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<section id="random-sampling">
<span id="module-numpy.random"></span><span id="numpyrandom"></span><h1>Random sampling<a class="headerlink" href="#random-sampling" title="Link to this heading">#</a></h1>
<section id="quick-start">
<span id="random-quick-start"></span><h2>Quick start<a class="headerlink" href="#quick-start" title="Link to this heading">#</a></h2>
<p>The <a class="reference internal" href="#module-numpy.random" title="numpy.random"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numpy.random</span></code></a> module implements pseudo-random number generators
(PRNGs or RNGs, for short) with the ability to draw samples from a variety of
probability distributions. In general, users will create 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
with <a class="reference internal" href="generator.html#numpy.random.default_rng" title="numpy.random.default_rng"><code class="xref py py-obj docutils literal notranslate"><span class="pre">default_rng</span></code></a> and call the various methods on it to obtain samples from
different distributions.</p>
<div class="try_examples_outer_container docutils container" id="44be5645-e064-4ab5-98bc-1aa1b150ab02">
<div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesShowIframe('44be5645-e064-4ab5-98bc-1aa1b150ab02','cf296516-a893-4ca8-80c8-1fadb762b739','096f4233-0162-4e19-af0e-18823dd7362f','../../lite/tree/../notebooks/index.html?path=2d9a2696_b8bd_42e6_8258_e05ccca6e1a7.ipynb','None')">Try it in your browser!</button></div><div class="try_examples_content docutils container">
<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">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</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>
</pre></div>
</div>
<p>Generate one random float uniformly distributed over the range <span class="math notranslate nohighlight">\([0, 1)\)</span>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">rng</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
<span class="go">0.06369197489564249 # may vary</span>
</pre></div>
</div>
<p>Generate an array of 10 numbers according to a unit Gaussian distribution:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">rng</span><span class="o">.</span><span class="n">standard_normal</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="go">array([-0.31018314, -1.8922078 , -0.3628523 , -0.63526532, 0.43181166, # may vary</span>
<span class="go"> 0.51640373, 1.25693945, 0.07779185, 0.84090247, -2.13406828])</span>
</pre></div>
</div>
<p>Generate an array of 5 integers uniformly over the range <span class="math notranslate nohighlight">\([0, 10)\)</span>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">rng</span><span class="o">.</span><span class="n">integers</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="go">array([8, 7, 6, 2, 0]) # may vary</span>
</pre></div>
</div>
</div>
</div>
<div id="096f4233-0162-4e19-af0e-18823dd7362f" class="try_examples_outer_iframe hidden"><div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesHideIframe('44be5645-e064-4ab5-98bc-1aa1b150ab02','096f4233-0162-4e19-af0e-18823dd7362f')">Go Back</button><button class="try_examples_button" onclick="window.openInNewTab('44be5645-e064-4ab5-98bc-1aa1b150ab02','096f4233-0162-4e19-af0e-18823dd7362f')">Open In Tab</button></div><div id="cf296516-a893-4ca8-80c8-1fadb762b739" class="jupyterlite_sphinx_iframe_container"></div></div><script>document.addEventListener("DOMContentLoaded", function() {window.loadTryExamplesConfig("../../try_examples.json");});</script><p>Our RNGs are deterministic sequences and can be reproduced by specifying a seed integer to
derive its initial state. By default, with no seed provided, <a class="reference internal" href="generator.html#numpy.random.default_rng" title="numpy.random.default_rng"><code class="xref py py-obj docutils literal notranslate"><span class="pre">default_rng</span></code></a> will
seed the RNG from nondeterministic data from the operating system and therefore
generate different numbers each time. The pseudo-random sequences will be
independent for all practical purposes, at least those purposes for which our
pseudo-randomness was good for in the first place.</p>
<div class="try_examples_outer_container docutils container" id="01707c0b-be7e-447d-be03-c998f873e3e8">
<div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesShowIframe('01707c0b-be7e-447d-be03-c998f873e3e8','4a6dd7b3-0d89-420c-a253-dbf00e47aed3','f800326d-a353-45a8-b7fb-3ba42dc78d28','../../lite/tree/../notebooks/index.html?path=94e1c70c_bf9f_420f_8462_d7445c6082e1.ipynb','None')">Try it in your browser!</button></div><div class="try_examples_content docutils container">
<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">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">rng1</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">rng1</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
<span class="go">0.6596288841243357 # may vary</span>
