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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;15.&#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;15.&#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<p>
62 f(x) =
63 </p>
64<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
65<li class="listitem">
66 1 / (upper - lower) for lower &lt; x &lt; upper
67 </li>
68<li class="listitem">
69 zero for x &lt; lower or x &gt; upper
70 </li>
71</ul></div>
72<p>
73 and in this implementation:
74 </p>
75<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
76 1 / (upper - lower) for x = lower or x = upper
77 </li></ul></div>
78<p>
79 The choice of x = lower or x = upper is made because statistical use of
80 this distribution judged is most likely: the method of maximum likelihood
81 uses this definition.
82 </p>
83<p>
84 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>.
85 </p>
86<p>
87 Parameters lower and upper can be any finite value.
88 </p>
89<p>
90 The <a href="http://en.wikipedia.org/wiki/Random_variate" target="_top">random variate</a>
91 x must also be finite, and is supported lower &lt;= x &lt;= upper.
92 </p>
93<p>
94 The lower parameter is also called the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda364.htm" target="_top">location
95 parameter</a>, <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">that
96 is where the origin of a plot will lie</a>, and (upper - lower) is
97 also called the <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale
98 parameter</a>.
99 </p>
100<p>
101 The following graph illustrates how the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
102 density function PDF</a> varies with the shape parameter:
103 </p>
104<p>
105 <span class="inlinemediaobject"><img src="../../../../graphs/uniform_pdf.svg" align="middle"></span>
106 </p>
107<p>
108 Likewise for the CDF:
109 </p>
110<p>
111 <span class="inlinemediaobject"><img src="../../../../graphs/uniform_cdf.svg" align="middle"></span>
112 </p>
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; 2006-2010, 2012-2014 Nikhar Agrawal,
354 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
355 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
356 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
357 Distributed under the Boost Software License, Version 1.0. (See accompanying
358 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>)
359 </p>
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