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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 |
| & 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> |
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| <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="../../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. |
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| <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 |
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