Joint probability of two correlated RVs I am trying to get the joint PDF of two RVs $X$ and $Y$ where $aX<Y<bX$, so I am stuck in calculating the probability of
$\mathbb{P}(X<x,Y<y|aX<Y<bX)$
any idea?
 A: Your question is quite general (specific cases may offer shortcuts), so I'll limit myself to suggesting strategies. Typically such a question involves constructing appropriate limits on integrals and trying to evaluate them by some means. Usually there will be a bivariate integral where the limits in the inner integral involve the variable in the outer integral. Sometimes the hardest part is simply writing the correct limits down; the general case will involve "min" and "max" functions on a problem like this. 
To make progress I strongly suggest you get into the habit of making diagrams of the region you're trying to integrate.
A couple of suggested strategies for approaching such a problem, by making slightly simpler problems you might see how to write integrals for. 
One approach: Let $Z = Y-aX$ and work out the joint probability in terms of $X$ and $Z$. 
Another approach: First, replace your $x$ and $y$ with $x_0$ and $y_0$ so I can use $x$ and $y$ as dummy variables in the integration. If $a\leq y_0/x_0\leq b$ then you have a region like this:
 
While you can actually write the integral by splitting it up into pieces (move a vertical line along the $x$-axis and split the integral where your line hits any 'corners' on the dark green region), you might otherwise evaluate it by working out the probability of being in the rectangle $0<X<x_0; 0<Y<y_0$ and then subtract the two triangles.
The other cases (the other arrangements of $x_0$ and $y_0$ relative to $a$ and $b$) might be worked out by drawing the relevant diagram in order to obtain the right limits on the integrals; in each case you'll do something similar, but sometimes you might not hit a corner.
With more details, more specific responses might be possible.
