Namespaces
Variants
Actions

Cauchy distribution

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


2020 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.

References

[F] W. Feller, "An introduction to probability theory and its applications", 2 , Wiley (1966)
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