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<section id="numpy-random-randomstate-standard-cauchy">
<h1>numpy.random.RandomState.standard_cauchy<a class="headerlink" href="#numpy-random-randomstate-standard-cauchy" title="Link to this heading">#</a></h1>
<p>method</p>
<dl class="py method">
<dt class="sig sig-object py" id="numpy.random.RandomState.standard_cauchy">
<span class="sig-prename descclassname"><span class="pre">random.RandomState.</span></span><span class="sig-name descname"><span class="pre">standard_cauchy</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.RandomState.standard_cauchy" title="Link to this definition">#</a></dt>
<dd><p>Draw samples from a standard Cauchy distribution with mode = 0.</p>
<p>Also known as the Lorentz distribution.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the
<a class="reference internal" href="numpy.random.Generator.standard_cauchy.html#numpy.random.Generator.standard_cauchy" title="numpy.random.Generator.standard_cauchy"><code class="xref py py-obj docutils literal notranslate"><span class="pre">standard_cauchy</span></code></a>
method of a <a class="reference internal" href="../generator.html#numpy.random.Generator" title="numpy.random.Generator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator</span></code></a> instance instead;
please see the <a class="reference internal" href="../index.html#random-quick-start"><span class="std std-ref">Quick start</span></a>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape. If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn. Default is None, in which case a
single value is returned.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>samples</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>The drawn samples.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.random.Generator.standard_cauchy.html#numpy.random.Generator.standard_cauchy" title="numpy.random.Generator.standard_cauchy"><code class="xref py py-obj docutils literal notranslate"><span class="pre">random.Generator.standard_cauchy</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The probability density function for the full Cauchy distribution is</p>
<div class="math notranslate nohighlight">
\[P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
(\frac{x-x_0}{\gamma})^2 \bigr] }\]</div>
<p>and the Standard Cauchy distribution just sets <span class="math notranslate nohighlight">\(x_0=0\)</span> and
<span class="math notranslate nohighlight">\(\gamma=1\)</span></p>
<p>The Cauchy distribution arises in the solution to the driven harmonic
oscillator problem, and also describes spectral line broadening. It
also describes the distribution of values at which a line tilted at
a random angle will cut the x axis.</p>
<p>When studying hypothesis tests that assume normality, seeing how the
tests perform on data from a Cauchy distribution is a good indicator of
their sensitivity to a heavy-tailed distribution, since the Cauchy looks
very much like a Gaussian distribution, but with heavier tails.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="rceffa9d7cf46-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>NIST/SEMATECH e-Handbook of Statistical Methods, “Cauchy
Distribution”,
<a class="reference external" href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm">https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm</a></p>
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<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>Weisstein, Eric W. “Cauchy Distribution.” From MathWorld–A
Wolfram Web Resource.
<a class="reference external" href="https://mathworld.wolfram.com/CauchyDistribution.html">https://mathworld.wolfram.com/CauchyDistribution.html</a></p>
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<span class="label"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></span>
<p>Wikipedia, “Cauchy distribution”
<a class="reference external" href="https://en.wikipedia.org/wiki/Cauchy_distribution">https://en.wikipedia.org/wiki/Cauchy_distribution</a></p>
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<p class="rubric">Examples</p>
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<p>Draw samples and plot the distribution:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<span class="gp">>>> </span><span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">standard_cauchy</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">s</span> <span class="o">=</span> <span class="n">s</span><span class="p">[(</span><span class="n">s</span><span class="o">>-</span><span class="mi">25</span><span class="p">)</span> <span class="o">&</span> <span class="p">(</span><span class="n">s</span><span class="o"><</span><span class="mi">25</span><span class="p">)]</span> <span class="c1"># truncate distribution so it plots well</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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