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<section id="numpy-random-generator-zipf">
<h1>numpy.random.Generator.zipf<a class="headerlink" href="#numpy-random-generator-zipf" title="Link to this heading">#</a></h1>
<p>method</p>
<dl class="py method">
<dt class="sig sig-object py" id="numpy.random.Generator.zipf">
<span class="sig-prename descclassname"><span class="pre">random.Generator.</span></span><span class="sig-name descname"><span class="pre">zipf</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <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.Generator.zipf" title="Link to this definition">#</a></dt>
<dd><p>Draw samples from a Zipf distribution.</p>
<p>Samples are drawn from a Zipf distribution with specified parameter
<em class="xref py py-obj">a</em> > 1.</p>
<p>The Zipf distribution (also known as the zeta distribution) is a
discrete probability distribution that satisfies Zipf’s law: the
frequency of an item is inversely proportional to its rank in a
frequency table.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>a</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Distribution parameter. Must be greater than 1.</p>
</dd>
<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. If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <code class="docutils literal notranslate"><span class="pre">a</span></code> is a scalar. Otherwise,
<code class="docutils literal notranslate"><span class="pre">np.array(a).size</span></code> samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized Zipf distribution.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.zipf.html#scipy.stats.zipf" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.stats.zipf</span></code></a></dt><dd><p>probability density function, distribution, or cumulative density function, etc.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The probability mass function (PMF) for the Zipf distribution is</p>
<div class="math notranslate nohighlight">
\[p(k) = \frac{k^{-a}}{\zeta(a)},\]</div>
<p>for integers <span class="math notranslate nohighlight">\(k \geq 1\)</span>, where <span class="math notranslate nohighlight">\(\zeta\)</span> is the Riemann Zeta
function.</p>
<p>It is named for the American linguist George Kingsley Zipf, who noted
that the frequency of any word in a sample of a language is inversely
proportional to its rank in the frequency table.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="ra33a5b77f548-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>Zipf, G. K., “Selected Studies of the Principle of Relative
Frequency in Language,” Cambridge, MA: Harvard Univ. Press,
1932.</p>
</div>
</div>
<p class="rubric">Examples</p>
<div class="try_examples_outer_container docutils container" id="12dcc620-104d-48cb-8f4c-286b1e7ed6ae">
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<p>Draw samples from the distribution:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="mf">4.0</span>
<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">20000</span>
<span class="gp">>>> </span><span class="n">rng</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">default_rng</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">s</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">zipf</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
</pre></div>
</div>
<p>Display the histogram of the samples, along with
the expected histogram based on the probability
density function:</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="kn">from</span><span class="w"> </span><span class="nn">scipy.special</span><span class="w"> </span><span class="kn">import</span> <span class="n">zeta</span>
</pre></div>
</div>
<p><a class="reference internal" href="../../generated/numpy.bincount.html#numpy.bincount" title="numpy.bincount"><code class="xref py py-obj docutils literal notranslate"><span class="pre">bincount</span></code></a> provides a fast histogram for small integers.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">count</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">bincount</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">k</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">s</span><span class="o">.</span><span class="n">max</span><span class="p">()</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">count</span><span class="p">[</span><span class="mi">1</span><span class="p">:],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'sample count'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">k</span><span class="p">,</span> <span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">k</span><span class="o">**-</span><span class="n">a</span><span class="p">)</span><span class="o">/</span><span class="n">zeta</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="s1">'k.-'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">label</span><span class="o">=</span><span class="s1">'expected count'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">semilogy</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">f</span><span class="s1">'Zipf sample, a=</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s1">, size=</span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s1">'</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>
</pre></div>
</div>
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