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+  <ul>
+<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code> --- 生成伪随机数</a><ul>
+<li><a class="reference internal" href="#bookkeeping-functions">簿记功能</a></li>
+<li><a class="reference internal" href="#functions-for-integers">整数用函数</a></li>
+<li><a class="reference internal" href="#functions-for-sequences">序列用函数</a></li>
+<li><a class="reference internal" href="#real-valued-distributions">实值分布</a></li>
+<li><a class="reference internal" href="#alternative-generator">替代生成器</a></li>
+<li><a class="reference internal" href="#notes-on-reproducibility">关于再现性的说明</a></li>
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+            
+  <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> --- 生成伪随机数<a class="headerlink" href="#module-random" title="永久链接至标题">¶</a></h1>
+<p><strong>源码：</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.8/Lib/random.py">Lib/random.py</a></p>
+<hr class="docutils" />
+<p>该模块实现了各种分布的伪随机数生成器。</p>
+<p>对于整数，从范围中有统一的选择。 对于序列，存在随机元素的统一选择、用于生成列表的随机排列的函数、以及用于随机抽样而无需替换的函数。</p>
+<p>在实数轴上，有计算均匀、正态（高斯）、对数正态、负指数、伽马和贝塔分布的函数。 为了生成角度分布，可以使用 von Mises 分布。</p>
+<p>几乎所有模块函数都依赖于基本函数 <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> ，它在半开放区间 [0.0,1.0) 内均匀生成随机浮点数。 Python 使用 Mersenne Twister 作为核心生成器。 它产生 53 位精度浮点数，周期为 2**19937-1 ，其在 C 中的底层实现既快又线程安全。 Mersenne Twister 是现存最广泛测试的随机数发生器之一。 但是，因为完全确定性，它不适用于所有目的，并且完全不适合加密目的。</p>
+<p>这个模块提供的函数实际上是 <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> 类的隐藏实例的绑定方法。 你可以实例化自己的 <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> 类实例以获取不共享状态的生成器。</p>
+<p>如果你想使用自己设计的不同基础生成器，类 <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> 也可以作为子类：在这种情况下，重载 <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> 以及 <code class="xref py py-meth docutils literal notranslate"><span class="pre">setstate()</span></code> 方法。可选地，新生成器可以提供 <code class="xref py py-meth docutils literal notranslate"><span class="pre">getrandbits()</span></code> 方法——这允许 <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> 在任意大的范围内产生选择。</p>
+<p><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> 模块还提供 <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> 类，它使用系统函数 <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> 从操作系统提供的源生成随机数。</p>
+<div class="admonition warning">
+<p class="admonition-title">警告</p>
+<p>不应将此模块的伪随机生成器用于安全目的。 有关安全性或加密用途，请参阅 <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> 模块。</p>
+</div>
+<div class="admonition seealso">
+<p class="admonition-title">参见</p>
+<p>M. Matsumoto and T. Nishimura, &quot;Mersenne Twister: A 623-dimensionally
+equidistributed uniform pseudorandom number generator&quot;, 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> 用于兼容的替代随机数发生器，具有长周期和相对简单的更新操作。</p>
+</div>
+<section id="bookkeeping-functions">
+<h2>簿记功能<a class="headerlink" href="#bookkeeping-functions" title="永久链接至标题">¶</a></h2>
+<dl class="function">
+<dt id="random.seed">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">seed</code><span class="sig-paren">(</span><em class="sig-param">a=None</em>, <em class="sig-param">version=2</em><span class="sig-paren">)</span><a class="headerlink" href="#random.seed" title="永久链接至目标">¶</a></dt>
+<dd><p>初始化随机数生成器。</p>
+<p>如果 <em>a</em> 被省略或为 <code class="docutils literal notranslate"><span class="pre">None</span></code> ，则使用当前系统时间。 如果操作系统提供随机源，则使用它们而不是系统时间（有关可用性的详细信息，请参阅 <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> 函数）。</p>
+<p>如果 <em>a</em> 是 int 类型，则直接使用。</p>
+<p>对于版本2（默认的），<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> 或 <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> 对象转换为 <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> 并使用它的所有位。</p>
+<p>对于版本1（用于从旧版本的Python再现随机序列），用于 <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> 和 <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> 的算法生成更窄的种子范围。</p>
+<div class="versionchanged">
+<p><span class="versionmodified changed">在 3.2 版更改: </span>已移至版本2方案，该方案使用字符串种子中的所有位。</p>
