Joint probability of $P(X<Y)$
My question is why are the bounds of integral from $0$ to $\infty$ and then from $0$ to $y$. I don't understand it.
Because
$$ \{(x,y)\in [0,\infty )^2:x< y\}=\{(x,y)\in [0,\infty )^2:0\leqslant x< y<\infty \}=\{(x,y)\in [0,\infty )^2:y\in[0,\infty )\text{ and }x\in[0,y)\} $$