1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Skew Normal Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="rayleigh.html" title="Rayleigh Distribution"> 10<link rel="next" href="students_t_dist.html" title="Students t Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="rayleigh.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="students_t_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h4 class="title"> 27<a name="math_toolkit.dist_ref.dists.skew_normal_dist"></a><a class="link" href="skew_normal_dist.html" title="Skew Normal Distribution">Skew 28 Normal Distribution</a> 29</h4></div></div></div> 30<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">skew_normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> 31<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> 32 33<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> 34 <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 21. 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> 35<span class="keyword">class</span> <span class="identifier">skew_normal_distribution</span><span class="special">;</span> 36 37<span class="keyword">typedef</span> <span class="identifier">skew_normal_distribution</span><span class="special"><></span> <span class="identifier">normal</span><span class="special">;</span> 38 39<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 40<span class="keyword">class</span> <span class="identifier">skew_normal_distribution</span> 41<span class="special">{</span> 42<span class="keyword">public</span><span class="special">:</span> 43 <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> 44 <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> 45 <span class="comment">// Constructor:</span> 46 <span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 47 <span class="comment">// Accessors:</span> 48 <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// mean if normal.</span> 49 <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// width, standard deviation if normal.</span> 50 <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// The distribution is right skewed if shape > 0 and is left skewed if shape < 0.</span> 51 <span class="comment">// The distribution is normal if shape is zero.</span> 52<span class="special">};</span> 53 54<span class="special">}}</span> <span class="comment">// namespaces</span> 55</pre> 56<p> 57 The skew normal distribution is a variant of the most well known Gaussian 58 statistical distribution. 59 </p> 60<p> 61 The skew normal distribution with shape zero resembles the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">Normal 62 Distribution</a>, hence the latter can be regarded as a special case 63 of the more generic skew normal distribution. 64 </p> 65<p> 66 If the standard (mean = 0, scale = 1) normal distribution probability density 67 function is 68 </p> 69<div class="blockquote"><blockquote class="blockquote"><p> 70 <span class="inlinemediaobject"><img src="../../../../equations/normal01_pdf.svg"></span> 71 72 </p></blockquote></div> 73<p> 74 and the cumulative distribution function 75 </p> 76<div class="blockquote"><blockquote class="blockquote"><p> 77 <span class="inlinemediaobject"><img src="../../../../equations/normal01_cdf.svg"></span> 78 79 </p></blockquote></div> 80<p> 81 then the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">PDF</a> 82 of the <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">skew 83 normal distribution</a> with shape parameter α, defined by O'Hagan and 84 Leonhard (1976) is 85 </p> 86<div class="blockquote"><blockquote class="blockquote"><p> 87 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf0.svg"></span> 88 89 </p></blockquote></div> 90<p> 91 Given <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">location</a> 92 ξ, <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale</a> 93 ω, and <a href="http://en.wikipedia.org/wiki/Shape_parameter" target="_top">shape</a> 94 α, it can be <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">transformed</a>, 95 to the form: 96 </p> 97<div class="blockquote"><blockquote class="blockquote"><p> 98 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span> 99 100 </p></blockquote></div> 101<p> 102 and <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" target="_top">CDF</a>: 103 </p> 104<div class="blockquote"><blockquote class="blockquote"><p> 105 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span> 106 107 </p></blockquote></div> 108<p> 109 where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, and <span class="emphasis"><em>Φ(x)</em></span> 110 is the normal distribution. 111 </p> 112<p> 113 The variation the PDF and CDF with its parameters is illustrated in the 114 following graphs: 115 </p> 116<div class="blockquote"><blockquote class="blockquote"><p> 117 <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_pdf.svg" align="middle"></span> 118 119 </p></blockquote></div> 120<div class="blockquote"><blockquote class="blockquote"><p> 121 <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_cdf.svg" align="middle"></span> 122 123 </p></blockquote></div> 124<h5> 125<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h0"></a> 126 <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.member_functions"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.member_functions">Member 127 Functions</a> 128 </h5> 129<pre class="programlisting"><span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 130</pre> 131<p> 132 Constructs a skew_normal distribution with location ξ, scale ω and shape α. 133 </p> 134<p> 135 Requires scale > 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 136 is called. 