+</div>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.getstate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">getstate</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.getstate" title="永久链接至目标">¶</a></dt>
+<dd><p>返回捕获生成器当前内部状态的对象。 这个对象可以传递给 <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> 来恢复状态。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.setstate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">setstate</code><span class="sig-paren">(</span><em class="sig-param">state</em><span class="sig-paren">)</span><a class="headerlink" href="#random.setstate" title="永久链接至目标">¶</a></dt>
+<dd><p><em>state</em> 应该是从之前调用 <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> 获得的，并且 <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> 将生成器的内部状态恢复到 <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> 被调用时的状态。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.getrandbits">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">getrandbits</code><span class="sig-paren">(</span><em class="sig-param">k</em><span class="sig-paren">)</span><a class="headerlink" href="#random.getrandbits" title="永久链接至目标">¶</a></dt>
+<dd><p>返回具有 <em>k</em> 个随机比特位的 Python 整数。 此方法随  Mersenne Twister 生成器一起提供，其他一些生成器也可能将其作为 API 的可选部分提供。 在可能的情况下，<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> 会启用 <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> 来处理任意大的区间。</p>
+</dd></dl>
+
+</section>
+<section id="functions-for-integers">
+<h2>整数用函数<a class="headerlink" href="#functions-for-integers" title="永久链接至标题">¶</a></h2>
+<dl class="function">
+<dt id="random.randrange">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">randrange</code><span class="sig-paren">(</span><em class="sig-param">stop</em><span class="sig-paren">)</span><a class="headerlink" href="#random.randrange" title="永久链接至目标">¶</a></dt>
+<dt>
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">randrange</code><span class="sig-paren">(</span><em class="sig-param">start</em>, <em class="sig-param">stop</em><span class="optional">[</span>, <em class="sig-param">step</em><span class="optional">]</span><span class="sig-paren">)</span></dt>
+<dd><p>从 <code class="docutils literal notranslate"><span class="pre">range(start,</span> <span class="pre">stop,</span> <span class="pre">step)</span></code> 返回一个随机选择的元素。 这相当于 <code class="docutils literal notranslate"><span class="pre">choice(range(start,</span> <span class="pre">stop,</span> <span class="pre">step))</span></code> ，但实际上并没有构建一个 range 对象。</p>
+<p>位置参数模式匹配 <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> 。不应使用关键字参数，因为该函数可能以意外的方式使用它们。</p>
+<div class="versionchanged">
+<p><span class="versionmodified changed">在 3.2 版更改: </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> 在生成均匀分布的值方面更为复杂。 以前它使用了像``int(random()*n)``这样的形式，它可以产生稍微不均匀的分布。</p>
+</div>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.randint">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">randint</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">b</em><span class="sig-paren">)</span><a class="headerlink" href="#random.randint" title="永久链接至目标">¶</a></dt>
+<dd><p>返回随机整数 <em>N</em> 满足 <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>。相当于 <code class="docutils literal notranslate"><span class="pre">randrange(a,</span> <span class="pre">b+1)</span></code>。</p>
+</dd></dl>
+
+</section>
+<section id="functions-for-sequences">
+<h2>序列用函数<a class="headerlink" href="#functions-for-sequences" title="永久链接至标题">¶</a></h2>
+<dl class="function">
+<dt id="random.choice">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">choice</code><span class="sig-paren">(</span><em class="sig-param">seq</em><span class="sig-paren">)</span><a class="headerlink" href="#random.choice" title="永久链接至目标">¶</a></dt>
