I was running into a problem where I need to evaluate some probability over a region $D$.
A toy example would be this probability $\Pr \left[ {\underbrace {Y < \frac{{5\left( {X + 7} \right)}}{{9X}}}_{Event1} \cap \underbrace {X < \frac{{3\left( {Y + 7} \right)}}{{5Y}}}_{Event2}} \right]$ where $X,Y$ are exponential random variable. In this case, it is straight forward to just plot the curve, find the intersection and then integrate.
My question is that what would be a systematic strategy to deal with a complicated region of integration, especially region that you cannot draw or visualize?
Particularly, my problem involve 4 exponential random variable $X,Y,Z,T$ and the probability that I am interest in is
$A=\Pr \left[ {\underbrace {Z > \frac{2}{X}}_{Event\,\,1} \cap \underbrace {Z > 5YT + \frac{3}{X}}_{Event\,2}} \right]$
Please support me on finding the correct integration limit for $A$. Thank you for your enthusiasm !