaussians, or they could be the same variable -- or anything in between.

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Ok. So in the example that I provided, since we know X and Y are Gaussian with a certain correlation coefficient, we can find their joint PDF, right? Thank you. Things are clearer now –  Damian Apr 24 '12 at 23:05
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Only if the correlation coefficient happens to be $\pm 1$. Otherwise there are always different possible joint distribution. For example, zero correlation can either arise when $X$ and $Y$ are independent, or when $X$ is Gaussian and $Y=X\cdot Z$ where $Z$ is independent of $X$ and uniformly distributed in $\{-1,1\}$. (Or in at least continuum many other ways). –  Henning Makholm Apr 24 '12 at 23:10

For example, if we let $X$ and $Y$ be Gaussian random variables with known mean, standard deviation and correlation coeffient, we could write their joint PDF by using the bivariate Gaussian PDF, right?

No, not necessarily. Look at this answer on stats.SE which illustrates that two standard Gaussian random variables with positive correlation need not have a bivariate Gaussian pdf. In fact, the joint pdf given there is zero in the second and fourth quadrants. Or they could have a bivariate joint Gaussian pdf, or something in between as Henning Makholm points out.

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