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 27<a name="document_to_test_formatting.lists_and_tables"></a><a class="link" href="lists_and_tables.html" title="Lists and Tables">Lists and
 28    Tables</a>
 29</h2></div></div></div>
 30<div class="toc"><dl>
 31<dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.lists">Lists</a></span></dt>
 32<dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.variable_lists">Variable
 33      Lists</a></span></dt>
 34<dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.tables">Tables</a></span></dt>
 35</dl></div>
 36<div class="section">
 37<div class="titlepage"><div><div><h3 class="title">
 38<a name="document_to_test_formatting.lists_and_tables.lists"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.lists" title="Lists">Lists</a>
 39</h3></div></div></div>
 40<p>
 41        A numbered list:
 42      </p>
 43<div class="orderedlist"><ol class="orderedlist" type="1">
 44<li class="listitem">
 45            One
 46          </li>
 47<li class="listitem">
 48            Two
 49          </li>
 50<li class="listitem">
 51            Three
 52            <div class="orderedlist"><ol class="orderedlist" type="a">
 53<li class="listitem">
 54                  Three.a
 55                </li>
 56<li class="listitem">
 57                  Three.b
 58                </li>
 59<li class="listitem">
 60                  Three.c
 61                </li>
 62</ol></div>
 63          </li>
 64<li class="listitem">
 65            Four
 66            <div class="orderedlist"><ol class="orderedlist" type="a"><li class="listitem">
 67                  Four.a
 68                  <div class="orderedlist"><ol class="orderedlist" type="i">
 69<li class="listitem">
 70                        Four.a.i
 71                      </li>
 72<li class="listitem">
 73                        Four.a.ii
 74                      </li>
 75</ol></div>
 76                </li></ol></div>
 77          </li>
 78<li class="listitem">
 79            Five
 80          </li>
 81</ol></div>
 82<p>
 83        An unordered list:
 84      </p>
 85<div class="itemizedlist"><ul class="itemizedlist" type="disc">
 86<li class="listitem">
 87            First
 88          </li>
 89<li class="listitem">
 90            Second
 91          </li>
 92<li class="listitem">
 93            Third
 94          </li>
 95</ul></div>
 96<p>
 97        A mixture of the two:
 98      </p>
 99<div class="orderedlist"><ol class="orderedlist" type="1">
100<li class="listitem">
101            1
102            <div class="itemizedlist"><ul class="itemizedlist" type="disc">
103<li class="listitem">
104                  1.a
105                  <div class="orderedlist"><ol class="orderedlist" type="a">
106<li class="listitem">
107                        1.a.1
108                      </li>
109<li class="listitem">
110                        1.a.2
111                      </li>
112</ol></div>
113                </li>
114<li class="listitem">
115                  1.b
116                </li>
117</ul></div>
118          </li>
119<li class="listitem">
120            2
121            <div class="itemizedlist"><ul class="itemizedlist" type="disc">
122<li class="listitem">
123                  2.a
124                </li>
125<li class="listitem">
126                  2.b
127                  <div class="orderedlist"><ol class="orderedlist" type="a">
128<li class="listitem">
129                        2.b.1
130                      </li>
131<li class="listitem">
132                        2.b.2
133                        <div class="itemizedlist"><ul class="itemizedlist" type="circle">
134<li class="listitem">
135                              2.b.2.a
136                            </li>
137<li class="listitem">
138                              2.b.2.b
139                            </li>
140</ul></div>
141                      </li>
142</ol></div>
143                </li>
144</ul></div>
145          </li>
146</ol></div>
147</div>
148<div class="section">
149<div class="titlepage"><div><div><h3 class="title">
150<a name="document_to_test_formatting.lists_and_tables.variable_lists"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.variable_lists" title="Variable Lists">Variable
151      Lists</a>
152</h3></div></div></div>
153<div class="variablelist">
154<p class="title"><b>A Variable List</b></p>
155<dl>
156<dt><span class="term">term 1</span></dt>
157<dd><p>
158              The definition of term 1
159            </p></dd>
160<dt><span class="term">term 2</span></dt>
161<dd><p>
162              The definition of term 2
163            </p></dd>
164<dt><span class="term">term 3</span></dt>
165<dd><p>
166              The definition of term 3
167            </p></dd>
168</dl>
169</div>
170</div>
171<div class="section">
172<div class="titlepage"><div><div><h3 class="title">
173<a name="document_to_test_formatting.lists_and_tables.tables"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.tables" title="Tables">Tables</a>
174</h3></div></div></div>
175<p>
176        Here's a big table with code and other tricky things:
177      </p>
178<div class="table">
179<a name="id769721"></a><p class="title"><b>Table&#160;1.&#160;Notes on the Implementation of the Beta Distribution</b></p>
180<div class="table-contents"><table class="table" summary="Notes on the Implementation of the Beta Distribution">
181<colgroup>
182<col>
183<col>
184</colgroup>
185<thead><tr>
186<th>
187                <p>
188                  Function
189                </p>
190              </th>
191<th>
192                <p>
193                  Implementation Notes
194                </p>
195              </th>
196</tr></thead>
197<tbody>
198<tr>
199<td>
200                <p>
201                  pdf
202                </p>
203              </td>
204<td>
205                <p>
206                  f(x;&#945;,&#946;) = x<sup>&#945; - 1</sup> (1 - x)<sup>&#946; -1</sup> / B(&#945;, &#946;)
