<title>Inverse Gaussian (or Inverse Normal) 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">inverse_gaussian_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
variate x, the inverse_gaussian distribution is defined by the probability
density function (PDF):
</p>
-<p>
-    f(x;μ, λ) = √(λ/2πx<sup>3</sup>) e<sup>-λ(x-μ)²/2μ²x</sup>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">f(x;μ, λ) = √(λ/2πx<sup>3</sup>) e<sup>-λ(x-μ)²/2μ²x</sup> </span>
+ </p></blockquote></div>
<p>
and Cumulative Density Function (CDF):
</p>
-<p>
-    F(x;μ, λ) = Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)} + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)}
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">F(x;μ, λ) = Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)}
+ + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)} </span>
+ </p></blockquote></div>
<p>
where Φ is the standard normal distribution CDF.
</p>
The following graphs illustrate how the PDF and CDF of the inverse_gaussian
distribution varies for a few values of parameters μ and λ:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
Tweedie also provided 3 other parameterisations where (μ and λ) are replaced
by their ratio φ = λ/μ and by 1/μ: these forms may be more suitable for Bayesian