Probability Distributions of 2 variables

Question

Consider two continuous random variables X and Y with joint p.d.f.

$f_{X,Y}(x,y) = \frac{x+2y}{24}, 0

Find the probability distribution of $W=X+Y$.

All I want to understand are the bounds for my double integral to find the probability distribution as a p.d.f. I understand that $0

I don't need a whole solution, just want to understand the bounds. Thank you!

Answer 1

Indeed the support is $0

For any $0

$$\max\{0,w-3\}< X< \min\{2, w-0\} \cap 0

Thusly

$$\begin{align}\Pr(W\leq w) & = \Pr(X+Y\leq w) \\[2ex] & = \mathbf 1_{0