321 lines
26 KiB
HTML
321 lines
26 KiB
HTML
<!DOCTYPE html>
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<html lang="zh" dir="ltr" class="client-nojs" xmlns="http://www.w3.org/1999/xhtml" xml:lang="zh">
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<head><meta http-equiv="x-ua-compatible" content="ie=edge">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<title>std::student_t_distribution</title>
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<link rel="stylesheet" href="ext.css" />
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<link rel="stylesheet" href="site_modules.css" />
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_random_student_t_distribution skin-cppreference2 action-view cpp-navbar">
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<div id="cpp-content-base">
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<div id="content"><a id="top"></a>
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>student_t_distribution</h1>
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<div id="bodyContent">
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<div id="contentSub"><span class="subpages">< <a href="cpp.html">cpp</a>‎ | <a href="cpp-numeric.html">numeric</a>‎ | <a href="cpp-numeric-random.html">random</a></span></div>
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<div id="mw-content-text" lang="zh" dir="ltr" class="mw-content-ltr" xml:lang="zh">
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<table class="t-dcl-begin">
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<tbody>
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<tr class="t-dsc-header">
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<td>
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<div>定义于头文件 <code><a href="cpp-header-random.html"><random></a></code></div>
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</td>
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<td></td>
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<td></td>
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<tr class="t-dcl t-since-cxx11">
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<div><span class="mw-geshi cpp source-cpp"><span class="kw1">template</span><span class="sy1"><</span> <span class="kw1">class</span> RealType <span class="sy1">=</span> <span class="kw4">double</span> <span class="sy1">></span><br />
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<span class="kw1">class</span> student_t_distribution<span class="sy4">;</span></span></div>
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</td>
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<td class="t-dcl-nopad"></td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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<tr class="t-dcl-sep">
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<td></td>
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<td></td>
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<td></td>
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</tr>
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</tbody>
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</table>
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<p>生成随机浮点值 <span class="texhtml" style="white-space: nowrap;">x</span> ，分布服从概率密度函数：</p>
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<dl>
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<dd><span class="mjax" style="display:none">\(p(x|n) = \frac{1}{\sqrt{n\pi} } \cdot \frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})} \cdot (1+\frac{x^2}{n})^{-\frac{n+1}{2} } \)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">p(x|n) = <span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>1</td>
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</tr>
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<tr>
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<td><span class="t-mrad"><span>√</span><span>nπ</span></span></td>
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</table>
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· <span class="t-mfrac"></span>
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<table>
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<tr>
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<td>Γ(<span class="t-mfrac"></span>
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<td>n+1</td>
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<td>2</td>
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</tr>
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</table>
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)</td>
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</tr>
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<tr>
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<td>Γ(<span class="t-mfrac"></span>
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<table>
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<tr>
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<td>n</td>
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<tr>
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<td>2</td>
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</tr>
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</table>
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)</td>
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</tr>
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</table>
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· <span class="t-mparen"><span style="font-size: 1.5em">⎛<br />
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⎜<br />
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⎝</span><span style="">1+<span class="t-mfrac"></span></span></span>
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<table>
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<tr>
