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统计学
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基础
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</span>
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</a>
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统计量
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</span>
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</a>
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抽样
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</span>
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</a>
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假设检验
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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推断
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</span>
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</a>
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<span class="md-ellipsis">
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概率论
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</span>
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<span class="md-nav__icon md-icon"></span>
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概率论
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</label>
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计数
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|
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</span>
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|
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|
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</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../02.%20probability%20concepts/" class="md-nav__link">
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概率概念
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
|
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<span class="md-ellipsis">
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分布
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</span>
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<span class="md-nav__icon md-icon"></span>
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分布
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|
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|
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</span>
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<span class="md-nav__icon md-icon"></span>
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目录
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<li class="md-nav__item">
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<a href="#colab" class="md-nav__link">
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编程练习（使用CoLab或笔记本）
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</span>
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</a>
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</li>
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</ul>
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</nav>
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贝叶斯
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</span>
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</a>
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</li>
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信息论
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</span>
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|
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</nav>
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<span class="md-ellipsis">
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|
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机器学习
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|
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</span>
|
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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机器学习
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<ul class="md-nav__list" data-md-scrollfix>
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<span class="md-ellipsis">
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经典机器学习
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<span class="md-ellipsis">
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梯度机器学习
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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<li class="md-nav__item">
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<a href="../../chapter%2006%3A%20machine%20learning/03.%20deep%20learning/" class="md-nav__link">
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<span class="md-ellipsis">
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深度学习
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
|
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<li class="md-nav__item">
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<a href="../../chapter%2006%3A%20machine%20learning/04.%20reinforcement%20learning/" class="md-nav__link">
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|
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<span class="md-ellipsis">
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|
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强化学习
|
||
|
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|
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|
||
</span>
|
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|
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|
||
|
||
</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter%2006%3A%20machine%20learning/05.%20distributed%20deep%20learning/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
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|
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|
