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<title>MADlib: prob.sql_in File Reference</title>
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<div class="header">
  <div class="summary">
<a href="#func-members">Functions</a>  </div>
  <div class="headertitle">
<div class="title">prob.sql_in File Reference</div>  </div>
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<div class="contents">

<p>SQL functions for evaluating probability functions.  
<a href="#details">More...</a></p>
<table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="func-members"></a>
Functions</h2></td></tr>
<tr class="memitem:aea21a931dc5578a570e3370af3d8d43a"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#aea21a931dc5578a570e3370af3d8d43a">bernoulli_cdf</a> (float8 x, float8 sp)</td></tr>
<tr class="memdesc:aea21a931dc5578a570e3370af3d8d43a"><td class="mdescLeft">&#160;</td><td class="mdescRight">Bernoulli cumulative distribution function.  <a href="#aea21a931dc5578a570e3370af3d8d43a">More...</a><br /></td></tr>
<tr class="separator:aea21a931dc5578a570e3370af3d8d43a"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a434b3ad1f3964835834dc2a942b820ef"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a434b3ad1f3964835834dc2a942b820ef">bernoulli_pmf</a> (int4 x, float8 sp)</td></tr>
<tr class="memdesc:a434b3ad1f3964835834dc2a942b820ef"><td class="mdescLeft">&#160;</td><td class="mdescRight">Bernoulli probability mass function.  <a href="#a434b3ad1f3964835834dc2a942b820ef">More...</a><br /></td></tr>
<tr class="separator:a434b3ad1f3964835834dc2a942b820ef"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a7133c2e86fd2f6384416ee0e4fd3a60b"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a7133c2e86fd2f6384416ee0e4fd3a60b">bernoulli_quantile</a> (float8 p, float8 sp)</td></tr>
<tr class="memdesc:a7133c2e86fd2f6384416ee0e4fd3a60b"><td class="mdescLeft">&#160;</td><td class="mdescRight">Bernoulli quantile function.  <a href="#a7133c2e86fd2f6384416ee0e4fd3a60b">More...</a><br /></td></tr>
<tr class="separator:a7133c2e86fd2f6384416ee0e4fd3a60b"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a72e1cca872da35592075dbcfb18aed3f"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a72e1cca872da35592075dbcfb18aed3f">beta_cdf</a> (float8 x, float8 alpha, float8 beta)</td></tr>
<tr class="memdesc:a72e1cca872da35592075dbcfb18aed3f"><td class="mdescLeft">&#160;</td><td class="mdescRight">Beta cumulative distribution function.  <a href="#a72e1cca872da35592075dbcfb18aed3f">More...</a><br /></td></tr>
<tr class="separator:a72e1cca872da35592075dbcfb18aed3f"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aa105049e6e3bb9b3891b0ed1b343e28e"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#aa105049e6e3bb9b3891b0ed1b343e28e">beta_pdf</a> (float8 x, float8 alpha, float8 beta)</td></tr>
<tr class="memdesc:aa105049e6e3bb9b3891b0ed1b343e28e"><td class="mdescLeft">&#160;</td><td class="mdescRight">Beta probability density function.  <a href="#aa105049e6e3bb9b3891b0ed1b343e28e">More...</a><br /></td></tr>
<tr class="separator:aa105049e6e3bb9b3891b0ed1b343e28e"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a32433aa742c0504d33e98e28a3e2f190"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a32433aa742c0504d33e98e28a3e2f190">beta_quantile</a> (float8 p, float8 alpha, float8 beta)</td></tr>
<tr class="memdesc:a32433aa742c0504d33e98e28a3e2f190"><td class="mdescLeft">&#160;</td><td class="mdescRight">Beta quantile function.  <a href="#a32433aa742c0504d33e98e28a3e2f190">More...</a><br /></td></tr>
<tr class="separator:a32433aa742c0504d33e98e28a3e2f190"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aa5000bad6e2e4af1c8cbfec7ea884476"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#aa5000bad6e2e4af1c8cbfec7ea884476">binomial_cdf</a> (float8 x, int4 n, float8 sp)</td></tr>
<tr class="memdesc:aa5000bad6e2e4af1c8cbfec7ea884476"><td class="mdescLeft">&#160;</td><td class="mdescRight">Binomial cumulative distribution function.  <a href="#aa5000bad6e2e4af1c8cbfec7ea884476">More...</a><br /></td></tr>
<tr class="separator:aa5000bad6e2e4af1c8cbfec7ea884476"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aa0614475b8685bf8e37533d2ac5bb116"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#aa0614475b8685bf8e37533d2ac5bb116">binomial_pmf</a> (int4 x, int4 n, float8 sp)</td></tr>
<tr class="memdesc:aa0614475b8685bf8e37533d2ac5bb116"><td class="mdescLeft">&#160;</td><td class="mdescRight">Binomial probability mass function.  <a href="#aa0614475b8685bf8e37533d2ac5bb116">More...</a><br /></td></tr>
<tr class="separator:aa0614475b8685bf8e37533d2ac5bb116"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a49f421c58d2e1abd63b83d71af9edf21"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a49f421c58d2e1abd63b83d71af9edf21">binomial_quantile</a> (float8 p, int4 n, float8 sp)</td></tr>
<tr class="memdesc:a49f421c58d2e1abd63b83d71af9edf21"><td class="mdescLeft">&#160;</td><td class="mdescRight">Binomial quantile function.  <a href="#a49f421c58d2e1abd63b83d71af9edf21">More...</a><br /></td></tr>
<tr class="separator:a49f421c58d2e1abd63b83d71af9edf21"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a2d8874c2a5679403a473bfedb14467a4"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a2d8874c2a5679403a473bfedb14467a4">cauchy_cdf</a> (float8 x, float8 location, float8 scale)</td></tr>
<tr class="memdesc:a2d8874c2a5679403a473bfedb14467a4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Cauchy cumulative distribution function.  <a href="#a2d8874c2a5679403a473bfedb14467a4">More...</a><br /></td></tr>
<tr class="separator:a2d8874c2a5679403a473bfedb14467a4"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aebfad9365a7fc7a553c3b5c7931f2450"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#aebfad9365a7fc7a553c3b5c7931f2450">cauchy_pdf</a> (float8 x, float8 location, float8 scale)</td></tr>
<tr class="memdesc:aebfad9365a7fc7a553c3b5c7931f2450"><td class="mdescLeft">&#160;</td><td class="mdescRight">Cauchy probability density function.  <a href="#aebfad9365a7fc7a553c3b5c7931f2450">More...</a><br /></td></tr>
<tr class="separator:aebfad9365a7fc7a553c3b5c7931f2450"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae8aa9b741e89c8d9236a682d218006e0"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#ae8aa9b741e89c8d9236a682d218006e0">cauchy_quantile</a> (float8 p, float8 location, float8 scale)</td></tr>
<tr class="memdesc:ae8aa9b741e89c8d9236a682d218006e0"><td class="mdescLeft">&#160;</td><td class="mdescRight">Cauchy quantile function.  <a href="#ae8aa9b741e89c8d9236a682d218006e0">More...</a><br /></td></tr>
<tr class="separator:ae8aa9b741e89c8d9236a682d218006e0"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a230513b6b549d5b445cbacbdbab42c15"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a230513b6b549d5b445cbacbdbab42c15">chi_squared_cdf</a> (float8 x, float8 df)</td></tr>
<tr class="memdesc:a230513b6b549d5b445cbacbdbab42c15"><td class="mdescLeft">&#160;</td><td class="mdescRight">Chi-squared cumulative distribution function.  <a href="#a230513b6b549d5b445cbacbdbab42c15">More...</a><br /></td></tr>
<tr class="separator:a230513b6b549d5b445cbacbdbab42c15"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a90bccc717d7052e83bafd7f160a783b1"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a90bccc717d7052e83bafd7f160a783b1">chi_squared_pdf</a> (float8 x, float8 df)</td></tr>
<tr class="memdesc:a90bccc717d7052e83bafd7f160a783b1"><td class="mdescLeft">&#160;</td><td class="mdescRight">Chi-squared distribution probability density function.  <a href="#a90bccc717d7052e83bafd7f160a783b1">More...</a><br /></td></tr>
<tr class="separator:a90bccc717d7052e83bafd7f160a783b1"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ad125307fe65a33b60f6dd524037d4548"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#ad125307fe65a33b60f6dd524037d4548">chi_squared_quantile</a> (float8 p, float8 df)</td></tr>
<tr class="memdesc:ad125307fe65a33b60f6dd524037d4548"><td class="mdescLeft">&#160;</td><td class="mdescRight">Chi-squared distribution quantile function.  <a href="#ad125307fe65a33b60f6dd524037d4548">More...</a><br /></td></tr>
<tr class="separator:ad125307fe65a33b60f6dd524037d4548"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a6d1bf6816f56b8e5ba6bf6ca94752f46"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a6d1bf6816f56b8e5ba6bf6ca94752f46">exponential_cdf</a> (float8 x, float8 lambda)</td></tr>
<tr class="memdesc:a6d1bf6816f56b8e5ba6bf6ca94752f46"><td class="mdescLeft">&#160;</td><td class="mdescRight">Exponential cumulative distribution function.  <a href="#a6d1bf6816f56b8e5ba6bf6ca94752f46">More...</a><br /></td></tr>
