Cauchy Distribution

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Cauchy Distribution</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="98%" align="left" valign="top"><p class="word_list"> <a href="#Cauchy distribution">Cauchy distribution</a><p class="word_list"> <a href="#standard Cauchy">standard Cauchy distribution</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table5"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> </table> <p class="body">Suppose a random variable<i> Y </i>is defined <a name="[1]">as</a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cauch_distribution_p1[2.gif" width="48" height="34"></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">where <img border="0" src="../symbols/X_1.gif" align="absbottom" width="15" height="15"> and <img border="0" src="../symbols/X_2.gif" align="absbottom" width="17" height="14"> are independent <a href="../articles/normal_distribution.htm">standard normal</a> random variables. How is <i>Y</i> distributed? The answer is that it has a <b><a name="Cauchy distribution">Cauchy distribution</a></b>.</p> <p class="body">The Cauchy distribution is specified with two parameters: <i>a</i> and <i>b</i>. We denote it <img border="0" src="../formulas/cauchy_distribution_C_a_b.gif" align="absbottom" width="39" height="15">. The random variable <i>Y</i> defined in [<a href="#[1]">1</a>] above actually has a <i>C</i>(0,1) distribution, which is called the <b> <a name="standard Cauchy">standard Cauchy</a></b> distribution. </p> <p class="body">The Cauchy distribution has probability density function (PDF)</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cauchy_distribution_{2].gif" width="161" height="70"></td> <td width="5%" align="right">[2]</td> </tr> </table> <p class="body">This is graphed in Exhibit 1.</p> <div align="center"> <center> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table8"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header"><i>C</i>(<i>a</i>,<i>b</i>) Probability Density Function</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_cauchy_distribution.gif" width="300" height="149"></td> </tr> <tr> <td> <p class="exhibit_legend">The <i>C</i>(<i>a</i>,<i>b</i>) distribution has a symmetric "bell shaped" probability density function, but it is more peaked at the center and has fatter tails than a normal distribution.</td> </tr> </table> </center> </div> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="135" align="left" id="table9"> <tr> <td width="125" height="10"> <font size="2"> </font></td> <td width="10" height="10"></td> </tr> <tr> <td width="125"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> <td width="10"> </td> </tr> </table> <p class="body">The PDF is more peaked in the middle and has fatter tails than a <a href="../articles/normal_distribution.htm">normal distribution</a>. For this reason, we might want to call the distribution <a href="../articles/kurtosis.htm">leptokurtic</a>. That would technically not be correct because the distribution's <a href="../articles/kurtosis.htm"> kurtosis</a> is undefined. Actually, its <a href="../articles/mean.htm"> mean</a>, <a href="../articles/standard_deviation.htm">standard deviation</a> and <a href="../articles/skewness.htm">skewness</a> are also not defined. As Exhibit 1 indicates, the distribution's median and mode both occur at <i>a</i>. The parameter <i>b</i> is a dispersion parameter, playing a role similar to that of a standard deviation.</p> <p class="body">The cumulative distribution function is</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cauchy_distribution_[3].gif" width="177" height="47"></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">Any Cauchy random variable X ~ <img border="0" src="../formulas/cauchy_distribution_C_a_b.gif" align="absbottom" width="39" height="15"> can be expressed in terms of a standard Cauchy variable <i>Z</i> ~ <i>C</i>(0,1) as</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table11"> <tr> <td width="100%" align="center"> <i>X = bZ + a</i></td> <td width="5%" align="right">[4]</td> </tr> </table> <p class="body">The Cauchy distribution is a <a href="../articles/stable_paretian_distributions.htm">stable Paretian</a> distribution, so a sum of Cauchy random variables is itself Cauchy. More precisely, consider <i>n</i> independent random variables <img border="0" src="../formulas/cauchy_distribution_abc.gif" align="absbottom" width="90" height="15">, and let <i>Y</i> equal their sum. Then</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table12"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cauchy_distribution_[5].gif" width="130" height="56"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p class="body">The Cauchy distribution has characteristic function</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" height="36" id="table13"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/cauchy_distribution_[6].gif" width="221" height="49"></td> <td width="5%" align="right">[6]</td> </tr> </table> <p class="body">The standard Cauchy distribution is a special case of the student <i>t</i> distribution with one degree of freedom.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" 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