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  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;15.&#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;15.&#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;15.&#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;15.&#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 <p>
 276         <span class="inlinemediaobject"><img src="../../../equations/hazard.svg"></span>
 277       </p>
 278 <div class="caution"><table border="0" summary="Caution">
 279 <tr>
 280 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
 281 <th align="left">Caution</th>
 282 </tr>
 283 <tr><td align="left" valign="top"><p>
 284           Some authors refer to this as the conditional failure density function
 285           rather than the hazard function.
 286         </p></td></tr>
 287 </table></div>
 288 <h5>
 289 <a name="math_toolkit.dist_ref.nmp.h5"></a>
 290         <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
 291         Hazard Function</a>
 292       </h5>
 293 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 294 <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;15.&#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>
 295 </pre>
 296 <p>
 297         Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard
 298         Function</a> of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>.
 299       </p>
 300 <p>
 301         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 302         if the random variable is outside the defined range for the distribution.
 303       </p>
 304 <p>
 305         <span class="inlinemediaobject"><img src="../../../equations/chf.svg"></span>
 306       </p>
 307 <div class="caution"><table border="0" summary="Caution">
 308 <tr>
 309 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
 310 <th align="left">Caution</th>
 311 </tr>
 312 <tr><td align="left" valign="top"><p>
 313           Some authors refer to this as simply the "Hazard Function".
 314         </p></td></tr>
 315 </table></div>
 316 <h5>
 317 <a name="math_toolkit.dist_ref.nmp.h6"></a>
 318         <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>
 319       </h5>
 320 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 321 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 322 </pre>
 323 <p>
 324         Returns the mean of the distribution <span class="emphasis"><em>dist</em></span>.
 325       </p>
 326 <p>
 327         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 328         if the distribution does not have a defined mean (for example the Cauchy
 329         distribution).
 330       </p>
 331 <h5>
 332 <a name="math_toolkit.dist_ref.nmp.h7"></a>
 333         <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>
 334       </h5>
 335 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 336 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 337 </pre>
 338 <p>
 339         Returns the median of the distribution <span class="emphasis"><em>dist</em></span>.
 340       </p>
 341 <h5>
 342 <a name="math_toolkit.dist_ref.nmp.h8"></a>
 343         <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>
 344       </h5>
 345 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 346 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 347 </pre>
 348 <p>
 349         Returns the mode of the distribution <span class="emphasis"><em>dist</em></span>.
 350       </p>
 351 <p>
 352         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 353         if the distribution does not have a defined mode.
 354       </p>
 355 <h5>
 356 <a name="math_toolkit.dist_ref.nmp.h9"></a>
 357         <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
 358         Density Function</a>
 359       </h5>
 360 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 361 <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;15.&#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>
 362 </pre>
 363 <p>
 364         For a continuous function, the probability density function (pdf) returns
 365         the probability that the variate has the value x. Since for continuous distributions
 366         the probability at a single point is actually zero, the probability is better
 367         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
 368         Distribution Function</a>.
 369       </p>
 370 <p>
 371         For a discrete distribution, the pdf is the probability that the variate
 372         takes the value x.
 373       </p>
 374 <p>
 375         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 376         if the random variable is outside the defined range for the distribution.
 377       </p>
 378 <p>
 379         For example, for a standard normal distribution the pdf looks like this:
 380       </p>
 381 <p>
 382         <span class="inlinemediaobject"><img src="../../../graphs/pdf.png"></span>
 383       </p>
 384 <h5>
 385 <a name="math_toolkit.dist_ref.nmp.h10"></a>
 386         <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>
 387       </h5>
 388 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 389 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 390 </pre>
 391 <p>
 392         Returns the valid range of the random variable over distribution <span class="emphasis"><em>dist</em></span>.
 393       </p>
 394 <h5>
 395 <a name="math_toolkit.dist_ref.nmp.h11"></a>
 396         <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>
 397       </h5>
 398 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 399 <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;15.&#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>
 400 </pre>
 401 <p>
 402         The quantile is best viewed as the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
 403         Distribution Function</a>, it returns a value <span class="emphasis"><em>x</em></span> such
 404         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>
 405         <span class="identifier">p</span></code>.
 406       </p>
 407 <p>
 408         This is also known as the <span class="emphasis"><em>percent point function</em></span>, or
 409         <span class="emphasis"><em>percentile</em></span>, or <span class="emphasis"><em>fractile</em></span>, it is
 410         also the same as calculating the <span class="emphasis"><em>lower critical value</em></span>
 411         of a distribution.
 412       </p>
 413 <p>
 414         This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 415         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>
 416         if there is no finite value that has the specified probability.