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分布式深度学习
|
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|
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|
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|
||
</span>
|
||
|
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|
||
|
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</a>
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</ul>
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</nav>
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</li>
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|
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|
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|
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<li class="md-nav__item md-nav__item--nested">
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<span class="md-ellipsis">
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|
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计算语言学
|
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|
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|
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|
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</span>
|
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|
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|
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|
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<span class="md-nav__icon md-icon"></span>
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计算语言学
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|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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|
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<span class="md-ellipsis">
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|
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语言学基础
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
|
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</li>
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<a href="../../chapter%2007%3A%20computational%20linguistics/02.%20text%20processing%20and%20classic%20NLP/" class="md-nav__link">
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|
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|
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<span class="md-ellipsis">
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|
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|
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文本处理与经典 NLP
|
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|
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|
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|
||
</span>
|
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|
||
|
||
|
||
</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter%2007%3A%20computational%20linguistics/03.%20embeddings%20and%20sequence%20models/" class="md-nav__link">
|
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|
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|
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<span class="md-ellipsis">
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|
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|
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嵌入与序列模型
|
||
|
||
|
||
|
||
</span>
|
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|
||
|
||
|
||
</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter%2007%3A%20computational%20linguistics/04.%20transformers%20and%20language%20models/" class="md-nav__link">
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|
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|
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<span class="md-ellipsis">
|
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|
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|
||
Transformer 与语言模型
|
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|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
||
</a>
|
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</li>
|
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|
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<li class="md-nav__item">
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<a href="../../chapter%2007%3A%20computational%20linguistics/05.%20advanced%20text%20generation/" class="md-nav__link">
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|
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|
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<span class="md-ellipsis">
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|
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|
||
高级文本生成
|
||
|
||
|
||
|
||
</span>
|
||
|
||
|
||
|
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</a>
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</li>
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</ul>
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</nav>
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|
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|
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|
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|
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计算机视觉
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图像基础
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目标检测与分割
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ViT 与生成模型
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视频与 3D 视觉
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音频与语音
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数字信号处理
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</a>
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<span class="md-ellipsis">
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自动语音识别
