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<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"/><title>9 Probability Distributions</title><link rel="stylesheet" type="text/css" href="../scribble.css" title="default"/><link rel="stylesheet" type="text/css" href="../racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-style.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../manual-racket.css" title="default"/><link rel="stylesheet" type="text/css" href="../doc-site.css" title="default"/><script type="text/javascript" src="../scribble-common.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../manual-racket.js"></script><script type="text/javascript" src="../doc-site.js"></script><script type="text/javascript" src="../local-redirect/local-redirect.js"></script><script type="text/javascript" src="../local-redirect/local-user-redirect.js"></script><!--[if IE 6]><style type="text/css">.SIEHidden { overflow: hidden; }</style><![endif]--></head><body id="doc-racket-lang-org"><div class="tocset"><div class="tocview"><div class="tocviewlist tocviewlisttopspace"><div class="tocviewtitle"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_0");">►</a></td><td></td><td><a href="index.html" class="tocviewlink" data-pltdoc="x">Math Library</a></td></tr></table></div><div class="tocviewsublisttop" style="display: none;" id="tocview_0"><table cellspacing="0" cellpadding="0"><tr><td align="right">1 </td><td><a href="base.html" class="tocviewlink" data-pltdoc="x">Constants and Elementary Functions</a></td></tr><tr><td align="right">2 </td><td><a href="flonum.html" class="tocviewlink" data-pltdoc="x">Flonums</a></td></tr><tr><td align="right">3 </td><td><a href="special.html" class="tocviewlink" data-pltdoc="x">Special Functions</a></td></tr><tr><td align="right">4 </td><td><a href="number-theory.html" class="tocviewlink" data-pltdoc="x">Number Theory</a></td></tr><tr><td align="right">5 </td><td><a href="bigfloat.html" class="tocviewlink" data-pltdoc="x">Arbitrary-<wbr></wbr>Precision Floating-<wbr></wbr>Point Numbers (<a name="(tech._bigfloat)"></a><span style="font-style: italic">Bigfloats</span>)</a></td></tr><tr><td align="right">6 </td><td><a href="array.html" class="tocviewlink" data-pltdoc="x">Arrays</a></td></tr><tr><td align="right">7 </td><td><a href="matrices.html" class="tocviewlink" data-pltdoc="x">Matrices and Linear Algebra</a></td></tr><tr><td align="right">8 </td><td><a href="stats.html" class="tocviewlink" data-pltdoc="x">Statistics Functions</a></td></tr><tr><td align="right">9 </td><td><a href="" class="tocviewselflink" data-pltdoc="x">Probability Distributions</a></td></tr><tr><td align="right">10 </td><td><a href="utils.html" class="tocviewlink" data-pltdoc="x">Stuff That Doesn’t Belong Anywhere Else</a></td></tr></table></div></div><div class="tocviewlist"><table cellspacing="0" cellpadding="0"><tr><td style="width: 1em;"><a href="javascript:void(0);" title="Expand/Collapse" class="tocviewtoggle" onclick="TocviewToggle(this,"tocview_1");">▼</a></td><td>9 </td><td><a href="" class="tocviewselflink" data-pltdoc="x">Probability Distributions</a></td></tr></table><div class="tocviewsublistbottom" style="display: block;" id="tocview_1"><table cellspacing="0" cellpadding="0"><tr><td align="right">9.1 </td><td><a href="dist_dist-objects.html" class="tocviewlink" data-pltdoc="x">Distribution Objects</a></td></tr><tr><td align="right">9.2 </td><td><a href="Distribution_Types_and_Operations.html" class="tocviewlink" data-pltdoc="x">Distribution