<title>Students t 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">students_t_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">students_t_distribution</span><span class="special"><></span> <span class="identifier">students_t</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">students_t_distribution</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>
<p>
Given N independent measurements, let
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/students_t_dist.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../../equations/students_t_dist.svg"></span>
- </p>
-<p>
- where <span class="emphasis"><em>M</em></span> is the population mean, <span class="emphasis"><em>μ</em></span>
- is the sample mean, and <span class="emphasis"><em>s</em></span> is the sample variance.
+ where <span class="emphasis"><em>M</em></span> is the population mean, μ is the sample mean,
+ and <span class="emphasis"><em>s</em></span> is the sample variance.
</p>
<p>
<a href="https://en.wikipedia.org/wiki/Student%27s_t-distribution" target="_top">Student's
t-distribution</a> is defined as the distribution of the random variable
- t which is - very loosely - the "best" that we can do not knowing
- the true standard deviation of the sample. It has the PDF:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../equations/students_t_ref1.svg"></span>
+ t which is - very loosely - the "best" that we can do while not
+ knowing the true standard deviation of the sample. It has the PDF:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/students_t_ref1.svg"></span>
+
+ </p></blockquote></div>
<p>
The Student's t-distribution takes a single parameter: the number of degrees
of freedom of the sample. When the degrees of freedom is <span class="emphasis"><em>one</em></span>
the normal-distribution. The following graph illustrates how the PDF varies
with the degrees of freedom ν:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/students_t_pdf.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/students_t_pdf.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.students_t_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.students_t_dist.member_functions"></a></span><a class="link" href="students_t_dist.html#math_toolkit.dist_ref.dists.students_t_dist.member_functions">Member
</td>
<td>
<p>
- Using the relation: pdf = (v / (v + t<sup>2</sup>))<sup>(1+v)/2 </sup> / (sqrt(v) *
- <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>(v/2,
- 0.5))
+ Using the relation: <span class="serif_italic">pdf = (v / (v
+
+ t<sup>2</sup>))<sup>(1+v)/2 </sup> / (sqrt(v) * <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>(v/2,
+ 0.5))</span>
</p>
</td>
</tr>
Using the relations:
</p>
<p>
- p = 1 - z <span class="emphasis"><em>iff t > 0</em></span>
+ <span class="serif_italic">p = 1 - z <span class="emphasis"><em>iff t > 0</em></span></span>
</p>
<p>
- p = z <span class="emphasis"><em>otherwise</em></span>
+ <span class="serif_italic">p = z <span class="emphasis"><em>otherwise</em></span></span>
</p>
<p>
where z is given by:
</td>
<td>
<p>
- Using the relation: t = sign(p - 0.5) * sqrt(v * y / x)
+ Using the relation: <span class="serif_italic">t = sign(p -
+ 0.5) * sqrt(v * y / x)</span>
</p>
<p>
where:
</p>
<p>
- x = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v
- / 2, 0.5, 2 * min(p, q))
+
<span class="serif_italic">x = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v
+ / 2, 0.5, 2 * min(p, q)) </span>
</p>
<p>
- y = 1 - x
+ <span class="serif_italic">y = 1 - x</span>
</p>
<p>
The quantities <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span>
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