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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.dist_ref.nmp"></a><a class="link" href="nmp.html" title="Non-Member Properties">Non-Member Properties</a>
  28 </h3></div></div></div>
  29 <p>
  30         Properties that are common to all distributions are accessed via non-member
  31         getter functions: non-membership allows more of these functions to be added
  32         over time, as the need arises. Unfortunately the literature uses many different
  33         and confusing names to refer to a rather small number of actual concepts;
  34         refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">concept
  35         index</a> to find the property you want by the name you are most familiar
  36         with. Or use the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">function
  37         index</a> to go straight to the function you want if you already know
  38         its name.
  39       </p>
  40 <h5>
  41 <a name="math_toolkit.dist_ref.nmp.h0"></a>
  42         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.function_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">Function
  43         Index</a>
  44       </h5>
  45 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  46 <li class="listitem">
  47             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
  48             Function</a>.
  49           </li>
  50 <li class="listitem">
  51             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
  52             Distribution Function</a>.
  53           </li>
  54 <li class="listitem">
  55             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
  56           </li>
  57 <li class="listitem">
  58             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>.
  59           </li>
  60 <li class="listitem">
  61             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
  62           </li>
  63 <li class="listitem">
  64             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
  65           </li>
  66 <li class="listitem">
  67             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
  68           </li>
  69 <li class="listitem">
  70             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
  71           </li>
  72 <li class="listitem">
  73             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
  74           </li>
  75 <li class="listitem">
  76             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
  77           </li>
  78 <li class="listitem">
  79             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
  80           </li>
  81 <li class="listitem">
  82             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
  83           </li>
  84 <li class="listitem">
  85             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
  86             complement of the probability</a>.
  87           </li>
  88 <li class="listitem">
  89             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
  90           </li>
  91 <li class="listitem">
  92             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>.
  93           </li>
  94 <li class="listitem">
  95             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
  96           </li>
  97 <li class="listitem">
  98             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
  99           </li>
 100 </ul></div>
 101 <h5>
 102 <a name="math_toolkit.dist_ref.nmp.h1"></a>
 103         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.concept_index"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">Conceptual
 104         Index</a>
 105       </h5>
 106 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 107 <li class="listitem">
 108             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
 109             Distribution Function</a>.
 110           </li>
 111 <li class="listitem">
 112             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
 113             Function</a>.
 114           </li>
 115 <li class="listitem">
 116             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
 117           </li>
 118 <li class="listitem">
 119             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse Cumulative
 120             Distribution Function</a>.
 121           </li>
 122 <li class="listitem">
 123             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse Survival
 124             Function</a>.
 125           </li>
 126 <li class="listitem">
 127             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
 128           </li>
 129 <li class="listitem">
 130             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower Critical
 131             Value</a>.
 132           </li>
 133 <li class="listitem">
 134             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
 135           </li>
 136 <li class="listitem">
 137             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
 138           </li>
 139 <li class="listitem">
 140             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
 141           </li>
 142 <li class="listitem">
 143             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
 144           </li>
 145 <li class="listitem">
 146             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
 147           </li>
 148 <li class="listitem">
 149             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P</a>.
 150           </li>
 151 <li class="listitem">
 152             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent Point Function</a>.
 153           </li>
 154 <li class="listitem">
 155             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
 156           </li>
 157 <li class="listitem">
 158             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability Mass Function</a>.
 159           </li>
 160 <li class="listitem">
 161             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
 162           </li>
 163 <li class="listitem">
 164             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">Q</a>.
 165           </li>
 166 <li class="listitem">
 167             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
 168           </li>
 169 <li class="listitem">
 170             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
 171             complement of the probability</a>.
 172           </li>
 173 <li class="listitem">
 174             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
 175           </li>
 176 <li class="listitem">
 177             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>
 178           </li>
 179 <li class="listitem">
 180             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival Function</a>.
