Put trivariate PDF in terms of bivariate PDFs Let there be the random variables $X$, $Y$, and $Z$. Let all the bivariate PDFs $f_{X, Y}$, $f_{X, Z}$, and $f_{Y, Z}$ be known.
Can we write the unknown trivariate PDF $f_{X, Y, Z}$ in terms of the known bivariate PDFs?

Let $f(x,y,z)=1$ when $x,y,z\in [0,1]$, and be zero otherwise. Then $f_{X,Y}$ is uniform on the unit square, and same for $f_{X,Z}$ and $f_{Y,Z}$. On the other hand, let
$$
g(x,y,z)=1+\sin(2\pi (x+y+z)) \hspace{1cm}\text{for }x,y,z\in [0,1]
$$
Then $g_{X,Y},g_{X,Z}$ and $g_{Y,Z}$ are also uniform on the unit square. 