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</span>
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</a>
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</li>
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<a href="../../chapter%2009%3A%20audio%20and%20speech/03.%20text%20to%20speech%20and%20voice/" class="md-nav__link">
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<span class="md-ellipsis">
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语音合成
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</span>
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</a>
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</li>
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<span class="md-ellipsis">
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说话人与音频分析
|
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|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<span class="md-ellipsis">
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源分离与降噪
|
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|
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</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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<span class="md-ellipsis">
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多模态学习
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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多模态学习
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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多模态表征
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</span>
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</a>
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视觉语言模型
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</span>
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</a>
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<span class="md-ellipsis">
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图像与视频 Token 化
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</span>
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|
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|
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|
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</a>
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</li>
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跨模态生成
|
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</span>
|
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2010%3A%20multimodal%20learning/05.%20unified%20multimodal%20architectures/" class="md-nav__link">
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<span class="md-ellipsis">
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统一多模态架构
|
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|
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|
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|
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</span>
|
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|
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|
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|
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</a>
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</ul>
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</nav>
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<li class="md-nav__item md-nav__item--nested">
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自主系统
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</span>
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<span class="md-nav__icon md-icon"></span>
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<span class="md-nav__icon md-icon"></span>
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自主系统
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感知
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</span>
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</a>
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机器人学习
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</span>
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</a>
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<span class="md-ellipsis">
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视觉-语言-动作模型
|
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</span>
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|
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|
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|
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</a>
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<a href="../../chapter%2011%3A%20autonomous%20systems/04.%20self-driving/" class="md-nav__link">
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自动驾驶
|
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</span>
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|
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|
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|
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</a>
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</li>
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<a href="../../chapter%2011%3A%20autonomous%20systems/05.%20space%20and%20extreme%20robotics/" class="md-nav__link">
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<span class="md-ellipsis">
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太空与极端机器人
|
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|
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|
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|
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</span>
|
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|
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|
||
|
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</a>
