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<section id="numpy-random-randomstate-normal">
<h1>numpy.random.RandomState.normal<a class="headerlink" href="#numpy-random-randomstate-normal" title="Link to this heading">#</a></h1>
<p>method</p>
<dl class="py method">
<dt class="sig sig-object py" id="numpy.random.RandomState.normal">
<span class="sig-prename descclassname"><span class="pre">random.RandomState.</span></span><span class="sig-name descname"><span class="pre">normal</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">loc</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">scale</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</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.RandomState.normal" title="Link to this definition">#</a></dt>
<dd><p>Draw random samples from a normal (Gaussian) distribution.</p>
<p>The probability density function of the normal distribution, first
derived by De Moivre and 200 years later by both Gauss and Laplace
independently <a class="reference internal" href="#ra2e838c5ea87-2" id="id1">[2]</a>, is often called the bell curve because of
its characteristic shape (see the example below).</p>
<p>The normal distributions occurs often in nature. For example, it
describes the commonly occurring distribution of samples influenced
by a large number of tiny, random disturbances, each with its own
unique distribution <a class="reference internal" href="#ra2e838c5ea87-2" id="id2">[2]</a>.</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.normal.html#numpy.random.Generator.normal" title="numpy.random.Generator.normal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">normal</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>loc</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Mean (“centre”) of the distribution.</p>
</dd>
<dt><strong>scale</strong><span class="classifier">float or array_like of floats</span></dt><dd><p>Standard deviation (spread or “width”) of the distribution. Must be
non-negative.</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">loc</span></code> and <code class="docutils literal notranslate"><span class="pre">scale</span></code> are both scalars.
Otherwise, <code class="docutils literal notranslate"><span class="pre">np.broadcast(loc,</span> <span class="pre">scale).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 normal 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.norm.html#scipy.stats.norm" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.stats.norm</span></code></a></dt><dd><p>probability density function, distribution or cumulative density function, etc.</p>
</dd>
<dt><a class="reference internal" href="numpy.random.Generator.normal.html#numpy.random.Generator.normal" title="numpy.random.Generator.normal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">random.Generator.normal</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 for the Gaussian distribution is</p>
<div class="math notranslate nohighlight">
\[p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },\]</div>
<p>where <span class="math notranslate nohighlight">\(\mu\)</span> is the mean and <span class="math notranslate nohighlight">\(\sigma\)</span> the standard
deviation. The square of the standard deviation, <span class="math notranslate nohighlight">\(\sigma^2\)</span>,
is called the variance.</p>
<p>The function has its peak at the mean, and its “spread” increases with
the standard deviation (the function reaches 0.607 times its maximum at
<span class="math notranslate nohighlight">\(x + \sigma\)</span> and <span class="math notranslate nohighlight">\(x - \sigma\)</span> <a class="reference internal" href="#ra2e838c5ea87-2" id="id3">[2]</a>). This implies that
normal is more likely to return samples lying close to the mean, rather
than those far away.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="ra2e838c5ea87-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>Wikipedia, “Normal distribution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Normal_distribution">https://en.wikipedia.org/wiki/Normal_distribution</a></p>
</div>
<div class="citation" id="ra2e838c5ea87-2" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id1">1</a>,<a role="doc-backlink" href="#id2">2</a>,<a role="doc-backlink" href="#id3">3</a>)</span>
<p>P. R. Peebles Jr., “Central Limit Theorem” in “Probability,
Random Variables and Random Signal Principles”, 4th ed., 2001,
pp. 51, 51, 125.</p>
</div>
</div>
<p class="rubric">Examples</p>
<div class="try_examples_outer_container docutils container" id="095b9ff8-d0ca-405a-a025-535e015cff2e">
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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">mu</span><span class="p">,</span> <span class="n">sigma</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span> <span class="c1"># mean and standard deviation</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">normal</span><span class="p">(</span><span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
</pre></div>
</div>
<p>Verify the mean and the standard deviation:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">abs</span><span class="p">(</span><span class="n">mu</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">s</span><span class="p">))</span>
<span class="go">0.0 # may vary</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">abs</span><span class="p">(</span><span class="n">sigma</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">ddof</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
<span class="go">0.0 # may vary</span>
</pre></div>
</div>
<p>Display the histogram of the samples, along with
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="n">count</span><span class="p">,</span> <span class="n">bins</span><span class="p">,</span> <span class="n">ignored</span> <span class="o">=</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="mi">30</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</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">bins</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">sigma</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span> <span class="o">*</span>
<span class="gp">... </span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span> <span class="o">-</span> <span class="p">(</span><span class="n">bins</span> <span class="o">-</span> <span class="n">mu</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">sigma</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="p">),</span>
<span class="gp">... </span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</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>
<figure class="align-default">
<img alt="../../../_images/numpy-random-RandomState-normal-1_00_00.png" class="plot-directive" src="../../../_images/numpy-random-RandomState-normal-1_00_00.png" />
</figure>
<p>Two-by-four array of samples from the normal distribution with
mean 3 and standard deviation 2.5:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="go">array([[-4.49401501, 4.00950034, -1.81814867, 7.29718677], # random</span>
<span class="go"> [ 0.39924804, 4.68456316, 4.99394529, 4.84057254]]) # random</span>
</pre></div>
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