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<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="id1067212"></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="id1067377"></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="id1067482"></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="id1067501"></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="id1068337"></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>
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<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 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>
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