# Joint Density function from marginal density functions

This there anyway to find the joint density function of random variables X and Y. Nothing is given about they being independent. So we have to solve by assuming that they are not independent.The following are the marginal density functions.

$f_X(x) = \left\{ \begin{array}{l l} 2x/(1+x)^3 & \quad \text{if$x \geq 0$}\\ 0 & \quad \text{if$x<0$} \end{array} \right.$

$f_Y(x) = \left\{ \begin{array}{l l} 1/(1+y)^2 & \quad \text{if$y \geq 0$}\\ 0 & \quad \text{if$y<0$} \end{array} \right.$

I want to find the joint density function corresponding the above marginal density functions. Thanks in advances :)

• "I want to find the joint density function corresponding [to] the above marginal density functions" This is absurd, there are tons of joint density functions corresponding to these marginal density functions. There are even (X,Y) with these marginal density functions that have no joint density. – Did Oct 2 '14 at 21:46

Here is an instructive example, if only one takes pain to fully analyze it: consider some i.i.d. random variables $U$ and $V$ uniform on $(0,1)$, and define $$X=\max\left\{\frac{U}{1-U},\frac{V}{1-V}\right\},\qquad Y=\frac{1-U}U.$$ Then $X$ has the density $f_X$ in the post, $Y$ has the density $f_Y$ in the post, and it happens that $$P(XY=1)=\frac12,$$ a property which shows that $(X,Y)$ can have no density.