joint/marginal pdfs over parallelogram integration limits

Question

I am TERRIBLE at figuring out the limits of integration when finding PDFs.

Say $a, b$ are uniformly distributed over the parallelogram with vertices $(0, 0), (1, 0), (2, 1), (1, 1)$. Find the joint PDF of a and b as well as the marginal PDFs of both variables:

Since we are told that $a, b$ are uniformly distributed over the parallelogram, I would think that their joint PDF would be $1$ when $a, b \in$ parallelogram, and $0$ elsewhere.

I could really use some help in computing the marginal PDFs. I have some intuition that we would need to split up each integral, and that a and b are not independent. I am not sure how to progress!

Answer 1

• $b\in(0,1)$ fixed $\implies a\in(b,b+1)$ a.s.
• $a\in(0,2)$ fixed $\implies b\in(\max\{0,a-1\},\min\{1,a\})$ a.s.

Thus $p_b(t)=1$ and $p_a(t)=\min\{1,t\}-\max\{0,t-1\}=\min\{t,2-t\}$.