Types and Operations</a></td></tr><tr><td align="right">9.3 </td><td><a href="Finite_Distribution_Families.html" class="tocviewlink" data-pltdoc="x">Finite Distribution Families</a></td></tr><tr><td align="right">9.4 </td><td><a href="Integer_Distribution_Families.html" class="tocviewlink" data-pltdoc="x">Integer Distribution Families</a></td></tr><tr><td align="right">9.5 </td><td><a href="Real_Distribution_Families.html" class="tocviewlink" data-pltdoc="x">Real Distribution Families</a></td></tr><tr><td align="right">9.6 </td><td><a href="dist_flonum.html" class="tocviewlink" data-pltdoc="x">Low-<wbr></wbr>Level Distribution Functions</a></td></tr></table></div></div></div></div><div class="maincolumn"><div class="main"><div class="versionbox"><span class="version">6.7</span></div><div class="navsettop"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter a search string to search the manuals" onkeypress="return DoSearchKey(event, this, "6.7", "../");" onfocus="this.style.color="black"; this.style.textAlign="left"; if (this.value == "...search manuals...") this.value="";" onblur="if (this.value.match(/^ *$/)) { this.style.color="#888"; this.style.textAlign="center"; this.value="...search manuals..."; }"/></form> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.7");">top</a></span><span class="navright"> <a href="stats.html" title="backward to "8 Statistics Functions"" data-pltdoc="x">← prev</a> <a href="index.html" title="up to "Math Library"" data-pltdoc="x">up</a> <a href="dist_dist-objects.html" title="forward to "9.1 Distribution Objects"" data-pltdoc="x">next →</a></span> </div><h3 x-source-module="(lib "math/scribblings/math.scrbl")" x-source-pkg="math-doc" x-part-tag=""dist"">9<tt> </tt><a name="(part._dist)"></a><a name="(mod-path._math/distributions)"></a>Probability Distributions</h3><div class="SAuthorListBox"><span class="SAuthorList"><p class="author">Neil Toronto <<a href="mailto:ntoronto@racket-lang.org">ntoronto@racket-lang.org</a>></p></span></div><p><table cellspacing="0" cellpadding="0" class="defmodule"><tr><td align="left"><span class="hspace"> </span><span class="RktPn">(</span><span class="RktSym"><a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=reference&rel=require.html%23%2528form._%2528%2528lib._racket%252Fprivate%252Fbase..rkt%2529._require%2529%2529&version=6.7" class="RktStxLink Sq" data-pltdoc="x">require</a></span><span class="stt"> </span><a href="" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/distributions</span></a><span class="RktPn">)</span></td><td align="right"><span class="RpackageSpec"><span class="Smaller"> package:</span> <span class="stt">math-lib</span></span></td></tr></table></p><p><div class="SIntrapara">The <a href="" class="RktModLink" data-pltdoc="x"><span class="RktSym">math/distributions</span></a> module exports the following:
</div><div class="SIntrapara"><ol><li><p><a href="dist_dist-objects.html#%28tech._distribution._object%29" class="techoutside" data-pltdoc="x"><span class="techinside">Distribution objects</span></a>, which represent probability distributions</p></li><li><p>Functions that operate on distribution objects</p></li><li><p>The low-level flonum functions used to define distribution objects</p></li></ol></div></p><p><span style="font-weight: bold">Performance Warning:</span> Using distribution objects in untyped Racket is currently 25-50 times
slower than using them in Typed Racket, due to the overhead of checking higher-order contracts.
