I am trying to find the joint CDF $F(x,y)$ for the following function.
The following model answer was given.
My query is why we need to consider the first case $0<x<y$ when $f_{X,Y}$ is only defined above $0<y<x$? Furthermore I do not understand what the variables $u$ and $v$ refer to in the integration. Do they refer to some new axis $u,v$? Do small $x$ and small $y$ represent any $x, y$ value, or do they refer to a point on the line $Y=X$?
When $0<x<y$, even though the pdf $f_{X,Y}(x,y)=0$, the region that must be integrated over to get the cdf still contains a sub-region where the pdf is non-zero, so the integration must be performed. In addition, the region arising from the intersection of the cdf region ($0<u<x,0<v<y$) and the support of the pdf has different forms depending on whether $x<y$ or $y<x$, so we cannot combine the calculations of the two cases together.
$u,v$ are merely integration variables used to avoid conflicting with $x,y$, which are the upper bounds for the cdf region.