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<title>random — Generate pseudo-random numbers &#8212; Python 3.7.4 documentation</title>
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<div class="section" id="module-random">
<span id="random-generate-pseudo-random-numbers"></span><h1><a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code></a> — Generate pseudo-random numbers<a class="headerlink" href="#module-random" title="Permalink to this headline"></a></h1>
<p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.7/Lib/random.py">Lib/random.py</a></p>
<hr class="docutils" />
<p>This module implements pseudo-random number generators for various
distributions.</p>
<p>For integers, there is uniform selection from a range. For sequences, there is
uniform selection of a random element, a function to generate a random
permutation of a list in-place, and a function for random sampling without
replacement.</p>
<p>On the real line, there are functions to compute uniform, normal (Gaussian),
lognormal, negative exponential, gamma, and beta distributions. For generating
distributions of angles, the von Mises distribution is available.</p>
<p>Almost all module functions depend on the basic function <a class="reference internal" href="#random.random" title="random.random"><code class="xref py py-func docutils literal notranslate"><span class="pre">random()</span></code></a>, which
generates a random float uniformly in the semi-open range [0.0, 1.0). Python
uses the Mersenne Twister as the core generator. It produces 53-bit precision
floats and has a period of 2**19937-1. The underlying implementation in C is
both fast and threadsafe. The Mersenne Twister is one of the most extensively
tested random number generators in existence. However, being completely
deterministic, it is not suitable for all purposes, and is completely unsuitable
for cryptographic purposes.</p>
<p>The functions supplied by this module are actually bound methods of a hidden
instance of the <a class="reference internal" href="#random.Random" title="random.Random"><code class="xref py py-class docutils literal notranslate"><span class="pre">random.Random</span></code></a> class. You can instantiate your own
instances of <a class="reference internal" href="#random.Random" title="random.Random"><code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code></a> to get generators that don’t share state.</p>
<p>Class <a class="reference internal" href="#random.Random" title="random.Random"><code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code></a> can also be subclassed if you want to use a different
basic generator of your own devising: in that case, override the <code class="xref py py-meth docutils literal notranslate"><span class="pre">random()</span></code>,
<code class="xref py py-meth docutils literal notranslate"><span class="pre">seed()</span></code>, <code class="xref py py-meth docutils literal notranslate"><span class="pre">getstate()</span></code>, and <code class="xref py py-meth docutils literal notranslate"><span class="pre">setstate()</span></code> methods.
Optionally, a new generator can supply a <code class="xref py py-meth docutils literal notranslate"><span class="pre">getrandbits()</span></code> method — this
allows <a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randrange()</span></code></a> to produce selections over an arbitrarily large range.</p>
<p>The <a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code></a> module also provides the <a class="reference internal" href="#random.SystemRandom" title="random.SystemRandom"><code class="xref py py-class docutils literal notranslate"><span class="pre">SystemRandom</span></code></a> class which
uses the system function <a class="reference internal" href="os.html#os.urandom" title="os.urandom"><code class="xref py py-func docutils literal notranslate"><span class="pre">os.urandom()</span></code></a> to generate random numbers
from sources provided by the operating system.</p>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>The pseudo-random generators of this module should not be used for
security purposes. For security or cryptographic uses, see the
<a class="reference internal" href="secrets.html#module-secrets" title="secrets: Generate secure random numbers for managing secrets."><code class="xref py py-mod docutils literal notranslate"><span class="pre">secrets</span></code></a> module.</p>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p>M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally
equidistributed uniform pseudorandom number generator”, ACM Transactions on
Modeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998.</p>
<p><a class="reference external" href="https://code.activestate.com/recipes/576707/">Complementary-Multiply-with-Carry recipe</a> for a compatible alternative
random number generator with a long period and comparatively simple update
operations.</p>
</div>
<div class="section" id="bookkeeping-functions">
<h2>Bookkeeping functions<a class="headerlink" href="#bookkeeping-functions" title="Permalink to this headline"></a></h2>
<dl class="function">
<dt id="random.seed">
<code class="descclassname">random.</code><code class="descname">seed</code><span class="sig-paren">(</span><em>a=None</em>, <em>version=2</em><span class="sig-paren">)</span><a class="headerlink" href="#random.seed" title="Permalink to this definition"></a></dt>
<dd><p>Initialize the random number generator.</p>
<p>If <em>a</em> is omitted or <code class="docutils literal notranslate"><span class="pre">None</span></code>, the current system time is used. If
randomness sources are provided by the operating system, they are used
instead of the system time (see the <a class="reference internal" href="os.html#os.urandom" title="os.urandom"><code class="xref py py-func docutils literal notranslate"><span class="pre">os.urandom()</span></code></a> function for details
on availability).</p>
<p>If <em>a</em> is an int, it is used directly.</p>
<p>With version 2 (the default), a <a class="reference internal" href="stdtypes.html#str" title="str"><code class="xref py py-class docutils literal notranslate"><span class="pre">str</span></code></a>, <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytes</span></code></a>, or <a class="reference internal" href="stdtypes.html#bytearray" title="bytearray"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytearray</span></code></a>
