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In this paper (page 6) I'm reading, the author has a uniform random variable $\zeta$ which takes on values between 0 and 1. He computes

$ 2 \arccos( \sqrt{ 1 - \zeta } ) $

But isn't that the same as computing $2 \arccos( \sqrt{ \zeta } )$, since $\zeta$ is uniform on [0..1]? Or is there a reason he may be doing that I'm missing?

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    @Willie Wong: +1 I thought of an expectation sign there (though, I see that there were no reasons for it). On the other hand my answer is still correct besides the phrase "You're right".2011-08-11

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You are right, if you consider $\eta = 1-\zeta$ then the distribution of $\eta$ is also uniform on $[0,1].$