1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Normal (Gaussian) 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="nc_t_dist.html" title="Noncentral T Distribution"> 10<link rel="next" href="pareto.html" title="Pareto 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="nc_t_dist.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="pareto.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.normal_dist"></a><a class="link" href="normal_dist.html" title="Normal (Gaussian) Distribution">Normal (Gaussian) 28 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">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">normal_distribution</span><span class="special">;</span> 36 37<span class="keyword">typedef</span> <span class="identifier">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">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">// Construct:</span> 46 <span class="identifier">normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> 47 <span class="comment">// Accessors:</span> 48 <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// location.</span> 49 <span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// scale.</span> 50 <span class="comment">// Synonyms, provided to allow generic use of find_location and find_scale.</span> 51 <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 52 <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 53<span class="special">};</span> 54 55<span class="special">}}</span> <span class="comment">// namespaces</span> 56</pre> 57<p> 58 The normal distribution is probably the most well known statistical distribution: 59 it is also known as the Gaussian Distribution. A normal distribution with 60 mean zero and standard deviation one is known as the <span class="emphasis"><em>Standard 61 Normal Distribution</em></span>. 62 </p> 63<p> 64 Given mean μ and standard deviation σ it has the PDF: 65 </p> 66<div class="blockquote"><blockquote class="blockquote"><p> 67 <span class="inlinemediaobject"><img src="../../../../equations/normal_ref1.svg"></span> 68 69 </p></blockquote></div> 70<p> 71 The variation the PDF with its parameters is illustrated in the following 72 graph: 73 </p> 74<div class="blockquote"><blockquote class="blockquote"><p> 75 <span class="inlinemediaobject"><img src="../../../../graphs/normal_pdf.svg" align="middle"></span> 76 77 </p></blockquote></div> 78<p> 79 The cumulative distribution function is given by 80 </p> 81<div class="blockquote"><blockquote class="blockquote"><p> 82 <span class="inlinemediaobject"><img src="../../../../equations/normal_cdf.svg"></span> 83 84 </p></blockquote></div> 85<p> 86 and illustrated by this graph 87 </p> 88<div class="blockquote"><blockquote class="blockquote"><p> 89 <span class="inlinemediaobject"><img src="../../../../graphs/normal_cdf.svg" align="middle"></span> 90 91 </p></blockquote></div> 92<h5> 93<a name="math_toolkit.dist_ref.dists.normal_dist.h0"></a> 94 <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.member_functions"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.member_functions">Member 95 Functions</a> 96 </h5> 97<pre class="programlisting"><span class="identifier">normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> 98</pre> 99<p> 100 Constructs a normal distribution with mean <span class="emphasis"><em>mean</em></span> and 101 standard deviation <span class="emphasis"><em>sd</em></span>. 102 </p> 103<p> 104 Requires <span class="emphasis"><em>sd</em></span> > 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 105 is called. 106 </p> 107<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 108<span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 109</pre> 110<p> 111 both return the <span class="emphasis"><em>mean</em></span> of this distribution. 112 </p> 113<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 114<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 115</pre> 116<p> 117 both return the <span class="emphasis"><em>standard deviation</em></span> of this distribution. 118 (Redundant location and scale function are provided to match other similar 119 distributions, allowing the functions find_location and find_scale to be 120 used generically). 121 </p> 122<h5> 123<a name="math_toolkit.dist_ref.dists.normal_dist.h1"></a> 124 <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.non_member_accessors"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.non_member_accessors">Non-member 125 Accessors</a> 126 </h5> 127<p> 128 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 129 functions</a> that are generic to all distributions are supported: 130 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 131 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 132 <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>, 133 <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>, 134 <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>, 135 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 136 <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>, 137 <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>. 