object gets converted to an <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> and all of its bits are used.</p>
<p>With version 1 (provided for reproducing random sequences from older versions
of Python), the algorithm for <a class="reference internal" href="stdtypes.html#str" title="str"><code class="xref py py-class docutils literal notranslate"><span class="pre">str</span></code></a> and <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytes</span></code></a> generates a
narrower range of seeds.</p>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 3.2: </span>Moved to the version 2 scheme which uses all of the bits in a string seed.</p>
</div>
</dd></dl>
<dl class="function">
<dt id="random.getstate">
<code class="descclassname">random.</code><code class="descname">getstate</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.getstate" title="Permalink to this definition"></a></dt>
<dd><p>Return an object capturing the current internal state of the generator. This
object can be passed to <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">setstate()</span></code></a> to restore the state.</p>
</dd></dl>
<dl class="function">
<dt id="random.setstate">
<code class="descclassname">random.</code><code class="descname">setstate</code><span class="sig-paren">(</span><em>state</em><span class="sig-paren">)</span><a class="headerlink" href="#random.setstate" title="Permalink to this definition"></a></dt>
<dd><p><em>state</em> should have been obtained from a previous call to <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">getstate()</span></code></a>, and
<a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">setstate()</span></code></a> restores the internal state of the generator to what it was at
the time <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">getstate()</span></code></a> was called.</p>
</dd></dl>
<dl class="function">
<dt id="random.getrandbits">
<code class="descclassname">random.</code><code class="descname">getrandbits</code><span class="sig-paren">(</span><em>k</em><span class="sig-paren">)</span><a class="headerlink" href="#random.getrandbits" title="Permalink to this definition"></a></dt>
<dd><p>Returns a Python integer with <em>k</em> random bits. This method is supplied with
the MersenneTwister generator and some other generators may also provide it
as an optional part of the API. When available, <a class="reference internal" href="#random.getrandbits" title="random.getrandbits"><code class="xref py py-meth docutils literal notranslate"><span class="pre">getrandbits()</span></code></a> enables
<a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randrange()</span></code></a> to handle arbitrarily large ranges.</p>
</dd></dl>
</div>
<div class="section" id="functions-for-integers">
<h2>Functions for integers<a class="headerlink" href="#functions-for-integers" title="Permalink to this headline"></a></h2>
<dl class="function">
<dt id="random.randrange">
<code class="descclassname">random.</code><code class="descname">randrange</code><span class="sig-paren">(</span><em>stop</em><span class="sig-paren">)</span><a class="headerlink" href="#random.randrange" title="Permalink to this definition"></a></dt>
<dt>
<code class="descclassname">random.</code><code class="descname">randrange</code><span class="sig-paren">(</span><em>start</em>, <em>stop</em><span class="optional">[</span>, <em>step</em><span class="optional">]</span><span class="sig-paren">)</span></dt>
<dd><p>Return a randomly selected element from <code class="docutils literal notranslate"><span class="pre">range(start,</span> <span class="pre">stop,</span> <span class="pre">step)</span></code>. This is
equivalent to <code class="docutils literal notranslate"><span class="pre">choice(range(start,</span> <span class="pre">stop,</span> <span class="pre">step))</span></code>, but doesn’t actually build a
range object.</p>
<p>The positional argument pattern matches that of <a class="reference internal" href="stdtypes.html#range" title="range"><code class="xref py py-func docutils literal notranslate"><span class="pre">range()</span></code></a>. Keyword arguments
should not be used because the function may use them in unexpected ways.</p>
<div class="versionchanged">
<p><span class="versionmodified changed">Changed in version 3.2: </span><a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randrange()</span></code></a> is more sophisticated about producing equally distributed
values. Formerly it used a style like <code class="docutils literal notranslate"><span class="pre">int(random()*n)</span></code> which could produce
slightly uneven distributions.</p>
</div>
</dd></dl>
<dl class="function">
<dt id="random.randint">
<code class="descclassname">random.</code><code class="descname">randint</code><span class="sig-paren">(</span><em>a</em>, <em>b</em><span class="sig-paren">)</span><a class="headerlink" href="#random.randint" title="Permalink to this definition"></a></dt>
<dd><p>Return a random integer <em>N</em> such that <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">b</span></code>. Alias for
<code class="docutils literal notranslate"><span class="pre">randrange(a,</span> <span class="pre">b+1)</span></code>.</p>
</dd></dl>
</div>
<div class="section" id="functions-for-sequences">
<h2>Functions for sequences<a class="headerlink" href="#functions-for-sequences" title="Permalink to this headline"></a></h2>
<dl class="function">
<dt id="random.choice">
<code class="descclassname">random.</code><code class="descname">choice</code><span class="sig-paren">(</span><em>seq</em><span class="sig-paren">)</span><a class="headerlink" href="#random.choice" title="Permalink to this definition"></a></dt>
<dd><p>Return a random element from the non-empty sequence <em>seq</em>. If <em>seq</em> is empty,
raises <a class="reference internal" href="exceptions.html#IndexError" title="IndexError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">IndexError</span></code></a>.</p>
</dd></dl>
<dl class="function">
<dt id="random.choices">
<code class="descclassname">random.</code><code class="descname">choices</code><span class="sig-paren">(</span><em>population</em>, <em>weights=None</em>, <em>*</em>, <em>cum_weights=None</em>, <em>k=1</em><span class="sig-paren">)</span><a class="headerlink" href="#random.choices" title="Permalink to this definition"></a></dt>
<dd><p>Return a <em>k</em> sized list of elements chosen from the <em>population</em> with replacement.