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<td>x<span class="t-su">2<br /></span></td>
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<td>n</td>
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</table>
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<span style="font-size: 1.5em">⎞<br />
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⎟<br />
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⎠</span> <span class="t-su">-<span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>n+1</td>
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</tr>
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<tr>
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<td>2</td>
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</tr>
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</table>
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<br /></dd>
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</dl>
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<p>其中 <span class="texhtml" style="white-space: nowrap;">n</span> 为<i>自由度</i>数。在给定 <span class="texhtml" style="white-space: nowrap;">n+1</span> 个独立测量，每个都带标准差未知的加性误差，如在物理测量中估计未知正态分布值的<i>平均数</i>时使用此分布。或者，在给定 <span class="texhtml" style="white-space: nowrap;">n+1</span> 个样本，估计拥有未知标准差的正态分布的未知平均值时，使用此分布。</p>
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<p><code>std::student_t_distribution</code> 满足<a href="cpp-named_req-RandomNumberDistribution.html"><span style="font-style:italic">随机数分布</span> <span style="font-family: Georgia, 'DejaVu Serif', serif; font-style:italic">(RandomNumberDistribution)</span></a> 的所有要求。</p>
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<h3><span class="mw-headline" id=".E6.A8.A1.E6.9D.BF.E5.BD.A2.E5.8F.82">模板形参</span></h3>
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<table class="t-par-begin">
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<tr class="t-par">
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<td>RealType</td>
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<td>-</td>
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<td>生成器所生成的结果类型。若它不是 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">float</span></span></span> 、 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 或 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> 之一则效果未定义。</td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E6.88.90.E5.91.98.E7.B1.BB.E5.9E.8B">成员类型</span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc-hitem">
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<td>成员类型</td>
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<td>定义</td>
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</tr>
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<tr class="t-dsc">
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<td><code>result_type</code></td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">RealType</span></span></td>
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</tr>
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<tr class="t-dsc">
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<td><code>param_type</code><span class="t-mark-rev t-since-cxx11">(C++11)</span></td>
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<td>参数集的类型，见<a href="cpp-named_req-RandomNumberDistribution.html"><span style="font-style:italic">随机数分布</span> <span style="font-family: Georgia, 'DejaVu Serif', serif; font-style:italic">(RandomNumberDistribution)</span></a> 。</td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E6.88.90.E5.91.98.E5.87.BD.E6.95.B0">成员函数</span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div t-dsc-member-nobold-div">
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<div><a href="cpp-numeric-random-student_t_distribution-student_t_distribution.html"><span class="t-lines"><span>(构造函数)</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>构造新分布<br />
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<span class="t-mark">(公开成员函数)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-reset.html"><span class="t-lines"><span>reset</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>重置分布的内部状态<br />
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<span class="t-mark">(公开成员函数)</span></td>
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</tr>
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<tr>
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<td colspan="2">
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<h5><span class="mw-headline" id=".E7.94.9F.E6.88.90">生成</span></h5>
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</td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-operator().html"><span class="t-lines"><span>operator()</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>生成分布中的下个随机数<br />
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<span class="t-mark">(公开成员函数)</span></td>
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</tr>
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<tr>
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<td colspan="2">
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<h5><span class="mw-headline" id=".E7.89.B9.E5.BE.81">特征</span></h5>
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</td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-n.html"><span class="t-lines"><span>n</span></span></a></div>
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</div>
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</td>
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<td>返回 <i>n</i> 分布参数（自由度）<br />
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<span class="t-mark">(公开成员函数)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-param.html"><span class="t-lines"><span>param</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>获取或设置随机参数对象<br />
