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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.uniform_dist"></a><a class="link" href="uniform_dist.html" title="Uniform Distribution">Uniform Distribution</a>
  28 </h4></div></div></div>
  29 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span></pre>
  30 <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>
  31  <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  32            <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  33  <span class="keyword">class</span> <span class="identifier">uniform_distribution</span><span class="special">;</span>
  34
  35  <span class="keyword">typedef</span> <span class="identifier">uniform_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">uniform</span><span class="special">;</span>
  36
  37  <span class="keyword">template</span> <span class="special">&lt;</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="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  38  <span class="keyword">class</span> <span class="identifier">uniform_distribution</span>
  39  <span class="special">{</span>
  40  <span class="keyword">public</span><span class="special">:</span>
  41     <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  42
  43     <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>
  44        <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>
  45     <span class="comment">// Accessor functions.</span>
  46     <span class="identifier">RealType</span> <span class="identifier">lower</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  47     <span class="identifier">RealType</span> <span class="identifier">upper</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  48  <span class="special">};</span> <span class="comment">// class uniform_distribution</span>
  49
  50 <span class="special">}}</span> <span class="comment">// namespaces</span>
  51 </pre>
  52 <p>
  53           The uniform distribution, also known as a rectangular distribution, is
  54           a probability distribution that has constant probability.
  55         </p>
  56 <p>
  57           The <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">continuous
  58           uniform distribution</a> is a distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
  59           density function</a>:
  60         </p>
  61 <div class="blockquote"><blockquote class="blockquote"><p>
  62             <span class="serif_italic">f(x) =1 / (upper - lower) &#8198; for lower &lt;
  63             x &lt; upper</span>
  64           </p></blockquote></div>
  65 <div class="blockquote"><blockquote class="blockquote"><p>
  66             <span class="serif_italic">f(x) =zero &#8198; for x &lt; lower or x &gt; upper</span>
  67           </p></blockquote></div>
  68 <p>
  69           and in this implementation:
  70         </p>
  71 <div class="blockquote"><blockquote class="blockquote"><p>
  72             <span class="serif_italic">1 / (upper - lower) &#8198; for x = lower or x =
  73             upper</span>
  74           </p></blockquote></div>
  75 <p>
  76           The choice of <span class="emphasis"><em>x = lower</em></span> or <span class="emphasis"><em>x = upper</em></span>
  77           is made because statistical use of this distribution judged is most likely:
  78           the method of maximum likelihood uses this definition.
  79         </p>
  80 <p>
  81           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>.
  82         </p>
  83 <p>
  84           Parameters lower and upper can be any finite value.
  85         </p>
  86 <p>
  87           The <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a>
  88           <span class="emphasis"><em>x</em></span> must also be finite, and is supported <span class="emphasis"><em>lower
  89           &lt;= x &lt;= upper</em></span>.
  90         </p>
  91 <p>
  92           The lower parameter is also called the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm" target="_top">location
  93           parameter</a>, <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">that
  94           is where the origin of a plot will lie</a>, and (upper - lower) is
  95           also called the <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale
  96           parameter</a>.
  97         </p>
  98 <p>
  99           The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
 100           density function PDF</a> varies with the shape parameter:
 101         </p>
 102 <div class="blockquote"><blockquote class="blockquote"><p>
 103             <span class="inlinemediaobject"><img src="../../../../graphs/uniform_pdf.svg" align="middle"></span>
 104
 105           </p></blockquote></div>
 106 <p>
 107           Likewise for the CDF:
 108         </p>
 109 <div class="blockquote"><blockquote class="blockquote"><p>
 110             <span class="inlinemediaobject"><img src="../../../../graphs/uniform_cdf.svg" align="middle"></span>
 111
 112           </p></blockquote></div>
 113 <h5>
 114 <a name="math_toolkit.dist_ref.dists.uniform_dist.h0"></a>
 115           <span class="phrase"><a name="math_toolkit.dist_ref.dists.uniform_dist.member_functions"></a></span><a class="link" href="uniform_dist.html#math_toolkit.dist_ref.dists.uniform_dist.member_functions">Member
 116           Functions</a>
 117         </h5>
 118 <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>
 119 </pre>
 120 <p>
 121           Constructs a <a href="http://en.wikipedia.org/wiki/uniform_distribution" target="_top">uniform
 122           distribution</a> with lower <span class="emphasis"><em>lower</em></span> (a) and upper
 123           <span class="emphasis"><em>upper</em></span> (b).
