A point P is randomly chosen on the triangle with sides' length 1. The triangle is spun randomly (uniformly) about its vertex (0,0). Let (X, Y) denote P's coordinate. Find the joint density of (X, Y).
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In polar coordinates $(R,\Theta)$, obviously $\Theta$ is uniformly distributed on $[0,2\pi)$, and $R$ is independent of $\Theta$. We only need to know the distribution of $R$. Two of the sides of the triangle give uniform distributions on $[0,1]$. I'll let you do the other one.