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24</div>
25<div class="section">
26<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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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