<tr class="separator:a6d1bf6816f56b8e5ba6bf6ca94752f46"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a18a5458c4bc85f0c4ea321317f90bdbb"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a18a5458c4bc85f0c4ea321317f90bdbb">exponential_pdf</a> (float8 x, float8 lambda)</td></tr>
<tr class="memdesc:a18a5458c4bc85f0c4ea321317f90bdbb"><td class="mdescLeft">&#160;</td><td class="mdescRight">Exponential probability density function.  <a href="#a18a5458c4bc85f0c4ea321317f90bdbb">More...</a><br /></td></tr>
<tr class="separator:a18a5458c4bc85f0c4ea321317f90bdbb"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae3687b8e69a402154b829a6531b1b279"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#ae3687b8e69a402154b829a6531b1b279">exponential_quantile</a> (float8 p, float8 lambda)</td></tr>
<tr class="memdesc:ae3687b8e69a402154b829a6531b1b279"><td class="mdescLeft">&#160;</td><td class="mdescRight">Exponential quantile function.  <a href="#ae3687b8e69a402154b829a6531b1b279">More...</a><br /></td></tr>
<tr class="separator:ae3687b8e69a402154b829a6531b1b279"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:acffffe04c15eccd2e88cdac250bccc68"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#acffffe04c15eccd2e88cdac250bccc68">extreme_value_cdf</a> (float8 x, float8 location, float8 scale)</td></tr>
<tr class="memdesc:acffffe04c15eccd2e88cdac250bccc68"><td class="mdescLeft">&#160;</td><td class="mdescRight">Extreme Value cumulative distribution function.  <a href="#acffffe04c15eccd2e88cdac250bccc68">More...</a><br /></td></tr>
<tr class="separator:acffffe04c15eccd2e88cdac250bccc68"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a03a3494462f4cb8c9fb6212e72b0b2e9"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a03a3494462f4cb8c9fb6212e72b0b2e9">extreme_value_pdf</a> (float8 x, float8 location, float8 scale)</td></tr>
<tr class="memdesc:a03a3494462f4cb8c9fb6212e72b0b2e9"><td class="mdescLeft">&#160;</td><td class="mdescRight">Extreme Value probability density function.  <a href="#a03a3494462f4cb8c9fb6212e72b0b2e9">More...</a><br /></td></tr>
<tr class="separator:a03a3494462f4cb8c9fb6212e72b0b2e9"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aeb5a7d295b83a891774a4fb0ef27c458"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#aeb5a7d295b83a891774a4fb0ef27c458">extreme_value_quantile</a> (float8 p, float8 location, float8 scale)</td></tr>
<tr class="memdesc:aeb5a7d295b83a891774a4fb0ef27c458"><td class="mdescLeft">&#160;</td><td class="mdescRight">Extreme Value quantile function.  <a href="#aeb5a7d295b83a891774a4fb0ef27c458">More...</a><br /></td></tr>
<tr class="separator:aeb5a7d295b83a891774a4fb0ef27c458"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a6c5b3e35531e44098f9d0cbef14cb8a6"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a6c5b3e35531e44098f9d0cbef14cb8a6">fisher_f_cdf</a> (float8 x, float8 df1, float8 df2)</td></tr>
<tr class="memdesc:a6c5b3e35531e44098f9d0cbef14cb8a6"><td class="mdescLeft">&#160;</td><td class="mdescRight">Fisher F cumulative distribution function.  <a href="#a6c5b3e35531e44098f9d0cbef14cb8a6">More...</a><br /></td></tr>
<tr class="separator:a6c5b3e35531e44098f9d0cbef14cb8a6"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a1c7937426379a8913519a6abc5a38ac2"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a1c7937426379a8913519a6abc5a38ac2">fisher_f_pdf</a> (float8 x, float8 df1, float8 df2)</td></tr>
<tr class="memdesc:a1c7937426379a8913519a6abc5a38ac2"><td class="mdescLeft">&#160;</td><td class="mdescRight">Fisher F probability density function.  <a href="#a1c7937426379a8913519a6abc5a38ac2">More...</a><br /></td></tr>
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<tr class="memitem:ab6ed888a5338a0bee9c55edf4d33847f"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#ab6ed888a5338a0bee9c55edf4d33847f">fisher_f_quantile</a> (float8 p, float8 df1, float8 df2)</td></tr>
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<tr class="memitem:a7370b797bf450f9aa54d4fea4d64d611"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a7370b797bf450f9aa54d4fea4d64d611">lognormal_pdf</a> (float8 x, float8 location, float8 scale)</td></tr>
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<tr class="memitem:ab9cbc30424eba30f2df2a32a7e45f138"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#ab9cbc30424eba30f2df2a32a7e45f138">negative_binomial_pmf</a> (int4 x, float8 r, float8 sp)</td></tr>
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<tr class="memdesc:abf9c7d870bcfe68cacaa421749bbdf35"><td class="mdescLeft">&#160;</td><td class="mdescRight">Triangular cumulative distribution function.  <a href="#abf9c7d870bcfe68cacaa421749bbdf35">More...</a><br /></td></tr>
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<tr class="memdesc:a0c511b9748b2f7a21fe56aaf5f66d188"><td class="mdescLeft">&#160;</td><td class="mdescRight">Triangular probability density function.  <a href="#a0c511b9748b2f7a21fe56aaf5f66d188">More...</a><br /></td></tr>
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<tr class="memdesc:a4777540ab1b003ff92d484c4bc26af27"><td class="mdescLeft">&#160;</td><td class="mdescRight">Triangular quantile function.  <a href="#a4777540ab1b003ff92d484c4bc26af27">More...</a><br /></td></tr>
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<tr class="memdesc:aa3a05f4f2e0ef9eb65e828261ecfbed9"><td class="mdescLeft">&#160;</td><td class="mdescRight">Uniform cumulative distribution function.  <a href="#aa3a05f4f2e0ef9eb65e828261ecfbed9">More...</a><br /></td></tr>
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<tr class="memitem:ab90fa34d90a9c75747a34c3f210df239"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#ab90fa34d90a9c75747a34c3f210df239">uniform_pdf</a> (float8 x, float8 lower, float8 upper)</td></tr>
<tr class="memdesc:ab90fa34d90a9c75747a34c3f210df239"><td class="mdescLeft">&#160;</td><td class="mdescRight">Uniform probability density function.  <a href="#ab90fa34d90a9c75747a34c3f210df239">More...</a><br /></td></tr>
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<tr class="memitem:a629587a0fdefb588d28b15517ae5cc04"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a629587a0fdefb588d28b15517ae5cc04">uniform_quantile</a> (float8 p, float8 lower, float8 upper)</td></tr>
<tr class="memdesc:a629587a0fdefb588d28b15517ae5cc04"><td class="mdescLeft">&#160;</td><td class="mdescRight">Uniform quantile function.  <a href="#a629587a0fdefb588d28b15517ae5cc04">More...</a><br /></td></tr>
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<tr class="memitem:a50e4a1883588cd7a4c1ff1017399e4af"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a50e4a1883588cd7a4c1ff1017399e4af">weibull_cdf</a> (float8 x, float8 shape, float8 scale)</td></tr>
<tr class="memdesc:a50e4a1883588cd7a4c1ff1017399e4af"><td class="mdescLeft">&#160;</td><td class="mdescRight">Weibull cumulative distribution function.  <a href="#a50e4a1883588cd7a4c1ff1017399e4af">More...</a><br /></td></tr>
<tr class="separator:a50e4a1883588cd7a4c1ff1017399e4af"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a81a876ae2b8598f060dadb179b9324d2"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#a81a876ae2b8598f060dadb179b9324d2">weibull_pdf</a> (float8 x, float8 shape, float8 scale)</td></tr>
<tr class="memdesc:a81a876ae2b8598f060dadb179b9324d2"><td class="mdescLeft">&#160;</td><td class="mdescRight">Weibull probability density function.  <a href="#a81a876ae2b8598f060dadb179b9324d2">More...</a><br /></td></tr>
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<tr class="memitem:aa544631ddeb7a5c06b995b4383c3b612"><td class="memItemLeft" align="right" valign="top">float8&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="prob_8sql__in.html#aa544631ddeb7a5c06b995b4383c3b612">weibull_quantile</a> (float8 p, float8 shape, float8 scale)</td></tr>
<tr class="memdesc:aa544631ddeb7a5c06b995b4383c3b612"><td class="mdescLeft">&#160;</td><td class="mdescRight">Weibull quantile function.  <a href="#aa544631ddeb7a5c06b995b4383c3b612">More...</a><br /></td></tr>
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</table>
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><dl class="section see"><dt>See also</dt><dd>For an overview of probability functions, see the module description <a class="el" href="group__grp__prob.html">Probability Functions</a>. </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="aea21a931dc5578a570e3370af3d8d43a"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aea21a931dc5578a570e3370af3d8d43a">&#9670;&nbsp;</a></span>bernoulli_cdf()</h2>

