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 <div class="title">Probability Functions</div>  </div>
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 <dl class="user"><dt><b>About:</b></dt><dd></dd></dl>
 <p>The Probability Functions module provides cumulative distribution, density/mass, and quantile functions for a wide range of probability distributions.</p>
 <p>Unless otherwise documented, all of these functions are wrappers around functionality provided by the boost C++ library [1, “<a href="http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/math_toolkit/dist.html">Statistical Distributions and Functions</a>”].</p>
 <p>For convenience, all cumulative distribution and density/mass functions (CDFs and PDF/PMFs in short) are defined over the range of all floating-point numbers including infinity. Inputs that are <code>NULL</code> or <code>NaN</code> (not a number) will always produce a <code>NULL</code> or <code>NaN</code> result, respectively. Inputs that are plus or minus infinity will return the respective limits.</p>
 <p>A quantile function for a probability distrution with CDF \( F \) takes a probability argument \( p \in [0,1] \) and returns the value \( x \) so that \( F(x) = p \), provided such an \( x \) exists and it is unique. If it does not, the result will be \( \sup \{ x \in D \mid F(x) \leq p \} \) (interpreted as 0 if the supremum is over an empty set) if \( p &lt; 0.5 \), and \( \inf \{ x \in D \mid F(x) \geq p \} \) if \( p \geq 0.5 \). Here \( D \) denotes the domain of the distribution, which is the set of reals \( \mathbb R \) for continuous and the set of nonnegative integers \( \mathbb N_0 \) for discrete distributions.</p>
 <p>Intuitively, the formulas in the previous paragraph deal with the following special cases. The 0-quantile will always be the “left end” of the support, and the 1-quantile will be the “right end” of the support of the distribution. For discrete distributions, most values of \( p \in [0,1] \) do not admit an \( x \) with \( F(x) = p \). Instead, there is an \( x \in \mathbb N_0 \) so that \( F(x) &lt; p &lt; F(x + 1) \). The above formulas mean that the value returned as \( p \)-quantile is \( x \) if \( p &lt; 0.5 \), and it is \( x + 1 \) if \( p \geq 0.5 \). (As a special case, in order to ensure that quantiles are always within the support, the \( p \)-quantile will be 0 if \( p &lt; F(0) \)).</p>
 <p>The rationale for choosing this behavior is that \(p\)-quantiles for \( p &lt; 0.5 \) are typically requested when interested in the value \( x \) such that with confidence level <b>at least</b> \( 1 - p \) a random variable will be \( &gt; x \) (or equivalently, with probability <b>at most</b> \( p \), it will be \( \leq x \)). Likewise, \(p\)-quantiles for \( p \geq 0.5 \) are typically requested when interested in the value \( x \) such that with confidence level <b>at least</b> \( p \) a random variable will be \( \leq x \). See also [1, “<a href="http://www.boost.org/doc/libs/1_46_1/libs/math/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html">Understanding Quantiles of Discrete Distributions</a>”].</p>
 <dl class="user"><dt><b>Usage:</b></dt><dd></dd></dl>
 <ul>
 <li>Cumulative distribution functions: <pre>SELECT <em>distribution</em>_cdf(<em>random variate</em>[, <em>parameter1</em> [, <em>parameter2</em> [, <em>parameter3</em>] ] ])</pre></li>
 <li>Probability density/mass functions: <pre>SELECT <em>distribution</em>_{pdf|pmf}(<em>random variate</em>[, <em>parameter1</em> [, <em>parameter2</em> [, <em>parameter3</em>] ] ])</pre></li>
 <li>Quantile functions: <pre>SELECT <em>distribution</em>_quantile(<em>probability</em>[, <em>parameter1</em> [, <em>parameter2</em> [, <em>parameter3</em>] ] ])</pre></li>
 </ul>
 <p>For concrete function signatures, see <a class="el" href="prob_8sql__in.html">prob.sql_in</a>.</p>
 <dl class="user"><dt><b>Examples:</b></dt><dd></dd></dl>
 <div class="fragment"><pre class="fragment">
 sql&gt; SELECT normal_cdf(0);
  normal_cdf
 ------------
         0.5

 sql&gt; SELECT normal_quantile(0.5, 0, 1);
  normal_quantile
 -----------------
                0
 (1 row)
 </pre></div><dl class="user"><dt><b>Literature:</b></dt><dd></dd></dl>
 <p>[1] John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg: <em>Boost Math Toolkit</em>, Version 1.49, available at: <a href="http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/index.html">http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/index.html</a></p>
 <dl class="see"><dt><b>See also:</b></dt><dd>File <a class="el" href="prob_8sql__in.html" title="SQL functions for evaluating probability functions.">prob.sql_in</a> documenting the SQL functions. </dd></dl>