207                </p>
208                <p>
209                  Implemented using ibeta_derivative(a, b, x).
210                </p>
211              </td>
212</tr>
213<tr>
214<td>
215                <p>
216                  cdf
217                </p>
218              </td>
219<td>
220                <p>
221                  Using the incomplete beta function ibeta(a, b, x)
222                </p>
223              </td>
224</tr>
225<tr>
226<td>
227                <p>
228                  cdf complement
229                </p>
230              </td>
231<td>
232                <p>
233                  ibetac(a, b, x)
234                </p>
235              </td>
236</tr>
237<tr>
238<td>
239                <p>
240                  quantile
241                </p>
242              </td>
243<td>
244                <p>
245                  Using the inverse incomplete beta function ibeta_inv(a, b, p)
246                </p>
247              </td>
248</tr>
249<tr>
250<td>
251                <p>
252                  quantile from the complement
253                </p>
254              </td>
255<td>
256                <p>
257                  ibetac_inv(a, b, q)
258                </p>
259              </td>
260</tr>
261<tr>
262<td>
263                <p>
264                  mean
265                </p>
266              </td>
267<td>
268                <p>
269                  <code class="computeroutput"><span class="identifier">a</span><span class="special">/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)</span></code>
270                </p>
271              </td>
272</tr>
273<tr>
274<td>
275                <p>
276                  variance
277                </p>
278              </td>
279<td>
280                <p>
281                  <code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span>
282                  <span class="identifier">b</span> <span class="special">/</span>
283                  <span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)^</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span>
284                  <span class="identifier">b</span> <span class="special">+</span>
285                  <span class="number">1</span><span class="special">)</span></code>
286                </p>
287              </td>
288</tr>
289<tr>
290<td>
291                <p>
292                  mode
293                </p>
294              </td>
295<td>
296                <p>
297                  <code class="computeroutput"><span class="special">(</span><span class="identifier">a</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">a</span>
298                  <span class="special">+</span> <span class="identifier">b</span>
299                  <span class="special">+</span> <span class="number">2</span><span class="special">)</span></code>
300                </p>
301              </td>
302</tr>
303<tr>
304<td>
305                <p>
306                  skewness
307                </p>
308              </td>
309<td>
310                <p>
311                  <code class="computeroutput"><span class="number">2</span> <span class="special">(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span>
312                  <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">1</span><span class="special">)/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span>
313                  <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code>
314                </p>
315              </td>
316</tr>
317<tr>
318<td>
319                <p>
320                  kurtosis excess
321                </p>
322              </td>
323<td>
324                <p>
325                  <span class="inlinemediaobject"><img src="../images/beta_dist_kurtosis.png" alt="beta_dist_kurtosis"></span>
326                </p>
327              </td>
328</tr>
329<tr>
330<td>
331                <p>
332                  kurtosis
333                </p>
334              </td>
335<td>
336                <p>
337                  <code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span>
338                  <span class="number">3</span></code>
339                </p>
340              </td>
341</tr>
342<tr>
343<td>
344                <p>
345                  parameter estimation
346                </p>
347              </td>
348<td>
349              </td>
350</tr>
351<tr>
352<td>
353                <p>
354                  alpha
355                </p>
356                <p>
357                  from mean and variance
358                </p>
359              </td>
360<td>
361                <p>
362                  <code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span>
363                  <span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span>
364                  <span class="special">(</span><span class="number">1</span>
365                  <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">variance</span><span class="special">)-</span>
366                  <span class="number">1</span><span class="special">)</span></code>
367                </p>
368              </td>
369</tr>
370<tr>
371<td>
372                <p>
373                  beta
374                </p>
375                <p>
376                  from mean and variance
377                </p>
378              </td>
379<td>
380                <p>
381                  <code class="computeroutput"><span class="special">(</span><span class="number">1</span>
382                  <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span> <span class="special">(((</span><span class="identifier">mean</span>
383                  <span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span>
384                  <span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code>
385                </p>
386              </td>
387</tr>
388<tr>
389<td>
390                <p>
391                  The member functions <code class="computeroutput"><span class="identifier">estimate_alpha</span></code>
392                  and <code class="computeroutput"><span class="identifier">estimate_beta</span></code>
393                </p>
394                <p>
395                  from cdf and probability x
396                </p>
397                <p>
398                  and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code>
399                  or <code class="computeroutput"><span class="identifier">beta</span></code>
400                </p>
401              </td>
402<td>
403                <p>
404                  Implemented in terms of the inverse incomplete beta functions
405                </p>
406                <p>
407                  ibeta_inva, and ibeta_invb respectively.
408                </p>
409              </td>
410</tr>
411<tr>
412<td>
413                <p>
414                  <code class="computeroutput"><span class="identifier">estimate_alpha</span></code>
415                </p>
416              </td>
417<td>
418                <p>
419                  <code class="computeroutput"><span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">beta</span><span class="special">,</span>
420                  <span class="identifier">x</span><span class="special">,</span>
421                  <span class="identifier">probability</span><span class="special">)</span></code>
422                </p>
423              </td>
424</tr>
425<tr>
426<td>
427                <p>
428                  <code class="computeroutput"><span class="identifier">estimate_beta</span></code>
429                </p>
430              </td>
431<td>
432                <p>
433                  <code class="computeroutput"><span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">,</span>
434                  <span class="identifier">x</span><span class="special">,</span>
435                  <span class="identifier">probability</span><span class="special">)</span></code>
436                </p>
437              </td>
438</tr>
439</tbody>
440</table></div>
441</div>
442<br class="table-break">
443</div>
444</div>
445<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
446<td align="left"></td>
447<td align="right"><div class="copyright-footer">Copyright &#169; 2007 John Maddock, Joel de Guzman, Eric Niebler and Matias
448      Capeletto<p>
449        Distributed under the Boost Software License, Version 1.0. (See accompanying
450        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>)
451      </p>
452</div></td>
453</tr></table>
454<hr>
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