I Have that (X,Y) are random variables that has uniform density over $\Omega=((x,y):x\geq 0, y\geq0, x+y\leq 1) $.
Using the steps given in this related question: Joint PDF of two random variables in a triangle , i find that the pdf to be: $f_{X,Y}= \begin{cases} 2; x,y \in \Omega \\ 0; \text{otherwise}\ \end{cases}$
My problem is though that i need to find the joint CDF, i.e $F_{X,Y}(x,y)$. I know that I have to integrate but I have problems finding the bounds. I think there is also going to be some different cases.
Any help / hints would be much appriciated.
For finding $P(X\leq t, Y\leq s)$ you have to integrate the density function over the region $\{(x,y): x,y \geq 0, x+y \leq 1, x\leq t, y\leq s\}$. You can integrate first w.r.t. $x$ from $0$ to $\min \{1-y,t\}$ and then integrate w.r.t. $y$ from $0$ to $s$. You can also reverse the order: integrate first w.r.t. $y$ from $0$ to $\min \{1-x,s\}$ and then integrate w.r.t. $x$ from $0$ to $t$.