If the <em>population</em> is empty, raises <a class="reference internal" href="exceptions.html#IndexError" title="IndexError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">IndexError</span></code></a>.</p>
<p>If a <em>weights</em> sequence is specified, selections are made according to the
relative weights. Alternatively, if a <em>cum_weights</em> sequence is given, the
selections are made according to the cumulative weights (perhaps computed
using <a class="reference internal" href="itertools.html#itertools.accumulate" title="itertools.accumulate"><code class="xref py py-func docutils literal notranslate"><span class="pre">itertools.accumulate()</span></code></a>). For example, the relative weights
<code class="docutils literal notranslate"><span class="pre">[10,</span> <span class="pre">5,</span> <span class="pre">30,</span> <span class="pre">5]</span></code> are equivalent to the cumulative weights
<code class="docutils literal notranslate"><span class="pre">[10,</span> <span class="pre">15,</span> <span class="pre">45,</span> <span class="pre">50]</span></code>. Internally, the relative weights are converted to
cumulative weights before making selections, so supplying the cumulative
weights saves work.</p>
<p>If neither <em>weights</em> nor <em>cum_weights</em> are specified, selections are made
with equal probability. If a weights sequence is supplied, it must be
the same length as the <em>population</em> sequence. It is a <a class="reference internal" href="exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a>
to specify both <em>weights</em> and <em>cum_weights</em>.</p>
<p>The <em>weights</em> or <em>cum_weights</em> can use any numeric type that interoperates
with the <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> values returned by <a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-func docutils literal notranslate"><span class="pre">random()</span></code></a> (that includes
integers, floats, and fractions but excludes decimals).</p>
<p>For a given seed, the <a class="reference internal" href="#random.choices" title="random.choices"><code class="xref py py-func docutils literal notranslate"><span class="pre">choices()</span></code></a> function with equal weighting
typically produces a different sequence than repeated calls to
<a class="reference internal" href="#random.choice" title="random.choice"><code class="xref py py-func docutils literal notranslate"><span class="pre">choice()</span></code></a>. The algorithm used by <a class="reference internal" href="#random.choices" title="random.choices"><code class="xref py py-func docutils literal notranslate"><span class="pre">choices()</span></code></a> uses floating
point arithmetic for internal consistency and speed. The algorithm used
by <a class="reference internal" href="#random.choice" title="random.choice"><code class="xref py py-func docutils literal notranslate"><span class="pre">choice()</span></code></a> defaults to integer arithmetic with repeated selections
to avoid small biases from round-off error.</p>
<div class="versionadded">
<p><span class="versionmodified added">New in version 3.6.</span></p>
</div>
</dd></dl>
<dl class="function">
<dt id="random.shuffle">
<code class="descclassname">random.</code><code class="descname">shuffle</code><span class="sig-paren">(</span><em>x</em><span class="optional">[</span>, <em>random</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.shuffle" title="Permalink to this definition"></a></dt>
<dd><p>Shuffle the sequence <em>x</em> in place.</p>
<p>The optional argument <em>random</em> is a 0-argument function returning a random
float in [0.0, 1.0); by default, this is the function <a class="reference internal" href="#random.random" title="random.random"><code class="xref py py-func docutils literal notranslate"><span class="pre">random()</span></code></a>.</p>
<p>To shuffle an immutable sequence and return a new shuffled list, use
<code class="docutils literal notranslate"><span class="pre">sample(x,</span> <span class="pre">k=len(x))</span></code> instead.</p>
<p>Note that even for small <code class="docutils literal notranslate"><span class="pre">len(x)</span></code>, the total number of permutations of <em>x</em>
can quickly grow larger than the period of most random number generators.