+<dd><p>从非空序列 <em>seq</em> 返回一个随机元素。 如果 <em>seq</em> 为空，则引发 <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="sig-prename descclassname">random.</code><code class="sig-name descname">choices</code><span class="sig-paren">(</span><em class="sig-param">population</em>, <em class="sig-param">weights=None</em>, <em class="sig-param">*</em>, <em class="sig-param">cum_weights=None</em>, <em class="sig-param">k=1</em><span class="sig-paren">)</span><a class="headerlink" href="#random.choices" title="永久链接至目标">¶</a></dt>
+<dd><p>从 <em>population</em> 中有重复地随机选取元素，返回大小为 <em>k</em> 的元素列表。 如果 <em>population</em> 为空，则引发 <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>如果指定了 <em>weight</em> 序列，则根据相对权重进行选择。 或者，如果给出 <em>cum_weights</em> 序列，则根据累积权重（可能使用 <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> 计算）进行选择。 例如，相对权重``[10, 5, 30, 5]``相当于累积权重``[10, 15, 45, 50]``。 在内部，相对权重在进行选择之前会转换为累积权重，因此提供累积权重可以节省工作量。</p>
+<p>如果既未指定 <em>weight</em> 也未指定 <em>cum_weights</em> ，则以相等的概率进行选择。 如果提供了权重序列，则它必须与 <em>population</em> 序列的长度相同。 一个 <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> 指定了 <em>weights</em> 和*cum_weights*。</p>
+<p><em>weights</em> 或 <em>cum_weights</em> 可以使用任何与 <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> 所返回的 <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> 值互操作的数值类型（包括整数、浮点数和分数但不包括十进制小数）。 权重假定为非负数。</p>
+<p>对于给定的种子，具有相等加权的 <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> 函数通常产生与重复调用 <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> 不同的序列。 <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> 使用的算法使用浮点运算来实现内部一致性和速度。 <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>  使用的算法默认为重复选择的整数运算，以避免因舍入误差引起的小偏差。</p>
+<div class="versionadded">
+<p><span class="versionmodified added">3.6 新版功能.</span></p>
+</div>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.shuffle">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">shuffle</code><span class="sig-paren">(</span><em class="sig-param">x</em><span class="optional">[</span>, <em class="sig-param">random</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.shuffle" title="永久链接至目标">¶</a></dt>
+<dd><p>将序列 <em>x</em> 随机打乱位置。</p>
+<p>可选参数 <em>random</em> 是一个0参数函数，在 [0.0, 1.0) 中返回随机浮点数；默认情况下，这是函数 <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>要改变一个不可变的序列并返回一个新的打乱列表，请使用``sample(x, k=len(x))``。</p>
+<p>请注意，即使对于小的 <code class="docutils literal notranslate"><span class="pre">len(x)</span></code>，<em>x</em> 的排列总数也可以快速增长，大于大多数随机数生成器的周期。 这意味着长序列的大多数排列永远不会产生。 例如，长度为2080的序列是可以在 Mersenne Twister 随机数生成器的周期内拟合的最大序列。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.sample">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">sample</code><span class="sig-paren">(</span><em class="sig-param">population</em>, <em class="sig-param">k</em><span class="sig-paren">)</span><a class="headerlink" href="#random.sample" title="永久链接至目标">¶</a></dt>
+<dd><p>返回从总体序列或集合中选择的唯一元素的 <em>k</em> 长度列表。 用于无重复的随机抽样。</p>
+<p>返回包含来自总体的元素的新列表，同时保持原始总体不变。 结果列表按选择顺序排列，因此所有子切片也将是有效的随机样本。 这允许抽奖获奖者（样本）被划分为大奖和第二名获胜者（子切片）。</p>
+<p>总体成员不必是 <a class="reference internal" href="../glossary.html#term-hashable"><span class="xref std std-term">hashable</span></a> 或 unique 。 如果总体包含重复，则每次出现都是样本中可能的选择。</p>
+<p>要从一系列整数中选择样本，请使用 <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> 对象作为参数。 对于从大量人群中采样，这种方法特别快速且节省空间：<code class="docutils literal notranslate"><span class="pre">sample(range(10000000),</span> <span class="pre">k=60)</span></code> 。</p>
+<p>如果样本大小大于总体大小，则引发 <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> 。</p>
+</dd></dl>
+
+</section>
+<section id="real-valued-distributions">
+<h2>实值分布<a class="headerlink" href="#real-valued-distributions" title="永久链接至标题">¶</a></h2>
+<p>以下函数生成特定的实值分布。如常用数学实践中所使用的那样, 函数参数以分布方程中的相应变量命名;大多数这些方程都可以在任何统计学教材中找到。</p>
+<dl class="function">
+<dt id="random.random">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">random</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.random" title="永久链接至目标">¶</a></dt>
+<dd><p>返回 [0.0, 1.0) 范围内的下一个随机浮点数。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.uniform">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">uniform</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">b</em><span class="sig-paren">)</span><a class="headerlink" href="#random.uniform" title="永久链接至目标">¶</a></dt>