<span class="gp">>>> </span><span class="n">rng2</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">rng2</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
<span class="go">0.11885628817151628 # may vary</span>
</pre></div>
</div>
</div>
</div>
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<p class="admonition-title">Warning</p>
<p>The pseudo-random number generators implemented in this module are designed
for statistical modeling and simulation. They are not suitable for security
or cryptographic purposes. See the <a class="reference external" href="https://docs.python.org/3/library/secrets.html#module-secrets" title="(in Python v3.13)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">secrets</span></code></a> module from the
standard library for such use cases.</p>
</div>
<p id="recommend-secrets-randbits">Seeds should be large positive integers. <a class="reference internal" href="generator.html#numpy.random.default_rng" title="numpy.random.default_rng"><code class="xref py py-obj docutils literal notranslate"><span class="pre">default_rng</span></code></a> can take positive
integers of any size. We recommend using very large, unique numbers to ensure
that your seed is different from anyone else’s. This is good practice to ensure
that your results are statistically independent from theirs unless you are
intentionally <em>trying</em> to reproduce their result. A convenient way to get
such a seed number is to use <a class="reference external" href="https://docs.python.org/3/library/secrets.html#secrets.randbits" title="(in Python v3.13)"><code class="xref py py-func docutils literal notranslate"><span class="pre">secrets.randbits</span></code></a> to get an
arbitrary 128-bit integer.</p>
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<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">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">secrets</span>
<span class="gp">>>> </span><span class="n">secrets</span><span class="o">.</span><span class="n">randbits</span><span class="p">(</span><span class="mi">128</span><span class="p">)</span>
<span class="go">122807528840384100672342137672332424406 # may vary</span>
<span class="gp">>>> </span><span class="n">rng1</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="mi">122807528840384100672342137672332424406</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">rng1</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
<span class="go">0.5363922081269535</span>
<span class="gp">>>> </span><span class="n">rng2</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="mi">122807528840384100672342137672332424406</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">rng2</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
<span class="go">0.5363922081269535</span>
</pre></div>
</div>
</div>
</div>
<div id="2ba76822-a7bb-4985-b58f-7ae40da069cd" class="try_examples_outer_iframe hidden"><div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesHideIframe('385c0d58-6049-4f12-aec4-984f20a5bc23','2ba76822-a7bb-4985-b58f-7ae40da069cd')">Go Back</button><button class="try_examples_button" onclick="window.openInNewTab('385c0d58-6049-4f12-aec4-984f20a5bc23','2ba76822-a7bb-4985-b58f-7ae40da069cd')">Open In Tab</button></div><div id="54c9f003-be2b-45e8-be24-d847a74fc614" class="jupyterlite_sphinx_iframe_container"></div></div><script>document.addEventListener("DOMContentLoaded", function() {window.loadTryExamplesConfig("../../try_examples.json");});</script><p>See the documentation on <a class="reference internal" href="generator.html#numpy.random.default_rng" title="numpy.random.default_rng"><code class="xref py py-obj docutils literal notranslate"><span class="pre">default_rng</span></code></a> and <a class="reference internal" href="bit_generators/generated/numpy.random.SeedSequence.html#numpy.random.SeedSequence" title="numpy.random.SeedSequence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SeedSequence</span></code></a> for more advanced
options for controlling the seed in specialized scenarios.</p>
<p><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> and its associated infrastructure was introduced in NumPy version
1.17.0. There is still a lot of code that uses the older <a class="reference internal" href="legacy.html#numpy.random.RandomState" title="numpy.random.RandomState"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RandomState</span></code></a> and the
functions in <a class="reference internal" href="#module-numpy.random" title="numpy.random"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.random</span></code></a>. While there are no plans to remove them at this
time, we do recommend transitioning to <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> as you can. The algorithms
are faster, more flexible, and will receive more improvements in the future.
For the most part, <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> can be used as a replacement for <a class="reference internal" href="legacy.html#numpy.random.RandomState" title="numpy.random.RandomState"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RandomState</span></code></a>.