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图神经网络
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</span>
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<span class="md-nav__icon md-icon"></span>
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图神经网络
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</label>
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几何深度学习
|
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<span class="md-ellipsis">
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|
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硬件基础
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</a>
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ARM 与 NEON
|
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x86 与 AVX
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Triton、TPU 与 Pallas
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AI 推理
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量化
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边缘推理
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扩缩与部署
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ML 系统设计
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应用 AI
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前沿 AI
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<h1 id="_1">概率分布<a class="headerlink" href="#_1" title="Permanent link">¶</a></h1>
|
||
<p><em>概率分布描述了随机结果如何在可能取值上分布。本文档整理了关键的离散和连续分布：伯努利分布、二项分布、泊松分布、高斯分布、指数分布、贝塔分布等，给出了各自的公式、直观理解及其在机器学习中的应用（损失函数、先验、噪声模型）。</em></p>
|
||
<ul>
|
||
<li>
|
||
<p>在第4章中，我们介绍了随机变量、PMF、PDF和CDF。本章列出你在机器学习和统计学中最常遇到的重要概率分布，给出每个分布的直观理解、公式、均值和方差。</p>
|
||
</li>
|
||
<li>
|
||
<p>三种核心函数的快速回顾（完整定义见第4章）：</p>
|
||
<ul>
|
||
<li><strong>PMF</strong> <span class="arithmatex">\(P(X = x)\)</span>：给出每个离散结果的概率。即条形图中每个条形的高度。</li>
|
||
<li><strong>PDF</strong> <span class="arithmatex">\(f(x)\)</span>：给出连续变量在每个点上的密度。两点之间曲线下的面积即为概率。</li>
|
||
<li><strong>CDF</strong> <span class="arithmatex">\(F(x) = P(X \le x)\)</span>：累积到 <span class="arithmatex">\(x\)</span> 为止的概率。取值范围始终从0到1且单调不减。</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>分布的<strong>支撑集</strong>是指PMF或PDF取正值的集合。对掷骰子而言，支撑集为 <span class="arithmatex">\(\{1,2,3,4,5,6\}\)</span>。对正态分布而言，支撑集为全体实数 <span class="arithmatex">\((-\infty, \infty)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>分布清晰地分为两个家族：离散分布（结果可数，使用PMF）和连续分布（结果不可数，使用PDF）。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>伯努利分布</strong>：最简单的分布。单次试验有两种结果：成功（1）的概率为 <span class="arithmatex">\(p\)</span>，失败（0）的概率为 <span class="arithmatex">\(1-p\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[P(X = x) = p^x (1 - p)^{1-x}, \quad x \in \{0, 1\}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = p\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = p(1-p)\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>每一次抛硬币、每一个是/否分类、每一个二元结果都是伯努利试验。在机器学习中，sigmoid函数的输出正是伯努利分布的参数 <span class="arithmatex">\(p\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>二项分布</strong>：计算 <span class="arithmatex">\(n\)</span> 次独立伯努利试验中成功的次数，每次试验的成功概率 <span class="arithmatex">\(p\)</span> 相同。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>二项式系数 <span class="arithmatex">\(\binom{n}{k}\)</span>（见文件01）计算了 <span class="arithmatex">\(k\)</span> 次成功在 <span class="arithmatex">\(n\)</span> 次试验中的排列方式数量。</p>
|
||
</li>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = np\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = np(1-p)\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="伯努利分布作为单一条形图与二项分布作为计数上的分布对比" src="../../images/bernoulli_binomial.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p>示例：抛一枚有偏硬币（<span class="arithmatex">\(p = 0.7\)</span>）八次。恰好得到6次正面的概率为 <span class="arithmatex">\(\binom{8}{6}(0.7)^6(0.3)^2 = 28 \times 0.1176 \times 0.09 \approx 0.296\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>泊松分布</strong>：在固定的时间或空间区间内，以已知的平均速率 <span class="arithmatex">\(\lambda\)</span> 计算事件发生的次数。适用于事件稀少且相互独立的情形。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \ldots\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = \lambda\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = \lambda\)</span>。均值等于方差是其标志性特征。</p>
|
||
</li>
|
||
<li>
|
||
<p>示例：每小时收到的邮件数（<span class="arithmatex">\(\lambda = 5\)</span>）、每页的错别字数、每秒的服务器请求数。在机器学习中，泊松回归用于建模计数数据，而线性模型可能会预测出负的计数值。</p>
|
||
</li>
|
||
<li>
|
||
<p>当 <span class="arithmatex">\(n \to \infty\)</span> 且 <span class="arithmatex">\(p \to 0\)</span>，且 <span class="arithmatex">\(np = \lambda\)</span> 保持不变时，二项分布 Binomial<span class="arithmatex">\((n,p)\)</span> 收敛于泊松分布 Poisson<span class="arithmatex">\((\lambda)\)</span>。这就是泊松分布适用于大总体中稀有事件的原因。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>几何分布</strong>：计算直到首次成功所需的试验次数。"我要抛多少次硬币才能第一次得到正面？"</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[P(X = k) = (1-p)^{k-1} p, \quad k = 1, 2, 3, \ldots\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = 1/p\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = (1-p)/p^2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>几何分布具有<strong>无记忆性</strong>：再等待 <span class="arithmatex">\(k\)</span> 次试验才成功的概率与你已经等待了多少次试验无关。这使得它在离散分布中非常特殊。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>负二项分布</strong>：推广了几何分布，计算直到第 <span class="arithmatex">\(r\)</span> 次成功所需的试验次数（几何分布是 <span class="arithmatex">\(r=1\)</span> 的特殊情形）。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[P(X = k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}, \quad k = r, r+1, r+2, \ldots\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = r/p\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = r(1-p)/p^2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>负二项分布在实践中也用于建模过度离散的计数数据（方差超过均值的情形），这是泊松分布无法处理的。</p>
|
||
</li>
|
||
<li>
|
||