137 </p> 138<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 139</pre> 140<p> 141 returns the location ξ of this distribution, 142 </p> 143<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 144</pre> 145<p> 146 returns the scale ω of this distribution, 147 </p> 148<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 149</pre> 150<p> 151 returns the shape α of this distribution. 152 </p> 153<p> 154 (Location and scale function match other similar distributions, allowing 155 the functions <code class="computeroutput"><span class="identifier">find_location</span></code> 156 and <code class="computeroutput"><span class="identifier">find_scale</span></code> to be used 157 generically). 158 </p> 159<div class="note"><table border="0" summary="Note"> 160<tr> 161<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> 162<th align="left">Note</th> 163</tr> 164<tr><td align="left" valign="top"> 165<p> 166 While the shape parameter may be chosen arbitrarily (finite), the resulting 167 <span class="bold"><strong>skewness</strong></span> of the distribution is in fact 168 limited to about (-1, 1); strictly, the interval is (-0.9952717, 0.9952717). 169 </p> 170<p> 171 A parameter δ is related to the shape α by δ = α / (1 + α²), and used in the expression 172 for skewness 173 </p> 174<div class="blockquote"><blockquote class="blockquote"><p> 175 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span> 176 177 </p></blockquote></div> 178</td></tr> 179</table></div> 180<h5> 181<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h1"></a> 182 <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.references"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.references">References</a> 183 </h5> 184<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 185<li class="listitem"> 186 <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal Probability 187 Distribution</a> for many links and bibliography. 188 </li> 189<li class="listitem"> 190 <a href="http://azzalini.stat.unipd.it/SN/Intro/intro.html" target="_top">A very 191 brief introduction to the skew-normal distribution</a> by Adelchi 192 Azzalini (2005-11-2). 193 </li> 194<li class="listitem"> 195 See a <a href="http://www.tri.org.au/azzalini.html" target="_top">skew-normal 196 function animation</a>. 197 </li> 198</ul></div> 199<h5> 200<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h2"></a> 201 <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.non_member_accessors"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.non_member_accessors">Non-member 202 Accessors</a> 203 </h5> 204<p> 205 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 206 functions</a> that are generic to all distributions are supported: 207 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 208 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 209 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, 210 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, 211 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, 212 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 213 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, 214 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. 215 </p> 216<p> 217 The domain of the random variable is <span class="emphasis"><em>-[max_value], +[min_value]</em></span>. 218 Infinite values are not supported. 219 </p> 220<p> 221 There are no <a href="http://en.wikipedia.org/wiki/Closed-form_expression" target="_top">closed-form 222 expression</a> known for the mode and median, but these are computed 223 for the 224 </p> 225<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 226<li class="listitem"> 227 mode - by finding the maximum of the PDF. 228 </li> 229<li class="listitem"> 230 median - by computing <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="number">1</span><span class="special">/</span><span class="number">2</span><span class="special">)</span></code>. 231 </li> 232</ul></div> 233<p> 234 The maximum of the PDF is sought through searching the root of f'(x)=0. 235 </p> 236<p> 237 Both involve iterative methods that will have lower accuracy than other 238 estimates. 239 </p> 240<h5> 241<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h3"></a> 242 <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.testing"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.testing">Testing</a> 243 </h5> 244<p> 245 <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a> 246 using library(sn) described at <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal 247 Probability Distribution</a>, and at <a href="http://cran.r-project.org/web/packages/sn/sn.pd" target="_top">R 248 skew-normal(sn) package</a>. 249 </p> 250<p> 251 Package sn provides functions related to the skew-normal (SN) and the skew-t 252 (ST) probability distributions, both for the univariate and for the the 253 multivariate case, including regression models. 254 </p> 255<p> 256 <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram 257 Mathematica</a> was also used to generate some more accurate spot test 258 data. 259 </p> 260<h5> 261<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h4"></a> 262 <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.accuracy"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.accuracy">Accuracy</a> 263 </h5> 264<p> 265 The skew_normal distribution with shape = zero is implemented as a special 266 case, equivalent to the normal distribution in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">error 267 function</a>, and therefore should have excellent accuracy. 