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<span class="t-mark">(公开成员函数)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-min.html"><span class="t-lines"><span>min</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>返回最小的潜在生成值<br />
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<span class="t-mark">(公开成员函数)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-max.html"><span class="t-lines"><span>max</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>返回最大的潜在生成值<br />
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<span class="t-mark">(公开成员函数)</span></td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E9.9D.9E.E6.88.90.E5.91.98.E5.87.BD.E6.95.B0">非成员函数</span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-operator_cmp.html"><span class="t-lines"><span>operator==</span><span>operator!=</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span><span class="t-mark-rev t-until-cxx20">(C++20 中移除)</span></span></span></div>
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</div>
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</td>
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<td>比较两个分布对象<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><a href="cpp-numeric-random-student_t_distribution-operator_ltltgtgt.html"><span class="t-lines"><span>operator<<</span><span>operator>></span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>执行伪随机数分布的流输入和输出<br />
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<span class="t-mark">(函数模板)</span></td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">示例</span></h3>
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<div class="t-example">
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<div class="t-example-live-link"></div>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="cpp source-cpp">
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<pre class="de1"><span class="co2">#include <algorithm></span>
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<span class="co2">#include <cmath></span>
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<span class="co2">#include <iomanip></span>
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<span class="co2">#include <iostream></span>
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<span class="co2">#include <map></span>
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<span class="co2">#include <random></span>
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<span class="co2">#include <vector></span>
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<span class="kw1">template</span> <span class="sy1"><</span><span class="kw4">int</span> Height <span class="sy1">=</span> <span class="nu0">5</span>, <span class="kw4">int</span> BarWidth <span class="sy1">=</span> <span class="nu0">1</span>, <span class="kw4">int</span> Padding <span class="sy1">=</span> <span class="nu0">1</span>, <span class="kw4">int</span> Offset <span class="sy1">=</span> <span class="nu0">0</span>,
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<span class="kw4">bool</span> DrawMinMax <span class="sy1">=</span> <span class="kw2">true</span>, <span class="kw1">class</span> Sample<span class="sy1">></span>
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<span class="kw4">void</span> draw_vbars<span class="br0">(</span>Sample <span class="kw4">const</span><span class="sy3">&</span> s<span class="br0">)</span> <span class="br0">{</span>
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static_assert<span class="br0">(</span><span class="br0">(</span>Height <span class="sy1">></span> <span class="nu0">0</span><span class="br0">)</span> <span class="sy3">&&</span> <span class="br0">(</span>BarWidth <span class="sy1">></span> <span class="nu0">0</span><span class="br0">)</span> <span class="sy3">&&</span> <span class="br0">(</span>Padding <span class="sy1">>=</span> <span class="nu0">0</span><span class="br0">)</span> <span class="sy3">&&</span> <span class="br0">(</span>Offset <span class="sy1">>=</span> <span class="nu0">0</span><span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="kw4">auto</span> cout_n <span class="sy1">=</span> <span class="br0">[</span><span class="br0">]</span><span class="br0">(</span><span class="kw4">auto</span> <span class="kw4">const</span><span class="sy3">&</span> v, <span class="kw4">int</span> n<span class="br0">)</span> <span class="br0">{</span> <span class="kw1">while</span> <span class="br0">(</span>n<span class="sy2">--</span> <span class="sy1">></span> <span class="nu0">0</span><span class="br0">)</span> <span class="br0">{</span> <a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> v<span class="sy4">;</span> <span class="br0">}</span> <span class="br0">}</span><span class="sy4">;</span>
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<span class="kw4">const</span> <span class="kw4">auto</span> <span class="br0">[</span>min, max<span class="br0">]</span> <span class="sy1">=</span> <a href="cpp-algorithm-minmax_element.html"><span class="kw1644">std::<span class="me2">minmax_element</span></span></a><span class="br0">(</span><a href="cpp-iterator-begin.html"><span class="kw677">std::<span class="me2">cbegin</span></span></a><span class="br0">(</span>s<span class="br0">)</span>, <a href="cpp-iterator-end.html"><span class="kw681">std::<span class="me2">cend</span></span></a><span class="br0">(</span>s<span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-container-vector.html"><span class="kw1269">std::<span class="me2">vector</span></span></a><span class="sy1"><</span><a href="cpp-numeric-math-div.html"><span class="kw1307">std::<span class="me2">div_t</span></span></a><span class="sy1">></span> qr<span class="sy4">;</span>
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<span class="kw1">for</span> <span class="br0">(</span><span class="kw4">float</span> e <span class="sy4">:</span> s<span class="br0">)</span> <span class="br0">{</span>
