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Numeric and Mathematical Modules</a> &#187;</li> <li class="right"> <div class="inline-search" style="display: none" role="search"> <form class="inline-search" action="../search.html" method="get"> <input placeholder="Quick search" type="text" name="q" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> <script type="text/javascript">$('.inline-search').show(0);</script> | </li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body" role="main"> <div class="section" id="module-random"> <span id="random-generate-pseudo-random-numbers"></span><h1>9.6. <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/3.6/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 <code class="xref py py-class docutils literal notranslate"><span class="pre">random.Random</span></code> class. You can instantiate your own instances of <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> to get generators that don’t share state.</p> <p>Class <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> 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="first admonition-title">Warning</p> <p class="last">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="first 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 class="last"><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>9.6.1. 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 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>9.6.2. 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 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>9.6.3. 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> <div class="versionadded"> <p><span class="versionmodified">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>9.6.4. 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>9.6.5. Alternative Generator<a class="headerlink" href="#alternative-generator" title="Permalink to this headline">¶</a></h2> <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>9.6.6. 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>If a new seeding method is added, then a backward compatible seeder will be offered.</li> <li>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.</li> </ul> </div> <div class="section" id="examples-and-recipes"> <span id="random-examples"></span><h2>9.6.7. 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="n">trial</span> <span class="o">=</span> <span class="k">lambda</span><span class="p">:</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">&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="n">trial</span> <span class="o">=</span> <span class="k">lambda</span> <span class="p">:</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">&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="first 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 class="last"><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> </div> </div> </div> </div> </div> </div> <div class="sphinxsidebar" role="navigation" aria-label="main navigation"> <div class="sphinxsidebarwrapper"> <h3><a href="../contents.html">Table Of Contents</a></h3> <ul> <li><a class="reference internal" href="#">9.6. <code class="docutils literal notranslate"><span class="pre">random</span></code> — Generate pseudo-random numbers</a><ul> <li><a class="reference internal" href="#bookkeeping-functions">9.6.1. Bookkeeping functions</a></li> <li><a class="reference internal" href="#functions-for-integers">9.6.2. Functions for integers</a></li> <li><a class="reference internal" href="#functions-for-sequences">9.6.3. Functions for sequences</a></li> <li><a class="reference internal" href="#real-valued-distributions">9.6.4. Real-valued distributions</a></li> <li><a class="reference internal" href="#alternative-generator">9.6.5. Alternative Generator</a></li> <li><a class="reference internal" href="#notes-on-reproducibility">9.6.6. Notes on Reproducibility</a></li> <li><a class="reference internal" href="#examples-and-recipes">9.6.7. 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