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<title>Discrete Probability Distributions</title>
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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.stat_tut.dist_params"></a><a class="link" href="dist_params.html" title="Discrete Probability Distributions">Discrete Probability
Distributions</a>
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<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.
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
<th align="left">Caution</th>
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<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="../pol_overview.html" title="Policy Overview">Policies</a>.
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
Quantiles of Discrete Distributions</a> before using the quantile function
on a discrete distribution. The <a class="link" href="../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
docs</a> describe how to change the rounding policy for these distributions.
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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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