268 </p> 269<p> 270 The PDF and mean, variance, skewness and kurtosis are also accurately evaluated 271 using <a href="http://en.wikipedia.org/wiki/Analytical_expression" target="_top">analytical 272 expressions</a>. The CDF requires <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owen's 273 T function</a> that is evaluated using a Boost C++ <a class="link" href="../../owens_t.html" title="Owen's T function">Owens 274 T</a> implementation of the algorithms of M. Patefield and D. Tandy, 275 Journal of Statistical Software, 5(5), 1-25 (2000); the complicated accuracy 276 of this function is discussed in detail at <a class="link" href="../../owens_t.html" title="Owen's T function">Owens 277 T</a>. 278 </p> 279<p> 280 The median and mode are calculated by iterative root finding, and both 281 will be less accurate. 282 </p> 283<h5> 284<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h5"></a> 285 <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.implementation"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.implementation">Implementation</a> 286 </h5> 287<p> 288 In the following table, ξ is the location of the distribution, and ω is its 289 scale, and α is its shape. 290 </p> 291<div class="informaltable"><table class="table"> 292<colgroup> 293<col> 294<col> 295</colgroup> 296<thead><tr> 297<th> 298 <p> 299 Function 300 </p> 301 </th> 302<th> 303 <p> 304 Implementation Notes 305 </p> 306 </th> 307</tr></thead> 308<tbody> 309<tr> 310<td> 311 <p> 312 pdf 313 </p> 314 </td> 315<td> 316 <p> 317 Using: 318 </p> 319 <div class="blockquote"><blockquote class="blockquote"><p> 320 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span> 321 322 </p></blockquote></div> 323 </td> 324</tr> 325<tr> 326<td> 327 <p> 328 cdf 329 </p> 330 </td> 331<td> 332 <p> 333 Using: 334 </p> 335 <div class="blockquote"><blockquote class="blockquote"><p> 336 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span> 337 338 </p></blockquote></div> 339 <p> 340 <br> where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, 341 and <span class="emphasis"><em>Φ(x)</em></span> is the normal distribution. 342 </p> 343 </td> 344</tr> 345<tr> 346<td> 347 <p> 348 cdf complement 349 </p> 350 </td> 351<td> 352 <p> 353 Using: complement of normal distribution + 2 * Owens_t 354 </p> 355 </td> 356</tr> 357<tr> 358<td> 359 <p> 360 quantile 361 </p> 362 </td> 363<td> 364 <p> 365 Maximum of the pdf is sought through searching the root of f'(x)=0 366 </p> 367 </td> 368</tr> 369<tr> 370<td> 371 <p> 372 quantile from the complement 373 </p> 374 </td> 375<td> 376 <p> 377 -quantile(SN(-location ξ, scale ω, -shapeα), p) 378 </p> 379 </td> 380</tr> 381<tr> 382<td> 383 <p> 384 location 385 </p> 386 </td> 387<td> 388 <p> 389 location ξ 390 </p> 391 </td> 392</tr> 393<tr> 394<td> 395 <p> 396 scale 397 </p> 398 </td> 399<td> 400 <p> 401 scale ω 402 </p> 403 </td> 404</tr> 405<tr> 406<td> 407 <p> 408 shape 409 </p> 410 </td> 411<td> 412 <p> 413 shape α 414 </p> 415 </td> 416</tr> 417<tr> 418<td> 419 <p> 420 median 421 </p> 422 </td> 423<td> 424 <p> 425 quantile(1/2) 426 </p> 427 </td> 428</tr> 429<tr> 430<td> 431 <p> 432 mean 433 </p> 434 </td> 435<td> 436 <div class="blockquote"><blockquote class="blockquote"><p> 437 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_mean.svg"></span> 438 439 </p></blockquote></div> 440 </td> 441</tr> 442<tr> 443<td> 444 <p> 445 mode 446 </p> 447 </td> 448<td> 449 <p> 450 Maximum of the pdf is sought through searching the root of f'(x)=0 451 </p> 452 </td> 453</tr> 454<tr> 455<td> 456 <p> 457 variance 458 </p> 459 </td> 460<td> 461 <div class="blockquote"><blockquote class="blockquote"><p> 462 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_variance.svg"></span> 463 464 </p></blockquote></div> 465 </td> 466</tr> 467<tr> 468<td> 469 <p> 470 skewness 471 </p> 472 </td> 473<td> 474 <div class="blockquote"><blockquote class="blockquote"><p> 475 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span> 476 477 </p></blockquote></div> 478 </td> 479</tr> 480<tr> 481<td> 482 <p> 483 kurtosis 484 </p> 485 </td> 486<td> 487 <p> 488 kurtosis excess-3 489 </p> 490 </td> 491</tr> 492<tr> 493<td> 494 <p> 495 kurtosis excess 496 </p> 497 </td> 498<td> 499 <div class="blockquote"><blockquote class="blockquote"><p> 500 <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_kurt_ex.svg"></span> 501 502 </p></blockquote></div> 503 </td> 504</tr> 505</tbody> 506</table></div> 507</div> 508<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 509<td align="left"></td> 510<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 511 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 512 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 513 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 514 Daryle Walker and Xiaogang Zhang<p> 515 Distributed under the Boost Software License, Version 1.0. (See accompanying 516 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>) 517 </p> 518</div></td> 519</tr></table> 520<hr> 521<div class="spirit-nav"> 522<a accesskey="p" href="rayleigh.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="students_t_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 523</div> 524</body> 525</html> 526
