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| <div class="titlepage"><div><div><h5 class="title"> |
| <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 Chi Squared 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">inverse_chi_squared</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">inverse_chi_squared_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="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> |
| |
| <span class="special">}}</span> <span class="comment">// namespace boost // namespace math |
| </span></pre> |
| <p> |
| The inverse chi squared distribution is a continuous probability distribution |
| of the <span class="bold"><strong>reciprocal</strong></span> of a variable distributed |
| according to the chi squared distribution. |
| </p> |
| <p> |
| 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 |
| scale = 1. |
| </p> |
| <p> |
| 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> |
| (geoR) with default scale = 1/degrees of freedom. |
| </p> |
| <p> |
| See |
| </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 special case of a inverse_gamma |
| distribution with ν (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, the (<span class="bold"><strong>unscaled</strong></span>) |
| inverse chi_squared distribution is defined by the probability density |
| function (PDF): |
| </p> |
| <p> |
|   f(x;ν) = 2<sup>-ν/2</sup> x<sup>-ν/2-1</sup> e<sup>-1/2x</sup> / Γ(ν/2) |
| </p> |
| <p> |
| and Cumulative Density Function (CDF) |
| </p> |
| <p> |
|   F(x;ν) = Γ(ν/2, 1/2x) / Γ(ν/2) |
| </p> |
| <p> |
| For degrees of freedom parameter ν and scale parameter ξ, the <span class="bold"><strong>scaled</strong></span> inverse chi_squared distribution is defined |
| by the probability density function (PDF): |
| </p> |
| <p> |
|   f(x;ν, ξ) = (ξν/2)<sup>ν/2</sup> e<sup>-νξ/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="id1028429"></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/df. |
| </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="id1028601"></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="id1028705"></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="id1028746"></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> |
| νξ / (ν - 2) 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="id1029075"></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> |
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| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
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