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qr.<span class="me1">push_back</span><span class="br0">(</span><a href="cpp-numeric-math-div.html"><span class="kw1303">std::<span class="me2">div</span></span></a><span class="br0">(</span>std<span class="sy4">::</span><span class="me2">lerp</span><span class="br0">(</span><span class="nu0">0</span>.<span class="me1">f</span>, Height<span class="sy2">*</span><span class="nu0">8</span>, <span class="br0">(</span>e <span class="sy2">-</span> <span class="sy2">*</span>min<span class="br0">)</span><span class="sy2">/</span><span class="br0">(</span><span class="sy2">*</span>max <span class="sy2">-</span> <span class="sy2">*</span>min<span class="br0">)</span><span class="br0">)</span>, <span class="nu0">8</span><span class="br0">)</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="br0">}</span>
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<span class="kw1">for</span> <span class="br0">(</span><span class="kw4">auto</span> h<span class="br0">{</span>Height<span class="br0">}</span><span class="sy4">;</span> h<span class="sy2">--</span> <span class="sy1">></span> <span class="nu0">0</span> <span class="sy4">;</span><span class="br0">)</span> <span class="br0">{</span>
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cout_n<span class="br0">(</span><span class="st0">' '</span>, Offset<span class="br0">)</span><span class="sy4">;</span>
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<span class="kw1">for</span> <span class="br0">(</span><span class="kw4">auto</span> <span class="br0">[</span>q, r<span class="br0">]</span> <span class="sy4">:</span> qr<span class="br0">)</span> <span class="br0">{</span>
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<span class="kw4">char</span> d<span class="br0">[</span><span class="br0">]</span> <span class="sy1">=</span> <span class="st0">"█"</span><span class="sy4">;</span> <span class="co1">// == { 0xe2, 0x96, 0x88, 0 }</span>
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q <span class="sy1"><</span> h <span class="sy4">?</span> d<span class="br0">[</span><span class="nu0">0</span><span class="br0">]</span> <span class="sy1">=</span> <span class="st0">' '</span>, d<span class="br0">[</span><span class="nu0">1</span><span class="br0">]</span> <span class="sy1">=</span> <span class="st0">'<span class="es5">\0</span>'</span> <span class="sy4">:</span> q <span class="sy1">==</span> h <span class="sy4">?</span> d<span class="br0">[</span><span class="nu0">2</span><span class="br0">]</span> <span class="sy2">-</span><span class="sy1">=</span> <span class="br0">(</span><span class="nu0">7</span> <span class="sy2">-</span> r<span class="br0">)</span> <span class="sy4">:</span> <span class="nu0">0</span><span class="sy4">;</span>
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cout_n<span class="br0">(</span>d, BarWidth<span class="br0">)</span><span class="sy4">;</span>
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cout_n<span class="br0">(</span><span class="st0">' '</span>, Padding<span class="br0">)</span><span class="sy4">;</span>
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<span class="br0">}</span>
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<span class="kw1">if</span> <span class="br0">(</span>DrawMinMax <span class="sy3">&&</span> Height <span class="sy1">></span> <span class="nu0">1</span><span class="br0">)</span>
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h <span class="sy1">==</span> Height <span class="sy2">-</span> <span class="nu0">1</span> <span class="sy4">?</span> <a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"┬ "</span> <span class="sy1"><<</span> <span class="sy2">*</span>max<span class="sy4">:</span>
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h <span class="sy3">!</span><span class="sy1">=</span> <span class="nu0">0</span> <span class="sy4">?</span> <a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"│"</span>
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<span class="sy4">:</span> <a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"┴ "</span> <span class="sy1"><<</span> <span class="sy2">*</span>min<span class="sy4">;</span>
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cout_n<span class="br0">(</span><span class="st0">'<span class="es1">\n</span>'</span>, <span class="nu0">1</span><span class="br0">)</span><span class="sy4">;</span>
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<span class="br0">}</span>
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<span class="br0">}</span>
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<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span> <span class="br0">{</span>
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<a href="cpp-numeric-random-random_device.html"><span class="kw1486">std::<span class="me2">random_device</span></span></a> rd<span class="br0">{</span><span class="br0">}</span><span class="sy4">;</span>
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<a href="cpp-numeric-random-mersenne_twister_engine.html"><span class="kw1489">std::<span class="me2">mt19937</span></span></a> gen<span class="br0">{</span>rd<span class="br0">(</span><span class="br0">)</span><span class="br0">}</span><span class="sy4">;</span>
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std<span class="sy4">::</span><span class="me2">student_t_distribution</span><span class="sy1"><></span> d<span class="br0">{</span><span class="nu17">10.0f</span><span class="br0">}</span><span class="sy4">;</span>
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<span class="kw4">const</span> <span class="kw4">int</span> norm <span class="sy1">=</span> <span class="nu0">10</span><span class="st0">'000;
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const float cutoff = 0.000'</span><span class="nu17">3f</span><span class="sy4">;</span>
