1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Uniform Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="triangular_dist.html" title="Triangular Distribution"> 10<link rel="next" href="weibull_dist.html" title="Weibull Distribution"> 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="triangular_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="weibull_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 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"><</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> 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"><</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 21. 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<></a> <span class="special">></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"><></span> <span class="identifier">uniform</span><span class="special">;</span> 36 37 <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="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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) for lower < 63 x < upper</span> 64 </p></blockquote></div> 65<div class="blockquote"><blockquote class="blockquote"><p> 66 <span class="serif_italic">f(x) =zero for x < lower or x > 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) 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 <= x <= 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> <= x <= <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 < a, 1 / (b - a) for a <= 205 x <= b, 0 for x > 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 < a, (x - a) / (b - a) for 218 a <= x <= b, 1 for x > 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">Wikipedia 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 © 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 Råde, 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>) 360 </p> 361</div></td> 362</tr></table> 363<hr> 364<div class="spirit-nav"> 365<a accesskey="p" href="triangular_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.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="weibull_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 366</div> 367</body> 368</html> 369