 124         </p>
 125 <p>
 126           Requires that the <span class="emphasis"><em>lower</em></span> and <span class="emphasis"><em>upper</em></span>
 127           parameters are both finite; otherwise if infinity or NaN then calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 128         </p>
 129 <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>
 130 </pre>
 131 <p>
 132           Returns the <span class="emphasis"><em>lower</em></span> parameter of this distribution.
 133         </p>
 134 <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>
 135 </pre>
 136 <p>
 137           Returns the <span class="emphasis"><em>upper</em></span> parameter of this distribution.
 138         </p>
 139 <h5>
 140 <a name="math_toolkit.dist_ref.dists.uniform_dist.h1"></a>
 141           <span class="phrase"><a name="math_toolkit.dist_ref.dists.uniform_dist.non_member_accessors"></a></span><a class="link" href="uniform_dist.html#math_toolkit.dist_ref.dists.uniform_dist.non_member_accessors">Non-member
 142           Accessors</a>
 143         </h5>
 144 <p>
 145           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 146           functions</a> that are generic to all distributions are supported:
 147           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 148           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 149           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 150           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 151           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 152           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 153           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 154           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 155         </p>
 156 <p>
 157           The domain of the random variable is any finite value, but the supported
 158           range is only <span class="emphasis"><em>lower</em></span> &lt;= x &lt;= <span class="emphasis"><em>upper</em></span>.
 159         </p>
 160 <h5>
 161 <a name="math_toolkit.dist_ref.dists.uniform_dist.h2"></a>
 162           <span class="phrase"><a name="math_toolkit.dist_ref.dists.uniform_dist.accuracy"></a></span><a class="link" href="uniform_dist.html#math_toolkit.dist_ref.dists.uniform_dist.accuracy">Accuracy</a>
 163         </h5>
 164 <p>
 165           The uniform distribution is implemented with simple arithmetic operators
 166           and so should have errors within an epsilon or two.
 167         </p>
 168 <h5>
 169 <a name="math_toolkit.dist_ref.dists.uniform_dist.h3"></a>
 170           <span class="phrase"><a name="math_toolkit.dist_ref.dists.uniform_dist.implementation"></a></span><a class="link" href="uniform_dist.html#math_toolkit.dist_ref.dists.uniform_dist.implementation">Implementation</a>
 171         </h5>
 172 <p>
 173           In the following table a is the <span class="emphasis"><em>lower</em></span> parameter of
 174           the distribution, b is the <span class="emphasis"><em>upper</em></span> parameter, <span class="emphasis"><em>x</em></span>
 175           is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
 176           = 1-p</em></span>.