138 </p> 139<p> 140 The domain of the random variable is [-[max_value], +[min_value]]. However, 141 the pdf of +∞ and -∞ = 0 is also supported, and cdf at -∞ = 0, cdf at +∞ = 1, and 142 complement cdf -∞ = 1 and +∞ = 0, if RealType permits. 143 </p> 144<h5> 145<a name="math_toolkit.dist_ref.dists.normal_dist.h2"></a> 146 <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.accuracy"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.accuracy">Accuracy</a> 147 </h5> 148<p> 149 The normal distribution is implemented in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">error 150 function</a>, and as such should have very low error rates. 151 </p> 152<h5> 153<a name="math_toolkit.dist_ref.dists.normal_dist.h3"></a> 154 <span class="phrase"><a name="math_toolkit.dist_ref.dists.normal_dist.implementation"></a></span><a class="link" href="normal_dist.html#math_toolkit.dist_ref.dists.normal_dist.implementation">Implementation</a> 155 </h5> 156<p> 157 In the following table <span class="emphasis"><em>m</em></span> is the mean of the distribution, 158 and <span class="emphasis"><em>s</em></span> is its standard deviation. 159 </p> 160<div class="informaltable"><table class="table"> 161<colgroup> 162<col> 163<col> 164</colgroup> 165<thead><tr> 166<th> 167 <p> 168 Function 169 </p> 170 </th> 171<th> 172 <p> 173 Implementation Notes 174 </p> 175 </th> 176</tr></thead> 177<tbody> 178<tr> 179<td> 180 <p> 181 pdf 182 </p> 183 </td> 184<td> 185 <p> 186 Using the relation: pdf = e<sup>-(x-m)<sup>2</sup>/(2s<sup>2</sup>)</sup> / (s * sqrt(2*pi)) 187 </p> 188 </td> 189</tr> 190<tr> 191<td> 192 <p> 193 cdf 194 </p> 195 </td> 196<td> 197 <p> 198 Using the relation: p = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">erfc</a>(-(x-m)/(s*sqrt(2))) 199 </p> 200 </td> 201</tr> 202<tr> 203<td> 204 <p> 205 cdf complement 206 </p> 207 </td> 208<td> 209 <p> 210 Using the relation: q = 0.5 * <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">erfc</a>((x-m)/(s*sqrt(2))) 211 </p> 212 </td> 213</tr> 214<tr> 215<td> 216 <p> 217 quantile 218 </p> 219 </td> 220<td> 221 <p> 222 Using the relation: x = m - s * sqrt(2) * <a class="link" href="../../sf_erf/error_inv.html" title="Error Function Inverses">erfc_inv</a>(2*p) 223 </p> 224 </td> 225</tr> 226<tr> 227<td> 228 <p> 229 quantile from the complement 230 </p> 231 </td> 232<td> 233 <p> 234 Using the relation: x = m + s * sqrt(2) * <a class="link" href="../../sf_erf/error_inv.html" title="Error Function Inverses">erfc_inv</a>(2*p) 235 </p> 236 </td> 237</tr> 238<tr> 239<td> 240 <p> 241 mean and standard deviation 242 </p> 243 </td> 244<td> 245 <p> 246 The same as <code class="computeroutput"><span class="identifier">dist</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span></code> and <code class="computeroutput"><span class="identifier">dist</span><span class="special">.</span><span class="identifier">standard_deviation</span><span class="special">()</span></code> 247 </p> 248 </td> 249</tr> 250<tr> 251<td> 252 <p> 253 mode 254 </p> 255 </td> 256<td> 257 <p> 258 The same as the mean. 259 </p> 260 </td> 261</tr> 262<tr> 263<td> 264 <p> 265 median 266 </p> 267 </td> 268<td> 269 <p> 270 The same as the mean. 271 </p> 272 </td> 273</tr> 274<tr> 275<td> 276 <p> 277 skewness 278 </p> 279 </td> 280<td> 281 <p> 282 0 283 </p> 284 </td> 285</tr> 286<tr> 287<td> 288 <p> 289 kurtosis 290 </p> 291 </td> 292<td> 293 <p> 294 3 295 </p> 296 </td> 297</tr> 298<tr> 299<td> 300 <p> 301 kurtosis excess 302 </p> 303 </td> 304<td> 305 <p> 306 0 307 </p> 308 </td> 309</tr> 310</tbody> 311</table></div> 312</div> 313<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 314<td align="left"></td> 315<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 316 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 317 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 318 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 319 Daryle Walker and Xiaogang Zhang<p> 320 Distributed under the Boost Software License, Version 1.0. (See accompanying 321 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>) 322 </p> 323</div></td> 324</tr></table> 325<hr> 326<div class="spirit-nav"> 327<a accesskey="p" href="nc_t_dist.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="pareto.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 328</div> 329</body> 330</html> 331