This implies that most permutations of a long sequence can never be
generated. For example, a sequence of length 2080 is the largest that
can fit within the period of the Mersenne Twister random number generator.</p>
</dd></dl>
<dl class="function">
<dt id="random.sample">
<code class="descclassname">random.</code><code class="descname">sample</code><span class="sig-paren">(</span><em>population</em>, <em>k</em><span class="sig-paren">)</span><a class="headerlink" href="#random.sample" title="Permalink to this definition"></a></dt>
<dd><p>Return a <em>k</em> length list of unique elements chosen from the population sequence
or set. Used for random sampling without replacement.</p>
<p>Returns a new list containing elements from the population while leaving the
original population unchanged. The resulting list is in selection order so that
all sub-slices will also be valid random samples. This allows raffle winners
(the sample) to be partitioned into grand prize and second place winners (the
subslices).</p>
<p>Members of the population need not be <a class="reference internal" href="../glossary.html#term-hashable"><span class="xref std std-term">hashable</span></a> or unique. If the population
contains repeats, then each occurrence is a possible selection in the sample.</p>
<p>To choose a sample from a range of integers, use a <a class="reference internal" href="stdtypes.html#range" title="range"><code class="xref py py-func docutils literal notranslate"><span class="pre">range()</span></code></a> object as an
argument. This is especially fast and space efficient for sampling from a large
population: <code class="docutils literal notranslate"><span class="pre">sample(range(10000000),</span> <span class="pre">k=60)</span></code>.</p>
<p>If the sample size is larger than the population size, a <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a>
is raised.</p>
</dd></dl>
</div>
<div class="section" id="real-valued-distributions">
<h2>Real-valued distributions<a class="headerlink" href="#real-valued-distributions" title="Permalink to this headline"></a></h2>
<p>The following functions generate specific real-valued distributions. Function
parameters are named after the corresponding variables in the distribution’s
equation, as used in common mathematical practice; most of these equations can
be found in any statistics text.</p>
<dl class="function">
<dt id="random.random">
<code class="descclassname">random.</code><code class="descname">random</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.random" title="Permalink to this definition"></a></dt>
<dd><p>Return the next random floating point number in the range [0.0, 1.0).</p>
</dd></dl>
<dl class="function">
<dt id="random.uniform">
<code class="descclassname">random.</code><code class="descname">uniform</code><span class="sig-paren">(</span><em>a</em>, <em>b</em><span class="sig-paren">)</span><a class="headerlink" href="#random.uniform" title="Permalink to this definition"></a></dt>
<dd><p>Return a random floating point number <em>N</em> such that <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">b</span></code> for
<code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&lt;=</span> <span class="pre">b</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">a</span></code> for <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">&lt;</span> <span class="pre">a</span></code>.</p>
<p>The end-point value <code class="docutils literal notranslate"><span class="pre">b</span></code> may or may not be included in the range
depending on floating-point rounding in the equation <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">(b-a)</span> <span class="pre">*</span> <span class="pre">random()</span></code>.</p>
</dd></dl>
<dl class="function">
<dt id="random.triangular">
<code class="descclassname">random.</code><code class="descname">triangular</code><span class="sig-paren">(</span><em>low</em>, <em>high</em>, <em>mode</em><span class="sig-paren">)</span><a class="headerlink" href="#random.triangular" title="Permalink to this definition"></a></dt>
<dd><p>Return a random floating point number <em>N</em> such that <code class="docutils literal notranslate"><span class="pre">low</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">high</span></code> and
with the specified <em>mode</em> between those bounds. The <em>low</em> and <em>high</em> bounds
default to zero and one. The <em>mode</em> argument defaults to the midpoint
between the bounds, giving a symmetric distribution.</p>
</dd></dl>
<dl class="function">
<dt id="random.betavariate">
<code class="descclassname">random.</code><code class="descname">betavariate</code><span class="sig-paren">(</span><em>alpha</em>, <em>beta</em><span class="sig-paren">)</span><a class="headerlink" href="#random.betavariate" title="Permalink to this definition"></a></dt>
<dd><p>Beta distribution. Conditions on the parameters are <code class="docutils literal notranslate"><span class="pre">alpha</span> <span class="pre">&gt;</span> <span class="pre">0</span></code> and
<code class="docutils literal notranslate"><span class="pre">beta</span> <span class="pre">&gt;</span> <span class="pre">0</span></code>. Returned values range between 0 and 1.</p>
</dd></dl>
<dl class="function">
<dt id="random.expovariate">
<code class="descclassname">random.</code><code class="descname">expovariate</code><span class="sig-paren">(</span><em>lambd</em><span class="sig-paren">)</span><a class="headerlink" href="#random.expovariate" title="Permalink to this definition"></a></dt>
<dd><p>Exponential distribution. <em>lambd</em> is 1.0 divided by the desired
mean. It should be nonzero. (The parameter would be called
“lambda”, but that is a reserved word in Python.) Returned values
range from 0 to positive infinity if <em>lambd</em> is positive, and from
negative infinity to 0 if <em>lambd</em> is negative.</p>
</dd></dl>