<p>接下来我们进入连续分布。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>均匀分布</strong>：区间 <span class="arithmatex">\([a, b]\)</span> 内的所有值等可能。其PDF是一个平坦的矩形。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(x) = \frac{1}{b - a}, \quad a \le x \le b\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = \frac{a+b}{2}\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = \frac{(b-a)^2}{12}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>随机数生成器以生成均匀分布 Uniform(0,1) 样本为起点。其他分布通过对这些均匀样本进行变换得到。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>正态（高斯）分布</strong>：统计学中最重要的分布。它由中心极限定理（见第4章）自然导出：大量独立随机变量的平均值趋于正态分布，无论原始分布是什么。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = \mu\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = \sigma^2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>标准正态分布</strong>的 <span class="arithmatex">\(\mu = 0\)</span> 且 <span class="arithmatex">\(\sigma = 1\)</span>。任意正态变量 <span class="arithmatex">\(X\)</span> 可通过 <span class="arithmatex">\(Z = (X - \mu)/\sigma\)</span> 标准化为标准正态变量 <span class="arithmatex">\(Z\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="带有68-95-99.7经验法则区域的钟形曲线" src="../../images/normal_empirical.svg" /></p>
|
||
<ul>
|
||
<li>
|
||
<p><strong>经验法则</strong>（68-95-99.7法则）指出：</p>
|
||
<ul>
|
||
<li>约68%的数据落在均值 <span class="arithmatex">\(\pm 1\sigma\)</span> 范围内</li>
|
||
<li>约95%的数据落在 <span class="arithmatex">\(\pm 2\sigma\)</span> 范围内</li>
|
||
<li>约99.7%的数据落在 <span class="arithmatex">\(\pm 3\sigma\)</span> 范围内</li>
|
||
</ul>
|
||
</li>
|
||
<li>
|
||
<p>在机器学习中，正态分布无处不在：权重初始化、数据增强中的噪声、MSE损失背后的假设（其隐含假设高斯误差）、以及变分自编码器中的重参数化技巧。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>指数分布</strong>：模拟泊松过程中事件之间的时间间隔。如果事件以速率 <span class="arithmatex">\(\lambda\)</span> 到达，则它们之间的等待时间服从指数分布 Exponential<span class="arithmatex">\((\lambda)\)</span>。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(x) = \lambda e^{-\lambda x}, \quad x \ge 0\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = 1/\lambda\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = 1/\lambda^2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>与离散变量中的几何分布类似，指数分布也具有<strong>无记忆性</strong>：<span class="arithmatex">\(P(X > s + t | X > s) = P(X > t)\)</span>。再等待 <span class="arithmatex">\(t\)</span> 个时间单位的概率与你已经等待了多长时间无关。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>伽马分布</strong>：推广了指数分布。它模拟泊松过程中第 <span class="arithmatex">\(\alpha\)</span> 个事件发生的时间（指数分布是 <span class="arithmatex">\(\alpha = 1\)</span> 的特殊情形）。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}, \quad x > 0\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>这里 <span class="arithmatex">\(\alpha\)</span>（形状参数）控制形状，<span class="arithmatex">\(\beta\)</span>（速率参数）控制尺度。<span class="arithmatex">\(\Gamma(\alpha)\)</span> 是伽马函数，它将阶乘推广到实数：对正整数 <span class="arithmatex">\(n\)</span> 有 <span class="arithmatex">\(\Gamma(n) = (n-1)!\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = \alpha/\beta\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = \alpha/\beta^2\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>贝塔分布</strong>：定义在区间 <span class="arithmatex">\([0, 1]\)</span> 上，非常适合对概率、比例和比率进行建模。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(x) = \frac{x^{\alpha - 1}(1 - x)^{\beta - 1}}{B(\alpha, \beta)}, \quad 0 \le x \le 1\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>分母 <span class="arithmatex">\(B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}\)</span> 是贝塔函数，起到归一化常数的作用。</p>
|
||
</li>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = \frac{\alpha}{\alpha + \beta}\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p>贝塔分布是伯努利和二项似然函数的共轭先验。这意味着如果先验是贝塔分布且数据服从伯努利分布，则后验也是贝塔分布，这使得贝叶斯更新在解析上易于处理。我们将在文件04中使用这一性质。</p>
|
||
</li>
|
||
</ul>
|
||
<p><img alt="四种常见的分布形状：均匀分布、指数分布、贝塔分布、泊松分布" src="../../images/common_distributions.svg" /></p>
|
||
<ul>
|
||
<li><strong>卡方分布</strong>（<span class="arithmatex">\(\chi^2\)</span>）：如果你取 <span class="arithmatex">\(k\)</span> 个独立的标准正态随机变量并求其平方和，结果服从自由度为 <span class="arithmatex">\(k\)</span> 的 <span class="arithmatex">\(\chi^2\)</span> 分布。</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(x) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2 - 1} e^{-x/2}, \quad x > 0\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = k\)</span>。方差：<span class="arithmatex">\(\text{Var}(X) = 2k\)</span>。</p>
|
||
</li>
|
||
<li>
|
||
<p><span class="arithmatex">\(\chi^2\)</span> 分布实际上是伽马分布的特殊情形，其中 <span class="arithmatex">\(\alpha = k/2\)</span> 且 <span class="arithmatex">\(\beta = 1/2\)</span>。它出现在假设检验（第4章中的卡方检验）、拟合优度检验以及方差置信区间的计算中。</p>
|
||
</li>
|
||
<li>
|
||
<p><strong>学生t分布</strong>：形状类似于正态分布但尾部更重。当你使用小样本且总体方差未知时，对正态分布总体的均值进行估计时就会出现t分布。</p>
|
||
</li>
|
||
</ul>
|
||
<div class="arithmatex">\[f(x) = \frac{\Gamma\!\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi}\,\Gamma\!\left(\frac{\nu}{2}\right)} \left(1 + \frac{x^2}{\nu}\right)^{-(\nu+1)/2}\]</div>
|
||
<ul>
|
||
<li>
|
||
<p>参数 <span class="arithmatex">\(\nu\)</span>（自由度）。当 <span class="arithmatex">\(\nu \to \infty\)</span> 时，t分布收敛于标准正态分布。当 <span class="arithmatex">\(\nu\)</span> 较小时，更重的尾部赋予极端值更高的概率，反映了小样本带来的额外不确定性。</p>
|
||
</li>
|
||
<li>
|
||
<p>均值：<span class="arithmatex">\(E[X] = 0\)</span>（当 <span class="arithmatex">\(\nu > 1\)</span> 时）。方差：<span class="arithmatex">\(\text{Var}(X) = \frac{\nu}{\nu - 2}\)</span>（当 <span class="arithmatex">\(\nu > 2\)</span> 时）。</p>
|
||
</li>
|
||
<li>
|
||
<p>t分布用于t检验（第4章），并出现在贝叶斯推断中，作为在积分消去未知方差时的边缘分布。</p>
|
||
</li>
|
||
<li>
|
||
<p>关键分布总结：</p>
|
||
</li>
|
||
</ul>
|
||
<table>
|
||
<thead>
|
||
<tr>
|
||
<th>分布</th>
|
||
<th>类型</th>
|
||
<th>支撑集</th>
|
||
<th>均值</th>
|
||
<th>方差</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>Bernoulli<span class="arithmatex">\((p)\)</span></td>
|
||
<td>离散</td>
|
||
<td><span class="arithmatex">\(\{0,1\}\)</span></td>
|
||
<td><span class="arithmatex">\(p\)</span></td>
|
||
<td><span class="arithmatex">\(p(1-p)\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Binomial<span class="arithmatex">\((n,p)\)</span></td>