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          <td class="memname">float8 bernoulli_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( p \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Bernoulli-distributed random variable with success probability \( \mathit{sp} \) </dd></dl>

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</div>
<a id="a434b3ad1f3964835834dc2a942b820ef"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a434b3ad1f3964835834dc2a942b820ef">&#9670;&nbsp;</a></span>bernoulli_pmf()</h2>

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          <td class="memname">float8 bernoulli_pmf </td>
          <td>(</td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability mass function of a Bernoulli-distributed random variable with success probability \( \mathit{sp} \) </dd></dl>

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</div>
<a id="a7133c2e86fd2f6384416ee0e4fd3a60b"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a7133c2e86fd2f6384416ee0e4fd3a60b">&#9670;&nbsp;</a></span>bernoulli_quantile()</h2>

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          <td class="memname">float8 bernoulli_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>0 if \( p \leq 1 - \mathit{sp} \) and 1 otherwise </dd></dl>

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</div>
<a id="a72e1cca872da35592075dbcfb18aed3f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a72e1cca872da35592075dbcfb18aed3f">&#9670;&nbsp;</a></span>beta_cdf()</h2>

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          <td class="memname">float8 beta_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>alpha</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>beta</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">alpha</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">beta</td><td>Shape \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a beta distributed random variable with shape parameters \( \alpha \) and \( \beta \) </dd></dl>

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<a id="aa105049e6e3bb9b3891b0ed1b343e28e"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa105049e6e3bb9b3891b0ed1b343e28e">&#9670;&nbsp;</a></span>beta_pdf()</h2>

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          <td class="memname">float8 beta_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>alpha</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>beta</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">alpha</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">beta</td><td>Shape \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a beta random variable with shape parameters \( \alpha \) and \( \beta \) </dd></dl>

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<a id="a32433aa742c0504d33e98e28a3e2f190"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a32433aa742c0504d33e98e28a3e2f190">&#9670;&nbsp;</a></span>beta_quantile()</h2>

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          <td class="memname">float8 beta_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>alpha</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>beta</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">alpha</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">beta</td><td>Shape \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is beta distribution random variable with shape parameters \( \alpha \) and \( \beta \) </dd></dl>

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</div>
<a id="aa5000bad6e2e4af1c8cbfec7ea884476"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa5000bad6e2e4af1c8cbfec7ea884476">&#9670;&nbsp;</a></span>binomial_cdf()</h2>

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          <td class="memname">float8 binomial_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>n</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">n</td><td>The number of trials \( n \in \mathbb N_0 \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a binomially distributed random variable with \( n \) trials and success probability \( \mathit{sp} \) </dd></dl>

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<a id="aa0614475b8685bf8e37533d2ac5bb116"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa0614475b8685bf8e37533d2ac5bb116">&#9670;&nbsp;</a></span>binomial_pmf()</h2>

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          <td class="memname">float8 binomial_pmf </td>
          <td>(</td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>n</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">n</td><td>The number of trials \( n \in \mathbb N_0 \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability mass function of a binomially distributed random variable with \( n \) trials and success probability \( \mathit{sp} \) </dd></dl>

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<a id="a49f421c58d2e1abd63b83d71af9edf21"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a49f421c58d2e1abd63b83d71af9edf21">&#9670;&nbsp;</a></span>binomial_quantile()</h2>

