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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.

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    The closest point to the origin on the third side is at distance $\sqrt{3}/2$. For \sqrt{3}/2 < r < 1, the part of the third side with distance from the origin has length $2 \sqrt{r^2 - 3/4}$2012-07-29