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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.poisson_dist"></a><a class="link" href="poisson_dist.html" title="Poisson Distribution">Poisson Distribution</a>
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</h4></div></div></div>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">poisson</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">poisson_distribution</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">poisson_distribution</span><span class="special"><></span> <span class="identifier">poisson</span><span class="special">;</span>
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<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 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">poisson_distribution</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
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<span class="identifier">poisson_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Constructor.</span>
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<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Accessor.</span>
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<span class="special">}</span>
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<span class="special">}}</span> <span class="comment">// namespaces boost::math</span>
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</pre>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Poisson_distribution" target="_top">Poisson
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distribution</a> is a well-known statistical discrete distribution.
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It expresses the probability of a number of events (or failures, arrivals,
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occurrences ...) occurring in a fixed period of time, provided these events
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occur with a known mean rate λ
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(events/time), and are independent of the
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time since the last event.
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</p>
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<p>
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The distribution was discovered by Siméon-Denis Poisson (1781 to 1840).
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</p>
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<p>
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It has the Probability Mass Function:
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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/poisson_ref1.svg"></span>
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</p></blockquote></div>
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<p>
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for k events, with an expected number of events λ.
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</p>
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<p>
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The following graph illustrates how the PDF varies with the parameter λ:
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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="../../../../graphs/poisson_pdf_1.svg" align="middle"></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">
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<p>
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The Poisson distribution is a discrete distribution: internally, functions
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like the <code class="computeroutput"><span class="identifier">cdf</span></code> and <code class="computeroutput"><span class="identifier">pdf</span></code> are treated "as if" they
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are continuous functions, but in reality the results returned from these
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functions only have meaning if an integer value is provided for the random
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variate argument.
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</p>
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<p>
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The quantile function will by default return an integer result that has
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been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles
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(where the probability is less than 0.5) are rounded downward, and upper
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quantiles (where the probability is greater than 0.5) are rounded upwards.
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This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at
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least</em></span> the requested coverage will be present in the central
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region, and <span class="emphasis"><em>no more than</em></span> the requested coverage
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will be present in the tails.
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</p>
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<p>
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This behaviour can be changed so that the quantile functions are rounded
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differently, or even return a real-valued result using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
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It is strongly recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
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Quantiles of Discrete Distributions</a> before using the quantile
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function on the Poisson distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
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docs</a> describe how to change the rounding policy for these distributions.
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</p>
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</td></tr>
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</table></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.poisson_dist.h0"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.member_functions"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_dist.member_functions">Member
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Functions</a>
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</h5>
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<pre class="programlisting"><span class="identifier">poisson_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
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</pre>
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<p>
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Constructs a poisson distribution with mean <span class="emphasis"><em>mean</em></span>.
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</p>
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<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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</pre>
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<p>
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Returns the <span class="emphasis"><em>mean</em></span> of this distribution.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.poisson_dist.h1"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.non_member_accessors"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_dist.non_member_accessors">Non-member
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Accessors</a>
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</h5>
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<p>
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All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
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functions</a> that are generic to all distributions are supported:
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
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</p>
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<p>
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The domain of the random variable is [0, ∞].
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.poisson_dist.h2"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.accuracy"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_dist.accuracy">Accuracy</a>
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</h5>
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<p>
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The Poisson distribution is implemented in terms of the incomplete gamma
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functions <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a> and
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<a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and as such
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should have low error rates: but refer to the documentation of those functions
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for more information. The quantile and its complement use the inverse gamma
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functions and are therefore probably slightly less accurate: this is because
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the inverse gamma functions are implemented using an iterative method with
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a lower tolerance to avoid excessive computation.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.poisson_dist.h3"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.poisson_dist.implementation"></a></span><a class="link" href="poisson_dist.html#math_toolkit.dist_ref.dists.poisson_dist.implementation">Implementation</a>
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</h5>
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<p>
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In the following table λ is the mean of the distribution, <span class="emphasis"><em>k</em></span>
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is the random variable, <span class="emphasis"><em>p</em></span> is the probability and
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<span class="emphasis"><em>q = 1-p</em></span>.
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Function
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</p>
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</th>
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<th>
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<p>
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Implementation Notes
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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pdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: pdf = e<sup>-λ</sup> λ<sup>k</sup> / k!
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: p = Γ(k+1, λ) / k! = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(k+1,
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λ)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf complement
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</p>
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</td>
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<td>
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<p>
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Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(k+1,
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λ)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile
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</p>
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</td>
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<td>
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<p>
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Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inva</a>(λ,
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p) - 1
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile from the complement
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</p>
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</td>
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<td>
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<p>
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Using the relation: k = <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inva</a>(λ,
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q) - 1
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mean
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</p>
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</td>
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<td>
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<p>
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λ
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mode
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</p>
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</td>
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<td>
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<p>
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floor (λ) or ⌊λ⌋
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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skewness
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</p>
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</td>
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<td>
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<p>
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1/√λ
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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kurtosis
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</p>
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</td>
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<td>
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<p>
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3 + 1/λ
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</p>
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</td>
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||
</tr>
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<tr>
|
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<td>
|
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<p>
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kurtosis excess
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</p>
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</td>
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<td>
|
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<p>
|
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1/λ
|
||
</p>
|
||
</td>
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||
</tr>
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||
</tbody>
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||
</table></div>
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</div>
|
||
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, 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>)
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</p>
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</div>
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<hr>
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