+<dd><p>返回一个随机浮点数 <em>N</em> ，当 <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&lt;=</span> <span class="pre">b</span></code> 时 <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> ，当 <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">&lt;</span> <span class="pre">a</span></code> 时 <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> 。</p>
+<p>取决于等式 <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> 中的浮点舍入，终点 <code class="docutils literal notranslate"><span class="pre">b</span></code> 可以包括或不包括在该范围内。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.triangular">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">triangular</code><span class="sig-paren">(</span><em class="sig-param">low</em>, <em class="sig-param">high</em>, <em class="sig-param">mode</em><span class="sig-paren">)</span><a class="headerlink" href="#random.triangular" title="永久链接至目标">¶</a></dt>
+<dd><p>返回一个随机浮点数 <em>N</em> ，使得 <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> 并在这些边界之间使用指定的 <em>mode</em> 。 <em>low</em> 和 <em>high</em> 边界默认为零和一。 <em>mode</em> 参数默认为边界之间的中点，给出对称分布。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.betavariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">betavariate</code><span class="sig-paren">(</span><em class="sig-param">alpha</em>, <em class="sig-param">beta</em><span class="sig-paren">)</span><a class="headerlink" href="#random.betavariate" title="永久链接至目标">¶</a></dt>
+<dd><p>Beta 分布。 参数的条件是 <code class="docutils literal notranslate"><span class="pre">alpha</span> <span class="pre">&gt;</span> <span class="pre">0</span></code> 和 <code class="docutils literal notranslate"><span class="pre">beta</span> <span class="pre">&gt;</span> <span class="pre">0</span></code>。 返回值的范围介于 0 和 1 之间。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.expovariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">expovariate</code><span class="sig-paren">(</span><em class="sig-param">lambd</em><span class="sig-paren">)</span><a class="headerlink" href="#random.expovariate" title="永久链接至目标">¶</a></dt>
+<dd><p>指数分布。 <em>lambd</em> 是 1.0 除以所需的平均值，它应该是非零的。 （该参数本应命名为 “lambda” ，但这是 Python 中的保留字。）如果 <em>lambd</em> 为正，则返回值的范围为 0 到正无穷大；如果 <em>lambd</em> 为负，则返回值从负无穷大到 0。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.gammavariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">gammavariate</code><span class="sig-paren">(</span><em class="sig-param">alpha</em>, <em class="sig-param">beta</em><span class="sig-paren">)</span><a class="headerlink" href="#random.gammavariate" title="永久链接至目标">¶</a></dt>
+<dd><p>Gamma 分布。 （ <em>不是</em> gamma 函数！ ） 参数的条件是 <code class="docutils literal notranslate"><span class="pre">alpha</span> <span class="pre">&gt;</span> <span class="pre">0</span></code> 和 <code class="docutils literal notranslate"><span class="pre">beta</span> <span class="pre">&gt;</span> <span class="pre">0</span></code>。</p>
+<p>概率分布函数是:</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="sig-prename descclassname">random.</code><code class="sig-name descname">gauss</code><span class="sig-paren">(</span><em class="sig-param">mu</em>, <em class="sig-param">sigma</em><span class="sig-paren">)</span><a class="headerlink" href="#random.gauss" title="永久链接至目标">¶</a></dt>
+<dd><p>高斯分布。 <em>mu</em> 是平均值，<em>sigma</em> 是标准差。 这比下面定义的 <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> 函数略快。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.lognormvariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">lognormvariate</code><span class="sig-paren">(</span><em class="sig-param">mu</em>, <em class="sig-param">sigma</em><span class="sig-paren">)</span><a class="headerlink" href="#random.lognormvariate" title="永久链接至目标">¶</a></dt>
