support on a triangle and uniform marginal distributions Suppose $X$ and $Y$ are jointly distributed on the support $\operatorname{conv} \{(0,0),(0,1),(1,0)\}$ with the joint PDF $f>0$ everywhere on the support. Is it possible to find $f$ such that the marginal PDFs are given by
$f_X(x)= 1$ for all $x\in[0,1]$ and
$f_Y(y)=1$ for all $y\in [0,1]$?
Thanks

You cannot have an everywhere positive density on the triangle support with the given marginal distributions.  To see this, suppse such a distribution exists, and then consider the small triangle to the right of $x=1/2$.  The x marginal distribution implies that the probability that x is greater than 1/2 is 1/2.  Similarly, the probability that y is greater than 1/2 is 1/2.  These 2 small triangles are disjoint, so the probability of being in one or the other is 1.  But that means that for the square where x and y are bounded by 0 and 1/2, the distribution will be zero almost everywhere.  So such a distribution does not exist.