See <a class="reference internal" href="legacy.html#legacy"><span class="std std-ref">Legacy random generation</span></a> for information on the legacy infrastructure,
<a class="reference internal" href="new-or-different.html#new-or-different"><span class="std std-ref">What’s new or different</span></a> for information on transitioning, and <a class="reference external" href="https://numpy.org/neps/nep-0019-rng-policy.html#nep19" title="(in NumPy Enhancement Proposals)"><span class="xref std std-ref">NEP 19</span></a> for some of the reasoning for the transition.</p>
</section>
<section id="design">
<h2>Design<a class="headerlink" href="#design" title="Link to this heading">#</a></h2>
<p>Users primarily interact with <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> instances. Each <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
owns a <a class="reference internal" href="bit_generators/generated/numpy.random.BitGenerator.html#numpy.random.BitGenerator" title="numpy.random.BitGenerator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">BitGenerator</span></code></a> instance that implements the core RNG algorithm. The
<a class="reference internal" href="bit_generators/generated/numpy.random.BitGenerator.html#numpy.random.BitGenerator" title="numpy.random.BitGenerator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">BitGenerator</span></code></a> has a limited set of responsibilities. It manages state and
provides functions to produce random doubles and random unsigned 32- and 64-bit
values.</p>
<p>The <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> takes the bit generator-provided stream and transforms them
into more useful distributions, e.g., simulated normal random values. This
structure allows alternative bit generators to be used with little code
duplication.</p>
<p>NumPy implements several different <a class="reference internal" href="bit_generators/generated/numpy.random.BitGenerator.html#numpy.random.BitGenerator" title="numpy.random.BitGenerator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">BitGenerator</span></code></a> classes implementing
different RNG algorithms. <a class="reference internal" href="generator.html#numpy.random.default_rng" title="numpy.random.default_rng"><code class="xref py py-obj docutils literal notranslate"><span class="pre">default_rng</span></code></a> currently uses <a class="reference internal" href="bit_generators/pcg64.html#numpy.random.PCG64" title="numpy.random.PCG64"><code class="xref py py-obj docutils literal notranslate"><span class="pre">PCG64</span></code></a> as the
default <a class="reference internal" href="bit_generators/generated/numpy.random.BitGenerator.html#numpy.random.BitGenerator" title="numpy.random.BitGenerator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">BitGenerator</span></code></a>. It has better statistical properties and performance
than the <a class="reference internal" href="bit_generators/mt19937.html#numpy.random.MT19937" title="numpy.random.MT19937"><code class="xref py py-obj docutils literal notranslate"><span class="pre">MT19937</span></code></a> algorithm used in the legacy <a class="reference internal" href="legacy.html#numpy.random.RandomState" title="numpy.random.RandomState"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RandomState</span></code></a>. See
<a class="reference internal" href="bit_generators/index.html#random-bit-generators"><span class="std std-ref">Bit generators</span></a> for more details on the supported BitGenerators.</p>
<p><a class="reference internal" href="generator.html#numpy.random.default_rng" title="numpy.random.default_rng"><code class="xref py py-obj docutils literal notranslate"><span class="pre">default_rng</span></code></a> and BitGenerators delegate the conversion of seeds into RNG
states to <a class="reference internal" href="bit_generators/generated/numpy.random.SeedSequence.html#numpy.random.SeedSequence" title="numpy.random.SeedSequence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SeedSequence</span></code></a> internally. <a class="reference internal" href="bit_generators/generated/numpy.random.SeedSequence.html#numpy.random.SeedSequence" title="numpy.random.SeedSequence"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SeedSequence</span></code></a> implements a sophisticated
algorithm that intermediates between the user’s input and the internal
implementation details of each <a class="reference internal" href="bit_generators/generated/numpy.random.BitGenerator.html#numpy.random.BitGenerator" title="numpy.random.BitGenerator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">BitGenerator</span></code></a> algorithm, each of which can
require different amounts of bits for its state. Importantly, it lets you use
arbitrary-sized integers and arbitrary sequences of such integers to mix
together into the RNG state. This is a useful primitive for constructing
a <a class="reference internal" href="parallel.html#seedsequence-spawn"><span class="std std-ref">flexible pattern for parallel RNG streams</span></a>.</p>
<p>For backward compatibility, we still maintain the legacy <a class="reference internal" href="legacy.html#numpy.random.RandomState" title="numpy.random.RandomState"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RandomState</span></code></a> class.