+<dd><p>对数正态分布。 如果你采用这个分布的自然对数，你将得到一个正态分布，平均值为 <em>mu</em> 和标准差为 <em>sigma</em> 。 <em>mu</em> 可以是任何值，<em>sigma</em> 必须大于零。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.normalvariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">normalvariate</code><span class="sig-paren">(</span><em class="sig-param">mu</em>, <em class="sig-param">sigma</em><span class="sig-paren">)</span><a class="headerlink" href="#random.normalvariate" title="永久链接至目标">¶</a></dt>
+<dd><p>正态分布。 <em>mu</em> 是平均值，<em>sigma</em> 是标准差。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.vonmisesvariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">vonmisesvariate</code><span class="sig-paren">(</span><em class="sig-param">mu</em>, <em class="sig-param">kappa</em><span class="sig-paren">)</span><a class="headerlink" href="#random.vonmisesvariate" title="永久链接至目标">¶</a></dt>
+<dd><p>冯·米塞斯分布。 <em>mu</em> 是平均角度，以弧度表示，介于0和 2*<em>pi</em> 之间，<em>kappa</em> 是浓度参数，必须大于或等于零。 如果 <em>kappa</em> 等于零，则该分布在 0 到 2*<em>pi</em> 的范围内减小到均匀的随机角度。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.paretovariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">paretovariate</code><span class="sig-paren">(</span><em class="sig-param">alpha</em><span class="sig-paren">)</span><a class="headerlink" href="#random.paretovariate" title="永久链接至目标">¶</a></dt>
+<dd><p>帕累托分布。 <em>alpha</em> 是形状参数。</p>
+</dd></dl>
+
+<dl class="function">
+<dt id="random.weibullvariate">
+<code class="sig-prename descclassname">random.</code><code class="sig-name descname">weibullvariate</code><span class="sig-paren">(</span><em class="sig-param">alpha</em>, <em class="sig-param">beta</em><span class="sig-paren">)</span><a class="headerlink" href="#random.weibullvariate" title="永久链接至目标">¶</a></dt>
+<dd><p>威布尔分布。 <em>alpha</em> 是比例参数，<em>beta</em> 是形状参数。</p>
+</dd></dl>
+
+</section>
+<section id="alternative-generator">
+<h2>替代生成器<a class="headerlink" href="#alternative-generator" title="永久链接至标题">¶</a></h2>
+<dl class="class">
+<dt id="random.Random">
+<em class="property">class </em><code class="sig-prename descclassname">random.</code><code class="sig-name descname">Random</code><span class="sig-paren">(</span><span class="optional">[</span><em class="sig-param">seed</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.Random" title="永久链接至目标">¶</a></dt>
+<dd><p>该类实现了 <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> 模块所用的默认伪随机数生成器。</p>
+</dd></dl>
+
+<dl class="class">
+<dt id="random.SystemRandom">
+<em class="property">class </em><code class="sig-prename descclassname">random.</code><code class="sig-name descname">SystemRandom</code><span class="sig-paren">(</span><span class="optional">[</span><em class="sig-param">seed</em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.SystemRandom" title="永久链接至目标">¶</a></dt>
+<dd><p>使用 <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> 函数的类，用从操作系统提供的源生成随机数。 这并非适用于所有系统。 也不依赖于软件状态，序列不可重现。 因此，<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> 方法没有效果而被忽略。 <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> 和 <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> 方法如果被调用则引发 <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>。</p>
+</dd></dl>
+
+</section>
+<section id="notes-on-reproducibility">
+<h2>关于再现性的说明<a class="headerlink" href="#notes-on-reproducibility" title="永久链接至标题">¶</a></h2>
+<p>有时能够重现伪随机数生成器给出的序列是有用的。 通过重新使用种子值，只要多个线程没有运行，相同的序列就可以在两次不同运行之间重现。</p>
+<p>大多数随机模块的算法和种子函数都会在 Python 版本中发生变化，但保证两个方面不会改变：</p>
+<ul class="simple">
+<li><p>如果添加了新的播种方法，则将提供向后兼容的播种机。</p></li>
+<li><p>当兼容的播种机被赋予相同的种子时，生成器的 <code class="xref py py-meth docutils literal notranslate"><span class="pre">random()</span></code> 方法将继续产生相同的序列。</p></li>
+</ul>
+</section>
+<section id="examples-and-recipes">
+<span id="random-examples"></span><h2>例子和配方<a class="headerlink" href="#examples-and-recipes" title="永久链接至标题">¶</a></h2>
+<p>基本示例:</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>模拟:</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">10_000</span><span class="p">))</span> <span class="o">/</span> <span class="mi">10_000</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">2_500</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">10_000</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">7_500</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">10_000</span><span class="p">))</span> <span class="o">/</span> <span class="mi">10_000</span>
