# joint/marginal pdfs over parallelogram integration limits

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!

• $$b\in(0,1)$$ 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 $$p_a(t)=\min\{1,t\}-\max\{0,t-1\}=\min\{t,2-t\}$$.
• oh man! The $b \in (0, 1)$ leading to the possibilities is so smart, I follow this solution fully. could I use a similar approach to compute the expected value of AB (from the joint PDF), or would I have to somehow integrate over the parallelogram?
• I would integrate, with substitution $(a,b)=(u+v,v)$ transforming $\iint_{\Pi}$ into $\iint_{[0,1]^2}$. Commented Dec 4, 2018 at 21:51