The Cauchy distribution, named after Augustin-Louis Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution ${\displaystyle f(x;x_{0},\gamma )}$ is the distribution of the x-intercept of a ray issuing from ${\displaystyle (x_{0},\gamma )}$ with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

The Cauchy distribution is often used in statistics as the canonical example of a "pathological" distribution since both its expected value and its variance are undefined (but see § Moments below). The Cauchy distribution does not have finite moments of order greater than or equal to one; only fractional absolute moments exist. The Cauchy distribution has no moment generating function.

In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane.

It is one of the few stable distributions with a probability density function that can be expressed analytically, the others being the normal distribution and the Lévy distribution.

Here are the most important constructions.

If one stands in front of a line and kicks a ball 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 ${\displaystyle (x_{0},\gamma )}$ in the x-y plane, and select a line passing through the point, with its direction (angle with the ${\displaystyle x}$-axis) chosen uniformly (between −180° and 0°) at random. The intersection of the line with the x-axis follows a Cauchy distribution with location ${\displaystyle x_{0}}$ and scale ${\displaystyle \gamma }$.

This definition gives a simple way to sample from the standard Cauchy distribution. Let ${\displaystyle u}$ be a sample from a uniform distribution from ${\displaystyle [0,1]}$, then we can generate a sample, ${\displaystyle x}$ from the standard Cauchy distribution using