 181           </li>
 182 <li class="listitem">
 183             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 184           </li>
 185 <li class="listitem">
 186             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper Critical
 187             Value</a>.
 188           </li>
 189 <li class="listitem">
 190             <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
 191           </li>
 192 </ul></div>
 193 <h5>
 194 <a name="math_toolkit.dist_ref.nmp.h2"></a>
 195         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 196         Distribution Function</a>
 197       </h5>
 198 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 199 <span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
 200 </pre>
 201 <p>
 202         The <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
 203         Function</a> is the probability that the variable takes a value less than
 204         or equal to x. It is equivalent to the integral from -infinity to x of the
 205         <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
 206       </p>
 207 <p>
 208         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 209         if the random variable is outside the defined range for the distribution.
 210       </p>
 211 <p>
 212         For example, the following graph shows the cdf for the normal distribution:
 213       </p>
 214 <p>
 215         <span class="inlinemediaobject"><img src="../../../graphs/cdf.png"></span>
 216       </p>
 217 <h5>
 218 <a name="math_toolkit.dist_ref.nmp.h3"></a>
 219         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.ccdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement
 220         of the Cumulative Distribution Function</a>
 221       </h5>
 222 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">&gt;</span>
 223 <span class="identifier">RealType</span> <span class="identifier">cdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special">&lt;</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;&amp;</span> <span class="identifier">comp</span><span class="special">);</span>
 224 </pre>
 225 <p>
 226         The complement of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 227         Distribution Function</a> is the probability that the variable takes a
 228         value greater than x. It is equivalent to the integral from x to infinity
 229         of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
 230         Function</a>, or 1 minus the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 231         Distribution Function</a> of x.
 232       </p>
 233 <p>
 234         This is also known as the survival function.
 235       </p>
 236 <p>
 237         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 238         if the random variable is outside the defined range for the distribution.
 239       </p>
 240 <p>
 241         In this library, it is obtained by wrapping the arguments to the <code class="computeroutput"><span class="identifier">cdf</span></code> function in a call to <code class="computeroutput"><span class="identifier">complement</span></code>, for example:
 242       </p>
 243 <pre class="programlisting"><span class="comment">// standard normal distribution object:</span>
 244 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span> <span class="identifier">norm</span><span class="special">;</span>
 245 <span class="comment">// print survival function for x=2.0:</span>
 246 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">norm</span><span class="special">,</span> <span class="number">2.0</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 247 </pre>
 248 <p>
 249         For example, the following graph shows the __complement of the cdf for the
 250         normal distribution:
 251       </p>
 252 <p>
 253         <span class="inlinemediaobject"><img src="../../../graphs/survival.png"></span>
 254       </p>
 255 <p>
 256         See <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a> for why the complement
 257         is useful and when it should be used.
 258       </p>
 259 <h5>
 260 <a name="math_toolkit.dist_ref.nmp.h4"></a>
 261         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.hazard"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard
 262         Function</a>
 263       </h5>
 264 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 265 <span class="identifier">RealType</span> <span class="identifier">hazard</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
 266 </pre>
 267 <p>
 268         Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
 269         of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>.
 270       </p>
 271 <p>
 272         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 273         if the random variable is outside the defined range for the distribution.
 274       </p>
 275 <div class="blockquote"><blockquote class="blockquote"><p>
 276           <span class="inlinemediaobject"><img src="../../../equations/hazard.svg"></span>
 277
 278         </p></blockquote></div>
 279 <div class="caution"><table border="0" summary="Caution">
 280 <tr>
 281 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
 282 <th align="left">Caution</th>
 283 </tr>
 284 <tr><td align="left" valign="top"><p>
 285           Some authors refer to this as the conditional failure density function
 286           rather than the hazard function.
 287         </p></td></tr>
 288 </table></div>
 289 <h5>
 290 <a name="math_toolkit.dist_ref.nmp.h5"></a>
 291         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.chf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative
 292         Hazard Function</a>
 293       </h5>
 294 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 295 <span class="identifier">RealType</span> <span class="identifier">chf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
 296 </pre>
 297 <p>
 298         Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard
 299         Function</a> of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>.
 300       </p>
 301 <p>
 302         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 303         if the random variable is outside the defined range for the distribution.