<dl class="function">
<dt id="random.gammavariate">
<code class="descclassname">random.</code><code class="descname">gammavariate</code><span class="sig-paren">(</span><em>alpha</em>, <em>beta</em><span class="sig-paren">)</span><a class="headerlink" href="#random.gammavariate" title="Permalink to this definition"></a></dt>
<dd><p>Gamma distribution. (<em>Not</em> the gamma function!) Conditions on the
parameters are <code class="docutils literal notranslate"><span class="pre">alpha</span> <span class="pre">&gt;</span> <span class="pre">0</span></code> and <code class="docutils literal notranslate"><span class="pre">beta</span> <span class="pre">&gt;</span> <span class="pre">0</span></code>.</p>
<p>The probability distribution function is:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span> <span class="n">x</span> <span class="o">**</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">/</span> <span class="n">beta</span><span class="p">)</span>
<span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">=</span> <span class="o">--------------------------------------</span>
<span class="n">math</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span> <span class="o">*</span> <span class="n">beta</span> <span class="o">**</span> <span class="n">alpha</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="random.gauss">
<code class="descclassname">random.</code><code class="descname">gauss</code><span class="sig-paren">(</span><em>mu</em>, <em>sigma</em><span class="sig-paren">)</span><a class="headerlink" href="#random.gauss" title="Permalink to this definition"></a></dt>
<dd><p>Gaussian distribution. <em>mu</em> is the mean, and <em>sigma</em> is the standard
deviation. This is slightly faster than the <a class="reference internal" href="#random.normalvariate" title="random.normalvariate"><code class="xref py py-func docutils literal notranslate"><span class="pre">normalvariate()</span></code></a> function
defined below.</p>
</dd></dl>
<dl class="function">
<dt id="random.lognormvariate">
<code class="descclassname">random.</code><code class="descname">lognormvariate</code><span class="sig-paren">(</span><em>mu</em>, <em>sigma</em><span class="sig-paren">)</span><a class="headerlink" href="#random.lognormvariate" title="Permalink to this definition"></a></dt>
<dd><p>Log normal distribution. If you take the natural logarithm of this
distribution, you’ll get a normal distribution with mean <em>mu</em> and standard
deviation <em>sigma</em>. <em>mu</em> can have any value, and <em>sigma</em> must be greater than
zero.</p>
</dd></dl>
<dl class="function">
<dt id="random.normalvariate">
<code class="descclassname">random.</code><code class="descname">normalvariate</code><span class="sig-paren">(</span><em>mu</em>, <em>sigma</em><span class="sig-paren">)</span><a class="headerlink" href="#random.normalvariate" title="Permalink to this definition"></a></dt>
<dd><p>Normal distribution. <em>mu</em> is the mean, and <em>sigma</em> is the standard deviation.</p>
</dd></dl>
<dl class="function">
<dt id="random.vonmisesvariate">
<code class="descclassname">random.</code><code class="descname">vonmisesvariate</code><span class="sig-paren">(</span><em>mu</em>, <em>kappa</em><span class="sig-paren">)</span><a class="headerlink" href="#random.vonmisesvariate" title="Permalink to this definition"></a></dt>
<dd><p><em>mu</em> is the mean angle, expressed in radians between 0 and 2*<em>pi</em>, and <em>kappa</em>
is the concentration parameter, which must be greater than or equal to zero. If
<em>kappa</em> is equal to zero, this distribution reduces to a uniform random angle
over the range 0 to 2*<em>pi</em>.</p>
</dd></dl>
<dl class="function">
<dt id="random.paretovariate">
<code class="descclassname">random.</code><code class="descname">paretovariate</code><span class="sig-paren">(</span><em>alpha</em><span class="sig-paren">)</span><a class="headerlink" href="#random.paretovariate" title="Permalink to this definition"></a></dt>
<dd><p>Pareto distribution. <em>alpha</em> is the shape parameter.</p>
</dd></dl>
<dl class="function">
<dt id="random.weibullvariate">
<code class="descclassname">random.</code><code class="descname">weibullvariate</code><span class="sig-paren">(</span><em>alpha</em>, <em>beta</em><span class="sig-paren">)</span><a class="headerlink" href="#random.weibullvariate" title="Permalink to this definition"></a></dt>
<dd><p>Weibull distribution. <em>alpha</em> is the scale parameter and <em>beta</em> is the shape
parameter.</p>
</dd></dl>
</div>
<div class="section" id="alternative-generator">
<h2>Alternative Generator<a class="headerlink" href="#alternative-generator" title="Permalink to this headline"></a></h2>
<dl class="class">
<dt id="random.Random">
<em class="property">class </em><code class="descclassname">random.</code><code class="descname">Random</code><span class="sig-paren">(</span><span class="optional">[</span><em>seed</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.Random" title="Permalink to this definition"></a></dt>
<dd><p>Class that implements the default pseudo-random number generator used by the
<a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code></a> module.</p>
</dd></dl>
<dl class="class">
<dt id="random.SystemRandom">
<em class="property">class </em><code class="descclassname">random.</code><code class="descname">SystemRandom</code><span class="sig-paren">(</span><span class="optional">[</span><em>seed</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.SystemRandom" title="Permalink to this definition"></a></dt>
<dd><p>Class that uses the <a class="reference internal" href="os.html#os.urandom" title="os.urandom"><code class="xref py py-func docutils literal notranslate"><span class="pre">os.urandom()</span></code></a> function for generating random numbers
from sources provided by the operating system. Not available on all systems.
