# CDF and Inverse CDF of Wrapped Cauchy Distribution

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?

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.