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          <td class="memname">float8 binomial_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>n</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">n</td><td>The number of trials \( n \in \mathbb N_0 \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>If \( p &lt; 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a binomially distributed random variable with \( n \) trials and success probability \( \mathit{sp} \). </dd></dl>

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</div>
<a id="a2d8874c2a5679403a473bfedb14467a4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a2d8874c2a5679403a473bfedb14467a4">&#9670;&nbsp;</a></span>cauchy_cdf()</h2>

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          <td class="memname">float8 cauchy_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( x_0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \gamma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Cauchy-distributed random variable with location and scale parameters \( x_0 \) and \( \gamma \), respectively </dd></dl>

</div>
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<a id="aebfad9365a7fc7a553c3b5c7931f2450"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aebfad9365a7fc7a553c3b5c7931f2450">&#9670;&nbsp;</a></span>cauchy_pdf()</h2>

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          <td class="memname">float8 cauchy_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( x_0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \gamma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a Cauchy-distributed random variable with location and scale parameters \( x_0 \) and \( \gamma \), respectively </dd></dl>

</div>
</div>
<a id="ae8aa9b741e89c8d9236a682d218006e0"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae8aa9b741e89c8d9236a682d218006e0">&#9670;&nbsp;</a></span>cauchy_quantile()</h2>

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          <td class="memname">float8 cauchy_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( x_0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \gamma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Cauchy-distributed random variable with location and scale parameters \( x_0 \) and \( \gamma \), respectively </dd></dl>

</div>
</div>
<a id="a230513b6b549d5b445cbacbdbab42c15"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a230513b6b549d5b445cbacbdbab42c15">&#9670;&nbsp;</a></span>chi_squared_cdf()</h2>

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          <td class="memname">float8 chi_squared_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a chi-squared distributed random variable with \( \nu \) degrees of freedom </dd></dl>

</div>
</div>
<a id="a90bccc717d7052e83bafd7f160a783b1"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a90bccc717d7052e83bafd7f160a783b1">&#9670;&nbsp;</a></span>chi_squared_pdf()</h2>

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          <td class="memname">float8 chi_squared_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a chi-squared distributed random variable with \( \nu \) degrees of freedom </dd></dl>

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<a id="ad125307fe65a33b60f6dd524037d4548"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad125307fe65a33b60f6dd524037d4548">&#9670;&nbsp;</a></span>chi_squared_quantile()</h2>

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          <td class="memname">float8 chi_squared_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \mu &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a chi-squared distributed random variable with \( \nu \) degrees of freedom </dd></dl>

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<a id="a6d1bf6816f56b8e5ba6bf6ca94752f46"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a6d1bf6816f56b8e5ba6bf6ca94752f46">&#9670;&nbsp;</a></span>exponential_cdf()</h2>

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          <td class="memname">float8 exponential_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lambda</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">lambda</td><td>Rate parameter \( \lambda &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is an exponentially distributed random variable with rate parameter \( \lambda \) </dd></dl>

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<a id="a18a5458c4bc85f0c4ea321317f90bdbb"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a18a5458c4bc85f0c4ea321317f90bdbb">&#9670;&nbsp;</a></span>exponential_pdf()</h2>

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          <td class="memname">float8 exponential_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lambda</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">lambda</td><td>Rate parameter \( \lambda &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of exponentially distributed random variable with rate parameter \( \lambda \) </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#ae3687b8e69a402154b829a6531b1b279">&#9670;&nbsp;</a></span>exponential_quantile()</h2>

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          <td class="memname">float8 exponential_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lambda</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">lambda</td><td>Rate parameter \( \lambda &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a exponentially distributed random variable with rate parameter \( \lambda \) </dd></dl>

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<a id="acffffe04c15eccd2e88cdac250bccc68"></a>
<h2 class="memtitle"><span class="permalink"><a href="#acffffe04c15eccd2e88cdac250bccc68">&#9670;&nbsp;</a></span>extreme_value_cdf()</h2>

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          <td class="memname">float8 extreme_value_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( \alpha \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is an extreme-value distributed random variable with location and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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<a id="a03a3494462f4cb8c9fb6212e72b0b2e9"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a03a3494462f4cb8c9fb6212e72b0b2e9">&#9670;&nbsp;</a></span>extreme_value_pdf()</h2>

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          <td class="memname">float8 extreme_value_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( \alpha \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of an extreme-value distributed random variable with location and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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<a id="aeb5a7d295b83a891774a4fb0ef27c458"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aeb5a7d295b83a891774a4fb0ef27c458">&#9670;&nbsp;</a></span>extreme_value_quantile()</h2>

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          <td class="memname">float8 extreme_value_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( \alpha \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is an extreme-value distributed random variable with location and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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<a id="a6c5b3e35531e44098f9d0cbef14cb8a6"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a6c5b3e35531e44098f9d0cbef14cb8a6">&#9670;&nbsp;</a></span>fisher_f_cdf()</h2>

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          <td class="memname">float8 fisher_f_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df1</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df2</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df1</td><td>Degrees of freedom in numerator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">df2</td><td>Degrees of freedom in denominator \( \nu_1 &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Fisher F-distributed random variable with parameters \( \nu_1 \) and \( \nu_2 \) </dd></dl>

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<a id="a1c7937426379a8913519a6abc5a38ac2"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a1c7937426379a8913519a6abc5a38ac2">&#9670;&nbsp;</a></span>fisher_f_pdf()</h2>

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          <td class="memname">float8 fisher_f_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df1</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df2</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df1</td><td>Degrees of freedom in numerator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">df2</td><td>Degrees of freedom in denominator \( \nu_1 &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a Fisher F-distributed random variable with parameters \( \nu_1 \) and \( \nu_2 \) </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#ab6ed888a5338a0bee9c55edf4d33847f">&#9670;&nbsp;</a></span>fisher_f_quantile()</h2>

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          <td class="memname">float8 fisher_f_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df1</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df2</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">df1</td><td>Degrees of freedom in numerator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">df2</td><td>Degrees of freedom in denominator \( \nu_1 &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Fisher F-distributed random variable with parameters \( \nu_1 \) and \( \nu_2 \) </dd></dl>

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</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ab6760b0486bad2f1ab0635eb59404e7c">&#9670;&nbsp;</a></span>gamma_cdf()</h2>

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          <td class="memname">float8 gamma_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( k &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \theta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a gamma distributed random variable with shape and scale parameters \( k \) and \( \theta \), respectively </dd></dl>

</div>
</div>
<a id="a6c37e3bda2596accbb46525321a328c4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a6c37e3bda2596accbb46525321a328c4">&#9670;&nbsp;</a></span>gamma_pdf()</h2>

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          <td class="memname">float8 gamma_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( k &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \theta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a gamma distributed random variable with shape and scale parameters \( k \) and \( \theta \), respectively </dd></dl>

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<a id="ac48bbd491bd34831415705c3a0b7bf29"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ac48bbd491bd34831415705c3a0b7bf29">&#9670;&nbsp;</a></span>gamma_quantile()</h2>

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          <td class="memname">float8 gamma_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( k &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \theta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a gamma distributed random variable with shape and scale parameters \( k \) and \( \theta \), respectively </dd></dl>

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</div>
<a id="a00879bdf7d48ceddedb3b4cc33511497"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a00879bdf7d48ceddedb3b4cc33511497">&#9670;&nbsp;</a></span>geometric_cdf()</h2>

