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A continuous probability distribution with density
 
A continuous probability distribution with density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208501.png" /></td> </tr></table>
+
$$
 +
p (x; \lambda , \mu )  = \
 +
{
 +
\frac{1} \pi
 +
}
 +
 
 +
\frac \lambda {\lambda  ^ {2} + (x - \mu )  ^ {2} }
 +
,\ \
 +
- \infty < x < \infty ,
 +
$$
  
 
and distribution function
 
and distribution function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208502.png" /></td> </tr></table>
+
$$
 +
F (x; \lambda , \mu )  = \
 +
{
 +
\frac{1}{2}
 +
} + {
 +
\frac{1} \pi
 +
} \
 +
\mathop{\rm arctan} 
 +
\frac{x - \mu } \lambda
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208504.png" /> are parameters. The Cauchy distribution is unimodal and symmetric about the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208505.png" />, which is its mode and median. No moments of positive order — including the expectation — exist. The characteristic function has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208506.png" />. The class of Cauchy distributions is closed under linear transformations: If a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208507.png" /> has the Cauchy distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c0208509.png" />, then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085010.png" /> also has a Cauchy distribution, with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085012.png" />. The class of Cauchy distributions is closed under convolution:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
p (x; \lambda _ {1} , \mu _ {1} )
 +
* \dots * p (x; \lambda _ {n} , \mu _ {n} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085014.png" /></td> </tr></table>
+
$$
 +
= \
 +
p (x; \lambda _ {1} + \dots + \lambda _ {n} , \mu _ {1} + \dots + \mu _ {n} ) ;
 +
$$
  
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|Stable distribution]]). The following property of Cauchy distributions is a corollary of (*): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085015.png" /> are independent random variables with the same Cauchy distribution, then their arithmetic mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085016.png" /> has the same distribution as each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085017.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085019.png" /> are independent and have the same Cauchy distribution, then the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085021.png" /> have the same Cauchy distribution. The Cauchy distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085023.png" /> is the Student <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085024.png" />-distribution with one degree of freedom. The Cauchy distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085025.png" /> is identical with the distribution of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085028.png" /> are independent and normally distributed with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085030.png" />, respectively. A random variable with this distribution is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085032.png" /> is a random variable uniformly distributed on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020850/c02085033.png" />. The Cauchy distribution is also defined in spaces of dimension greater than one. The concept was first investigated by A.L. Cauchy.
+
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|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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Wiley (1966)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''2''' , Wiley (1966)
 +
|}

Latest revision as of 15:35, 4 June 2020


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=20843
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article