Standard Cauchy

The standard Cauchy distribution, also known as the standard Lorentzian distribution, is a special case of the Cauchy distribution with location parameter $x_0=0$ and scale parameter $\gamma=1$. The PDF of the standard Cauchy distribution is given by:

$$f(x) = \frac{1}{\pi(1+x^2)}$$

where $x$ is an real number. The CDF (cumulative distribution function) of the standard Cauchy distribution is given by:

$$F(x) = \frac{1}{\pi}\tan^{-1}(x) + \frac{1}{2}$$