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          <td class="memname">float8 geometric_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a geometrically distributed random variable with success probability \( \mathit{sp} \). </dd></dl>

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<a id="a5e08db93bd448a1e2164e106ce5781a4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a5e08db93bd448a1e2164e106ce5781a4">&#9670;&nbsp;</a></span>geometric_pmf()</h2>

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          <td class="memname">float8 geometric_pmf </td>
          <td>(</td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability mass function of a geometrically distributed random variable with success probability \( \mathit{sp} \) </dd></dl>

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<a id="a62674ca958aec0533cdf0a74a1dadea9"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a62674ca958aec0533cdf0a74a1dadea9">&#9670;&nbsp;</a></span>geometric_quantile()</h2>

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          <td class="memname">float8 geometric_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in [0,1] \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>If \( p &lt; 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a geometrically distributed random variable with success probability \( \mathit{sp} \). </dd></dl>

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<a id="a5c48e7fa2fc7bcbc69c7f4da663d457f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a5c48e7fa2fc7bcbc69c7f4da663d457f">&#9670;&nbsp;</a></span>hypergeometric_cdf()</h2>

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          <td class="memname">float8 hypergeometric_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>r</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>n</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>N</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">r</td><td>Number \( r \in \{ 0, 1, \dots, N \} \) of items with distinct property (sometimes called the number of <em>success states</em> in population) </td></tr>
    <tr><td class="paramname">n</td><td>Number \( n \in \{ 0, 1, \dots, N \} \) of draws (without replacement) </td></tr>
    <tr><td class="paramname">N</td><td>Total number \( N \in \mathbb N \) of items </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a hypergeometrically distributed random variable with parameters \( r, n, N \) </dd></dl>

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<a id="afbd2f8d9fb30fb179f59cc14f1fd8d6d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#afbd2f8d9fb30fb179f59cc14f1fd8d6d">&#9670;&nbsp;</a></span>hypergeometric_pmf()</h2>

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          <td class="memname">float8 hypergeometric_pmf </td>
          <td>(</td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>r</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>n</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>N</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">r</td><td>Number \( r \in \{ 0, 1, \dots, N \} \) of items with distinct property (sometimes called the number of <em>success states</em> in population) </td></tr>
    <tr><td class="paramname">n</td><td>Number \( n \in \{ 0, 1, \dots, N \} \) of draws (without replacement) </td></tr>
    <tr><td class="paramname">N</td><td>Total number \( N \in \mathbb N \) of items </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability mass function of a hypergeometrically distributed random variable with parameters \( r, n, N \) </dd></dl>

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<a id="a813cc27fe097e797ed0fb6022c7bb79a"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a813cc27fe097e797ed0fb6022c7bb79a">&#9670;&nbsp;</a></span>hypergeometric_quantile()</h2>

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          <td class="memname">float8 hypergeometric_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>r</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>n</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>N</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">r</td><td>Number \( r \in \{ 0, 1, \dots, N \} \) of items with distinct property (sometimes called the number of <em>success states</em> in population) </td></tr>
    <tr><td class="paramname">n</td><td>Number \( n \in \{ 0, 1, \dots, N \} \) of draws (without replacement) </td></tr>
    <tr><td class="paramname">N</td><td>Total number \( N \in \mathbb N \) of items </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a hypergeometrically distributed random variable with parameters \( r, n, N \) </dd></dl>

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</div>
<a id="a85e9c16aa2c6973ddeb7883a5f153d93"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a85e9c16aa2c6973ddeb7883a5f153d93">&#9670;&nbsp;</a></span>inverse_gamma_cdf()</h2>

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          <td class="memname">float8 inverse_gamma_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is an inverse-gamma distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#a126211c2172a43a654288fa72a2349f9">&#9670;&nbsp;</a></span>inverse_gamma_pdf()</h2>

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          <td class="memname">float8 inverse_gamma_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of an inverse-gamma distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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</div>
<a id="a5876aae01f14729866d4fd52918a65ba"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a5876aae01f14729866d4fd52918a65ba">&#9670;&nbsp;</a></span>inverse_gamma_quantile()</h2>

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          <td class="memname">float8 inverse_gamma_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is an inverse-gamma distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#aeef43f74f583bdff17bd074d9c0d9607">&#9670;&nbsp;</a></span>kolmogorov_cdf()</h2>

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          <td class="memname">float8 kolmogorov_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em></td><td>)</td>
          <td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Kolmogorov distributed random variable</dd></dl>
<dl class="section see"><dt>See also</dt><dd>Kolmogorov-Smirnov test: ks_test() </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#a64e197de8da3761acdeec9db7e211703">&#9670;&nbsp;</a></span>laplace_cdf()</h2>

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          <td class="memname">float8 laplace_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( b &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Laplace-distributed random variable with mean \( \mu \) and variance \( 2 b^2 \) </dd></dl>

</div>
</div>
<a id="a750278ad29d514793f76e159b773f410"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a750278ad29d514793f76e159b773f410">&#9670;&nbsp;</a></span>laplace_pdf()</h2>

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          <td class="memname">float8 laplace_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( b &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a Laplace-distributed random variable with mean \( \mu \) and variance \( 2 b^2 \) </dd></dl>

</div>
</div>
<a id="a77f94fc43d4777fc4f68d18e29454a81"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a77f94fc43d4777fc4f68d18e29454a81">&#9670;&nbsp;</a></span>laplace_quantile()</h2>

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          <td class="memname">float8 laplace_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( b &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Laplace-distributed random variable with mean \( \mu \) and variance \( 2 b^2 \) </dd></dl>

</div>
</div>
<a id="a140f674876813d5e786a4d8ba8d75c87"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a140f674876813d5e786a4d8ba8d75c87">&#9670;&nbsp;</a></span>logistic_cdf()</h2>

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          <td class="memname">float8 logistic_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( s &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a logistically distributed random variable with mean \( \mu \) and scale parameter \( s \) </dd></dl>

</div>
</div>
<a id="afa38eb6c61d3c9825d5c172e6c17dbf7"></a>
<h2 class="memtitle"><span class="permalink"><a href="#afa38eb6c61d3c9825d5c172e6c17dbf7">&#9670;&nbsp;</a></span>logistic_pdf()</h2>

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          <td class="memname">float8 logistic_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( s &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a logistically distributed random variable with mean \( \mu \) and scale parameter \( s \) </dd></dl>

</div>
</div>
<a id="a5a77a0bc5884af2a914a955174892ae2"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a5a77a0bc5884af2a914a955174892ae2">&#9670;&nbsp;</a></span>logistic_quantile()</h2>

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          <td class="memname">float8 logistic_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( s &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a logistically distributed random variable with mean \( \mu \) and scale parameter \( s \) </dd></dl>

</div>
</div>
<a id="a4c05b347f8feb64e1236d21b850af61e"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a4c05b347f8feb64e1236d21b850af61e">&#9670;&nbsp;</a></span>lognormal_cdf()</h2>

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          <td class="memname">float8 lognormal_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( m \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( s &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a lognormally distributed random variable with location and scale parameters \( m \) and \( s \), respectively </dd></dl>

