| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Non-Member Properties</title> |
| <link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> |
| <link rel="home" href="../../../index.html" title="Math Toolkit"> |
| <link rel="up" href="../dist_ref.html" title="Statistical Distributions Reference"> |
| <link rel="prev" href="../dist_ref.html" title="Statistical Distributions Reference"> |
| <link rel="next" href="dists.html" title="Distributions"> |
| </head> |
| <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> |
| <table cellpadding="2" width="100%"><tr> |
| <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> |
| <td align="center"><a href="../../../../../../../../index.html">Home</a></td> |
| <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td> |
| <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> |
| <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> |
| <td align="center"><a href="../../../../../../../../more/index.htm">More</a></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="../dist_ref.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dist_ref.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="dists.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.dist.dist_ref.nmp"></a><a class="link" href="nmp.html" title="Non-Member Properties">Non-Member Properties</a> |
| </h4></div></div></div> |
| <p> |
| Properties that are common to all distributions are accessed via non-member |
| getter functions: non-membership allows more of these functions to be added |
| over time, as the need arises. Unfortunately the literature uses many different |
| and confusing names to refer to a rather small number of actual concepts; |
| refer to the <a class="link" href="nmp.html#concept_index">concept index</a> to find |
| the property you want by the name you are most familiar with. Or use the |
| <a class="link" href="nmp.html#function_index">function index</a> to go straight to |
| the function you want if you already know its name. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__function_index____function_index"></a><h5> |
| <a name="id1154234"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__function_index____function_index"><a name="function_index"></a> Function Index</a> |
| </h5> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| <a class="link" href="nmp.html#math.dist.cdf">cdf</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.ccdf">cdf complement</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.chf">chf</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.hazard">hazard</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.kurtosis">kurtosis</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a> |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.mean">mean</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.median">median</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.mode">mode</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.pdf">pdf</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.range">range</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.quantile">quantile</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.quantile_c">quantile from the complement</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.skewness">skewness</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.sd">standard_deviation</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.support">support</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.variance">variance</a>. |
| </li> |
| </ul></div> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__concept_index____conceptual_index"></a><h5> |
| <a name="id1154457"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__concept_index____conceptual_index"><a name="concept_index"></a> Conceptual Index</a> |
| </h5> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| <a class="link" href="nmp.html#math.dist.ccdf">Complement of the Cumulative Distribution |
| Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.cdf">Cumulative Distribution Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.chf">Cumulative Hazard Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#cdf_inv">Inverse Cumulative Distribution Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#survival_inv">Inverse Survival Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.hazard">Hazard Function</a> |
| </li> |
| <li> |
| <a class="link" href="nmp.html#lower_critical">Lower Critical Value</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.kurtosis">kurtosis</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a> |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.mean">mean</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.median">median</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.mode">mode</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#cdfPQ">P</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#percent">Percent Point Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.pdf">Probability Density Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#pmf">Probability Mass Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.range">range</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#cdfPQ">Q</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.quantile">Quantile</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.quantile_c">Quantile from the complement of |
| the probability</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.skewness">skewness</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.sd">standard deviation</a> |
| </li> |
| <li> |
| <a class="link" href="nmp.html#survival">Survival Function</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.support">support</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#upper_critical">Upper Critical Value</a>. |
| </li> |
| <li> |
| <a class="link" href="nmp.html#math.dist.variance">variance</a>. |
| </li> |
| </ul></div> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_cdf____cumulative_distribution_function"></a><h5> |
| <a name="id1154788"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_cdf____cumulative_distribution_function"><a name="math.dist.cdf"></a> Cumulative Distribution Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| The <a class="link" href="nmp.html#math.dist.cdf">Cumulative Distribution Function</a> |
| is the probability that the variable takes a value less than or equal to |
| x. It is equivalent to the integral from -infinity to x of the <a class="link" href="nmp.html#math.dist.pdf">Probability |