|
||
<td>离散</td>
|
||
<td><span class="arithmatex">\(\{0,\ldots,n\}\)</span></td>
|
||
<td><span class="arithmatex">\(np\)</span></td>
|
||
<td><span class="arithmatex">\(np(1-p)\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Poisson<span class="arithmatex">\((\lambda)\)</span></td>
|
||
<td>离散</td>
|
||
<td><span class="arithmatex">\(\{0,1,2,\ldots\}\)</span></td>
|
||
<td><span class="arithmatex">\(\lambda\)</span></td>
|
||
<td><span class="arithmatex">\(\lambda\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Geometric<span class="arithmatex">\((p)\)</span></td>
|
||
<td>离散</td>
|
||
<td><span class="arithmatex">\(\{1,2,3,\ldots\}\)</span></td>
|
||
<td><span class="arithmatex">\(1/p\)</span></td>
|
||
<td><span class="arithmatex">\((1-p)/p^2\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Uniform<span class="arithmatex">\((a,b)\)</span></td>
|
||
<td>连续</td>
|
||
<td><span class="arithmatex">\([a,b]\)</span></td>
|
||
<td><span class="arithmatex">\((a+b)/2\)</span></td>
|
||
<td><span class="arithmatex">\((b-a)^2/12\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Normal<span class="arithmatex">\((\mu,\sigma^2)\)</span></td>
|
||
<td>连续</td>
|
||
<td><span class="arithmatex">\((-\infty,\infty)\)</span></td>
|
||
<td><span class="arithmatex">\(\mu\)</span></td>
|
||
<td><span class="arithmatex">\(\sigma^2\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Exponential<span class="arithmatex">\((\lambda)\)</span></td>
|
||
<td>连续</td>
|
||
<td><span class="arithmatex">\([0,\infty)\)</span></td>
|
||
<td><span class="arithmatex">\(1/\lambda\)</span></td>
|
||
<td><span class="arithmatex">\(1/\lambda^2\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Gamma<span class="arithmatex">\((\alpha,\beta)\)</span></td>
|
||
<td>连续</td>
|
||
<td><span class="arithmatex">\((0,\infty)\)</span></td>
|
||
<td><span class="arithmatex">\(\alpha/\beta\)</span></td>
|
||
<td><span class="arithmatex">\(\alpha/\beta^2\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Beta<span class="arithmatex">\((\alpha,\beta)\)</span></td>
|
||
<td>连续</td>
|
||
<td><span class="arithmatex">\([0,1]\)</span></td>
|
||
<td><span class="arithmatex">\(\alpha/(\alpha+\beta)\)</span></td>
|
||
<td>见上文</td>
|
||
</tr>
|
||
<tr>
|
||
<td><span class="arithmatex">\(\chi^2(k)\)</span></td>
|
||
<td>连续</td>
|
||
<td><span class="arithmatex">\((0,\infty)\)</span></td>
|
||
<td><span class="arithmatex">\(k\)</span></td>
|
||
<td><span class="arithmatex">\(2k\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td>Student's <span class="arithmatex">\(t(\nu)\)</span></td>
|
||
<td>连续</td>
|
||
<td><span class="arithmatex">\((-\infty,\infty)\)</span></td>
|
||
<td><span class="arithmatex">\(0\)</span></td>
|
||
<td><span class="arithmatex">\(\nu/(\nu-2)\)</span></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h2 id="colab">编程练习（使用CoLab或笔记本）<a class="headerlink" href="#colab" title="Permanent link">¶</a></h2>
|
||
<ol>
|
||
<li>
|
||
<p>绘制 <span class="arithmatex">\(n=20\)</span> 时二项分布PMF在不同 <span class="arithmatex">\(p\)</span> 取值下的图像。观察形状如何从左偏变为对称再变为右偏。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="kn">from</span><span class="w"> </span><span class="nn">math</span><span class="w"> </span><span class="kn">import</span> <span class="n">comb</span>
|
||
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a>
|
||
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a><span class="n">n</span> <span class="o">=</span> <span class="mi">20</span>
|
||
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a><span class="n">ks</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a>
|
||
<a id="__codelineno-0-8" name="__codelineno-0-8" href="#__codelineno-0-8"></a><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">4</span><span class="p">),</span> <span class="n">sharey</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<a id="__codelineno-0-9" name="__codelineno-0-9" href="#__codelineno-0-9"></a><span class="k">for</span> <span class="n">ax</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">],</span> <span class="p">[</span><span class="s2">"#e74c3c"</span><span class="p">,</span> <span class="s2">"#3498db"</span><span class="p">,</span> <span class="s2">"#27ae60"</span><span class="p">]):</span>
|
||
<a id="__codelineno-0-10" name="__codelineno-0-10" href="#__codelineno-0-10"></a> <span class="n">pmf</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">comb</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">))</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">ks</span><span class="p">])</span>
|
||
<a id="__codelineno-0-11" name="__codelineno-0-11" href="#__codelineno-0-11"></a> <span class="n">ax</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">ks</span><span class="p">,</span> <span class="n">pmf</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
|
||
<a id="__codelineno-0-12" name="__codelineno-0-12" href="#__codelineno-0-12"></a> <span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Binomial(n=</span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s2">, p=</span><span class="si">{</span><span class="n">p</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-13" name="__codelineno-0-13" href="#__codelineno-0-13"></a> <span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"k"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-14" name="__codelineno-0-14" href="#__codelineno-0-14"></a><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"P(X = k)"</span><span class="p">)</span>
|
||
<a id="__codelineno-0-15" name="__codelineno-0-15" href="#__codelineno-0-15"></a><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
|
||
