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\begin{align*} D = \frac{\arcsin{(X \sin{\theta})} - \theta}{\sin{\theta}} \end{align*} and \begin{align*} X \sim \text{Uniform}[-1, 1], \hspace{0.5in} \theta \sim \text{Uniform}[0, 2 \pi] \end{align*}

How to find the PDF of $D$? or is there anyway to simplify the equation of $D$!?

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Isn't it that for $\theta<\pi$, $d$ becomes negative? – Fabian May 4 '11 at 11:59
Sorry .. forget about the P(d > 0) .. what about calculating the PDF?? – Osama Gamal May 4 '11 at 13:10

Expanding on Fabian's comment, for $\theta < \pi$, $d \ge 0$ iff $\arcsin(x \sin \theta) \ge \theta$ iff $x \sin \theta \ge \sin \theta$, that is, never for $x \le 1$.

And for $\pi < \theta < 2\pi$, $d \ge 0$ iff $\arcsin(x \sin \theta) \le \theta$ which is always true. Your probability is $\frac{1}{2}$.

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Thanks but I care more about calculating the PDF!! – Osama Gamal May 4 '11 at 13:11

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