Does not rely on software state, and sequences are not reproducible. Accordingly,
the <a class="reference internal" href="#random.seed" title="random.seed"><code class="xref py py-meth docutils literal notranslate"><span class="pre">seed()</span></code></a> method has no effect and is ignored.
The <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">getstate()</span></code></a> and <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">setstate()</span></code></a> methods raise
<a class="reference internal" href="exceptions.html#NotImplementedError" title="NotImplementedError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">NotImplementedError</span></code></a> if called.</p>
</dd></dl>
</div>
<div class="section" id="notes-on-reproducibility">
<h2>Notes on Reproducibility<a class="headerlink" href="#notes-on-reproducibility" title="Permalink to this headline"></a></h2>
<p>Sometimes it is useful to be able to reproduce the sequences given by a pseudo
random number generator. By re-using a seed value, the same sequence should be
reproducible from run to run as long as multiple threads are not running.</p>
<p>Most of the random module’s algorithms and seeding functions are subject to
change across Python versions, but two aspects are guaranteed not to change:</p>
<ul class="simple">
<li><p>If a new seeding method is added, then a backward compatible seeder will be
offered.</p></li>
<li><p>The generator’s <code class="xref py py-meth docutils literal notranslate"><span class="pre">random()</span></code> method will continue to produce the same
sequence when the compatible seeder is given the same seed.</p></li>
</ul>
</div>
<div class="section" id="examples-and-recipes">
<span id="random-examples"></span><h2>Examples and Recipes<a class="headerlink" href="#examples-and-recipes" title="Permalink to this headline"></a></h2>
<p>Basic examples:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">random</span><span class="p">()</span> <span class="c1"># Random float: 0.0 &lt;= x &lt; 1.0</span>
<span class="go">0.37444887175646646</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">uniform</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">)</span> <span class="c1"># Random float: 2.5 &lt;= x &lt; 10.0</span>
<span class="go">3.1800146073117523</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">expovariate</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="mi">5</span><span class="p">)</span> <span class="c1"># Interval between arrivals averaging 5 seconds</span>
<span class="go">5.148957571865031</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">randrange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span> <span class="c1"># Integer from 0 to 9 inclusive</span>
<span class="go">7</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">randrange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="c1"># Even integer from 0 to 100 inclusive</span>
<span class="go">26</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">choice</span><span class="p">([</span><span class="s1">&#39;win&#39;</span><span class="p">,</span> <span class="s1">&#39;lose&#39;</span><span class="p">,</span> <span class="s1">&#39;draw&#39;</span><span class="p">])</span> <span class="c1"># Single random element from a sequence</span>
<span class="go">&#39;draw&#39;</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">deck</span> <span class="o">=</span> <span class="s1">&#39;ace two three four&#39;</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">shuffle</span><span class="p">(</span><span class="n">deck</span><span class="p">)</span> <span class="c1"># Shuffle a list</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">deck</span>
<span class="go">[&#39;four&#39;, &#39;two&#39;, &#39;ace&#39;, &#39;three&#39;]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">sample</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">50</span><span class="p">],</span> <span class="n">k</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span> <span class="c1"># Four samples without replacement</span>
<span class="go">[40, 10, 50, 30]</span>
</pre></div>
</div>
<p>Simulations:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># Six roulette wheel spins (weighted sampling with replacement)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">choices</span><span class="p">([</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="s1">&#39;green&#39;</span><span class="p">],</span> <span class="p">[</span><span class="mi">18</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">k</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span>
<span class="go">[&#39;red&#39;, &#39;green&#39;, &#39;black&#39;, &#39;black&#39;, &#39;red&#39;, &#39;black&#39;]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Deal 20 cards without replacement from a deck of 52 playing cards</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># and determine the proportion of cards with a ten-value</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># (a ten, jack, queen, or king).</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">deck</span> <span class="o">=</span> <span class="n">collections</span><span class="o">.</span><span class="n">Counter</span><span class="p">(</span><span class="n">tens</span><span class="o">=</span><span class="mi">16</span><span class="p">,</span> <span class="n">low_cards</span><span class="o">=</span><span class="mi">36</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">seen</span> <span class="o">=</span> <span class="n">sample</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">deck</span><span class="o">.</span><span class="n">elements</span><span class="p">()),</span> <span class="n">k</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">seen</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s1">&#39;tens&#39;</span><span class="p">)</span> <span class="o">/</span> <span class="mi">20</span>
