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<title>Inverse Chi Squared Distribution</title>
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<div class="section" lang="en">
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<div class="titlepage"><div><div><h5 class="title">
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<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist"></a><a class="link" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse
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Chi Squared Distribution</a>
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</h5></div></div></div>
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<p>
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</p>
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<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">inverse_chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
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<p>
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</p>
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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>
|
||||
|
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<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">inverse_chi_squared_distribution</span>
|
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<span class="special">{</span>
|
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<span class="keyword">public</span><span class="special">:</span>
|
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Not explicitly scaled, default 1/df.
|
||||
</span> <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">/</span><span class="identifier">df</span><span class="special">);</span> <span class="comment">// Scaled.
|
||||
</span>
|
||||
<span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Default 1.
|
||||
</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Optional scale [xi] (variance), default 1/degrees_of_freedom.
|
||||
</span><span class="special">};</span>
|
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|
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<span class="special">}}</span> <span class="comment">// namespace boost // namespace math
|
||||
</span></pre>
|
||||
<p>
|
||||
The inverse chi squared distribution is a continuous probability distribution
|
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of the <span class="bold"><strong>reciprocal</strong></span> of a variable distributed
|
||||
according to the chi squared distribution.
|
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</p>
|
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<p>
|
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The sources below give confusingly different formulae using different
|
||||
symbols for the distribution pdf, but they are all the same, or related
|
||||
by a change of variable, or choice of scale.
|
||||
</p>
|
||||
<p>
|
||||
Two constructors are available to implement both the scaled and (implicitly)
|
||||
unscaled versions.
|
||||
</p>
|
||||
<p>
|
||||
The main version has an explicit scale parameter which implements the
|
||||
<a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">scaled
|
||||
inverse chi_squared distribution</a>.
|
||||
</p>
|
||||
<p>
|
||||
A second version has an implicit scale = 1/degrees of freedom and gives
|
||||
the 1st definition in the <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">Wikipedia
|
||||
inverse chi_squared distribution</a>. The 2nd Wikipedia inverse chi_squared
|
||||
distribution definition can be implemented by explicitly specifying a
|
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scale = 1.
|
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</p>
|
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<p>
|
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Both definitions are also available in Wolfram Mathematica and in <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>
|
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(geoR) with default scale = 1/degrees of freedom.
|
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</p>
|
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<p>
|
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See
|
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</p>
|
||||
<div class="itemizedlist"><ul type="disc">
|
||||
<li>
|
||||
Inverse chi_squared distribution <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Inverse-chi-square_distribution</a>
|
||||
</li>
|
||||
<li>
|
||||
Scaled inverse chi_squared distribution<a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution</a>
|
||||
</li>
|
||||
<li>
|
||||
R inverse chi_squared distribution functions <a href="http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html" target="_top">R
|
||||
</a>
|
||||
</li>
|
||||
<li>
|
||||
Inverse chi_squared distribution functions <a href="http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html" target="_top">Weisstein,
|
||||
Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A
|
||||
Wolfram Web Resource.</a>
|
||||
</li>
|
||||
<li>
|
||||
Inverse chi_squared distribution reference <a href="http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html" target="_top">Weisstein,
|
||||
Eric W. "Inverse Chi-Squared Distribution reference." From
|
||||
Wolfram Mathematica.</a>
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
The inverse_chi_squared distribution is used in <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
|
||||
statistics</a>: the scaled inverse chi-square is conjugate prior
|
||||
for the normal distribution with known mean, model parameter σ² (variance).
|
||||
</p>
|
||||
<p>
|
||||
See <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">conjugate
|
||||
priors including a table of distributions and their priors.</a>
|
||||
</p>
|
||||
<p>
|
||||
See also <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
|
||||
Gamma Distribution</a> and <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
|
||||
Squared Distribution</a>.
|
||||
</p>
|
||||
<p>
|
||||
The inverse_chi_squared distribution is a psecial case of a inverse_gamma
|
||||
distribution with nu (degrees_of_freedom) shape (α) and scale (β) where
|
||||
</p>
|
||||
<p>
|
||||
  α= ν /2 and β = ½.
