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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>
</h3></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="../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.
        </p>
</td></tr>
</table></div>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
      Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
      Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, 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>)
      </p>
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Commit	Line	Data
7c673cae FG	1	<html>
	2	<head>
	3	<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
	4	<title>Discrete Probability Distributions</title>
	5	<link rel="stylesheet" href="../../math.css" type="text/css">
	6	<meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
	7	<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
	8	<link rel="up" href="../stat_tut.html" title="Statistical Distributions Tutorial">
	9	<link rel="prev" href="variates.html" title="Random Variates and Distribution Parameters">
	10	<link rel="next" href="../dist_ref.html" title="Statistical Distributions Reference">
	11	</head>
	12	<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
	13	<table cellpadding="2" width="100%"><tr>
	14	<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
	15	<td align="center"><a href="../../../../../../index.html">Home</a></td>
	16	<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
	17	<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
	18	<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
	19	<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
	20	</tr></table>
	21	<hr>
	22	<div class="spirit-nav">
	23	<a accesskey="p" href="variates.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../stat_tut.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../dist_ref.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
	24	</div>
	25	<div class="section">
	26	<div class="titlepage"><div><div><h3 class="title">
	27	<a name="math_toolkit.stat_tut.dist_params"></a><a class="link" href="dist_params.html" title="Discrete Probability Distributions">Discrete Probability
	28	Distributions</a>
	29	</h3></div></div></div>
	30	<p>
	31	Note that the <a href="http://en.wikipedia.org/wiki/Discrete_probability_distribution" target="_top">discrete
	32	distributions</a>, including the binomial, negative binomial, Poisson
	33	& Bernoulli, are all mathematically defined as discrete functions: only
	34	integral values of the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random
	35	variate</a> are envisaged and the functions are only defined at these
	36	integral values. However because the method of calculation often uses continuous
	37	functions, it is convenient to treat them as if they were continuous functions,
	38	and permit non-integral values of their parameters.
	39	</p>
	40	<p>
	41	To enforce a strict mathematical model, users may use floor or ceil functions
	42	on the <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a>,
	43	prior to calling the distribution function, to enforce integral values.
	44	</p>
	45	<p>
	46	For similar reasons, in continuous distributions, parameters like degrees
	47	of freedom that might appear to be integral, are treated as real values (and
	48	are promoted from integer to floating-point if necessary). In this case however,
	49	that there are a small number of situations where non-integral degrees of
	50	freedom do have a genuine meaning.
	51	</p>
	52	<p>
	53	Generally speaking there is no loss of performance from allowing real-values
	54	parameters: the underlying special functions contain optimizations for integer-valued
	55	arguments when applicable.
	56	</p>
	57	<div class="caution"><table border="0" summary="Caution">
	58	<tr>
	59	<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
	60	<th align="left">Caution</th>
	61	</tr>
	62	<tr><td align="left" valign="top">
	63	<p>
	64	The quantile function of a discrete distribution will by default return
65	an integer result that has been <span class="emphasis"><em>rounded outwards</em></span>.
66	That is to say lower quantiles (where the probability is less than 0.5)
67	are rounded downward, and upper quantiles (where the probability is greater
68	than 0.5) are rounded upwards. This behaviour ensures that if an X% quantile
69	is requested, then <span class="emphasis"><em>at least</em></span> the requested coverage
70	will be present in the central region, and <span class="emphasis"><em>no more than</em></span>
71	the requested coverage will be present in the tails.
72	</p>
73	<p>
74	This behaviour can be changed so that the quantile functions are rounded
75	differently, or even return a real-valued result using <a class="link" href="../pol_overview.html" title="Policy Overview">Policies</a>.
76	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
77	Quantiles of Discrete Distributions</a> before using the quantile function
78	on a discrete distribution. The <a class="link" href="../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
79	docs</a> describe how to change the rounding policy for these distributions.
80	</p>
81	</td></tr>
82	</table></div>
83	</div>
84	<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
85	<td align="left"></td>
86	<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
87	Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
88	Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
89	Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
90	Distributed under the Boost Software License, Version 1.0. (See accompanying
91	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>)
92	</p>
93	</div></td>
94	</tr></table>
95	<hr>
96	<div class="spirit-nav">
97	<a accesskey="p" href="variates.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../stat_tut.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../dist_ref.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
98	</div>
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100	</html>