<span class="go">0.15</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Estimate the probability of getting 5 or more heads from 7 spins</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># of a biased coin that settles on heads 60% of the time.</span>
<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">trial</span><span class="p">():</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">choices</span><span class="p">(</span><span class="s1">&#39;HT&#39;</span><span class="p">,</span> <span class="n">cum_weights</span><span class="o">=</span><span class="p">(</span><span class="mf">0.60</span><span class="p">,</span> <span class="mf">1.00</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s1">&#39;H&#39;</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">5</span>
<span class="gp">...</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="n">trial</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10000</span><span class="p">))</span> <span class="o">/</span> <span class="mi">10000</span>
<span class="go">0.4169</span>
<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Probability of the median of 5 samples being in middle two quartiles</span>
<span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">trial</span><span class="p">():</span>
<span class="gp">... </span> <span class="k">return</span> <span class="mi">2500</span> <span class="o">&lt;=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">choices</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">10000</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="mi">5</span><span class="p">))[</span><span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">7500</span>
<span class="gp">...</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="n">trial</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10000</span><span class="p">))</span> <span class="o">/</span> <span class="mi">10000</span>
<span class="go">0.7958</span>
</pre></div>
</div>
<p>Example of <a class="reference external" href="https://en.wikipedia.org/wiki/Bootstrapping_(statistics)">statistical bootstrapping</a> using resampling
with replacement to estimate a confidence interval for the mean of a sample of
size five:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># http://statistics.about.com/od/Applications/a/Example-Of-Bootstrapping.htm</span>
<span class="kn">from</span> <span class="nn">statistics</span> <span class="k">import</span> <span class="n">mean</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="k">import</span> <span class="n">choices</span>
<span class="n">data</span> <span class="o">=</span> <span class="mi">1</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="mi">4</span><span class="p">,</span> <span class="mi">10</span>
<span class="n">means</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">mean</span><span class="p">(</span><span class="n">choices</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">5</span><span class="p">))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;The sample mean of {mean(data):.1f} has a 90</span><span class="si">% c</span><span class="s1">onfidence &#39;</span>
<span class="n">f</span><span class="s1">&#39;interval from </span><span class="si">{means[1]:.1f}</span><span class="s1"> to </span><span class="si">{means[-2]:.1f}</span><span class="s1">&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>Example of a <a class="reference external" href="https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests">resampling permutation test</a>
to determine the statistical significance or <a class="reference external" href="https://en.wikipedia.org/wiki/P-value">p-value</a> of an observed difference
between the effects of a drug versus a placebo:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Example from &quot;Statistics is Easy&quot; by Dennis Shasha and Manda Wilson</span>
<span class="kn">from</span> <span class="nn">statistics</span> <span class="k">import</span> <span class="n">mean</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="k">import</span> <span class="n">shuffle</span>
<span class="n">drug</span> <span class="o">=</span> <span class="p">[</span><span class="mi">54</span><span class="p">,</span> <span class="mi">73</span><span class="p">,</span> <span class="mi">53</span><span class="p">,</span> <span class="mi">70</span><span class="p">,</span> <span class="mi">73</span><span class="p">,</span> <span class="mi">68</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">65</span><span class="p">,</span> <span class="mi">65</span><span class="p">]</span>
<span class="n">placebo</span> <span class="o">=</span> <span class="p">[</span><span class="mi">54</span><span class="p">,</span> <span class="mi">51</span><span class="p">,</span> <span class="mi">58</span><span class="p">,</span> <span class="mi">44</span><span class="p">,</span> <span class="mi">55</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mi">47</span><span class="p">,</span> <span class="mi">58</span><span class="p">,</span> <span class="mi">46</span><span class="p">]</span>
<span class="n">observed_diff</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">drug</span><span class="p">)</span> <span class="o">-</span> <span class="n">mean</span><span class="p">(</span><span class="n">placebo</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">combined</span> <span class="o">=</span> <span class="n">drug</span> <span class="o">+</span> <span class="n">placebo</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="n">shuffle</span><span class="p">(</span><span class="n">combined</span><span class="p">)</span>