| Density Function</a>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| For example, the following graph shows the cdf for the normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/cdf.png" alt="cdf"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_ccdf____complement_of_the_cumulative_distribution_function"></a><h5> |
| <a name="id1154955"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_ccdf____complement_of_the_cumulative_distribution_function"><a name="math.dist.ccdf"></a> Complement of the Cumulative Distribution Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> |
| <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"><</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">>&</span> <span class="identifier">comp</span><span class="special">);</span> |
| </pre> |
| <p> |
| The complement of the <a class="link" href="nmp.html#math.dist.cdf">Cumulative Distribution |
| Function</a> is the probability that the variable takes a value greater |
| than x. It is equivalent to the integral from x to infinity of the <a class="link" href="nmp.html#math.dist.pdf">Probability Density Function</a>, or 1 minus |
| the <a class="link" href="nmp.html#math.dist.cdf">Cumulative Distribution Function</a> |
| of x. |
| </p> |
| <p> |
| This is also known as the survival function. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| 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: |
| </p> |
| <pre class="programlisting"><span class="comment">// standard normal distribution object: |
| </span><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> |
| <span class="comment">// print survival function for x=2.0: |
| </span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| For example, the following graph shows the __complement of the cdf for |
| the normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/survival.png" alt="survival"></span> |
| </p> |
| <p> |
| See <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a> for why the |
| complement is useful and when it should be used. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_hazard____hazard_function"></a><h5> |
| <a name="id1155447"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_hazard____hazard_function"><a name="math.dist.hazard"></a> Hazard Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the <a class="link" href="nmp.html#math.dist.hazard">Hazard Function</a> of |
| <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/hazard.png"></span> |
| </p> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| Some authors refer to this as the conditional failure density function |
| rather than the hazard function. |
| </p></td></tr> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_chf____cumulative_hazard_function"></a><h5> |
| <a name="id1155626"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_chf____cumulative_hazard_function"><a name="math.dist.chf"></a> Cumulative Hazard Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the <a class="link" href="nmp.html#math.dist.chf">Cumulative Hazard Function</a> |
| of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/chf.png"></span> |
| </p> |
| <div class="caution"><table border="0" summary="Caution"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../../doc/src/images/caution.png"></td> |
| <th align="left">Caution</th> |
| </tr> |
| <tr><td align="left" valign="top"><p> |
| Some authors refer to this as simply the "Hazard Function". |
| </p></td></tr> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_mean____mean"></a><h5> |
| <a name="id1155804"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_mean____mean"><a name="math.dist.mean"></a> mean</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the mean of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the distribution does not have a defined mean (for example the Cauchy |
| distribution). |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_median____median"></a><h5> |
| <a name="id1155921"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_median____median"><a name="math.dist.median"></a> median</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the median of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_mode____mode"></a><h5> |
| <a name="id1156029"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_mode____mode"><a name="math.dist.mode"></a> mode</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the mode of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the distribution does not have a defined mode. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_pdf____probability_density_function"></a><h5> |
| <a name="id1156140"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_pdf____probability_density_function"><a name="math.dist.pdf"></a> Probability Density Function</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">x</span><span class="special">);</span> |
| </pre> |
| <p> |
| For a continuous function, the probability density function (pdf) returns |
| the probability that the variate has the value x. Since for continuous |
| distributions the probability at a single point is actually zero, the probability |
| is better expressed as the integral of the pdf between two points: see |
| the <a class="link" href="nmp.html#math.dist.cdf">Cumulative Distribution Function</a>. |
| </p> |
| <p> |
| For a discrete distribution, the pdf is the probability that the variate |
| takes the value x. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the random variable is outside the defined range for the distribution. |
| </p> |
| <p> |
| For example, for a standard normal distribution the pdf looks like this: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/pdf.png" alt="pdf"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_range____range"></a><h5> |
| <a name="id1156306"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_range____range"><a name="math.dist.range"></a> Range</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">></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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the valid range of the random variable over distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_quantile____quantile"></a><h5> |
| <a name="id1156440"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_quantile____quantile"><a name="math.dist.quantile"></a> Quantile</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">p</span><span class="special">);</span> |
| </pre> |
| <p> |
| The quantile is best viewed as the inverse of the <a class="link" href="nmp.html#math.dist.cdf">Cumulative |
| Distribution Function</a>, it returns 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">dist</span><span class="special">,</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> |
| <span class="identifier">p</span></code>. |
| </p> |
| <p> |
