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Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the weighted sum of <math>H(Y|X=x)</math> for each possible value of <math>x</math>, using  <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}

Given [[Discrete random variable|discrete random variables]] <math>X</math> with image <math>\mathcal X</math> and <math>Y</math> with image <math>\mathcal Y</math>, the conditional entropy of <math>Y</math> given <math>X</math> is defined as the weighted sum of <math>H(Y|X=x)</math> for each possible value of <math>x</math>, using  <math>p(x)</math> as the weights:<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}

给定具有像<math>\mathcal X</math>的离散随机变量<math>X</math>和具有像<math>\mathcal Y</math>的<math>Y</math>，将给定<math>X</math>的<math>Y</math>的条件熵定义为<math>H(Y|X=x)</math>的权重之和，以<math>x</math>的每个可能值为准，并使用<math>p(x)</math>作为权重：<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}

给定具有像<math>\mathcal X</math>的离散随机变量<math>X</math>和具有像<math>\mathcal Y</math>的离散随机变量<math>Y</math>，将给定<math>X</math>的<math>Y</math>的条件熵定义为<math>H(Y|X=x)</math>的权重之和，以<math>x</math>的每个可能值为准，并使用<math>p(x)</math>作为权重，其表达式如下：<ref name=cover1991>{{cite book|isbn=0-471-06259-6|year=1991|authorlink1=Thomas M. Cover|author1=T. Cover|author2=J. Thomas|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|15}}

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\begin{align}

\begin{align}

\Eta(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,\Eta(Y|X=x)\\

H(Y|X)\ &\equiv \sum_{x\in\mathcal X}\,p(x)\,H(Y|X=x)\\

& =-\sum_{x\in\mathcal X} p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, p(y|x)\\

& =-\sum_{x\in\mathcal X} p(x)\sum_{y\in\mathcal Y}\,p(y|x)\,\log\, p(y|x)\\

& =-\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}\,p(x,y)\,\log\,p(y|x)\\

& =-\sum_{x\in\mathcal X}\sum_{y\in\mathcal Y}\,p(x,y)\,\log\,p(y|x)\\