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</div>
<a id="a7370b797bf450f9aa54d4fea4d64d611"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a7370b797bf450f9aa54d4fea4d64d611">&#9670;&nbsp;</a></span>lognormal_pdf()</h2>

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          <td class="memname">float8 lognormal_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( m \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( s &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a lognormally distributed random variable with location and scale parameters \( m \) and \( s \), respectively </dd></dl>

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</div>
<a id="aab3a6de990ae5a81834274a1cf9cad8f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aab3a6de990ae5a81834274a1cf9cad8f">&#9670;&nbsp;</a></span>lognormal_quantile()</h2>

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          <td class="memname">float8 lognormal_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>location</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">location</td><td>Location \( m \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( s &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a lognormally distributed random variable with location and scale parameters \( m \) and \( s \), respectively </dd></dl>

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</div>
<a id="ad0a7e4474f828869fb90e62f8e6f04d7"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad0a7e4474f828869fb90e62f8e6f04d7">&#9670;&nbsp;</a></span>negative_binomial_cdf()</h2>

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          <td class="memname">float8 negative_binomial_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>r</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">r</td><td>Total number \( r &gt; 0 \) of successes in \( x + r \) trials (assuming success in the last trial) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in (0,1] \) in each trial </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a negative-binomially distributed random variable with parameters \( r, \mathit{sp} \) </dd></dl>

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</div>
<a id="ab9cbc30424eba30f2df2a32a7e45f138"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ab9cbc30424eba30f2df2a32a7e45f138">&#9670;&nbsp;</a></span>negative_binomial_pmf()</h2>

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          <td class="memname">float8 negative_binomial_pmf </td>
          <td>(</td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>r</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">r</td><td>Total number \( r &gt; 0 \) of successes in \( x + r \) trials (assuming success in the last trial) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in (0,1] \) in each trial </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability mass function of a negative-binomially distributed random variable with parameters \( r, \mathit{sp} \) </dd></dl>

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</div>
<a id="ad9e541de8b41da2e7b7434f862db4845"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad9e541de8b41da2e7b7434f862db4845">&#9670;&nbsp;</a></span>negative_binomial_quantile()</h2>

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          <td class="memname">float8 negative_binomial_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>r</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">r</td><td>Total number \( r &gt; 0 \) of successes in \( x + r \) trials (assuming success in the last trial) </td></tr>
    <tr><td class="paramname">sp</td><td>Success probability \( \mathit{sp} \in (0,1] \) in each trial </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>If \( p &lt; 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a negative-binomially distributed random variable with parameters \( r, \mathit{sp} \) </dd></dl>

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</div>
<a id="a1361569bd86e41f796c70f8cb277010e"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a1361569bd86e41f796c70f8cb277010e">&#9670;&nbsp;</a></span>non_central_beta_cdf()</h2>

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          <td class="memname">float8 non_central_beta_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>alpha</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>beta</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">alpha</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">beta</td><td>Shape \( \beta &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>Noncentrality parameter \( \delta \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a noncentral-beta distributed random variable with shape parameters \( shape_1 \) and \( shape_2 \) and noncentrality parameter \( \delta \) </dd></dl>

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</div>
<a id="ad4a12c083054f0e2d316ae76c9aaeef7"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad4a12c083054f0e2d316ae76c9aaeef7">&#9670;&nbsp;</a></span>non_central_beta_pdf()</h2>

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          <td class="memname">float8 non_central_beta_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>alpha</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>beta</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">alpha</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">beta</td><td>Shape \( \beta &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>Noncentrality parameter \( \delta \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a noncentral-beta distributed random variable with shape parameters \( shape_1 \) and \( shape_2 \) and noncentrality parameter \( \delta \) </dd></dl>

</div>
</div>
<a id="a3073b409eaee3faa6d43df014662c279"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a3073b409eaee3faa6d43df014662c279">&#9670;&nbsp;</a></span>non_central_beta_quantile()</h2>

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          <td class="memname">float8 non_central_beta_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>alpha</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>beta</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">alpha</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">beta</td><td>Shape \( \beta &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>Noncentrality parameter \( \delta \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral-beta distributed random variable with shape parameters \( shape_1 \) and \( shape_2 \) and noncentrality parameter \( \delta \) </dd></dl>

</div>
</div>
<a id="ab4b7d2cf10bb031328dcc34c6ff494ad"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ab4b7d2cf10bb031328dcc34c6ff494ad">&#9670;&nbsp;</a></span>non_central_chi_squared_cdf()</h2>

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          <td class="memname">float8 non_central_chi_squared_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>The noncentrality parameter \( \lambda \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a noncentral-chi-squared distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \lambda \) </dd></dl>

</div>
</div>
<a id="aa7a563183224593d1e0d623a3c5489d8"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa7a563183224593d1e0d623a3c5489d8">&#9670;&nbsp;</a></span>non_central_chi_squared_pdf()</h2>

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          <td class="memname">float8 non_central_chi_squared_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>The noncentrality parameter \( \lambda \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a noncentral-chi-squared distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \lambda \) </dd></dl>

</div>
</div>
<a id="ad694e29187b629ae683ef1235d2b9270"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ad694e29187b629ae683ef1235d2b9270">&#9670;&nbsp;</a></span>non_central_chi_squared_quantile()</h2>

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          <td class="memname">float8 non_central_chi_squared_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>The noncentrality parameter \( \lambda \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral-chi-squared distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \lambda \) </dd></dl>

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</div>
<a id="a00051df630007b530ce86b4ab44a0434"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a00051df630007b530ce86b4ab44a0434">&#9670;&nbsp;</a></span>non_central_f_cdf()</h2>

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          <td class="memname">float8 non_central_f_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df1</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df2</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df1</td><td>Degrees of freedom in numerator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">df2</td><td>Degrees of freedom in denominator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>The noncentrality parameter \( \lambda \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a noncentral Fisher F-distributed random variable with parameters \( \nu_1, \nu_2, \lambda \) </dd></dl>

</div>
</div>
<a id="a3d94edcf90fca1fa52671293a9ea9c2f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a3d94edcf90fca1fa52671293a9ea9c2f">&#9670;&nbsp;</a></span>non_central_f_pdf()</h2>

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          <td class="memname">float8 non_central_f_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df1</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df2</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df1</td><td>Degrees of freedom in numerator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">df2</td><td>Degrees of freedom in denominator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>The noncentrality parameter \( \lambda \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a noncentral Fisher F-distributed random variable with parameters \( \nu_1, \nu_2, \lambda \) </dd></dl>

</div>
</div>
<a id="a92b2a978db480a6c78cfb708107ecb92"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a92b2a978db480a6c78cfb708107ecb92">&#9670;&nbsp;</a></span>non_central_f_quantile()</h2>

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          <td class="memname">float8 non_central_f_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df1</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df2</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">df1</td><td>Degrees of freedom in numerator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">df2</td><td>Degrees of freedom in denominator \( \nu_1 &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>The noncentrality parameter \( \lambda \geq 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral Fisher F-distributed random variable with parameters \( \nu_1, \nu_2, \lambda \) </dd></dl>

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<a id="afaf4374d2720b230a54713e21ecb1955"></a>
<h2 class="memtitle"><span class="permalink"><a href="#afaf4374d2720b230a54713e21ecb1955">&#9670;&nbsp;</a></span>non_central_t_cdf()</h2>

