I am looking at this problem, and get it up to the calculation of the joint PDF, which is 1/12. From that point, I do not get how they achieve the parallelogram. Could someone please tell me? Thank you.
2026-03-28 11:34:58.1774697698
How did they come up with joint PDF representation?
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Found the answer in this video.
From the inequality $\theta - 1 \leq x \leq \theta + 1$, two equations are obtained:
$\theta \leq x + 1$, and
$\theta \geq x - 1$.
Plot them on the $\theta$ vs $x$ plane, we got the shown graph.
The conditional expectation $\mathbf E[\Theta\mid X=x]$ of the uniformly distributed $\Theta$ is just the midpoint of that uniform distribution. This explain the plot of $\mathbf E[\Theta\mid X=x]$ inside the parallelogram.