+<span class="go">0.7958</span>
+</pre></div>
+</div>
+<p><a class="reference external" href="https://en.wikipedia.org/wiki/Bootstrapping_(statistics)">statistical bootstrapping</a> 的示例，使用重新采样和替换来估计一个样本的均值的置信区间:</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="kn">import</span> <span class="n">fmean</span> <span class="k">as</span> <span class="n">mean</span>
+<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">choices</span>
+
+<span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mi">41</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">29</span><span class="p">,</span> <span class="mi">37</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">73</span><span class="p">,</span> <span class="mi">63</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">35</span><span class="p">,</span> <span class="mi">68</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">95</span><span class="p">]</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="nb">len</span><span class="p">(</span><span class="n">data</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">100</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;The sample mean of </span><span class="si">{</span><span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1"> has a 90% confidence &#39;</span>
+      <span class="sa">f</span><span class="s1">&#39;interval from </span><span class="si">{</span><span class="n">means</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1"> to </span><span class="si">{</span><span class="n">means</span><span class="p">[</span><span class="mi">94</span><span class="p">]</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>使用 <a class="reference external" href="https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests">重新采样排列测试</a> 来确定统计学显著性或者使用  <a class="reference external" href="https://en.wikipedia.org/wiki/P-value">p-值</a> 来观察药物与安慰剂的作用之间差异的示例:</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="kn">import</span> <span class="n">fmean</span> <span class="k">as</span> <span class="n">mean</span>
+<span class="kn">from</span> <span class="nn">random</span> <span class="kn">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">10_000</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="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s1"> label reshufflings produced only </span><span class="si">{</span><span class="n">count</span><span class="si">}</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="sa">f</span><span class="s1">&#39;at least as extreme as the observed difference of </span><span class="si">{</span><span class="n">observed_diff</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">.&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;The one-sided p-value of </span><span class="si">{</span><span class="n">count</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">n</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1"> leads us to reject the null&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">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>多服务器队列的到达时间和服务交付模拟:</p>
+<div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">heapq</span> <span class="kn">import</span> <span class="n">heappush</span><span class="p">,</span> <span class="n">heappop</span>
+<span class="kn">from</span> <span class="nn">random</span> <span class="kn">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="kn">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">15.0</span>
+<span class="n">stdev_service_time</span> <span class="o">=</span> <span class="mf">3.5</span>
+<span class="n">num_servers</span> <span class="o">=</span> <span class="mi">3</span>
+
+<span class="n">waits</span> <span class="o">=</span> <span class="p">[]</span>