 417       </p>
 418 <p>
 419         The following graph shows the quantile function for a standard normal distribution:
 420       </p>
 421 <p>
 422         <span class="inlinemediaobject"><img src="../../../graphs/quantile.png"></span>
 423       </p>
 424 <h5>
 425 <a name="math_toolkit.dist_ref.nmp.h12"></a>
 426         <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
 427         from the complement of the probability.</a>
 428       </h5>
 429 <p>
 430         See also <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">complements</a>.
 431       </p>
 432 <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>
 433 <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>
 434 </pre>
 435 <p>
 436         This is the inverse of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement
 437         of the Cumulative Distribution Function</a>. It is calculated by wrapping
 438         the arguments in a call to the quantile function in a call to <span class="emphasis"><em>complement</em></span>.
 439         For example:
 440       </p>
 441 <pre class="programlisting"><span class="comment">// define a standard normal distribution:</span>
 442 <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>
 443 <span class="comment">// print the value of x for which the complement</span>
 444 <span class="comment">// of the probability is 0.05:</span>
 445 <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>
 446 </pre>
 447 <p>
 448         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>
 449         <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code> where
 450         <span class="emphasis"><em>q</em></span> is complement of the probability.
 451       </p>
 452 <p>
 453         <a class="link" href="../stat_tut/overview/complements.html#why_complements">Why complements?</a>
 454       </p>
 455 <p>
 456         This function is also called the inverse survival function, and is the same
 457         as calculating the <span class="emphasis"><em>upper critical value</em></span> of a distribution.
 458       </p>
 459 <p>
 460         This function returns a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 461         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>
 462         if there is no finite value that has the specified probability.
 463       </p>
 464 <p>
 465         The following graph show the inverse survival function for the normal distribution:
 466       </p>
 467 <p>
 468         <span class="inlinemediaobject"><img src="../../../graphs/survival_inv.png"></span>
 469       </p>
 470 <h5>
 471 <a name="math_toolkit.dist_ref.nmp.h13"></a>
 472         <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
 473         Deviation</a>
 474       </h5>
 475 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 476 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 477 </pre>
 478 <p>
 479         Returns the standard deviation of distribution <span class="emphasis"><em>dist</em></span>.
 480       </p>
 481 <p>
 482         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 483         if the distribution does not have a defined standard deviation.
 484       </p>
 485 <h5>
 486 <a name="math_toolkit.dist_ref.nmp.h14"></a>
 487         <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>
 488       </h5>
 489 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 490 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 491 </pre>
 492 <p>
 493         Returns the supported range of random variable over the distribution <span class="emphasis"><em>dist</em></span>.
 494       </p>
 495 <p>
 496         The distribution is said to be 'supported' over a range that is <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">"the
 497         smallest closed set whose complement has probability zero"</a>.
 498         Non-mathematicians might say it means the 'interesting' smallest range of
 499         random variate x that has the cdf going from zero to unity. Outside are uninteresting
 500         zones where the pdf is zero, and the cdf zero or unity.
 501       </p>
 502 <h5>
 503 <a name="math_toolkit.dist_ref.nmp.h15"></a>
 504         <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>
 505       </h5>
 506 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 507 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 508 </pre>
 509 <p>
 510         Returns the variance of the distribution <span class="emphasis"><em>dist</em></span>.
 511       </p>
 512 <p>
 513         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 514         if the distribution does not have a defined variance.
 515       </p>
 516 <h5>
 517 <a name="math_toolkit.dist_ref.nmp.h16"></a>
 518         <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>
 519       </h5>
 520 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 521 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 522 </pre>
 523 <p>
 524         Returns the skewness of the distribution <span class="emphasis"><em>dist</em></span>.
 525       </p>
 526 <p>
 527         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 528         if the distribution does not have a defined skewness.
 529       </p>
 530 <h5>
 531 <a name="math_toolkit.dist_ref.nmp.h17"></a>
 532         <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>
 533       </h5>
 534 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 535 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 536 </pre>
 537 <p>
 538         Returns the 'proper' kurtosis (normalized fourth moment) of the distribution
 539         <span class="emphasis"><em>dist</em></span>.
 540       </p>
 541 <p>
 542         kertosis = &#946;<sub>2</sub> &#160;= &#956;<sub>4</sub> &#160; / &#956;<sub>2</sub><sup>2</sup>
 543       </p>
 544 <p>
 545         Where &#956;<sub>i</sub> &#160; is the i'th central moment of the distribution, and in particular
 546         &#956;<sub>2</sub> &#160; is the variance of the distribution.
 547       </p>
 548 <p>
 549         The kurtosis is a measure of the "peakedness" of a distribution.
 550       </p>
 551 <p>
 552         Note that the literature definition of kurtosis is confusing. The definition
 553         used here is that used by for example <a href="http://mathworld.wolfram.com/Kurtosis.html" target="_top">Wolfram
 554         MathWorld</a> (that includes a table of formulae for kurtosis excess
 555         for various distributions) but NOT the definition of <a href="http://en.wikipedia.org/wiki/Kurtosis" target="_top">kurtosis
 556         used by Wikipedia</a> which treats "kurtosis" and "kurtosis
 557         excess" as the same quantity.