We are working on it.</p><p>For now, if you need speed, either use the <a href="https://download.racket-lang.org/docs/6.7/html/local-redirect/index.html?doc=ts-reference&rel=index.html&version=6.7" class="RktModLink Sq" data-pltdoc="x"><span class="RktSym">typed/racket</span></a> language, or use just the
low-level flonum functions, which are documented in <a href="dist_flonum.html" data-pltdoc="x">Low-Level Distribution Functions</a>.</p><table cellspacing="0" cellpadding="0"><tr><td><p><span class="hspace"> </span><a href="dist_dist-objects.html" class="toclink" data-pltdoc="x">9.1<span class="hspace"> </span>Distribution Objects</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Distribution_Types_and_Operations.html" class="toclink" data-pltdoc="x">9.2<span class="hspace"> </span>Distribution Types and Operations</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Finite_Distribution_Families.html" class="toclink" data-pltdoc="x">9.3<span class="hspace"> </span>Finite Distribution Families</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Finite_Distribution_Families.html#%28part._.Unordered_.Discrete_.Distributions%29" class="toclink" data-pltdoc="x">9.3.1<span class="hspace"> </span>Unordered Discrete Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html" class="toclink" data-pltdoc="x">9.4<span class="hspace"> </span>Integer Distribution Families</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Bernoulli_.Distributions%29" class="toclink" data-pltdoc="x">9.4.1<span class="hspace"> </span>Bernoulli Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Binomial_.Distributions%29" class="toclink" data-pltdoc="x">9.4.2<span class="hspace"> </span>Binomial Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Geometric_.Distributions%29" class="toclink" data-pltdoc="x">9.4.3<span class="hspace"> </span>Geometric Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Integer_Distribution_Families.html#%28part._.Poisson_.Distributions%29" class="toclink" data-pltdoc="x">9.4.4<span class="hspace"> </span>Poisson Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html" class="toclink" data-pltdoc="x">9.5<span class="hspace"> </span>Real Distribution Families</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Beta_.Distributions%29" class="toclink" data-pltdoc="x">9.5.1<span class="hspace"> </span>Beta Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Cauchy_.Distributions%29" class="toclink" data-pltdoc="x">9.5.2<span class="hspace"> </span>Cauchy Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Delta_.Distributions%29" class="toclink" data-pltdoc="x">9.5.3<span class="hspace"> </span>Delta Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Exponential_.Distributions%29" class="toclink" data-pltdoc="x">9.5.4<span class="hspace"> </span>Exponential Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Gamma_.Distributions%29" class="toclink" data-pltdoc="x">9.5.5<span class="hspace"> </span>Gamma Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Logistic_.Distributions%29" class="toclink" data-pltdoc="x">9.5.6<span class="hspace"> </span>Logistic Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Normal_.Distributions%29" class="toclink" data-pltdoc="x">9.5.7<span class="hspace"> </span>Normal Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Triangular_.Distributions%29" class="toclink" data-pltdoc="x">9.5.8<span class="hspace"> </span>Triangular Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Truncated_.Distributions%29" class="toclink" data-pltdoc="x">9.5.9<span class="hspace"> </span>Truncated Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="Real_Distribution_Families.html#%28part._.Uniform_.Distributions%29" class="toclink" data-pltdoc="x">9.5.10<span class="hspace"> </span>Uniform Distributions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist_flonum.html" class="toclink" data-pltdoc="x">9.6<span class="hspace"> </span>Low-Level Distribution Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist_flonum.html#%28part._.Integer_.Distribution_.Functions%29" class="toclink" data-pltdoc="x">9.6.1<span class="hspace"> </span>Integer Distribution Functions</a></p></td></tr><tr><td><p><span class="hspace"> </span><a href="dist_flonum.html#%28part._.Real_.Distribution_.Functions%29" class="toclink" data-pltdoc="x">9.6.2<span class="hspace"> </span>Real Distribution Functions</a></p></td></tr></table><div class="navsetbottom"><span class="navleft"><form class="searchform"><input class="searchbox" style="color: #888;" type="text" value="...search manuals..." title="Enter a search string to search the manuals" onkeypress="return DoSearchKey(event, this, "6.7", "../");" onfocus="this.style.color="black"; this.style.textAlign="left"; if (this.value == "...search manuals...") this.value="";" onblur="if (this.value.match(/^ *$/)) { this.style.color="#888"; this.style.textAlign="center"; this.value="...search manuals..."; }"/></form> <a href="../index.html" title="up to the documentation top" data-pltdoc="x" onclick="return GotoPLTRoot("6.7");">top</a></span><span class="navright"> <a href="stats.html" 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