!!abstract,linked gloses,internal links,content,dynamic examples,...
!set gl_author=Sophie, Lemaire
!set gl_keywords=continuous_probability_distribution
!set gl_title=Cauchy distribution
!set gl_level=U1,U2,U3
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:
:tool/stat/table.fr
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<div class="wims_defn"><h4>Definition</h4>
Let \(lambda) and \(s) be real numbers with \(s>0\).
The <strong>Cauchy distribution</strong> with
parameters \(lambda) and \(s) (denoted by \(C(\lambda , s)))
is a continuous distribution on \(\RR) with density function:

<div class="wimscenter">
\(x\mapsto \frac{1}{\pi s}\frac{1}{1+(\frac{x-\lambda}{s})^2} )
</div>
</div>
<p>
let \(X) be a random variable having the Cauchy distribution
\(C(0 , 1)). The random variable
\(Y=\lambda+s X) has the Cauchy distribution \(C(\lambda,s)).
The median of \(Y) is \(\lambda). The first and third quartiles are \(\lambda \pm s).<br/>
The Cauchy distributions do not have expectation.
</p>

<table class="wimsborder wimscenter">
<tr><th>Expectation</th><th>Variance</th><th>Characteristic function</th></tr>
<tr>
<td>&nbsp;</td><td>&nbsp;</td><td>
\(\exp(i t\lambda-s|t|))</td></tr></table>