 304       </p>
 305 <div class="blockquote"><blockquote class="blockquote"><p>
 306           <span class="inlinemediaobject"><img src="../../../equations/chf.svg"></span>
 307
 308         </p></blockquote></div>
 309 <div class="caution"><table border="0" summary="Caution">
 310 <tr>
 311 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
 312 <th align="left">Caution</th>
 313 </tr>
 314 <tr><td align="left" valign="top"><p>
 315           Some authors refer to this as simply the "Hazard Function".
 316         </p></td></tr>
 317 </table></div>
 318 <h5>
 319 <a name="math_toolkit.dist_ref.nmp.h6"></a>
 320         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mean"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>
 321       </h5>
 322 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 323 <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 324 </pre>
 325 <p>
 326         Returns the mean of the distribution <span class="emphasis"><em>dist</em></span>.
 327       </p>
 328 <p>
 329         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 330         if the distribution does not have a defined mean (for example the Cauchy
 331         distribution).
 332       </p>
 333 <h5>
 334 <a name="math_toolkit.dist_ref.nmp.h7"></a>
 335         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.median"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>
 336       </h5>
 337 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 338 <span class="identifier">RealType</span> <span class="identifier">median</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 339 </pre>
 340 <p>
 341         Returns the median of the distribution <span class="emphasis"><em>dist</em></span>.
 342       </p>
 343 <h5>
 344 <a name="math_toolkit.dist_ref.nmp.h8"></a>
 345         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.mode"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>
 346       </h5>
 347 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 348 <span class="identifier">RealType</span> <span class="identifier">mode</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 349 </pre>
 350 <p>
 351         Returns the mode of the distribution <span class="emphasis"><em>dist</em></span>.
 352       </p>
 353 <p>
 354         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 355         if the distribution does not have a defined mode.
 356       </p>
 357 <h5>
 358 <a name="math_toolkit.dist_ref.nmp.h9"></a>
 359         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pdf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
 360         Density Function</a>
 361       </h5>
 362 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 363 <span class="identifier">RealType</span> <span class="identifier">pdf</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">);</span>
 364 </pre>
 365 <p>
 366         For a continuous function, the probability density function (pdf) returns
 367         the probability that the variate has the value x. Since for continuous distributions
 368         the probability at a single point is actually zero, the probability is better
 369         expressed as the integral of the pdf between two points: see the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 370         Distribution Function</a>.
 371       </p>
 372 <p>
 373         For a discrete distribution, the pdf is the probability that the variate
 374         takes the value x.
 375       </p>
 376 <p>
 377         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 378         if the random variable is outside the defined range for the distribution.
 379       </p>
 380 <p>
 381         For example, for a standard normal distribution the pdf looks like this:
 382       </p>
 383 <p>
 384         <span class="inlinemediaobject"><img src="../../../graphs/pdf.png"></span>
 385       </p>
 386 <h5>
 387 <a name="math_toolkit.dist_ref.nmp.h10"></a>
 388         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.range"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">Range</a>
 389       </h5>
 390 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 391 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">range</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 392 </pre>
 393 <p>
 394         Returns the valid range of the random variable over distribution <span class="emphasis"><em>dist</em></span>.
 395       </p>
 396 <h5>
 397 <a name="math_toolkit.dist_ref.nmp.h11"></a>
 398         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>
 399       </h5>
 400 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 401 <span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">p</span><span class="special">);</span>
 402 </pre>
 403 <p>
 404         The quantile is best viewed as the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 405         Distribution Function</a>, it returns a value <span class="emphasis"><em>x</em></span> such
 406         that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span>
 407         <span class="identifier">p</span></code>.
 408       </p>
 409 <p>
 410         This is also known as the <span class="emphasis"><em>percent point function</em></span>, or
 411         <span class="emphasis"><em>percentile</em></span>, or <span class="emphasis"><em>fractile</em></span>, it is
 412         also the same as calculating the <span class="emphasis"><em>lower critical value</em></span>
 413         of a distribution.