<a id="__codelineno-0-16" name="__codelineno-0-16" href="#__codelineno-0-16"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>验证泊松分布对二项分布的近似。设 <span class="arithmatex">\(n = 1000\)</span>，<span class="arithmatex">\(p = 0.003\)</span>，比较二项分布 Binomial<span class="arithmatex">\((n, p)\)</span> 和泊松分布 Poisson<span class="arithmatex">\((\lambda = np)\)</span>。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a><span class="kn">from</span><span class="w"> </span><span class="nn">math</span><span class="w"> </span><span class="kn">import</span> <span class="n">comb</span><span class="p">,</span> <span class="n">factorial</span><span class="p">,</span> <span class="n">exp</span>
|
||
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a>
|
||
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">n</span><span class="p">,</span> <span class="n">p</span> <span class="o">=</span> <span class="mi">1000</span><span class="p">,</span> <span class="mf">0.003</span>
|
||
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a><span class="n">lam</span> <span class="o">=</span> <span class="n">n</span> <span class="o">*</span> <span class="n">p</span>
|
||
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="n">ks</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span>
|
||
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a>
|
||
<a id="__codelineno-1-9" name="__codelineno-1-9" href="#__codelineno-1-9"></a><span class="n">binom_pmf</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">comb</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">))</span> <span class="o">*</span> <span class="n">p</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="n">k</span><span class="p">)</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">ks</span><span class="p">])</span>
|
||
<a id="__codelineno-1-10" name="__codelineno-1-10" href="#__codelineno-1-10"></a><span class="n">poisson_pmf</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">lam</span><span class="o">**</span><span class="n">k</span> <span class="o">*</span> <span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">lam</span><span class="p">)</span> <span class="o">/</span> <span class="n">factorial</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">k</span><span class="p">))</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">ks</span><span class="p">])</span>
|
||
<a id="__codelineno-1-11" name="__codelineno-1-11" href="#__codelineno-1-11"></a>
|
||
<a id="__codelineno-1-12" name="__codelineno-1-12" href="#__codelineno-1-12"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<a id="__codelineno-1-13" name="__codelineno-1-13" href="#__codelineno-1-13"></a><span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">ks</span> <span class="o">-</span> <span class="mf">0.15</span><span class="p">,</span> <span class="n">binom_pmf</span><span class="p">,</span> <span class="n">width</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#3498db"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">"Binomial(</span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s2">,</span><span class="si">{</span><span class="n">p</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-14" name="__codelineno-1-14" href="#__codelineno-1-14"></a><span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">(</span><span class="n">ks</span> <span class="o">+</span> <span class="mf">0.15</span><span class="p">,</span> <span class="n">poisson_pmf</span><span class="p">,</span> <span class="n">width</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"#e74c3c"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.7</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">"Poisson(</span><span class="si">{</span><span class="n">lam</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-15" name="__codelineno-1-15" href="#__codelineno-1-15"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"k"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-16" name="__codelineno-1-16" href="#__codelineno-1-16"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"P(X = k)"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-17" name="__codelineno-1-17" href="#__codelineno-1-17"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"泊松分布对二项分布的近似"</span><span class="p">)</span>
|
||
<a id="__codelineno-1-18" name="__codelineno-1-18" href="#__codelineno-1-18"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<a id="__codelineno-1-19" name="__codelineno-1-19" href="#__codelineno-1-19"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>从正态分布中采样并验证经验法则。计算落在1、2和3个标准差内的样本比例。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
|
||
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="n">key</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">PRNGKey</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">mu</span><span class="p">,</span> <span class="n">sigma</span> <span class="o">=</span> <span class="mf">5.0</span><span class="p">,</span> <span class="mf">2.0</span>
|
||
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a><span class="n">samples</span> <span class="o">=</span> <span class="n">mu</span> <span class="o">+</span> <span class="n">sigma</span> <span class="o">*</span> <span class="n">jax</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">shape</span><span class="o">=</span><span class="p">(</span><span class="mi">100_000</span><span class="p">,))</span>
|
||
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a>
|
||
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">]:</span>
|
||
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a> <span class="n">within</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">samples</span> <span class="o">-</span> <span class="n">mu</span><span class="p">)</span> <span class="o"><=</span> <span class="n">k</span> <span class="o">*</span> <span class="n">sigma</span>
|
||