It continues to use the <a class="reference internal" href="bit_generators/mt19937.html#numpy.random.MT19937" title="numpy.random.MT19937"><code class="xref py py-obj docutils literal notranslate"><span class="pre">MT19937</span></code></a> algorithm by default, and old seeds continue
to reproduce the same results. The convenience <a class="reference internal" href="legacy.html#functions-in-numpy-random"><span class="std std-ref">Functions in numpy.random</span></a>
are still aliases to the methods on a single global <a class="reference internal" href="legacy.html#numpy.random.RandomState" title="numpy.random.RandomState"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RandomState</span></code></a> instance. See
<a class="reference internal" href="legacy.html#legacy"><span class="std std-ref">Legacy random generation</span></a> for the complete details. See <a class="reference internal" href="new-or-different.html#new-or-different"><span class="std std-ref">What’s new or different</span></a> for
a detailed comparison between <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> and <a class="reference internal" href="legacy.html#numpy.random.RandomState" title="numpy.random.RandomState"><code class="xref py py-obj docutils literal notranslate"><span class="pre">RandomState</span></code></a>.</p>
<section id="parallel-generation">
<h3>Parallel Generation<a class="headerlink" href="#parallel-generation" title="Link to this heading">#</a></h3>
<p>The included generators can be used in parallel, distributed applications in
a number of ways:</p>
<ul class="simple">
<li><p><a class="reference internal" href="parallel.html#seedsequence-spawn"><span class="std std-ref">SeedSequence spawning</span></a></p></li>
<li><p><a class="reference internal" href="parallel.html#sequence-of-seeds"><span class="std std-ref">Sequence of integer seeds</span></a></p></li>
<li><p><a class="reference internal" href="parallel.html#independent-streams"><span class="std std-ref">Independent streams</span></a></p></li>
<li><p><a class="reference internal" href="parallel.html#parallel-jumped"><span class="std std-ref">Jumping the BitGenerator state</span></a></p></li>
</ul>
<p>Users with a very large amount of parallelism will want to consult
<a class="reference internal" href="upgrading-pcg64.html#upgrading-pcg64"><span class="std std-ref">Upgrading PCG64 with PCG64DXSM</span></a>.</p>
</section>
</section>
<section id="concepts">
<h2>Concepts<a class="headerlink" href="#concepts" title="Link to this heading">#</a></h2>
<div class="toctree-wrapper compound">
<ul>
<li class="toctree-l1"><a class="reference internal" href="generator.html">Random <code class="docutils literal notranslate"><span class="pre">Generator</span></code></a></li>
<li class="toctree-l1"><a class="reference internal" href="legacy.html">Legacy Generator (RandomState)</a></li>
<li class="toctree-l1"><a class="reference internal" href="bit_generators/index.html">Bit generators</a></li>
<li class="toctree-l1"><a class="reference internal" href="bit_generators/index.html#seeding-and-entropy">Seeding and entropy</a></li>
<li class="toctree-l1"><a class="reference internal" href="upgrading-pcg64.html">Upgrading PCG64 with PCG64DXSM</a></li>
<li class="toctree-l1"><a class="reference internal" href="compatibility.html">Compatibility policy</a></li>
</ul>
</div>
</section>
<section id="features">
<h2>Features<a class="headerlink" href="#features" title="Link to this heading">#</a></h2>
<div class="toctree-wrapper compound">
<ul>
<li class="toctree-l1"><a class="reference internal" href="parallel.html">Parallel Applications</a><ul>
<li class="toctree-l2"><a class="reference internal" href="parallel.html#seedsequence-spawning"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SeedSequence</span></code> spawning</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel.html#sequence-of-integer-seeds">Sequence of integer seeds</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel.html#independent-streams">Independent streams</a></li>
<li class="toctree-l2"><a class="reference internal" href="parallel.html#jumping-the-bitgenerator-state">Jumping the BitGenerator state</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="multithreading.html">Multithreaded Generation</a></li>
<li class="toctree-l1"><a class="reference internal" href="new-or-different.html">What’s new or different</a></li>
<li class="toctree-l1"><a class="reference internal" href="performance.html">Comparing Performance</a><ul>
<li class="toctree-l2"><a class="reference internal" href="performance.html#recommendation">Recommendation</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance.html#timings">Timings</a></li>
<li class="toctree-l2"><a class="reference internal" href="performance.html#performance-on-different-operating-systems">Performance on different operating systems</a></li>
</ul>
</li>
<li class="toctree-l1"><a class="reference internal" href="c-api.html">C API for random</a></li>
<li class="toctree-l1"><a class="reference internal" href="extending.html">Examples of using Numba, Cython, CFFI</a><ul>
<li class="toctree-l2"><a class="reference internal" href="extending.html#numba">Numba</a></li>
<li class="toctree-l2"><a class="reference internal" href="extending.html#cython">Cython</a></li>
<li class="toctree-l2"><a class="reference internal" href="extending.html#id2">CFFI</a></li>
<li class="toctree-l2"><a class="reference internal" href="extending.html#new-bitgenerators">New BitGenerators</a></li>
<li class="toctree-l2"><a class="reference internal" href="extending.html#examples">Examples</a></li>
</ul>
</li>
</ul>
</div>
<section id="original-source-of-the-generator-and-bitgenerators">
<h3>Original Source of the Generator and BitGenerators<a class="headerlink" href="#original-source-of-the-generator-and-bitgenerators" title="Link to this heading">#</a></h3>
<p>This package was developed independently of NumPy and was integrated in version
1.17.0. The original repo is at <a class="github reference external" href="https://github.com/bashtage/randomgen">bashtage/randomgen</a>.</p>
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