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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>
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