<span class="n">new_diff</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">combined</span><span class="p">[:</span><span class="nb">len</span><span class="p">(</span><span class="n">drug</span><span class="p">)])</span> <span class="o">-</span> <span class="n">mean</span><span class="p">(</span><span class="n">combined</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">drug</span><span class="p">):])</span>
<span class="n">count</span> <span class="o">+=</span> <span class="p">(</span><span class="n">new_diff</span> <span class="o">&gt;=</span> <span class="n">observed_diff</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;</span><span class="si">{n}</span><span class="s1"> label reshufflings produced only </span><span class="si">{count}</span><span class="s1"> instances with a difference&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;at least as extreme as the observed difference of </span><span class="si">{observed_diff:.1f}</span><span class="s1">.&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;The one-sided p-value of {count / n:.4f} leads us to reject the null&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;hypothesis that there is no difference between the drug and the placebo.&#39;</span><span class="p">)</span>
</pre></div>
</div>
<p>Simulation of arrival times and service deliveries in a single server queue:</p>
<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">random</span> <span class="k">import</span> <span class="n">expovariate</span><span class="p">,</span> <span class="n">gauss</span>
<span class="kn">from</span> <span class="nn">statistics</span> <span class="k">import</span> <span class="n">mean</span><span class="p">,</span> <span class="n">median</span><span class="p">,</span> <span class="n">stdev</span>
<span class="n">average_arrival_interval</span> <span class="o">=</span> <span class="mf">5.6</span>
<span class="n">average_service_time</span> <span class="o">=</span> <span class="mf">5.0</span>
<span class="n">stdev_service_time</span> <span class="o">=</span> <span class="mf">0.5</span>
<span class="n">num_waiting</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">arrivals</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">starts</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">arrival</span> <span class="o">=</span> <span class="n">service_end</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">20000</span><span class="p">):</span>
<span class="k">if</span> <span class="n">arrival</span> <span class="o">&lt;=</span> <span class="n">service_end</span><span class="p">:</span>
<span class="n">num_waiting</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">arrival</span> <span class="o">+=</span> <span class="n">expovariate</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">average_arrival_interval</span><span class="p">)</span>
<span class="n">arrivals</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">arrival</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">num_waiting</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="n">service_start</span> <span class="o">=</span> <span class="n">service_end</span> <span class="k">if</span> <span class="n">num_waiting</span> <span class="k">else</span> <span class="n">arrival</span>
<span class="n">service_time</span> <span class="o">=</span> <span class="n">gauss</span><span class="p">(</span><span class="n">average_service_time</span><span class="p">,</span> <span class="n">stdev_service_time</span><span class="p">)</span>
<span class="n">service_end</span> <span class="o">=</span> <span class="n">service_start</span> <span class="o">+</span> <span class="n">service_time</span>
<span class="n">starts</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">service_start</span><span class="p">)</span>
<span class="n">waits</span> <span class="o">=</span> <span class="p">[</span><span class="n">start</span> <span class="o">-</span> <span class="n">arrival</span> <span class="k">for</span> <span class="n">arrival</span><span class="p">,</span> <span class="n">start</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">arrivals</span><span class="p">,</span> <span class="n">starts</span><span class="p">)]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;Mean wait: {mean(waits):.1f}. Stdev wait: {stdev(waits):.1f}.&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">f</span><span class="s1">&#39;Median wait: {median(waits):.1f}. Max wait: {max(waits):.1f}.&#39;</span><span class="p">)</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference external" href="https://www.youtube.com/watch?v=Iq9DzN6mvYA">Statistics for Hackers</a>
a video tutorial by
<a class="reference external" href="https://us.pycon.org/2016/speaker/profile/295/">Jake Vanderplas</a>
on statistical analysis using just a few fundamental concepts
including simulation, sampling, shuffling, and cross-validation.</p>
<p><a class="reference external" href="http://nbviewer.jupyter.org/url/norvig.com/ipython/Economics.ipynb">Economics Simulation</a>
a simulation of a marketplace by
<a class="reference external" href="http://norvig.com/bio.html">Peter Norvig</a> that shows effective
use of many of the tools and distributions provided by this module
(gauss, uniform, sample, betavariate, choice, triangular, and randrange).</p>
<p><a class="reference external" href="http://nbviewer.jupyter.org/url/norvig.com/ipython/Probability.ipynb">A Concrete Introduction to Probability (using Python)</a>
a tutorial by <a class="reference external" href="http://norvig.com/bio.html">Peter Norvig</a> covering
the basics of probability theory, how to write simulations, and
how to perform data analysis using Python.</p>
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<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code> — Generate pseudo-random numbers</a><ul>
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