 414       </p>
 415 <p>
 416         This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 417         if the probability lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
 418         if there is no finite value that has the specified probability.
 419       </p>
 420 <p>
 421         The following graph shows the quantile function for a standard normal distribution:
 422       </p>
 423 <p>
 424         <span class="inlinemediaobject"><img src="../../../graphs/quantile.png"></span>
 425       </p>
 426 <h5>
 427 <a name="math_toolkit.dist_ref.nmp.h12"></a>
 428         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.quantile_c"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile
 429         from the complement of the probability.</a>
 430       </h5>
 431 <p>
 432         See also <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">complements</a>.
 433       </p>
 434 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Distribution</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">&gt;</span>
 435 <span class="identifier">RealType</span> <span class="identifier">quantile</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Unspecified-Complement-Type</em></span><span class="special">&lt;</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;&amp;</span> <span class="identifier">comp</span><span class="special">);</span>
 436 </pre>
 437 <p>
 438         This is the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement
 439         of the Cumulative Distribution Function</a>. It is calculated by wrapping
 440         the arguments in a call to the quantile function in a call to <span class="emphasis"><em>complement</em></span>.
 441         For example:
 442       </p>
 443 <pre class="programlisting"><span class="comment">// define a standard normal distribution:</span>
 444 <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">normal</span> <span class="identifier">norm</span><span class="special">;</span>
 445 <span class="comment">// print the value of x for which the complement</span>
 446 <span class="comment">// of the probability is 0.05:</span>
 447 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">norm</span><span class="special">,</span> <span class="number">0.05</span><span class="special">))</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 448 </pre>
 449 <p>
 450         The function computes a value <span class="emphasis"><em>x</em></span> such that <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span>
 451         <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code> where
 452         <span class="emphasis"><em>q</em></span> is complement of the probability.
 453       </p>
 454 <p>
 455         <a class="link" href="../stat_tut/overview/complements.html#why_complements">Why complements?</a>
 456       </p>
 457 <p>
 458         This function is also called the inverse survival function, and is the same
 459         as calculating the <span class="emphasis"><em>upper critical value</em></span> of a distribution.
 460       </p>
 461 <p>
 462         This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 463         if the probablity lies outside [0,1]. The function may return an <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
 464         if there is no finite value that has the specified probability.
 465       </p>
 466 <p>
 467         The following graph show the inverse survival function for the normal distribution:
 468       </p>
 469 <p>
 470         <span class="inlinemediaobject"><img src="../../../graphs/survival_inv.png"></span>
 471       </p>
 472 <h5>
 473 <a name="math_toolkit.dist_ref.nmp.h13"></a>
 474         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.sd"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">Standard
 475         Deviation</a>
 476       </h5>
 477 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 478 <span class="identifier">RealType</span> <span class="identifier">standard_deviation</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 479 </pre>
 480 <p>
 481         Returns the standard deviation of distribution <span class="emphasis"><em>dist</em></span>.
 482       </p>
 483 <p>
 484         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 485         if the distribution does not have a defined standard deviation.
 486       </p>
 487 <h5>
 488 <a name="math_toolkit.dist_ref.nmp.h14"></a>
 489         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.support"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>
 490       </h5>
 491 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 492 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">&gt;</span> <span class="identifier">support</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 493 </pre>
 494 <p>
 495         Returns the supported range of random variable over the distribution <span class="emphasis"><em>dist</em></span>.
 496       </p>
 497 <p>
 498         The distribution is said to be 'supported' over a range that is <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">"the
 499         smallest closed set whose complement has probability zero"</a>.
 500         Non-mathematicians might say it means the 'interesting' smallest range of
 501         random variate x that has the cdf going from zero to unity. Outside are uninteresting
 502         zones where the pdf is zero, and the cdf zero or unity.
