Cauchy distribution
2020 Mathematics Subject Classification: Primary: 60E07 [MSN][ZBL]
A continuous probability distribution with density
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and distribution function
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where and
are parameters. The Cauchy distribution is unimodal and symmetric about the point
, which is its mode and median. No moments of positive order — including the expectation — exist. The characteristic function has the form
. The class of Cauchy distributions is closed under linear transformations: If a random variable
has the Cauchy distribution with parameters
and
, then the random variable
also has a Cauchy distribution, with parameters
and
. The class of Cauchy distributions is closed under convolution:
![]() | (*) |
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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 are independent random variables with the same Cauchy distribution, then their arithmetic mean
has the same distribution as each
. 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
and
are independent and have the same Cauchy distribution, then the random variables
and
have the same Cauchy distribution. The Cauchy distribution with parameters
and
is the Student
-distribution with one degree of freedom. The Cauchy distribution with parameters
is identical with the distribution of the random variable
, where
and
are independent and normally distributed with parameters
and
, respectively. A random variable with this distribution is the function
, where
is a random variable uniformly distributed on the interval
. The Cauchy distribution is also defined in spaces of dimension greater than one. The concept was first investigated by A.L. Cauchy.
References
[1] | W. Feller, "An introduction to probability theory and its applications", 2 , Wiley (1966) |
Cauchy distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_distribution&oldid=26375