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Within </span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">σ: </span><span class="si">{</span><span class="n">within</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> (expected: </span><span class="si">{</span><span class="p">[</span><span class="mf">0.6827</span><span class="p">,</span><span class="w"> </span><span class="mf">0.9545</span><span class="p">,</span><span class="w"> </span><span class="mf">0.9973</span><span class="p">][</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
<li>
|
||
<p>通过改变 <span class="arithmatex">\(\alpha\)</span> 和 <span class="arithmatex">\(\beta\)</span> 探索贝塔分布。绘制几种形状，观察分布如何从均匀变为偏斜再变为集中。
|
||
<div class="highlight"><pre><span></span><code><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
|
||
<a id="__codelineno-3-2" name="__codelineno-3-2" href="#__codelineno-3-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
|
||
<a id="__codelineno-3-3" name="__codelineno-3-3" href="#__codelineno-3-3"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
|
||
<a id="__codelineno-3-4" name="__codelineno-3-4" href="#__codelineno-3-4"></a>
|
||
<a id="__codelineno-3-5" name="__codelineno-3-5" href="#__codelineno-3-5"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.99</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
|
||
<a id="__codelineno-3-6" name="__codelineno-3-6" href="#__codelineno-3-6"></a>
|
||
<a id="__codelineno-3-7" name="__codelineno-3-7" href="#__codelineno-3-7"></a><span class="k">def</span><span class="w"> </span><span class="nf">beta_pdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
|
||
<a id="__codelineno-3-8" name="__codelineno-3-8" href="#__codelineno-3-8"></a> <span class="c1"># 未归一化，用于形状比较</span>
|
||
<a id="__codelineno-3-9" name="__codelineno-3-9" href="#__codelineno-3-9"></a> <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">b</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
|
||
<a id="__codelineno-3-10" name="__codelineno-3-10" href="#__codelineno-3-10"></a>
|
||
<a id="__codelineno-3-11" name="__codelineno-3-11" href="#__codelineno-3-11"></a><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
|
||
<a id="__codelineno-3-12" name="__codelineno-3-12" href="#__codelineno-3-12"></a><span class="n">params</span> <span class="o">=</span> <span class="p">[(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="s2">"均匀"</span><span class="p">),</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="s2">"左偏"</span><span class="p">),</span> <span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="s2">"右偏"</span><span class="p">),</span>
|
||
<a id="__codelineno-3-13" name="__codelineno-3-13" href="#__codelineno-3-13"></a> <span class="p">(</span><span class="mi">5</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="s2">"对称"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="s2">"U形"</span><span class="p">)]</span>
|
||
<a id="__codelineno-3-14" name="__codelineno-3-14" href="#__codelineno-3-14"></a><span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"#999"</span><span class="p">,</span> <span class="s2">"#e74c3c"</span><span class="p">,</span> <span class="s2">"#3498db"</span><span class="p">,</span> <span class="s2">"#27ae60"</span><span class="p">,</span> <span class="s2">"#9b59b6"</span><span class="p">]</span>
|
||
<a id="__codelineno-3-15" name="__codelineno-3-15" href="#__codelineno-3-15"></a>
|
||
<a id="__codelineno-3-16" name="__codelineno-3-16" href="#__codelineno-3-16"></a><span class="k">for</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">label</span><span class="p">),</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">params</span><span class="p">,</span> <span class="n">colors</span><span class="p">):</span>
|
||
<a id="__codelineno-3-17" name="__codelineno-3-17" href="#__codelineno-3-17"></a> <span class="n">y</span> <span class="o">=</span> <span class="n">beta_pdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
|
||
<a id="__codelineno-3-18" name="__codelineno-3-18" href="#__codelineno-3-18"></a> <span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">/</span> <span class="n">jnp</span><span class="o">.</span><span class="n">trapezoid</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="c1"># 归一化</span>
|
||
<a id="__codelineno-3-19" name="__codelineno-3-19" href="#__codelineno-3-19"></a> <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">f</span><span class="s2">"α=</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s2">, β=</span><span class="si">{</span><span class="n">b</span><span class="si">}</span><span class="s2"> (</span><span class="si">{</span><span class="n">label</span><span class="si">}</span><span class="s2">)"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<a id="__codelineno-3-20" name="__codelineno-3-20" href="#__codelineno-3-20"></a>
|
||
<a id="__codelineno-3-21" name="__codelineno-3-21" href="#__codelineno-3-21"></a><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"x"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-22" name="__codelineno-3-22" href="#__codelineno-3-22"></a><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"密度"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-23" name="__codelineno-3-23" href="#__codelineno-3-23"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"贝塔分布形状"</span><span class="p">)</span>
|
||
<a id="__codelineno-3-24" name="__codelineno-3-24" href="#__codelineno-3-24"></a><span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<a id="__codelineno-3-25" name="__codelineno-3-25" href="#__codelineno-3-25"></a><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
|
||
<a id="__codelineno-3-26" name="__codelineno-3-26" href="#__codelineno-3-26"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</code></pre></div></p>
|
||
</li>
|
||
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|
||
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