Overview

The Cauchy distribution, also called the Cauchy–Lorentz distribution, is a two-parameter continuous probability law studied in mathematics. It is named for Augustin-Louis Cauchy and Hendrik Lorentz. Physicists commonly refer to the same family as the Lorentz distribution; statisticians typically call it the Cauchy distribution. As a continuous probability distribution, it is specified by a location parameter (often called x0) and a positive scale parameter (often called γ or scale).

Probability density and standard form

The probability density function can be written in compact form. For location x0 and scale γ>0 the density at a real value x is proportional to γ/((x−x0)^2+γ^2). The normalized expression is given in more detail in references to the probability density function. When x0=0 and γ=1 one obtains the standard Cauchy density f(x)=1/[π(1+x^2)], a symmetric, bell-shaped curve with heavier tails than a normal law.

Key properties

The Cauchy distribution is notable for its heavy tails. Unlike many common distributions, its first and second moments do not exist: the usual arithmetic mean and variance are undefined (see mean and variance). Consequently, sample averages do not converge in the way guaranteed by the classical law of large numbers. However, the distribution has a well-defined median and mode, both equal to the location parameter, and it is closed under certain types of linear fractional transformations.

History and connections

Historically, the shape was studied in the 19th century and later adopted by physicists studying resonance phenomena. The same functional form describes many physical line shapes and resonances, which is why the distribution is often referenced in physics texts and applications by people working in spectroscopy and related fields. See entries on spectroscopy and resonance for physical contexts where this profile appears.

Uses and examples

Practical uses of the Cauchy distribution include modeling resonant line shapes and as a canonical example in theoretical statistics to illustrate limitations of moment-based inference. It is also identical to the Student's t-distribution with one degree of freedom, and it provides a simple heavy-tailed alternative to the normal distribution when extreme values are more likely than the normal model permits. In robust statistics and Bayesian modeling a Cauchy-like prior is sometimes used when large deviations must be accommodated.

Notable distinctions and summary

In summary, the Cauchy distribution is a compact, two-parameter family distinguished by heavy tails and the absence of finite mean and variance. These features make it both a useful model in physics and a standard counterexample in probability and statistics, emphasizing that familiar convergence results do not apply automatically to all continuous distributions.

  • Also called: Lorentz distribution (in physics) — physicists.
  • Standard form: x0=0, γ=1 — f(x)=1/[π(1+x^2)] (see pdf).
  • Statistical caution: mean and variance undefined (mean, variance); sample mean need not converge (law of large numbers).