is my selection of limits correct?

Question

I am given a question:

$$ f(x,y)= c(2x+y)\\ 2<x<6, 0<y<5 $$

I am asked to find the following things:

a) Value of c

b) P[X+Y>4]

Now I solved part a but I am stuck at b. Should it be like this?:

$$ \int_{4}^{6} \int_{4-x}^{5} c(2x+y) dy dx $$

Answer 1

No. Considering the cases $x>4$ and $x <4$ separately we get $\int_2^{4} \int_{4-x}^{5}c(2x+y)\, dy\,dx+\int_4^{6} \int_{0}^{5}c(2x+y)\,dy\, dx$

Answer 2

No, if $X>4$, then you are integrating the density over a region where $y<0$; there the density vanishes. Write the region you want to integrate over as a disjoint union of regions where $X\leq4$ and one where $X>4$. In the latter region, any value for $Y$ (with positive density) suffices for $X+Y>4$.