# Cauchy distribution

2010 Mathematics Subject Classification: Primary: 60E07 [MSN][ZBL]

A continuous probability distribution with density

$$p (x; \lambda , \mu ) = \ { \frac{1} \pi } \frac \lambda {\lambda ^ {2} + (x - \mu ) ^ {2} } ,\ \ - \infty < x < \infty ,$$

and distribution function

$$F (x; \lambda , \mu ) = \ { \frac{1}{2} } + { \frac{1} \pi } \ \mathop{\rm arctan} \frac{x - \mu } \lambda ,$$

where $- \infty < \mu < \infty$ and $\lambda > 0$ are parameters. The Cauchy distribution is unimodal and symmetric about the point $x = \mu$, which is its mode and median. No moments of positive order — including the expectation — exist. The characteristic function has the form $\mathop{\rm exp} ( i \mu t - \lambda | t | )$. The class of Cauchy distributions is closed under linear transformations: If a random variable $X$ has the Cauchy distribution with parameters $\lambda$ and $\mu$, then the random variable $Y = aX + b$ also has a Cauchy distribution, with parameters $\lambda ^ \prime = | a | \lambda$ and $\mu ^ \prime = a \mu + b$. The class of Cauchy distributions is closed under convolution:

$$\tag{* } p (x; \lambda _ {1} , \mu _ {1} ) * \dots * p (x; \lambda _ {n} , \mu _ {n} ) =$$

$$= \ p (x; \lambda _ {1} + \dots + \lambda _ {n} , \mu _ {1} + \dots + \mu _ {n} ) ;$$

in other words, a sum of independent random variables with Cauchy distributions is again a random variable with a Cauchy distribution. Thus, the Cauchy distribution, like the normal distribution, belongs to the class of stable distributions; to be precise: It is a symmetric stable distribution with index 1 (cf. Stable distribution). The following property of Cauchy distributions is a corollary of (*): If $X _ {1} \dots X _ {n}$ are independent random variables with the same Cauchy distribution, then their arithmetic mean $(X _ {1} + \dots + X _ {n} ) /n$ has the same distribution as each $X _ {k}$. One more property of Cauchy distributions: In the family of Cauchy distributions, the distribution of a sum of random variables may be given by (*) even if the variables are dependent. For example, if $X$ and $Y$ are independent and have the same Cauchy distribution, then the random variables $X + X$ and $X + Y$ have the same Cauchy distribution. The Cauchy distribution with parameters $\lambda = 1$ and $\mu = 0$ is the Student $t$- distribution with one degree of freedom. The Cauchy distribution with parameters $( \lambda , \mu )$ is identical with the distribution of the random variable $\mu + ( X/Y )$, where $X$ and $Y$ are independent and normally distributed with parameters $(0, \lambda ^ {2} )$ and $(0, 1)$, respectively. A random variable with this distribution is the function $\mu + \lambda \mathop{\rm tan} z$, where $z$ is a random variable uniformly distributed on the interval $[- \pi /2, \pi /2]$. The Cauchy distribution is also defined in spaces of dimension greater than one. The concept was first investigated by A.L. Cauchy.

How to Cite This Entry:
Cauchy distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_distribution&oldid=46277
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article