|
||||
</p>
|
||||
<div class="note"><table border="0" summary="Note">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td>
|
||||
<th align="left">Note</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top">
|
||||
<p>
|
||||
This distribution <span class="bold"><strong>does</strong></span> provide the
|
||||
typedef:
|
||||
</p>
|
||||
<p>
|
||||
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">inverse_chi_squared_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">inverse_chi_squared</span><span class="special">;</span></pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
|
||||
inverse_chi_squared distribution you can use
|
||||
</p>
|
||||
<p>
|
||||
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_chi_squared_distribution</span><span class="special"><></span></pre>
|
||||
<p>
|
||||
</p>
|
||||
<p>
|
||||
or you can write <code class="computeroutput"><span class="identifier">inverse_chi_squared</span>
|
||||
<span class="identifier">my_invchisqr</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span></code>
|
||||
</p>
|
||||
</td></tr>
|
||||
</table></div>
|
||||
<p>
|
||||
For degrees of freedom parameter ν and scale parameter ξ, it is defined
|
||||
by the probability density function (PDF):
|
||||
</p>
|
||||
<p>
|
||||
  f(x;ν, ξ) = 2 <sup>-ν/2</sup> e<sup>(-1/2x</sup> x <sup>(-1-ν/2)</sup> / Γ(ν/2)
|
||||
</p>
|
||||
<p>
|
||||
and Cumulative Density Function (CDF)
|
||||
</p>
|
||||
<p>
|
||||
  F(x;ν, ξ) = Γ( ν /2, νξ/2x) / Γ(ν /2)
|
||||
</p>
|
||||
<p>
|
||||
The following graphs illustrate how the PDF and CDF of the inverse chi_squared
|
||||
distribution varies for a few values of parameters ν and ξ:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../../../graphs/inverse_chi_squared_pdf.png" align="middle"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../../../graphs/inverse_chi_squared_cdf.png" align="middle"></span>
|
||||
</p>
|
||||
<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.member_functions"></a><h5>
|
||||
<a name="id1013289"></a>
|
||||
<a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.member_functions">Member
|
||||
Functions</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Implicitly scaled 1/df.
|
||||
</span><span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">);</span> <span class="comment">// Explicitly scaled.
|
||||
</span></pre>
|
||||
<p>
|
||||
Constructs an inverse chi_squared distribution with ν degrees of freedom
|
||||
<span class="emphasis"><em>df</em></span>, and scale <span class="emphasis"><em>scale</em></span> with default
|
||||
value 1<span class="emphasis"><em>df (1</em></span>ν.
|
||||
</p>
|
||||
<p>
|
||||
Requires that the degrees of freedom ν parameter is greater than zero,
|
||||
otherwise 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">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
Returns the degrees_of_freedom ν parameter of this distribution.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
Returns the scale ξ parameter of this distribution.
|
||||
</p>
|
||||
<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors"></a><h5>
|
||||
<a name="id1013464"></a>
|
||||
<a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_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 variate is [0,+∞].
|
||||
</p>
|
||||
<div class="note"><table border="0" summary="Note">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td>
|
||||
<th align="left">Note</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
Unlike some definitions, this implementation supports a random variate
|
||||
equal to zero as a special case, returning zero for both pdf and cdf.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.accuracy"></a><h5>
|
||||
<a name="id1013569"></a>
|
||||
<a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
The inverse gamma distribution is implemented in terms of the incomplete
|
||||
gamma functions like the <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
|
||||
Gamma Distribution</a> that use <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
|
||||
and <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>
|
||||
and their inverses <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
|
||||
and <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>:
|
||||
refer to the accuracy data for those functions for more information.
|
||||
But in general, gamma (and thus inverse gamma) results are often accurate
|
||||
to a few epsilon, >14 decimal digits accuracy for 64-bit double. unless
|
||||
iteration is involved, as for the estimation of degrees of freedom.
|
||||
</p>
|
||||
<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.implementation"></a><h5>
|
||||
<a name="id1013609"></a>
|
||||
<a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
In the following table ν is the degrees of freedom parameter and ξ is the
|
||||
scale parameter 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> its complement. Parameters α for shape and β for scale are
|
||||
used for the inverse gamma function: α = ν/2 and β = ν * ξ/2.
|
||||
</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 = <a class="link" href="../../../special/sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(α,
|
||||
β/ x, β) / x * x
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
cdf
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Using the relation: p = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(α,
|
||||
β / x)
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
cdf complement
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Using the relation: q = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(α,
|
||||
β / x)
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
quantile
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Using the relation: x = β ​/ <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(α,
|
||||
p)
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
quantile from the complement
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Using the relation: x = α ​/ <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(α,
|
||||
q)
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
mode
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
ν * ξ / (ν + 2)
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
median
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
no closed form analytic equation is known, but is evaluated
|
||||
as quantile(0.5)
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
mean
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
1 / (ν - 1) for ν > 2, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
variance
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
2 ν² ξ² / ((ν -2)² (ν -4)) for ν >4, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
skewness
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
4 √2 √(ν-4) /(ν-6) for ν >6, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
kurtosis_excess
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
12 * (5ν - 22) / (ν - 6) * (ν - 8) for ν >8, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
kurtosis
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
3 + 12 * (5ν - 22) / (ν - 6) * (ν-8) for ν >8, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.references"></a><h5>
|
||||
<a name="id1013938"></a>
|
||||
<a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.references">References</a>
|
||||
</h5>
|
||||
<div class="orderedlist"><ol type="1">
|
||||
<li>
|
||||
Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern,
|
||||
Donald B. Rubin, ISBN-13: 978-1584883883, Chapman & Hall; 2 edition
|
||||
(29 July 2003).
|
||||
</li>
|
||||
<li>
|
||||
Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973,
|
||||
Springer; 2nd ed. edition (10 Jun 2009)
|
||||
</li>
|
||||
</ol></div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"></td>
|
||||
<td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 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>
|
||||
</div></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="gamma_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="inverse_gamma_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
Reference in New Issue
Block a user