1
$\begingroup$

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?

$\endgroup$
  • 3
    $\begingroup$ No. Knowing the pairwise distributions between the variables is not sufficient to know the mutual distribution. $\endgroup$ – Graham Kemp May 22 at 3:19
  • $\begingroup$ Ok, thank you for that. Might you please provide an informal proof by contradiction with some example? $\endgroup$ – Tyler Collins May 22 at 3:29
3
$\begingroup$

Here is an example, obtained by tweaking a 2D counterexample: Let $(X,Y,Z)$ be such that $$ f_{X,Y,Z}(x,y,z)= \begin{cases} 2 \phi(x)\phi(y)\phi(z) & xyz>0\\ 0 & \text{otherwise} \end{cases} $$ where $\phi$ is the standard 1D Gaussian pdf. Then the bivariates $f_{X,Y}, f_{Y,Z}, f_{Z,X}$ are all standard 2D Gaussians, but of course $(X,Y,Z)$ is not Gaussian. Now both the standard 3D Gaussian and this $f$ give the same distribution of bivariates.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.