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I am trying to find the pdf of $Y=\cos(X)$ where $X$ is a random variable distributed uniformly in $[-\pi/2,\pi/2]$.

I tried to use the change of variable theorem but there is a detail in $[-\pi/2,\pi/2]$ since the cosine it's not monotone, and since $\cos(\pi/2)=0=\cos(\pi/2)$. I don't have a way to see the interval that corresponds to $Y$.

Thank you for helping!

StubbornAtom
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José Daniel Castilla del Valle
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    @StubbornAtom. Cos(x) in $[-\pi/2, \pi/2]$ is always nonnegative (but not monotone) while it has both positive and negative values in $[0,\pi]$ and is monotone. Being monotonous or not decides if the change of variables theorem can be directly applied or not. The problems are very different in that sense! – Xiaohai Zhang Nov 23 '19 at 17:26
  • @XiaohaiZhang You are right, it is not an exact duplicate. My bad. – StubbornAtom Nov 23 '19 at 18:46

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You can let $Z=\vert X \vert$. Then $Z$ is uniform on $[0, \pi/2].$ And $Y=cos(X)=cos(Z)$. Note the mapping from $Z$ to $Y$ is monotone, and you can apply the change of variable theorem there now.

Xiaohai Zhang
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