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!


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

  • $\begingroup$ 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? $\endgroup$
    – 0k33
    Commented Dec 4, 2018 at 21:23
  • $\begingroup$ I would integrate, with substitution $(a,b)=(u+v,v)$ transforming $\iint_{\Pi}$ into $\iint_{[0,1]^2}$. $\endgroup$
    – metamorphy
    Commented Dec 4, 2018 at 21:51
  • $\begingroup$ I did just that and it worked like a charm! Thank you so much! $\endgroup$
    – 0k33
    Commented Dec 5, 2018 at 1:14

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