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	<dl class="user"><dt><b>About:</b></dt><dd></dd></dl>
	<p>The Probability Functions module provides cumulative distribution, density/mass, and quantile functions for a wide range of probability distributions.</p>
	<p>Unless otherwise documented, all of these functions are wrappers around functionality provided by the boost C++ library [1, “<a href="http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/math_toolkit/dist.html">Statistical Distributions and Functions</a>”].</p>
	<p>For convenience, all cumulative distribution and density/mass functions (CDFs and PDF/PMFs in short) are defined over the range of all floating-point numbers including infinity. Inputs that are <code>NULL</code> or <code>NaN</code> (not a number) will always produce a <code>NULL</code> or <code>NaN</code> result, respectively. Inputs that are plus or minus infinity will return the respective limits.</p>
	<p>A quantile function for a probability distrution with CDF \( F \) takes a probability argument \( p \in [0,1] \) and returns the value \( x \) so that \( F(x) = p \), provided such an \( x \) exists and it is unique. If it does not, the result will be \( \sup \{ x \in D \mid F(x) \leq p \} \) (interpreted as 0 if the supremum is over an empty set) if \( p < 0.5 \), and \( \inf \{ x \in D \mid F(x) \geq p \} \) if \( p \geq 0.5 \). Here \( D \) denotes the domain of the distribution, which is the set of reals \( \mathbb R \) for continuous and the set of nonnegative integers \( \mathbb N_0 \) for discrete distributions.</p>
	<p>Intuitively, the formulas in the previous paragraph deal with the following special cases. The 0-quantile will always be the “left end” of the support, and the 1-quantile will be the “right end” of the support of the distribution. For discrete distributions, most values of \( p \in [0,1] \) do not admit an \( x \) with \( F(x) = p \). Instead, there is an \( x \in \mathbb N_0 \) so that \( F(x) < p < F(x + 1) \). The above formulas mean that the value returned as \( p \)-quantile is \( x \) if \( p < 0.5 \), and it is \( x + 1 \) if \( p \geq 0.5 \). (As a special case, in order to ensure that quantiles are always within the support, the \( p \)-quantile will be 0 if \( p < F(0) \)).</p>
	<p>The rationale for choosing this behavior is that \(p\)-quantiles for \( p < 0.5 \) are typically requested when interested in the value \( x \) such that with confidence level <b>at least</b> \( 1 - p \) a random variable will be \( > x \) (or equivalently, with probability <b>at most</b> \( p \), it will be \( \leq x \)). Likewise, \(p\)-quantiles for \( p \geq 0.5 \) are typically requested when interested in the value \( x \) such that with confidence level <b>at least</b> \( p \) a random variable will be \( \leq x \). See also [1, “<a href="http://www.boost.org/doc/libs/1_46_1/libs/math/doc/sf_and_dist/html/math_toolkit/policy/pol_tutorial/understand_dis_quant.html">Understanding Quantiles of Discrete Distributions</a>”].</p>
	<dl class="user"><dt><b>Usage:</b></dt><dd></dd></dl>
	<ul>
	<li>Cumulative distribution functions: <pre>SELECT <em>distribution</em>_cdf(<em>random variate</em>[, <em>parameter1</em> [, <em>parameter2</em> [, <em>parameter3</em>] ] ])</pre></li>
	<li>Probability density/mass functions: <pre>SELECT <em>distribution</em>_{pdf\|pmf}(<em>random variate</em>[, <em>parameter1</em> [, <em>parameter2</em> [, <em>parameter3</em>] ] ])</pre></li>
	<li>Quantile functions: <pre>SELECT <em>distribution</em>_quantile(<em>probability</em>[, <em>parameter1</em> [, <em>parameter2</em> [, <em>parameter3</em>] ] ])</pre></li>
	</ul>
	<p>For concrete function signatures, see <a class="el" href="prob_8sql__in.html">prob.sql_in</a>.</p>
	<dl class="user"><dt><b>Examples:</b></dt><dd></dd></dl>
	<div class="fragment"><pre class="fragment">
	sql> SELECT normal_cdf(0);
	normal_cdf
	------------
	0.5

	sql> SELECT normal_quantile(0.5, 0, 1);
	normal_quantile
	-----------------
	0
	(1 row)
	</pre></div><dl class="user"><dt><b>Literature:</b></dt><dd></dd></dl>
	<p>[1] John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg: <em>Boost Math Toolkit</em>, Version 1.49, available at: <a href="http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/index.html">http://www.boost.org/doc/libs/1_49_0/libs/math/doc/sf_and_dist/html/index.html</a></p>
	<dl class="see"><dt><b>See also:</b></dt><dd>File <a class="el" href="prob_8sql__in.html" title="SQL functions for evaluating probability functions.">prob.sql_in</a> documenting the SQL functions. </dd></dl>
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