+<span class="n">arrival_time</span> <span class="o">=</span> <span class="mf">0.0</span>
+<span class="n">servers</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">]</span> <span class="o">*</span> <span class="n">num_servers</span>  <span class="c1"># time when each server becomes available</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">100_000</span><span class="p">):</span>
+    <span class="n">arrival_time</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">next_server_available</span> <span class="o">=</span> <span class="n">heappop</span><span class="p">(</span><span class="n">servers</span><span class="p">)</span>
+    <span class="n">wait</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">next_server_available</span> <span class="o">-</span> <span class="n">arrival_time</span><span class="p">)</span>
+    <span class="n">waits</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">wait</span><span class="p">)</span>
+    <span class="n">service_duration</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_completed</span> <span class="o">=</span> <span class="n">arrival_time</span> <span class="o">+</span> <span class="n">wait</span> <span class="o">+</span> <span class="n">service_duration</span>
+    <span class="n">heappush</span><span class="p">(</span><span class="n">servers</span><span class="p">,</span> <span class="n">service_completed</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Mean wait: </span><span class="si">{</span><span class="n">mean</span><span class="p">(</span><span class="n">waits</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">.  Stdev wait: </span><span class="si">{</span><span class="n">stdev</span><span class="p">(</span><span class="n">waits</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">.&#39;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Median wait: </span><span class="si">{</span><span class="n">median</span><span class="p">(</span><span class="n">waits</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">.  Max wait: </span><span class="si">{</span><span class="nb">max</span><span class="p">(</span><span class="n">waits</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">.&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="admonition seealso">
+<p class="admonition-title">参见</p>
+<p><a class="reference external" href="https://www.youtube.com/watch?v=Iq9DzN6mvYA">Statistics for Hackers</a> <a class="reference external" href="https://us.pycon.org/2016/speaker/profile/295/">Jake Vanderplas</a> 撰写的视频教程，使用一些基本概念进行统计分析，包括模拟、抽样、改组和交叉验证。</p>
+<p><a class="reference external" href="http://nbviewer.jupyter.org/url/norvig.com/ipython/Economics.ipynb">Economics Simulation</a> <a class="reference external" href="http://norvig.com/bio.html">Peter Norvig</a> 编写的市场模拟，显示了该模块提供的许多工具和分布的有效使用（高斯、均匀、样本、beta变量、选择、三角和随机范围等）。</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 class="reference external" href="http://norvig.com/bio.html">Peter Norvig</a> 撰写的教程，涵盖了概率论基础知识，如何编写模拟，以及如何使用 Python 进行数据分析。</p>
+</div>
+</section>
+</section>
+
+
+          </div>
+        </div>
+      </div>
+      <div class="sphinxsidebar" role="navigation" aria-label="main navigation">
+        <div class="sphinxsidebarwrapper">
+  <h3><a href="../contents.html">目录</a></h3>
+  <ul>
+<li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code> --- 生成伪随机数</a><ul>
+<li><a class="reference internal" href="#bookkeeping-functions">簿记功能</a></li>
+<li><a class="reference internal" href="#functions-for-integers">整数用函数</a></li>
+<li><a class="reference internal" href="#functions-for-sequences">序列用函数</a></li>
+<li><a class="reference internal" href="#real-valued-distributions">实值分布</a></li>
+<li><a class="reference internal" href="#alternative-generator">替代生成器</a></li>
+<li><a class="reference internal" href="#notes-on-reproducibility">关于再现性的说明</a></li>
+<li><a class="reference internal" href="#examples-and-recipes">例子和配方</a></li>
+</ul>
+</li>
+</ul>
+
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