| This is also known as the <span class="emphasis"><em>percent point function</em></span>, |
| or <span class="emphasis"><em>percentile</em></span>, or <span class="emphasis"><em>fractile</em></span>, it |
| is also the same as calculating the <span class="emphasis"><em>lower critical value</em></span> |
| of a distribution. |
| </p> |
| <p> |
| This function returns a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the probability lies outside [0,1]. The function may return an <a class="link" href="../../main_overview/error_handling.html#overflow_error">overflow_error</a> if there is no finite value |
| that has the specified probability. |
| </p> |
| <p> |
| The following graph shows the quantile function for a standard normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/quantile.png" alt="quantile"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_quantile_c____quantile_from_the_complement_of_the_probability_"></a><h5> |
| <a name="id1156666"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_quantile_c____quantile_from_the_complement_of_the_probability_"><a name="math.dist.quantile_c"></a> Quantile from the complement of the probability.</a> |
| </h5> |
| <p> |
| <a class="link" href="../stat_tut/overview/complements.html#complements">complements</a> |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</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">></span> |
| <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"><</span><span class="identifier">Distribution</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">>&</span> <span class="identifier">comp</span><span class="special">);</span> |
| </pre> |
| <p> |
| This is the inverse of the <a class="link" href="nmp.html#math.dist.ccdf">Complement of |
| the Cumulative Distribution Function</a>. It is calculated by wrapping |
| the arguments in a call to the quantile function in a call to <span class="emphasis"><em>complement</em></span>. |
| For example: |
| </p> |
| <pre class="programlisting"><span class="comment">// define a standard normal distribution: |
| </span><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> |
| <span class="comment">// print the value of x for which the complement |
| </span><span class="comment">// of the probability is 0.05: |
| </span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</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"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| 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> <span class="identifier">x</span><span class="special">))</span> <span class="special">==</span> <span class="identifier">q</span></code> |
| where <span class="emphasis"><em>q</em></span> is complement of the probability. |
| </p> |
| <p> |
| <a class="link" href="../stat_tut/overview/complements.html#why_complements">Why complements?</a> |
| </p> |
| <p> |
| This function is also called the inverse survival function, and is the |
| same as calculating the <span class="emphasis"><em>upper critical value</em></span> of a |
| distribution. |
| </p> |
| <p> |
| This function returns a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the probablity lies outside [0,1]. The function may return an <a class="link" href="../../main_overview/error_handling.html#overflow_error">overflow_error</a> if there is no finite value |
| that has the specified probability. |
| </p> |
| <p> |
| The following graph show the inverse survival function for the normal distribution: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/survival_inv.png" alt="survival_inv"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_sd____standard_deviation"></a><h5> |
| <a name="id1157006"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_sd____standard_deviation"><a name="math.dist.sd"></a> Standard Deviation</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the standard deviation of distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the distribution does not have a defined standard deviation. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_support____support"></a><h5> |
| <a name="id1157124"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_support____support"><a name="math.dist.support"></a> support</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">,</span> <span class="identifier">RealType</span><span class="special">></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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the supported range of random variable over the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| The distribution is said to be 'supported' over a range that is <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">"the smallest |
| closed set whose complement has probability zero"</a>. Non-mathematicians |
| might say it means the 'interesting' smallest range of random variate x |
| that has the cdf going from zero to unity. Outside are uninteresting zones |
| where the pdf is zero, and the cdf zero or unity. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_variance____variance"></a><h5> |
| <a name="id1157267"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_variance____variance"><a name="math.dist.variance"></a> Variance</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the variance of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the distribution does not have a defined variance. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_skewness____skewness"></a><h5> |
| <a name="id1157385"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_skewness____skewness"><a name="math.dist.skewness"></a> Skewness</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the skewness of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the distribution does not have a defined skewness. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_kurtosis____kurtosis"></a><h5> |
| <a name="id1157502"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_kurtosis____kurtosis"><a name="math.dist.kurtosis"></a> Kurtosis</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the 'proper' kurtosis (normalized fourth moment) of the distribution |
| <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| kertosis = β<sub>2</sub>  = μ<sub>4</sub>   / μ<sub>2</sub><sup>2</sup> |
| </p> |
| <p> |
| Where μ<sub>i</sub>   is the i'th central moment of the distribution, and in particular |
| μ<sub>2</sub>   is the variance of the distribution. |
| </p> |
| <p> |
| The kurtosis is a measure of the "peakedness" of a distribution. |
| </p> |
| <p> |
| Note that the literature definition of kurtosis is confusing. The definition |
| used here is that used by for example <a href="http://mathworld.wolfram.com/Kurtosis.html" target="_top">Wolfram |