 558       </p>
 559 <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>
 560 </pre>
 561 <p>
 562         This subtraction of 3 is convenient so that the <span class="emphasis"><em>kurtosis excess</em></span>
 563         of a normal distribution is zero.
 564       </p>
 565 <p>
 566         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 567         if the distribution does not have a defined kurtosis.
 568       </p>
 569 <p>
 570         'Proper' kurtosis can have a value from zero to + infinity.
 571       </p>
 572 <h5>
 573 <a name="math_toolkit.dist_ref.nmp.h18"></a>
 574         <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
 575         excess</a>
 576       </h5>
 577 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
 578 <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;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;&amp;</span> <span class="identifier">dist</span><span class="special">);</span>
 579 </pre>
 580 <p>
 581         Returns the kurtosis excess of the distribution <span class="emphasis"><em>dist</em></span>.
 582       </p>
 583 <p>
 584         kurtosis excess = &#947;<sub>2</sub> &#160;= &#956;<sub>4</sub> &#160; / &#956;<sub>2</sub><sup>2</sup> &#160;- 3 = kurtosis - 3
 585       </p>
 586 <p>
 587         Where &#956;<sub>i</sub> &#160; is the i'th central moment of the distribution, and in particular
 588         &#956;<sub>2</sub> &#160; is the variance of the distribution.
 589       </p>
 590 <p>
 591         The kurtosis excess is a measure of the "peakedness" of a distribution,
 592         and is more widely used than the "kurtosis proper". It is defined
 593         so that the kurtosis excess of a normal distribution is zero.
 594       </p>
 595 <p>
 596         This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 597         if the distribution does not have a defined kurtosis excess.
 598       </p>
 599 <p>
 600         Kurtosis excess can have a value from -2 to + infinity.
 601       </p>
 602 <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>
 603 </pre>
 604 <p>
 605         The kurtosis excess of a normal distribution is zero.
 606       </p>
 607 <h5>
 608 <a name="math_toolkit.dist_ref.nmp.h19"></a>
 609         <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
 610         and Q</a>
 611       </h5>
 612 <p>
 613         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
 614         Distribution Function</a> and its <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">complement</a>
 615         respectively. Lowercase p and q are sometimes used to refer to the values
 616         returned by these functions.
 617       </p>
 618 <h5>
 619 <a name="math_toolkit.dist_ref.nmp.h20"></a>
 620         <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
 621         Point Function or Percentile</a>
 622       </h5>
 623 <p>
 624         The percent point function, also known as the percentile, is the same as
 625         the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
 626       </p>
 627 <h5>
 628 <a name="math_toolkit.dist_ref.nmp.h21"></a>
 629         <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
 630         CDF Function.</a>
 631       </h5>
 632 <p>
 633         The inverse of the cumulative distribution function, is the same as the
 634         <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
 635       </p>
 636 <h5>
 637 <a name="math_toolkit.dist_ref.nmp.h22"></a>
 638         <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
 639         Survival Function.</a>
 640       </h5>
 641 <p>
 642         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
 643         of the probability</a>.
 644       </p>
 645 <h5>
 646 <a name="math_toolkit.dist_ref.nmp.h23"></a>
 647         <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
 648         Mass Function</a>
 649       </h5>
 650 <p>
 651         The Probability Mass Function is the same as the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability
 652         Density Function</a>.
 653       </p>
 654 <p>
 655         The term Mass Function is usually applied to discrete distributions, while
 656         the term <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
 657         Function</a> applies to continuous distributions.
 658       </p>
 659 <h5>
 660 <a name="math_toolkit.dist_ref.nmp.h24"></a>
 661         <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
 662         Critical Value.</a>
 663       </h5>
 664 <p>
 665         The lower critical value calculates the value of the random variable given
 666         the area under the left tail of the distribution. It is equivalent to calculating
 667         the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
 668       </p>
 669 <h5>
 670 <a name="math_toolkit.dist_ref.nmp.h25"></a>
 671         <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
 672         Critical Value.</a>
 673       </h5>
 674 <p>
 675         The upper critical value calculates the value of the random variable given
 676         the area under the right tail of the distribution. It is equivalent to calculating
 677         the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">quantile from the
 678         complement of the probability</a>.
 679       </p>
 680 <h5>
 681 <a name="math_toolkit.dist_ref.nmp.h26"></a>
 682         <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
 683         Function</a>
 684       </h5>
 685 <p>
 686         Refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of
 687         the Cumulative Distribution Function</a>.
 688       </p>
 689 </div>
 690 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 691 <td align="left"></td>
 692 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 693       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 694       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 695       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 696         Distributed under the Boost Software License, Version 1.0. (See accompanying
 697         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>)
 698       </p>
 699 </div></td>
 700 </tr></table>
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