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 <div class="titlepage"><div><div><h4 class="title">
 <a name="math_toolkit.dist.stat_tut.dist_params"></a><a class="link" href="dist_params.html" title="Discrete Probability Distributions"> Discrete Probability
         Distributions</a>
 </h4></div></div></div>
 <p>
           Note that the <a href="http://en.wikipedia.org/wiki/Discrete_probability_distribution" target="_top">discrete
           distributions</a>, including the binomial, negative binomial, Poisson
           &amp; Bernoulli, are all mathematically defined as discrete functions:
           only integral values of the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
           variate</a> are envisaged and the functions are only defined at these
           integral values. However because the method of calculation often uses continuous
           functions, it is convenient to treat them as if they were continuous functions,
           and permit non-integral values of their parameters.
         </p>
 <p>
           To enforce a strict mathematical model, users may use floor or ceil functions
           on the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
           variate</a>, prior to calling the distribution function, to enforce
           integral values.
         </p>
 <p>
           For similar reasons, in continuous distributions, parameters like degrees
           of freedom that might appear to be integral, are treated as real values
           (and are promoted from integer to floating-point if necessary). In this
           case however, that there are a small number of situations where non-integral
           degrees of freedom do have a genuine meaning.
         </p>
 <p>
           Generally speaking there is no loss of performance from allowing real-values
           parameters: the underlying special functions contain optimizations for
           integer-valued arguments when applicable.
         </p>
 <div class="caution"><table border="0" summary="Caution">
 <tr>
 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td>
 <th align="left">Caution</th>
 </tr>
 <tr><td align="left" valign="top">
 <p>
             The quantile function of a discrete distribution will by default return
             an integer result that has been <span class="emphasis"><em>rounded outwards</em></span>.
             That is to say lower quantiles (where the probability is less than 0.5)
             are rounded downward, and upper quantiles (where the probability is greater
             than 0.5) are rounded upwards. This behaviour ensures that if an X% quantile
             is requested, then <span class="emphasis"><em>at least</em></span> the requested coverage
             will be present in the central region, and <span class="emphasis"><em>no more than</em></span>
             the requested coverage will be present in the tails.
           </p>
 <p>
             This behaviour can be changed so that the quantile functions are rounded
             differently, or even return a real-valued result using <a class="link" href="../../policy/pol_overview.html" title="Policy Overview">Policies</a>.
             It is strongly recommended that you read the tutorial <a class="link" href="../../policy/pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
             Quantiles of Discrete Distributions</a> before using the quantile
             function on a discrete distribution. The <a class="link" href="../../policy/pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
             docs</a> describe how to change the rounding policy for these distributions.
           </p>
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 <td align="right"><div class="copyright-footer">Copyright &#169; 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow,
       Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan R&#229;de, Gautam Sewani and
       Thijs van den Berg<p>
         Distributed under the Boost Software License, Version 1.0. (See accompanying
         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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	<title>Discrete Probability Distributions</title>
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	</div>
	<div class="section" lang="en">
	<div class="titlepage"><div><div><h4 class="title">
	<a name="math_toolkit.dist.stat_tut.dist_params"></a><a class="link" href="dist_params.html" title="Discrete Probability Distributions"> Discrete Probability
	Distributions</a>
	</h4></div></div></div>
	<p>
	Note that the <a href="http://en.wikipedia.org/wiki/Discrete_probability_distribution" target="_top">discrete
	distributions</a>, including the binomial, negative binomial, Poisson
	& Bernoulli, are all mathematically defined as discrete functions:
	only integral values of the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
	variate</a> are envisaged and the functions are only defined at these
	integral values. However because the method of calculation often uses continuous
	functions, it is convenient to treat them as if they were continuous functions,
	and permit non-integral values of their parameters.
	</p>
	<p>
	To enforce a strict mathematical model, users may use floor or ceil functions
	on the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
	variate</a>, prior to calling the distribution function, to enforce
	integral values.
	</p>
	<p>
	For similar reasons, in continuous distributions, parameters like degrees
	of freedom that might appear to be integral, are treated as real values
	(and are promoted from integer to floating-point if necessary). In this
	case however, that there are a small number of situations where non-integral
	degrees of freedom do have a genuine meaning.
	</p>
	<p>
	Generally speaking there is no loss of performance from allowing real-values
	parameters: the underlying special functions contain optimizations for
	integer-valued arguments when applicable.
	</p>
	<div class="caution"><table border="0" summary="Caution">
	<tr>
	<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td>
	<th align="left">Caution</th>
	</tr>
	<tr><td align="left" valign="top">
	<p>
	The quantile function of a discrete distribution will by default return
	an integer result that has been <span class="emphasis"><em>rounded outwards</em></span>.
	That is to say lower quantiles (where the probability is less than 0.5)
	are rounded downward, and upper quantiles (where the probability is greater
	than 0.5) are rounded upwards. This behaviour ensures that if an X% quantile
	is requested, then <span class="emphasis"><em>at least</em></span> the requested coverage
	will be present in the central region, and <span class="emphasis"><em>no more than</em></span>
	the requested coverage will be present in the tails.
	</p>
	<p>
	This behaviour can be changed so that the quantile functions are rounded
	differently, or even return a real-valued result using <a class="link" href="../../policy/pol_overview.html" title="Policy Overview">Policies</a>.
	It is strongly recommended that you read the tutorial <a class="link" href="../../policy/pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
	Quantiles of Discrete Distributions</a> before using the quantile
	function on a discrete distribution. The <a class="link" href="../../policy/pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
	docs</a> describe how to change the rounding policy for these distributions.
	</p>
	</td></tr>
	</table></div>
	</div>
	<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
	<td align="left"></td>
	<td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow,
	Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and
	Thijs van den Berg<p>
	Distributed under the Boost Software License, Version 1.0. (See accompanying
	file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
	</p>
	</div></td>
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