I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$.
For example, when I have $f(x,y) = \frac {x} {5}\ +\frac {y} {20}$, $0 < x < 1$ and $1 < y < 5$ , I know the boundaries of the integrals when I'm trying to find $P(X + Y > 3)$ should be $(1,5)$ for $dy$ and $(3-y, 1)$ for $dx$.
However, when I have $f(x,y) = xy$, $0 < x < 1$ and $0 < y < 2$, and the question is asking for $P(X + Y < 1)$, I need to change the boundaries of integrals of $dy$ to $(0, 1)$ if I am setting the boundaries of integration of the $dx$ part to be $(0, 1-y)$. Why does that happen? Why do I get a wrong answer if I leave the dy's boundaries as $(0, 2)$ but I still get the right answer when I leave it as $(1,5)$ for the queston above?
Thanks in advance.
In your second example, you can't integrate $y$ over $(0,2)$, because we want $X+Y<1$, and since $X\geq 0$, we must have $Y\leq 1$. Thus, $1$ is an upper bound for $Y$. You could also integrate $Y$ over $\mathbb R$ and $x$ over $(-\infty, 1-Y)$, but then you should use that $f(x,y)=0$ when $$(x,y)\not\in\{(a,b)|0\leq a\leq 1, \leq b\leq 2\}$$