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710 lines
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.dist_ref.nmp"></a><a class="link" href="nmp.html" title="Non-Member Properties">Non-Member Properties</a>
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</h3></div></div></div>
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<p>
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Properties that are common to all distributions are accessed via non-member
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getter functions: non-membership allows more of these functions to be added
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over time, as the need arises. Unfortunately the literature uses many different
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and confusing names to refer to a rather small number of actual concepts;
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refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.concept_index">concept
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index</a> to find the property you want by the name you are most familiar
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with. Or use the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.function_index">function
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index</a> to go straight to the function you want if you already know
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its name.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.nmp.h0"></a>
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<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
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Index</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
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Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
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Distribution Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
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complement of the probability</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
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</li>
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</ul></div>
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<h5>
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<a name="math_toolkit.dist_ref.nmp.h1"></a>
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<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
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Index</a>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">Complement of the Cumulative
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Distribution Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
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Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf_inv">Inverse Cumulative
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Distribution Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival_inv">Inverse Survival
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Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.lower_critical">Lower Critical
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Value</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.median">median</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">P</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.percent">Percent Point Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pmf">Probability Mass Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.range">range</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdfPQ">Q</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile_c">Quantile from the
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complement of the probability</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.survival">Survival Function</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.upper_critical">Upper Critical
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Value</a>.
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</li>
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<li class="listitem">
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>.
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</li>
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</ul></div>
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<h5>
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<a name="math_toolkit.dist_ref.nmp.h2"></a>
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<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
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Distribution Function</a>
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</h5>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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>
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</pre>
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<p>
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The <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution
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Function</a> is the probability that the variable takes a value less than
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or equal to x. It is equivalent to the integral from -infinity to x of the
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<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>.
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</p>
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<p>
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This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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if the random variable is outside the defined range for the distribution.
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</p>
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<p>
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For example, the following graph shows the cdf for the normal distribution:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/cdf.png"></span>
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.nmp.h3"></a>
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<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
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of the Cumulative Distribution Function</a>
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</h5>
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<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>
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<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>
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</pre>
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<p>
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The complement of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
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Distribution Function</a> is the probability that the variable takes a
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value greater than x. It is equivalent to the integral from x to infinity
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of the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density
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Function</a>, or 1 minus the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative
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Distribution Function</a> of x.
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</p>
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<p>
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This is also known as the survival function.
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</p>
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<p>
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This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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if the random variable is outside the defined range for the distribution.
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</p>
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<p>
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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:
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</p>
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<pre class="programlisting"><span class="comment">// standard normal distribution object:</span>
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<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>
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<span class="comment">// print survival function for x=2.0:</span>
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<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>
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</pre>
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<p>
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For example, the following graph shows the __complement of the cdf for the
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normal distribution:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/survival.png"></span>
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</p>
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<p>
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See <a class="link" href="../stat_tut/overview/complements.html#why_complements">why complements?</a> for why the complement
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is useful and when it should be used.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.nmp.h4"></a>
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<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
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Function</a>
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</h5>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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>
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</pre>
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<p>
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Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>
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of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>.
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</p>
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<p>
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This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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if the random variable is outside the defined range for the distribution.
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/hazard.svg"></span>
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</p></blockquote></div>
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<div class="caution"><table border="0" summary="Caution">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
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<th align="left">Caution</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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Some authors refer to this as the conditional failure density function
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rather than the hazard function.
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</p></td></tr>
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</table></div>
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<h5>
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<a name="math_toolkit.dist_ref.nmp.h5"></a>
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<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
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Hazard Function</a>
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</h5>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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>
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</pre>
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<p>
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Returns the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard
|
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Function</a> of <span class="emphasis"><em>x</em></span> and distibution <span class="emphasis"><em>dist</em></span>.
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</p>
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<p>
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This function may return a <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
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if the random variable is outside the defined range for the distribution.
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</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/chf.svg"></span>
|
|
|
|
</p></blockquote></div>
|
|
<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>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h6"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
if the distribution does not have a defined mean (for example the Cauchy
|
|
distribution).
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h7"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h8"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
if the distribution does not have a defined mode.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h9"></a>
|
|
<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
|
|
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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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_toolkit.dist_ref.nmp.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="../error_handling.html#math_toolkit.error_handling.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"></span>
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h10"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h11"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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_toolkit.dist_ref.nmp.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="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
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>
|
|
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"></span>
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h12"></a>
|
|
<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
|
|
from the complement of the probability.</a>
|
|
</h5>
|
|
<p>
|
|
See also <a class="link" href="../stat_tut/overview/complements.html" title="Complements are supported too - and when to use them">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_toolkit.dist_ref.nmp.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="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
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>
|
|
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"></span>
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h13"></a>
|
|
<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
|
|
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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
if the distribution does not have a defined standard deviation.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h14"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h15"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
if the distribution does not have a defined variance.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h16"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
|
|
if the distribution does not have a defined skewness.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h17"></a>
|
|
<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>
|
|
</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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="../error_handling.html#math_toolkit.error_handling.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>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h18"></a>
|
|
<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
|
|
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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="Chapter 20. Policies: Controlling Precision, Error Handling etc">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="../error_handling.html#math_toolkit.error_handling.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>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h19"></a>
|
|
<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
|
|
and Q</a>
|
|
</h5>
|
|
<p>
|
|
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
|
|
Distribution Function</a> and its <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.ccdf">complement</a>
|
|
respectively. Lowercase p and q are sometimes used to refer to the values
|
|
returned by these functions.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h20"></a>
|
|
<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
|
|
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_toolkit.dist_ref.nmp.quantile">Quantile</a>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h21"></a>
|
|
<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
|
|
CDF Function.</a>
|
|
</h5>
|
|
<p>
|
|
The inverse of the cumulative distribution function, is the same as the
|
|
<a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h22"></a>
|
|
<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
|
|
Survival Function.</a>
|
|
</h5>
|
|
<p>
|
|
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
|
|
of the probability</a>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h23"></a>
|
|
<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
|
|
Mass Function</a>
|
|
</h5>
|
|
<p>
|
|
The Probability Mass Function is the same as the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.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_toolkit.dist_ref.nmp.pdf">Probability Density
|
|
Function</a> applies to continuous distributions.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h24"></a>
|
|
<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
|
|
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_toolkit.dist_ref.nmp.quantile">Quantile</a>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h25"></a>
|
|
<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
|
|
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_toolkit.dist_ref.nmp.quantile_c">quantile from the
|
|
complement of the probability</a>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.dist_ref.nmp.h26"></a>
|
|
<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
|
|
Function</a>
|
|
</h5>
|
|
<p>
|
|
Refer to the <a class="link" href="nmp.html#math_toolkit.dist_ref.nmp.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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<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>
|
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