 503       </p>
 504 <h5>
 505 <a name="math_toolkit.dist_ref.nmp.h15"></a>
 506         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.variance"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">Variance</a>
 507       </h5>
 508 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 509 <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 510 </pre>
 511 <p>
 512         Returns the variance of the distribution <span class="emphasis"><em>dist</em></span>.
 513       </p>
 514 <p>
 515         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 516         if the distribution does not have a defined variance.
 517       </p>
 518 <h5>
 519 <a name="math_toolkit.dist_ref.nmp.h16"></a>
 520         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.skewness"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">Skewness</a>
 521       </h5>
 522 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 523 <span class="identifier">RealType</span> <span class="identifier">skewness</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 524 </pre>
 525 <p>
 526         Returns the skewness of the distribution <span class="emphasis"><em>dist</em></span>.
 527       </p>
 528 <p>
 529         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 530         if the distribution does not have a defined skewness.
 531       </p>
 532 <h5>
 533 <a name="math_toolkit.dist_ref.nmp.h17"></a>
 534         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">Kurtosis</a>
 535       </h5>
 536 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 537 <span class="identifier">RealType</span> <span class="identifier">kurtosis</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 538 </pre>
 539 <p>
 540         Returns the 'proper' kurtosis (normalized fourth moment) of the distribution
 541         <span class="emphasis"><em>dist</em></span>.
 542       </p>
 543 <p>
 544         kertosis = &#946;<sub>2</sub>= &#956;<sub>4</sub> / &#956;<sub>2</sub><sup>2</sup>
 545       </p>
 546 <p>
 547         Where &#956;<sub>i</sub> is the i'th central moment of the distribution, and in particular
 548         &#956;<sub>2</sub> is the variance of the distribution.
 549       </p>
 550 <p>
 551         The kurtosis is a measure of the "peakedness" of a distribution.
 552       </p>
 553 <p>
 554         Note that the literature definition of kurtosis is confusing. The definition
 555         used here is that used by for example <a href="http://mathworld.wolfram.com/Kurtosis.html" target="_top">Wolfram
 556         MathWorld</a> (that includes a table of formulae for kurtosis excess
 557         for various distributions) but NOT the definition of <a href="http://en.wikipedia.org/wiki/Kurtosis" target="_top">kurtosis
 558         used by Wikipedia</a> which treats "kurtosis" and "kurtosis
 559         excess" as the same quantity.
 560       </p>
 561 <pre class="programlisting"><span class="identifier">kurtosis_excess</span> <span class="special">=</span> <span class="char">'proper'</span> <span class="identifier">kurtosis</span> <span class="special">-</span> <span class="number">3</span>
 562 </pre>
 563 <p>
 564         This subtraction of 3 is convenient so that the <span class="emphasis"><em>kurtosis excess</em></span>
 565         of a normal distribution is zero.
 566       </p>
 567 <p>
 568         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 569         if the distribution does not have a defined kurtosis.
 570       </p>
 571 <p>
 572         'Proper' kurtosis can have a value from zero to + infinity.
 573       </p>
 574 <h5>
 575 <a name="math_toolkit.dist_ref.nmp.h18"></a>
 576         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.kurtosis_excess"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">Kurtosis
 577         excess</a>
 578       </h5>
 579 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 580 <span class="identifier">RealType</span> <span class="identifier">kurtosis_excess</span><span class="special">(</span><span class="keyword">const</span> <span class="emphasis"><em>Distribution-Type</em></span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 581 </pre>
 582 <p>
 583         Returns the kurtosis excess of the distribution <span class="emphasis"><em>dist</em></span>.
 584       </p>
 585 <p>
 586         kurtosis excess = &#947;<sub>2</sub>= &#956;<sub>4</sub> / &#956;<sub>2</sub><sup>2</sup>- 3 = kurtosis - 3
 587       </p>
 588 <p>
 589         Where &#956;<sub>i</sub> is the i'th central moment of the distribution, and in particular
 590         &#956;<sub>2</sub> is the variance of the distribution.
