The Lorenzian distribution (also known as Cauchy, Cauchy-Lorenz, Lorenz, and Breit-Wigner) is a continuous bell-shaped distribution having the indicated «mode», and with the second parameter, «shape», specifying the half-width at the half-maximum density.
It has uses in physics, especially in the study of resonance and spectroscopy where it describes the shape of spectral lines that are broadened through various resonances.
The standard form, in which «mode» = 0 and «shape» = 1, is known as the standard Cauchy distribution.
The Lorenz distribution has some unusual mathematical properties that are uncommon among the standard distributions. Its mean, variance and higher moments are all undefined. As a result, the law of large numbers does not apply to samples generated from a Lorenz distribution.