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          <td class="memname">float8 non_central_t_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>Noncentrality parameter \( \delta \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a noncentral Student's t-distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \delta \) </dd></dl>

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</div>
<a id="a4799e3bb68a496d9bc1ef1ea85265409"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a4799e3bb68a496d9bc1ef1ea85265409">&#9670;&nbsp;</a></span>non_central_t_pdf()</h2>

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          <td class="memname">float8 non_central_t_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>Noncentrality parameter \( \delta \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral Student's t-distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \delta \) </dd></dl>

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<a id="af50865aba2ece2e23b2af461a02f7d12"></a>
<h2 class="memtitle"><span class="permalink"><a href="#af50865aba2ece2e23b2af461a02f7d12">&#9670;&nbsp;</a></span>non_central_t_quantile()</h2>

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          <td class="memname">float8 non_central_t_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>ncp</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
    <tr><td class="paramname">ncp</td><td>Noncentrality parameter \( \delta \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a noncentral Student's t-distributed random variable with \( \nu \) degrees of freedom and noncentrality parameter \( \delta \) </dd></dl>

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</div>
<a id="aebcd34ad7b1ca4b31d9699112c9a3b90"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aebcd34ad7b1ca4b31d9699112c9a3b90">&#9670;&nbsp;</a></span>normal_cdf() <span class="overload">[1/3]</span></h2>

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          <td class="memname">float8 normal_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em> = <code>0</code>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sd</em> = <code>1</code>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">sd</td><td>Standard deviation \( \sigma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( T \) is a normally distributed random variable with mean \( \mu \) and variance \( \sigma^2 \) </dd></dl>

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</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a370e31a46781ed8832b31625a683d053">&#9670;&nbsp;</a></span>normal_cdf() <span class="overload">[2/3]</span></h2>

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          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">

</div>
</div>
<a id="a6c0a499faa80db26c0178f1e69cf7a50"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a6c0a499faa80db26c0178f1e69cf7a50">&#9670;&nbsp;</a></span>normal_cdf() <span class="overload">[3/3]</span></h2>

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          <td class="memname">float8 normal_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em></td><td>)</td>
          <td></td>
        </tr>
      </table>
</div><div class="memdoc">

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<h2 class="memtitle"><span class="permalink"><a href="#a63f555f36385d86e229cdca223e39567">&#9670;&nbsp;</a></span>normal_pdf() <span class="overload">[1/3]</span></h2>

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          <td class="memname">float8 normal_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em> = <code>0</code>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sd</em> = <code>1</code>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">sd</td><td>Standard deviation \( \sigma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a normally distributed random variable with mean \( \mu \) and variance \( \sigma^2 \) </dd></dl>

</div>
</div>
<a id="a4ee303e56677465989c30230c7f55004"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a4ee303e56677465989c30230c7f55004">&#9670;&nbsp;</a></span>normal_pdf() <span class="overload">[2/3]</span></h2>

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          <td class="memname">float8 normal_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">

</div>
</div>
<a id="a2836483a456c4bf2c9886763b270317e"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a2836483a456c4bf2c9886763b270317e">&#9670;&nbsp;</a></span>normal_pdf() <span class="overload">[3/3]</span></h2>

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          <td class="memname">float8 normal_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em></td><td>)</td>
          <td></td>
        </tr>
      </table>
</div><div class="memdoc">

</div>
</div>
<a id="a53d56b672fe4cd1277cb5eac5de5118f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a53d56b672fe4cd1277cb5eac5de5118f">&#9670;&nbsp;</a></span>normal_quantile() <span class="overload">[1/3]</span></h2>

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          <td class="memname">float8 normal_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em> = <code>0</code>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>sd</em> = <code>1</code>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">mean</td><td>Mean \( \mu \) </td></tr>
    <tr><td class="paramname">sd</td><td>Standard deviation \( \sigma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a normally distributed random variable with mean \( \mu \) and variance \( \sigma^2 \) </dd></dl>

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</div>
<a id="a1b1f9d685f262ade8863ab3a1632b8d6"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a1b1f9d685f262ade8863ab3a1632b8d6">&#9670;&nbsp;</a></span>normal_quantile() <span class="overload">[2/3]</span></h2>

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          <td class="memname">float8 normal_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">

</div>
</div>
<a id="a643bb4b6b880b96cf924c16e08b015d3"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a643bb4b6b880b96cf924c16e08b015d3">&#9670;&nbsp;</a></span>normal_quantile() <span class="overload">[3/3]</span></h2>

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          <td class="memname">float8 normal_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em></td><td>)</td>
          <td></td>
        </tr>
      </table>
</div><div class="memdoc">

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</div>
<a id="aa1a42ebd68f20f65bc1784b427721b5d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa1a42ebd68f20f65bc1784b427721b5d">&#9670;&nbsp;</a></span>pareto_cdf()</h2>

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          <td class="memname">float8 pareto_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Pareto-distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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</div>
<a id="a22c56a6e48bc442435b13afac2a1eb37"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a22c56a6e48bc442435b13afac2a1eb37">&#9670;&nbsp;</a></span>pareto_pdf()</h2>

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          <td class="memname">float8 pareto_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a Pareto-distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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<a id="a77779e2b5fa951189ccba6806c503c4d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a77779e2b5fa951189ccba6806c503c4d">&#9670;&nbsp;</a></span>pareto_quantile()</h2>

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          <td class="memname">float8 pareto_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Pareto-distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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<a id="ae0b4313d9fe730d6efb3f7c44206f345"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ae0b4313d9fe730d6efb3f7c44206f345">&#9670;&nbsp;</a></span>poisson_cdf()</h2>

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          <td class="memname">float8 poisson_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Average occurrence rate \( \lambda &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Poisson distributed random variable with mean \( \lambda \) </dd></dl>

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</div>
<a id="a82f1edc27261021c73cd080ff2677a9f"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a82f1edc27261021c73cd080ff2677a9f">&#9670;&nbsp;</a></span>poisson_pmf()</h2>

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          <td class="memname">float8 poisson_pmf </td>
          <td>(</td>
          <td class="paramtype">int4&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">mean</td><td>Average occurrence rate \( \lambda &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability mass function of a Poisson distributed random variable with mean \( \lambda \) </dd></dl>

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</div>
<a id="a032d26db18b2ee1034085f5521939c61"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a032d26db18b2ee1034085f5521939c61">&#9670;&nbsp;</a></span>poisson_quantile()</h2>

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          <td class="memname">float8 poisson_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mean</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">mean</td><td>Average occurrence rate \( \lambda &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>If \( p &lt; 0.5 \) the maximum \( x \) such that \( p \geq \Pr[X \leq x] \). If \( p \geq 0.5 \) the minimum \( x \) such that \( p \leq \Pr[X \leq x] \). Here, \( X \) is a Poisson distributed random variable with mean \( \lambda \) </dd></dl>

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</div>
<a id="aab0ddb8a5348cfa387d777043a3cb6d0"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aab0ddb8a5348cfa387d777043a3cb6d0">&#9670;&nbsp;</a></span>rayleigh_cdf()</h2>

