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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.f_dist"></a><a class="link" href="f_dist.html" title="F Distribution">F Distribution</a>
</h4></div></div></div>
<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">fisher_f</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<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">&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>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;14.&#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>
<span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">fisher_f_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">fisher_f</span><span class="special">;</span>

<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;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">fisher_f_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="comment">// Construct:</span>
   <span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">j</span><span class="special">);</span>

   <span class="comment">// Accessors:</span>
   <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
   <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">//namespaces</span>
</pre>
<p>
          The F distribution is a continuous distribution that arises when testing
          whether two samples have the same variance. If &#967;<sup>2</sup><sub>m</sub> &#160; and &#967;<sup>2</sup><sub>n</sub> &#160; are independent
          variates each distributed as Chi-Squared with <span class="emphasis"><em>m</em></span> and
          <span class="emphasis"><em>n</em></span> degrees of freedom, then the test statistic:
        </p>
<p>
          F<sub>n,m</sub> &#160; = (&#967;<sup>2</sup><sub>n</sub> &#160; / n) / (&#967;<sup>2</sup><sub>m</sub> &#160; / m)
        </p>
<p>
          Is distributed over the range [0, &#8734;] with an F distribution, and has the
          PDF:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
        </p>
<p>
          The following graph illustrates how the PDF varies depending on the two
          degrees of freedom parameters.
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../graphs/fisher_f_pdf.svg" align="middle"></span>
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.member_functions"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.member_functions">Member Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">df1</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">df2</span><span class="special">);</span>
</pre>
<p>
          Constructs an F-distribution with numerator degrees of freedom <span class="emphasis"><em>df1</em></span>
          and denominator degrees of freedom <span class="emphasis"><em>df2</em></span>.
        </p>
<p>
          Requires that <span class="emphasis"><em>df1</em></span> and <span class="emphasis"><em>df2</em></span> are
          both greater than zero, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
          is called.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the numerator degrees of freedom parameter of the distribution.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the denominator degrees of freedom parameter of the distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.non_member_accessors"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <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>,
          <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>,
          <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>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <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>,
          <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>.
        </p>
<p>
          The domain of the random variable is [0, +&#8734;].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.examples"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.examples">Examples</a>
        </h5>
<p>
          Various <a class="link" href="../../stat_tut/weg/f_eg.html" title="F Distribution Examples">worked examples</a>
          are available illustrating the use of the F Distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.accuracy"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The normal distribution is implemented in terms of the <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
          beta function</a> and its <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">inverses</a>,
          refer to those functions for accuracy data.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.implementation"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span>
          are the first and second degrees of freedom parameters of the distribution,
          <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>
                    The usual form of the PDF is given by:
                  </p>
                  <p>
                    <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
                  </p>
                  <p>
                    However, that form is hard to evaluate directly without incurring
                    problems with either accuracy or numeric overflow.
                  </p>
                  <p>
                    Direct differentiation of the CDF expressed in terms of the incomplete
                    beta function
                  </p>
                  <p>
                    led to the following two formulas:
                  </p>
                  <p>
                    f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v2
                    / 2, v1 / 2, v2 / (v2 + v1 * x))
                  </p>
                  <p>
                    with y = (v2 * v1) / ((v2 + v1 * x) * (v2 + v1 * x))
                  </p>
                  <p>
                    and
                  </p>
                  <p>
                    f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v1
                    / 2, v2 / 2, v1 * x / (v2 + v1 * x))
                  </p>
                  <p>
                    with y = (z * v1 - x * v1 * v1) / z<sup>2</sup>
                  </p>
                  <p>
                    and z = v2 + v1 * x
                  </p>
                  <p>
                    The first of these is used for v1 * x &gt; v2, otherwise the
                    second is used.
                  </p>
                  <p>
                    The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>
                    away from 1 to avoid rounding error.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relations:
                  </p>
                  <p>
                    p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v1
                    / 2, v2 / 2, v1 * x / (v2 + v1 * x))
                  </p>
                  <p>
                    and
                  </p>
                  <p>
                    p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v2
                    / 2, v1 / 2, v2 / (v2 + v1 * x))
                  </p>
                  <p>
                    The first is used for v1 * x &gt; v2, otherwise the second is
                    used.
                  </p>
                  <p>
                    The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well
                    away from 1 to avoid rounding error.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relations:
                  </p>
                  <p>
                    p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v1
                    / 2, v2 / 2, v1 * x / (v2 + v1 * x))
                  </p>
                  <p>
                    and
                  </p>
                  <p>
                    p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v2
                    / 2, v1 / 2, v2 / (v2 + v1 * x))
                  </p>
                  <p>
                    The first is used for v1 * x &lt; v2, otherwise the second is
                    used.
                  </p>
                  <p>
                    The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well
                    away from 1 to avoid rounding error.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation:
                  </p>
                  <p>
                    x = v2 * a / (v1 * b)
                  </p>
                  <p>
                    where:
                  </p>
                  <p>
                    a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v1
                    / 2, v2 / 2, p)
                  </p>
                  <p>
                    and
                  </p>
                  <p>
                    b = 1 - a
                  </p>
                  <p>
                    Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
                    are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>
                    without the subtraction implied above.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                  <p>
                    from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation:
                  </p>
                  <p>
                    x = v2 * a / (v1 * b)
                  </p>
                  <p>
                    where
                  </p>
                  <p>
                    a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(v1
                    / 2, v2 / 2, p)
                  </p>
                  <p>
                    and
                  </p>
                  <p>
                    b = 1 - a
                  </p>
                  <p>
                    Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
                    are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>
                    without the subtraction implied above.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    v2 / (v2 - 2)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    2 * v2<sup>2 </sup> * (v1 + v2 - 2) / (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    v2 * (v1 - 2) / (v1 * (v2 + 2))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) / (v1 * (v2 + v1 -
                    2))) / (v2 - 6)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis and kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    Refer to, <a href="http://mathworld.wolfram.com/F-Distribution.html" target="_top">Weisstein,
                    Eric W. "F-Distribution." From MathWorld--A Wolfram
                    Web Resource.</a>
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
      Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
      Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
      Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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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