<title>F Distribution</title>
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<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="Chapter 
19. 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>
+ <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 
20. 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>
<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"><></span> <span class="identifier">fisher_f</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="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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>
</pre>
<p>
The F distribution is a continuous distribution that arises when testing
- whether two samples have the same variance. If χ<sup>2</sup><sub>m</sub>   and χ<sup>2</sup><sub>n</sub>   are independent
+ whether two samples have the same variance. If χ<sup>2</sup><sub>m</sub> and χ<sup>2</sup><sub>n</sub> 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>   = (χ<sup>2</sup><sub>n</sub>   / n) / (χ<sup>2</sup><sub>m</sub>   / m)
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">F<sub>n,m</sub> = (χ<sup>2</sup><sub>n</sub> / n) / (χ<sup>2</sup><sub>m</sub> / m)</span>
+ </p></blockquote></div>
<p>
Is distributed over the range [0, ∞] with an F distribution, and has the
PDF:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
+
+ </p></blockquote></div>
<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>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/fisher_f_pdf.svg" align="middle"></span>
+
+ </p></blockquote></div>
<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>
<p>
The usual form of the PDF is given by:
</p>
- <p>
- <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
- </p>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
+
+ </p></blockquote></div>
<p>
However, that form is hard to evaluate directly without incurring
problems with either accuracy or numeric overflow.
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">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))</span>
+ </p></blockquote></div>
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">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))</span>
+ </p></blockquote></div>
<p>
with y = (z * v1 - x * v1 * v1) / z<sup>2</sup>
</p>
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">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))</span>
+ </p></blockquote></div>
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">:p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v2
+ / 2, v1 / 2, v2 / (v2 + v1 * x))</span>
+ </p></blockquote></div>
<p>
The first is used for v1 * x > v2, otherwise the second is
used.
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">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))</span>
+ </p></blockquote></div>
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v2
+ / 2, v1 / 2, v2 / (v2 + v1 * x))</span>
+ </p></blockquote></div>
<p>
The first is used for v1 * x < v2, otherwise the second is
used.
<p>
Using the relation:
</p>
- <p>
- x = v2 * a / (v1 * b)
- </p>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">x = v2 * a / (v1 * b)</span>
+ </p></blockquote></div>
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">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)</span>
+ </p></blockquote></div>
<p>
and
</p>
- <p>
- b = 1 - a
- </p>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">b = 1 - a</span>
+ </p></blockquote></div>
<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>
<p>
Using the relation:
</p>
- <p>
- x = v2 * a / (v1 * b)
- </p>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">x = v2 * a / (v1 * b)</span>
+ </p></blockquote></div>
<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>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+
<span class="serif_italic">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)</span>
+ </p></blockquote></div>
<p>
and
</p>
- <p>
- b = 1 - a
- </p>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">b = 1 - a</span>
+ </p></blockquote></div>
<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>
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