Can the joint PDF of two random variables be computed from their marginal PDFs? 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
 A: 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.
A: 
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.
