We recommend reading this documentation on
https://alexpghayes.github.io/distributions3/, where the math
will render with additional detail and much greater clarity.
In the following, let \(X\) be a Cauchy variable with mean
location =
\(x_0\) and scale
= \(\gamma\).
Support: \(R\), the set of all real numbers
Mean: Undefined.
Variance: Undefined.
Probability density function (p.d.f):
$$
f(x) = \frac{1}{\pi \gamma \left[1 + \left(\frac{x - x_0}{\gamma} \right)^2 \right]}
$$
Cumulative distribution function (c.d.f):
$$
F(t) = \frac{1}{\pi} \arctan \left( \frac{t - x_0}{\gamma} \right) +
\frac{1}{2}
$$
Moment generating function (m.g.f):
Does not exist.