102 lines
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102 lines
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Discrete Probability Distributions</title>
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<table cellpadding="2" width="100%"><tr>
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.stat_tut.dist_params"></a><a class="link" href="dist_params.html" title="Discrete Probability Distributions">Discrete Probability
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Distributions</a>
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</h3></div></div></div>
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<p>
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Note that the <a href="http://en.wikipedia.org/wiki/Discrete_probability_distribution" target="_top">discrete
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distributions</a>, including the binomial, negative binomial, Poisson
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& Bernoulli, are all mathematically defined as discrete functions: only
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integral values of the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
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variate</a> are envisaged and the functions are only defined at these
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integral values. However because the method of calculation often uses continuous
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functions, it is convenient to treat them as if they were continuous functions,
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and permit non-integral values of their parameters.
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</p>
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<p>
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To enforce a strict mathematical model, users may use floor or ceil functions
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on the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a>,
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prior to calling the distribution function, to enforce integral values.
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</p>
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<p>
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For similar reasons, in continuous distributions, parameters like degrees
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of freedom that might appear to be integral, are treated as real values (and
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are promoted from integer to floating-point if necessary). In this case however,
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that there are a small number of situations where non-integral degrees of
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freedom do have a genuine meaning.
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</p>
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<p>
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Generally speaking there is no loss of performance from allowing real-values
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parameters: the underlying special functions contain optimizations for integer-valued
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arguments when applicable.
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</p>
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<div class="caution"><table border="0" summary="Caution">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
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<th align="left">Caution</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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The quantile function of a discrete distribution will by default return
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an integer result that has been <span class="emphasis"><em>rounded outwards</em></span>.
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That is to say lower quantiles (where the probability is less than 0.5)
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are rounded downward, and upper quantiles (where the probability is greater
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than 0.5) are rounded upwards. This behaviour ensures that if an X% quantile
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is requested, then <span class="emphasis"><em>at least</em></span> the requested coverage
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will be present in the central region, and <span class="emphasis"><em>no more than</em></span>
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the requested coverage will be present in the tails.
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</p>
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<p>
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This behaviour can be changed so that the quantile functions are rounded
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differently, or even return a real-valued result using <a class="link" href="../pol_overview.html" title="Policy Overview">Policies</a>.
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It is strongly recommended that you read the tutorial <a class="link" href="../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
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Quantiles of Discrete Distributions</a> before using the quantile function
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on a discrete distribution. The <a class="link" href="../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
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docs</a> describe how to change the rounding policy for these distributions.
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</p>
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</td></tr>
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</table></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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