 591       </p>
 592 <p>
 593         The kurtosis excess is a measure of the "peakedness" of a distribution,
 594         and is more widely used than the "kurtosis proper". It is defined
 595         so that the kurtosis excess of a normal distribution is zero.
 596       </p>
 597 <p>
 598         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 599         if the distribution does not have a defined kurtosis excess.
 600       </p>
 601 <p>
 602         Kurtosis excess can have a value from -2 to + infinity.
 603       </p>
 604 <pre class="programlisting"><span class="identifier">kurtosis</span> <span class="special">=</span> <span class="identifier">kurtosis_excess</span> <span class="special">+</span><span class="number">3</span><span class="special">;</span>
 605 </pre>
 606 <p>
 607         The kurtosis excess of a normal distribution is zero.
 608       </p>
 609 <h5>
 610 <a name="math_toolkit.dist_ref.nmp.h19"></a>
 611         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdfPQ"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P
 612         and Q</a>
 613       </h5>
 614 <p>
 615         The terms P and Q are sometimes used to refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 616         Distribution Function</a> and its <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">complement</a>
 617         respectively. Lowercase p and q are sometimes used to refer to the values
 618         returned by these functions.
 619       </p>
 620 <h5>
 621 <a name="math_toolkit.dist_ref.nmp.h20"></a>
 622         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.percent"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent
 623         Point Function or Percentile</a>
 624       </h5>
 625 <p>
 626         The percent point function, also known as the percentile, is the same as
 627         the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
 628       </p>
 629 <h5>
 630 <a name="math_toolkit.dist_ref.nmp.h21"></a>
 631         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.cdf_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse
 632         CDF Function.</a>
 633       </h5>
 634 <p>
 635         The inverse of the cumulative distribution function, is the same as the
 636         <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
 637       </p>
 638 <h5>
 639 <a name="math_toolkit.dist_ref.nmp.h22"></a>
 640         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival_inv"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse
 641         Survival Function.</a>
 642       </h5>
 643 <p>
 644         The inverse of the survival function, is the same as computing the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the complement
 645         of the probability</a>.
 646       </p>
 647 <h5>
 648 <a name="math_toolkit.dist_ref.nmp.h23"></a>
 649         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.pmf"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability
 650         Mass Function</a>
 651       </h5>
 652 <p>
 653         The Probability Mass Function is the same as the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
 654         Density Function</a>.
 655       </p>
 656 <p>
 657         The term Mass Function is usually applied to discrete distributions, while
 658         the term <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
 659         Function</a> applies to continuous distributions.
 660       </p>
 661 <h5>
 662 <a name="math_toolkit.dist_ref.nmp.h24"></a>
 663         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.lower_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower
 664         Critical Value.</a>
 665       </h5>
 666 <p>
 667         The lower critical value calculates the value of the random variable given
 668         the area under the left tail of the distribution. It is equivalent to calculating
 669         the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
 670       </p>
 671 <h5>
 672 <a name="math_toolkit.dist_ref.nmp.h25"></a>
 673         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.upper_critical"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper
 674         Critical Value.</a>
 675       </h5>
 676 <p>
 677         The upper critical value calculates the value of the random variable given
 678         the area under the right tail of the distribution. It is equivalent to calculating
 679         the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the
 680         complement of the probability</a>.
 681       </p>
 682 <h5>
 683 <a name="math_toolkit.dist_ref.nmp.h26"></a>
 684         <span class="phrase"><a name="math_toolkit.dist_ref.nmp.survival"></a></span><a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival
 685         Function</a>
 686       </h5>
 687 <p>
 688         Refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of
 689         the Cumulative Distribution Function</a>.
 690       </p>
 691 </div>
 692 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 693 <td align="left"></td>
 694 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 695       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 696       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 697       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 698       Daryle Walker and Xiaogang Zhang<p>
 699         Distributed under the Boost Software License, Version 1.0. (See accompanying
 700         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>)
 701       </p>
 702 </div></td>
 703 </tr></table>
 704 <hr>
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