| MathWorld</a> (that includes a table of formulae for kurtosis excess |
| for various distributions) but NOT the definition of <a href="http://en.wikipedia.org/wiki/Kurtosis" target="_top">kurtosis |
| used by Wikipedia</a> which treats "kurtosis" and "kurtosis |
| excess" as the same quantity. |
| </p> |
| <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> |
| </pre> |
| <p> |
| This subtraction of 3 is convenient so that the <span class="emphasis"><em>kurtosis excess</em></span> |
| of a normal distribution is zero. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the distribution does not have a defined kurtosis. |
| </p> |
| <p> |
| 'Proper' kurtosis can have a value from zero to + infinity. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_kurtosis_excess____kurtosis_excess"></a><h5> |
| <a name="id1159507"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__math_dist_kurtosis_excess____kurtosis_excess"><a name="math.dist.kurtosis_excess"></a> Kurtosis excess</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <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"><</span><span class="identifier">RealType</span><span class="special">,</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">>&</span> <span class="identifier">dist</span><span class="special">);</span> |
| </pre> |
| <p> |
| Returns the kurtosis excess of the distribution <span class="emphasis"><em>dist</em></span>. |
| </p> |
| <p> |
| kurtosis excess = γ<sub>2</sub>  = μ<sub>4</sub>   / μ<sub>2</sub><sup>2</sup>  - 3 = kurtosis - 3 |
| </p> |
| <p> |
| Where μ<sub>i</sub>   is the i'th central moment of the distribution, and in particular |
| μ<sub>2</sub>   is the variance of the distribution. |
| </p> |
| <p> |
| The kurtosis excess is a measure of the "peakedness" of a distribution, |
| and is more widely used than the "kurtosis proper". It is defined |
| so that the kurtosis excess of a normal distribution is zero. |
| </p> |
| <p> |
| This function may return a <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| if the distribution does not have a defined kurtosis excess. |
| </p> |
| <p> |
| Kurtosis excess can have a value from -2 to + infinity. |
| </p> |
| <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> |
| </pre> |
| <p> |
| The kurtosis excess of a normal distribution is zero. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__cdfpq____p_and_q"></a><h5> |
| <a name="id1159694"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__cdfpq____p_and_q"><a name="cdfPQ"></a> P and Q</a> |
| </h5> |
| <p> |
| The terms P and Q are sometimes used to refer to the <a class="link" href="nmp.html#math.dist.cdf">Cumulative |
| Distribution Function</a> and its <a class="link" href="nmp.html#math.dist.ccdf">complement</a> |
| respectively. Lowercase p and q are sometimes used to refer to the values |
| returned by these functions. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__percent____percent_point_function_or_percentile"></a><h5> |
| <a name="id1159726"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__percent____percent_point_function_or_percentile"><a name="percent"></a> Percent Point Function or Percentile</a> |
| </h5> |
| <p> |
| The percent point function, also known as the percentile, is the same as |
| the <a class="link" href="nmp.html#math.dist.quantile">Quantile</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__cdf_inv____inverse_cdf_function_"></a><h5> |
| <a name="id1159750"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__cdf_inv____inverse_cdf_function_"><a name="cdf_inv"></a> Inverse CDF Function.</a> |
| </h5> |
| <p> |
| The inverse of the cumulative distribution function, is the same as the |
| <a class="link" href="nmp.html#math.dist.quantile">Quantile</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__survival_inv____inverse_survival_function_"></a><h5> |
| <a name="id1159774"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__survival_inv____inverse_survival_function_"><a name="survival_inv"></a> Inverse Survival Function.</a> |
| </h5> |
| <p> |
| The inverse of the survival function, is the same as computing the <a class="link" href="nmp.html#math.dist.quantile_c">quantile from the complement of the probability</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__pmf____probability_mass_function"></a><h5> |
| <a name="id1159798"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__pmf____probability_mass_function"><a name="pmf"></a> Probability Mass Function</a> |
| </h5> |
| <p> |
| The Probability Mass Function is the same as the <a class="link" href="nmp.html#math.dist.pdf">Probability |
| Density Function</a>. |
| </p> |
| <p> |
| The term Mass Function is usually applied to discrete distributions, while |
| the term <a class="link" href="nmp.html#math.dist.pdf">Probability Density Function</a> |
| applies to continuous distributions. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__lower_critical____lower_critical_value_"></a><h5> |
| <a name="id1159831"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__lower_critical____lower_critical_value_"><a name="lower_critical"></a> Lower Critical Value.</a> |
| </h5> |
| <p> |
| The lower critical value calculates the value of the random variable given |
| the area under the left tail of the distribution. It is equivalent to calculating |
| the <a class="link" href="nmp.html#math.dist.quantile">Quantile</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__upper_critical____upper_critical_value_"></a><h5> |
| <a name="id1159855"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__upper_critical____upper_critical_value_"><a name="upper_critical"></a> Upper Critical Value.</a> |
| </h5> |
| <p> |
| The upper critical value calculates the value of the random variable given |
| the area under the right tail of the distribution. It is equivalent to |
| calculating the <a class="link" href="nmp.html#math.dist.quantile_c">quantile from the |
| complement of the probability</a>. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.nmp._anchor_id__survival____survival_function"></a><h5> |
| <a name="id1159879"></a> |
| <a class="link" href="nmp.html#math_toolkit.dist.dist_ref.nmp._anchor_id__survival____survival_function"><a name="survival"></a> Survival Function</a> |
| </h5> |
| <p> |
| Refer to the <a class="link" href="nmp.html#math.dist.ccdf">Complement of the Cumulative |
| Distribution Function</a>. |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="../dist_ref.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dist_ref.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="dists.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
