The standard wrapped-up Cauchy distribution has the following probability density function:

$$f(x,p)= \frac{1-p^2}{2\pi(1+p^2)-2p\cos(x)}$$

where x is from $0$ to $2\pi$

Can anybody know, what is the CDF and InvCDF of this distribution?

Can you please also suggest any other wrapped distribution?


1 Answer 1


Wolfram Alpha nows the antiderivate of your PDF. Simply enter int((1-p^2)/(2*Pi*(1+p^2)-2*p*cos(x)), x) and get a complicated looking expession. $$\frac{(p^2 - 1) \tanh^{-1}\left(\frac{(π p^2 + p + \pi) \tan(x/2)}{\sqrt{p^2 - π^2 (p^2 + 1)^2}}\right)}{\sqrt{p^2 - \pi^2 (p^2 + 1)^2}} + c$$

But the dependence on $x$ is in a single $\tan(x/2)$ term, so it can be easily inverted.


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