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          <td class="memname">float8 rayleigh_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \sigma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Rayleigh-distributed random variable with parameter \( \sigma \) </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#a798541736d9255bdd5c0bd94924d47bc">&#9670;&nbsp;</a></span>rayleigh_pdf()</h2>

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          <td class="memname">float8 rayleigh_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \sigma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a Rayleigh-distributed random variable with parameter \( \sigma \) </dd></dl>

</div>
</div>
<a id="acd6757acab1683c735e2b57901494336"></a>
<h2 class="memtitle"><span class="permalink"><a href="#acd6757acab1683c735e2b57901494336">&#9670;&nbsp;</a></span>rayleigh_quantile()</h2>

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          <td class="memname">float8 rayleigh_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \sigma &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Rayleigh-distributed random variable with parameter \( \sigma \) </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#a5322531131074c23a2dbf067ee504ef7">&#9670;&nbsp;</a></span>students_t_cdf()</h2>

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          <td class="memname">float8 students_t_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a Student's t-distributed random variable with \( \nu \) degrees of freedom </dd></dl>

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<h2 class="memtitle"><span class="permalink"><a href="#a8815c21670fff9d31946553a84b845b1">&#9670;&nbsp;</a></span>students_t_pdf()</h2>

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          <td class="memname">float8 students_t_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a Stundent's t-distributed random variable with \( \nu \) degrees of freedom </dd></dl>

</div>
</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a7d64add02af21a95d73502b2dd466a75">&#9670;&nbsp;</a></span>students_t_quantile()</h2>

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          <td class="memname">float8 students_t_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>df</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">df</td><td>Degrees of freedom \( \nu &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a Student's t-distributed random variable with \( \nu \) degrees of freedom </dd></dl>

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<a id="abf9c7d870bcfe68cacaa421749bbdf35"></a>
<h2 class="memtitle"><span class="permalink"><a href="#abf9c7d870bcfe68cacaa421749bbdf35">&#9670;&nbsp;</a></span>triangular_cdf()</h2>

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          <td class="memname">float8 triangular_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lower</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mode</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>upper</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">lower</td><td>Lower bound \( a \) </td></tr>
    <tr><td class="paramname">mode</td><td>Mode \( c \geq a \) </td></tr>
    <tr><td class="paramname">upper</td><td>Upper bound \( b \geq c \), where \( b &gt; a \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a triangular distributed random variable with parameters \( a, b, c \) </dd></dl>

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<a id="a0c511b9748b2f7a21fe56aaf5f66d188"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a0c511b9748b2f7a21fe56aaf5f66d188">&#9670;&nbsp;</a></span>triangular_pdf()</h2>

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          <td class="memname">float8 triangular_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lower</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mode</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>upper</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">lower</td><td>Lower bound \( a \) </td></tr>
    <tr><td class="paramname">mode</td><td>Mode \( c \geq a \) </td></tr>
    <tr><td class="paramname">upper</td><td>Upper bound \( b \geq c \), where \( b &gt; a \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a triangular distributed random variable with parameters \( a, b, c \) </dd></dl>

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<a id="a4777540ab1b003ff92d484c4bc26af27"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a4777540ab1b003ff92d484c4bc26af27">&#9670;&nbsp;</a></span>triangular_quantile()</h2>

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          <td class="memname">float8 triangular_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lower</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>mode</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>upper</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">lower</td><td>Lower bound \( a \) </td></tr>
    <tr><td class="paramname">mode</td><td>Mode \( c \geq a \) </td></tr>
    <tr><td class="paramname">upper</td><td>Upper bound \( b \geq c \), where \( b &gt; a \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a trianbular distributed random variable with parameters \( a, b, c \) </dd></dl>

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<a id="aa3a05f4f2e0ef9eb65e828261ecfbed9"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa3a05f4f2e0ef9eb65e828261ecfbed9">&#9670;&nbsp;</a></span>uniform_cdf()</h2>

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          <td class="memname">float8 uniform_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lower</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>upper</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">lower</td><td>Lower bound \( a \) </td></tr>
    <tr><td class="paramname">upper</td><td>Upper bound \( b \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a uniform distributed random variable with support \( [a, b] \) </dd></dl>

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</div>
<a id="ab90fa34d90a9c75747a34c3f210df239"></a>
<h2 class="memtitle"><span class="permalink"><a href="#ab90fa34d90a9c75747a34c3f210df239">&#9670;&nbsp;</a></span>uniform_pdf()</h2>

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        <tr>
          <td class="memname">float8 uniform_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lower</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>upper</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">lower</td><td>Lower bound \( a \) </td></tr>
    <tr><td class="paramname">upper</td><td>Upper bound \( b \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a uniform distributed random variable with support \( [a, b] \) </dd></dl>

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<a id="a629587a0fdefb588d28b15517ae5cc04"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a629587a0fdefb588d28b15517ae5cc04">&#9670;&nbsp;</a></span>uniform_quantile()</h2>

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          <td class="memname">float8 uniform_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>lower</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>upper</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">lower</td><td>Lower bound \( a \) </td></tr>
    <tr><td class="paramname">upper</td><td>Upper bound \( b \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a uniform distributed random variable with support \( [a, b] \) </dd></dl>

</div>
</div>
<a id="a50e4a1883588cd7a4c1ff1017399e4af"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a50e4a1883588cd7a4c1ff1017399e4af">&#9670;&nbsp;</a></span>weibull_cdf()</h2>

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          <td class="memname">float8 weibull_cdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( \Pr[X \leq x] \) where \( X \) is a weibull distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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</div>
<a id="a81a876ae2b8598f060dadb179b9324d2"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a81a876ae2b8598f060dadb179b9324d2">&#9670;&nbsp;</a></span>weibull_pdf()</h2>

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          <td class="memname">float8 weibull_pdf </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>x</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">x</td><td>Random variate \( x \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( f(x) \) where \( f \) is the probability density function of a weibull distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

</div>
</div>
<a id="aa544631ddeb7a5c06b995b4383c3b612"></a>
<h2 class="memtitle"><span class="permalink"><a href="#aa544631ddeb7a5c06b995b4383c3b612">&#9670;&nbsp;</a></span>weibull_quantile()</h2>

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          <td class="memname">float8 weibull_quantile </td>
          <td>(</td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>p</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>shape</em>, </td>
        </tr>
        <tr>
          <td class="paramkey"></td>
          <td></td>
          <td class="paramtype">float8&#160;</td>
          <td class="paramname"><em>scale</em>&#160;</td>
        </tr>
        <tr>
          <td></td>
          <td>)</td>
          <td></td><td></td>
        </tr>
      </table>
</div><div class="memdoc">
<dl class="params"><dt>Parameters</dt><dd>
  <table class="params">
    <tr><td class="paramname">p</td><td>Probability \( p \in [0,1] \) </td></tr>
    <tr><td class="paramname">shape</td><td>Shape \( \alpha &gt; 0 \) </td></tr>
    <tr><td class="paramname">scale</td><td>Scale \( \beta &gt; 0 \) </td></tr>
  </table>
  </dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>\( x \) such that \( p = \Pr[X \leq x] \) where \( X \) is a weibull distributed random variable with shape and scale parameters \( \alpha \) and \( \beta \), respectively </dd></dl>

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