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How to calculate the PDF or the CDF of $D$ where:

$$D = \sqrt{1 - X^2 \sin^2{\theta}} - X \cos{\theta}$$

If $X$ is uniform distributed on $[-1, 1]$, $\theta$ is uniformly distributed on $[0, 2 \pi]$ and they are independent.

I know that: $$F_D(d) = \iint\limits_D \, f_\theta(\theta) f_X(x) \mathrm{d}x\,\mathrm{d}\theta $$

But I don't know how to find the ranges I should integrate on!

  • 0
    Do you mean $\{-1,1\}$, i.e., the atoms $-1$ and $+1$ or do you mean $[-1,1]$, i.e., the interval of real numbers $-1 \leq x \leq 1$? Same question for $\{0,2\pi\}$.2011-05-29
  • 0
    Yes I mean on the interval of real number $-1≤x≤1$2011-05-29

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