Editing Cauchy distribution (section)

=== Rotational symmetry ===
If one stands in front of a line and kicks a ball with at a uniformly distributed random angle towards the line, then the distribution of the point where the ball hits the line is a Cauchy distribution.

For example, consider a point at <math>(x_0, \gamma)</math> in the x-y plane, and select a line passing through the point, with its direction (angle with the <math>x</math>-axis) chosen uniformly (between −180° and 0°) at random. The intersection of the line with the x-axis follows a Cauchy distribution with location <math>x_0</math> and scale <math>\gamma</math>.

This definition gives a simple way to sample from the standard Cauchy distribution. Let <math> u </math> be a sample from a uniform distribution from <math>[0,1]</math>, then we can generate a sample, <math>x</math> from the standard Cauchy distribution using

<math display="block"> x = \tan\left(\pi(u-\tfrac{1}{2})\right) </math>
When <math>U</math> and <math>V</math> are two independent [[normal distribution|normally distributed]] [[random variable]]s with [[expected value]] 0 and [[variance]] 1, then the ratio <math>U/V</math> has the standard Cauchy distribution.

More generally, if <math>(U, V)</math> is a rotationally symmetric distribution on the plane, then the ratio <math>U/V</math> has the standard Cauchy distribution.