<!DOCTYPE html>

<html>
  <head>
    <meta charset="utf-8">
    
    <title>Poisson Distribution &mdash; SciPy v0.16.1 Reference Guide</title>
    
    <link rel="stylesheet" type="text/css" href="../../static_/css/spc-bootstrap.css">
    <link rel="stylesheet" type="text/css" href="../../static_/css/spc-extend.css">
    <link rel="stylesheet" href="../../static_/scipy.css" type="text/css" >
    <link rel="stylesheet" href="../../static_/pygments.css" type="text/css" >
    
    <script type="text/javascript">
      var DOCUMENTATION_OPTIONS = {
        URL_ROOT:    '../../',
        VERSION:     '0.16.1',
        COLLAPSE_INDEX: false,
        FILE_SUFFIX: '.html',
        HAS_SOURCE:  false
      };
    </script>
    <script type="text/javascript" src="../../static_/jquery.js"></script>
    <script type="text/javascript" src="../../static_/underscore.js"></script>
    <script type="text/javascript" src="../../static_/doctools.js"></script>
    <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
    <script type="text/javascript" src="../../static_/js/copybutton.js"></script>
    <link rel="top" title="SciPy v0.16.1 Reference Guide" href="../../index.html" > 
  </head>
  <body>

  <div class="container">
    <div class="header">
    </div>
  </div>


    <div class="container">
      <div class="main">
        
	<div class="row-fluid">
	  <div class="span12">
	    <div class="spc-navbar">
              
    <ul class="nav nav-pills pull-left">
	
        <li class="active"><a href="../../index.html">SciPy v0.16.1 Reference Guide</a></li>
	 
    </ul>
              
              
    <ul class="nav nav-pills pull-right">
      <li class="active">
        <a href="../../genindex.html" title="General Index"
           accesskey="I">index</a>
      </li>
      <li class="active">
        <a href="../../py-modindex.html" title="Python Module Index"
           >modules</a>
      </li>
      <li class="active">
        <a href="../../scipy-optimize-modindex.html" title="Python Module Index"
           >modules</a>
      </li>
    </ul>
              
	    </div>
	  </div>
	</div>
        

	<div class="row-fluid">
      <div class="spc-rightsidebar span3">
        <div class="sphinxsidebarwrapper">
            <p class="logo"><a href="../../index.html">
              <img class="logo" src="../../static_/scipyshiny_small.png" alt="Logo">
            </a></p>


        </div>
      </div>
          <div class="span9">
            
        <div class="bodywrapper">
          <div class="body" id="spc-section-body">
            
  <div class="section" id="poisson-distribution">
<span id="discrete-poisson"></span><h1>Poisson Distribution<a class="headerlink" href="#poisson-distribution" title="Permalink to this headline">¶</a></h1>
<p>The Poisson random variable counts the number of successes in <span class="math">\(n\)</span> independent Bernoulli trials in the limit as <span class="math">\(n\rightarrow\infty\)</span> and <span class="math">\(p\rightarrow0\)</span> where the probability of success in each trial is <span class="math">\(p\)</span> and <span class="math">\(np=\lambda\geq0\)</span> is a constant. It can be used to approximate the Binomial random
variable or in it&#8217;s own right to count the number of events that occur
in the interval <span class="math">\(\left[0,t\right]\)</span> for a process satisfying certain &#8220;sparsity &#8220;constraints. The functions are</p>
<div class="math">
\[ \begin{eqnarray*} p\left(k;\lambda\right) & = & e^{-\lambda}\frac{\lambda^{k}}{k!}\quad k\geq0,\\ F\left(x;\lambda\right) & = & \sum_{n=0}^{\left\lfloor x\right\rfloor }e^{-\lambda}\frac{\lambda^{n}}{n!}=\frac{1}{\Gamma\left(\left\lfloor x\right\rfloor +1\right)}\int_{\lambda}^{\infty}t^{\left\lfloor x\right\rfloor }e^{-t}dt,\\ \mu & = & \lambda\\ \mu_{2} & = & \lambda\\ \gamma_{1} & = & \frac{1}{\sqrt{\lambda}}\\ \gamma_{2} & = & \frac{1}{\lambda}.\end{eqnarray*}\]</div><div class="math">
\[ M\left(t\right)=\exp\left[\lambda\left(e^{t}-1\right)\right].\]</div><p>Implementation: <a class="reference internal" href="../../generated/scipy.stats.poisson.html#scipy.stats.poisson" title="scipy.stats.poisson"><tt class="xref py py-obj docutils literal"><span class="pre">scipy.stats.poisson</span></tt></a></p>
</div>


          </div>
        </div>
          </div>
        </div>
      </div>
    </div>

    <div class="container container-navbar-bottom">
      <div class="spc-navbar">
        
      </div>
    </div>
    <div class="container">
    <div class="footer">
    <div class="row-fluid">
    <ul class="inline pull-left">
      <li>
        &copy; Copyright 2008-2014, The Scipy community.
      </li>
      <li>
      Last updated on Oct 24, 2015.
      </li>
      <li>
      Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.2.1.
      </li>
    </ul>
    </div>
    </div>
    </div>
  </body>
</html>