 177         </p>
 178 <div class="informaltable"><table class="table">
 179 <colgroup>
 180 <col>
 181 <col>
 182 </colgroup>
 183 <thead><tr>
 184 <th>
 185                   <p>
 186                     Function
 187                   </p>
 188                 </th>
 189 <th>
 190                   <p>
 191                     Implementation Notes
 192                   </p>
 193                 </th>
 194 </tr></thead>
 195 <tbody>
 196 <tr>
 197 <td>
 198                   <p>
 199                     pdf
 200                   </p>
 201                 </td>
 202 <td>
 203                   <p>
 204                     Using the relation: pdf = 0 for x &lt; a, 1 / (b - a) for a &lt;=
 205                     x &lt;= b, 0 for x &gt; b
 206                   </p>
 207                 </td>
 208 </tr>
 209 <tr>
 210 <td>
 211                   <p>
 212                     cdf
 213                   </p>
 214                 </td>
 215 <td>
 216                   <p>
 217                     Using the relation: cdf = 0 for x &lt; a, (x - a) / (b - a) for
 218                     a &lt;= x &lt;= b, 1 for x &gt; b
 219                   </p>
 220                 </td>
 221 </tr>
 222 <tr>
 223 <td>
 224                   <p>
 225                     cdf complement
 226                   </p>
 227                 </td>
 228 <td>
 229                   <p>
 230                     Using the relation: q = 1 - p, (b - x) / (b - a)
 231                   </p>
 232                 </td>
 233 </tr>
 234 <tr>
 235 <td>
 236                   <p>
 237                     quantile
 238                   </p>
 239                 </td>
 240 <td>
 241                   <p>
 242                     Using the relation: x = p * (b - a) + a;
 243                   </p>
 244                 </td>
 245 </tr>
 246 <tr>
 247 <td>
 248                   <p>
 249                     quantile from the complement
 250                   </p>
 251                 </td>
 252 <td>
 253                   <p>
 254                     x = -q * (b - a) + b
 255                   </p>
 256                 </td>
 257 </tr>
 258 <tr>
 259 <td>
 260                   <p>
 261                     mean
 262                   </p>
 263                 </td>
 264 <td>
 265                   <p>
 266                     (a + b) / 2
 267                   </p>
 268                 </td>
 269 </tr>
 270 <tr>
 271 <td>
 272                   <p>
 273                     variance
 274                   </p>
 275                 </td>
 276 <td>
 277                   <p>
 278                     (b - a) <sup>2</sup> / 12
 279                   </p>
 280                 </td>
 281 </tr>
 282 <tr>
 283 <td>
 284                   <p>
 285                     mode
 286                   </p>
 287                 </td>
 288 <td>
 289                   <p>
 290                     any value in [a, b] but a is chosen. (Would NaN be better?)
 291                   </p>
 292                 </td>
 293 </tr>
 294 <tr>
 295 <td>
 296                   <p>
 297                     skewness
 298                   </p>
 299                 </td>
 300 <td>
 301                   <p>
 302                     0
 303                   </p>
 304                 </td>
 305 </tr>
 306 <tr>
 307 <td>
 308                   <p>
 309                     kurtosis excess
 310                   </p>
 311                 </td>
 312 <td>
 313                   <p>
 314                     -6/5 = -1.2 exactly. (kurtosis - 3)
 315                   </p>
 316                 </td>
 317 </tr>
 318 <tr>
 319 <td>
 320                   <p>
 321                     kurtosis
 322                   </p>
 323                 </td>
 324 <td>
 325                   <p>
 326                     9/5
 327                   </p>
 328                 </td>
 329 </tr>
 330 </tbody>
 331 </table></div>
 332 <h5>
 333 <a name="math_toolkit.dist_ref.dists.uniform_dist.h4"></a>
 334           <span class="phrase"><a name="math_toolkit.dist_ref.dists.uniform_dist.references"></a></span><a class="link" href="uniform_dist.html#math_toolkit.dist_ref.dists.uniform_dist.references">References</a>
 335         </h5>
 336 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 337 <li class="listitem">
 338               <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">Wikpedia
 339               continuous uniform distribution</a>
 340             </li>
 341 <li class="listitem">
 342               <a href="http://mathworld.wolfram.com/UniformDistribution.html" target="_top">Weisstein,
 343               Weisstein, Eric W. "Uniform Distribution." From MathWorld--A
 344               Wolfram Web Resource.</a>
 345             </li>
 346 <li class="listitem">
 347               <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>
 348             </li>
 349 </ul></div>
 350 </div>
 351 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 352 <td align="left"></td>
 353 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 354       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 355       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 356       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 357       Daryle Walker and Xiaogang Zhang<p>
 358         Distributed under the Boost Software License, Version 1.0. (See accompanying
 359         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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