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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.uniform_dist"></a><a class="link" href="uniform_dist.html" title="Uniform Distribution"> Uniform |
| Distribution</a> |
| </h5></div></div></div> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">uniform</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> |
| <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">uniform_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">uniform_distribution</span><span class="special"><></span> <span class="identifier">uniform</span><span class="special">;</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">uniform_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| |
| <span class="identifier">uniform_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Constructor. |
| </span> <span class="special">:</span> <span class="identifier">m_lower</span><span class="special">(</span><span class="identifier">lower</span><span class="special">),</span> <span class="identifier">m_upper</span><span class="special">(</span><span class="identifier">upper</span><span class="special">)</span> <span class="comment">// Default is standard uniform distribution. |
| </span> <span class="comment">// Accessor functions. |
| </span> <span class="identifier">RealType</span> <span class="identifier">lower</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">RealType</span> <span class="identifier">upper</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> <span class="comment">// class uniform_distribution |
| </span> |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| The uniform distribution, also known as a rectangular distribution, is |
| a probability distribution that has constant probability. |
| </p> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">continuous |
| uniform distribution</a> is a distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability |
| density function</a>: |
| </p> |
| <p> |
| f(x) = |
| </p> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| 1 / (upper - lower) for lower < x < upper |
| </li> |
| <li> |
| zero for x < lower or x > upper |
| </li> |
| </ul></div> |
| <p> |
| and in this implementation: |
| </p> |
| <div class="itemizedlist"><ul type="disc"><li> |
| 1 / (upper - lower) for x = lower or x = upper |
| </li></ul></div> |
| <p> |
| The choice of x = lower or x = upper is made because statistical use |
| of this distribution judged is most likely: the method of maximum likelihood |
| uses this definition. |
| </p> |
| <p> |
| There is also a <a href="http://en.wikipedia.org/wiki/Discrete_uniform_distribution" target="_top"><span class="bold"><strong>discrete</strong></span> uniform distribution</a>. |
| </p> |
| <p> |
| Parameters lower and upper can be any finite value. |
| </p> |
| <p> |
| The <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a> |
| x must also be finite, and is supported lower <= x <= upper. |
| </p> |
| <p> |
| The lower parameter is also called the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm" target="_top">location |
| parameter</a>, <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">that |
| is where the origin of a plot will lie</a>, and (upper - lower) is |
| also called the <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale |
| parameter</a>. |
| </p> |
| <p> |
| The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability |
| density function PDF</a> varies with the shape parameter: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/uniform_pdf.png" align="middle"></span> |
| </p> |
| <p> |
| Likewise for the CDF: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/uniform_cdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.uniform_dist.member_functions"></a><h5> |
| <a name="id1063915"></a> |
| <a class="link" href="uniform_dist.html#math_toolkit.dist.dist_ref.dists.uniform_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">uniform_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a <a href="http://en.wikipedia.org/wiki/uniform_distribution" target="_top">uniform |
| distribution</a> with lower <span class="emphasis"><em>lower</em></span> (a) and upper |
| <span class="emphasis"><em>upper</em></span> (b). |
| </p> |
| <p> |
| Requires that the <span class="emphasis"><em>lower</em></span> and <span class="emphasis"><em>upper</em></span> |
| parameters are both finite; otherwise if infinity or NaN then calls |
| <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">lower</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the <span class="emphasis"><em>lower</em></span> parameter of this distribution. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">upper</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the <span class="emphasis"><em>upper</em></span> parameter of this distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.uniform_dist.non_member_accessors"></a><h5> |
| <a name="id1064080"></a> |
| <a class="link" href="uniform_dist.html#math_toolkit.dist.dist_ref.dists.uniform_dist.non_member_accessors">Non-member |
| Accessors</a> |
| </h5> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member |
| accessor functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard |
| Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, |
| <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, |
| <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| The domain of the random variable is any finite value, but the supported |
| range is only <span class="emphasis"><em>lower</em></span> <= x <= <span class="emphasis"><em>upper</em></span>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.uniform_dist.accuracy"></a><h5> |
| <a name="id1064185"></a> |
| <a class="link" href="uniform_dist.html#math_toolkit.dist.dist_ref.dists.uniform_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The uniform distribution is implemented with simple arithmetic operators |
| and so should have errors within an epsilon or two. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.uniform_dist.implementation"></a><h5> |
| <a name="id1064204"></a> |
| <a class="link" href="uniform_dist.html#math_toolkit.dist.dist_ref.dists.uniform_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table a is the <span class="emphasis"><em>lower</em></span> parameter |
| of the distribution, b is the <span class="emphasis"><em>upper</em></span> parameter, |
| <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> |
| is the probability and <span class="emphasis"><em>q = 1-p</em></span>. |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Implementation Notes |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| pdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: pdf = 0 for x < a, 1 / (b - a) for a |
| <= x <= b, 0 for x > b |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: cdf = 0 for x < a, (x - a) / (b - a) |
| for a <= x <= b, 1 for x > b |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: q = 1 - p, (b - x) / (b - a) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = p * (b - a) + a; |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| x = -q * (b - a) + b |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| (a + b) / 2 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| (b - a) <sup>2</sup> / 12 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| any value in [a, b] but a is chosen. (Would NaN be better?) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| -6/5 = -1.2 exactly. (kurtosis - 3) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| 9/5 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.dists.uniform_dist.references"></a><h5> |
| <a name="id1064482"></a> |
| <a class="link" href="uniform_dist.html#math_toolkit.dist.dist_ref.dists.uniform_dist.references">References</a> |
| </h5> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">Wikpedia |
| continuous uniform distribution</a> |
| </li> |
| <li> |
| <a href="http://mathworld.wolfram.com/UniformDistribution.html" target="_top">Weisstein, |
| Weisstein, Eric W. "Uniform Distribution." From MathWorld--A |
| Wolfram Web Resource.</a> |
| </li> |
| <li> |
| <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm" target="_top">http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm</a> |
| </li> |
| </ul></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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