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<a href="cpp-container-map.html"><span class="kw1281">std::<span class="me2">map</span></span></a><span class="sy1"><</span><span class="kw4">int</span>, <span class="kw4">int</span><span class="sy1">></span> hist<span class="br0">{</span><span class="br0">}</span><span class="sy4">;</span>
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<span class="kw1">for</span><span class="br0">(</span><span class="kw4">int</span> n<span class="sy1">=</span><span class="nu0">0</span><span class="sy4">;</span> n<span class="sy1"><</span>norm<span class="sy4">;</span> <span class="sy2">++</span>n<span class="br0">)</span> <span class="br0">{</span> <span class="sy2">++</span>hist<span class="br0">[</span><a href="cpp-numeric-math-round.html"><span class="kw1353">std::<span class="me2">round</span></span></a><span class="br0">(</span>d<span class="br0">(</span>gen<span class="br0">)</span><span class="br0">)</span><span class="br0">]</span><span class="sy4">;</span> <span class="br0">}</span>
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<a href="cpp-container-vector.html"><span class="kw1269">std::<span class="me2">vector</span></span></a><span class="sy1"><</span><span class="kw4">float</span><span class="sy1">></span> bars<span class="sy4">;</span>
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<a href="cpp-container-vector.html"><span class="kw1269">std::<span class="me2">vector</span></span></a><span class="sy1"><</span><span class="kw4">int</span><span class="sy1">></span> indices<span class="sy4">;</span>
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<span class="kw1">for</span> <span class="br0">(</span><span class="kw4">const</span> <span class="kw4">auto</span> <span class="br0">[</span>n, p<span class="br0">]</span> <span class="sy4">:</span> hist<span class="br0">)</span> <span class="br0">{</span>
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<span class="kw1">if</span> <span class="br0">(</span><span class="kw4">float</span> x <span class="sy1">=</span> p <span class="sy2">*</span> <span class="br0">(</span><span class="nu17">1.0f</span> <span class="sy2">/</span> norm<span class="br0">)</span><span class="sy4">;</span> cutoff <span class="sy1"><</span> x<span class="br0">)</span> <span class="br0">{</span>
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bars.<span class="me1">push_back</span><span class="br0">(</span>x<span class="br0">)</span><span class="sy4">;</span>
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indices.<span class="me1">push_back</span><span class="br0">(</span>n<span class="br0">)</span><span class="sy4">;</span>
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<span class="br0">}</span>
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<span class="br0">}</span>
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draw_vbars<span class="sy1"><</span><span class="nu0">8</span>,<span class="nu0">5</span><span class="sy1">></span><span class="br0">(</span>bars<span class="br0">)</span><span class="sy4">;</span>
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<span class="kw1">for</span> <span class="br0">(</span><span class="kw4">int</span> n <span class="sy4">:</span> indices<span class="br0">)</span> <span class="br0">{</span> <a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">" "</span> <span class="sy1"><<</span> <a href="cpp-io-manip-setw.html"><span class="kw1796">std::<span class="me2">setw</span></span></a><span class="br0">(</span><span class="nu0">2</span><span class="br0">)</span> <span class="sy1"><<</span> n <span class="sy1"><<</span> <span class="st0">" "</span><span class="sy4">;</span> <span class="br0">}</span>
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<a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="br0">}</span></pre></div>
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</div>
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<p>可能的输出：</p>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="text source-text">
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<pre class="de1">
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█████ ┬ 0.3753
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█████ │
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▁▁▁▁▁ █████ │
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█████ █████ ▆▆▆▆▆ │
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█████ █████ █████ │
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█████ █████ █████ │
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▄▄▄▄▄ █████ █████ █████ ▄▄▄▄▄ │
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▁▁▁▁▁ ▃▃▃▃▃ █████ █████ █████ █████ █████ ▃▃▃▃▃ ▁▁▁▁▁ ▁▁▁▁▁ ┴ 0.0049
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-4 -3 -2 -1 0 1 2 3 4 5</pre></div>
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</div>
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</div>
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<h3><span class="mw-headline" id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5">外部链接</span></h3>
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<p><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Studentst-Distribution.html">Weisstein, Eric W. “学生 t 分布”</a>来自 MathWorld--A Wolfram Web Resource 。</p>
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</div>
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<div class="visualClear"></div>
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</div>
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</div>
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</div>
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</body>
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</html>
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