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The joint [[Shannon entropy]] (in [[bit]]s) of two discrete [[random variable|random variables]] <math>X</math> and <math>Y</math> with images <math>\mathcal X</math> and <math>\mathcal Y</math> is defined as<ref name=cover1991>{{cite book |author1=Thomas M. Cover |author2=Joy A. Thomas |title=Elements of Information Theory |publisher=Wiley |location=Hoboken, New Jersey |year= |isbn=0-471-24195-4}}</ref>{{rp|16}}

The joint [[Shannon entropy]] (in [[bit]]s) of two discrete [[random variable|random variables]] <math>X</math> and <math>Y</math> with images <math>\mathcal X</math> and <math>\mathcal Y</math> is defined as<ref name=cover1991>{{cite book |author1=Thomas M. Cover |author2=Joy A. Thomas |title=Elements of Information Theory |publisher=Wiley |location=Hoboken, New Jersey |year= |isbn=0-471-24195-4}}</ref>{{rp|16}}

具有像<math>\mathcal X</math>和<math>\mathcal Y</math>的两个离散随机变量<math>X</math>和<math>Y</math>的'''<font color="#ff8000"> 联合香农熵Shannon entropy </font>'''（以比特为单位）定义为：

联合香农熵Shannon entropy </font>'''的定义是：以比特为单位，具有<math>\mathcal X</math>和<math>\mathcal Y</math>的两个离散随机变量<math>X</math>和<math>Y</math>的'''<font color="#ff8000">  

 

{{Equation box 1

{{Equation box 1

where <math>x</math> and <math>y</math> are particular values of <math>X</math> and <math>Y</math>, respectively, <math>P(x,y)</math> is the [[joint probability]] of these values occurring together, and <math>P(x,y) \log_2[P(x,y)]</math> is defined to be 0 if <math>P(x,y)=0</math>.

where <math>x</math> and <math>y</math> are particular values of <math>X</math> and <math>Y</math>, respectively, <math>P(x,y)</math> is the [[joint probability]] of these values occurring together, and <math>P(x,y) \log_2[P(x,y)]</math> is defined to be 0 if <math>P(x,y)=0</math>.

其中<math>x</math>和<math>y</math>分别是<math>X</math>和<math>Y</math>的特定值，<math>P(x,y)</math>是这些值产生交集时的联合概率，如果<math>P(x,y)=0</math>则<math>P(x,y) \log_2[P(x,y)]</math>定义为0。

其中<math>x</math>和<math>y</math>分别是<math>X</math>和<math>Y</math>的特定值，<math>P(x,y)</math>是这些值产生交集时的联合概率，如果<math>P(x,y)=0</math>那么<math>P(x,y) \log_2[P(x,y)]</math>定义为0。

where <math>x_1,...,x_n</math> are particular values of <math>X_1,...,X_n</math>, respectively, <math>P(x_1, ..., x_n)</math> is the probability of these values occurring together, and <math>P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)]</math> is defined to be 0 if <math>P(x_1, ..., x_n)=0</math>.

where <math>x_1,...,x_n</math> are particular values of <math>X_1,...,X_n</math>, respectively, <math>P(x_1, ..., x_n)</math> is the probability of these values occurring together, and <math>P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)]</math> is defined to be 0 if <math>P(x_1, ..., x_n)=0</math>.

其中<math>x_1,...,x_n</math>分别是<math>X_1,...,X_n</math>的特定值，<math>P(x_1, ..., x_n)</math>是这些值产生交集时的概率，如果<math>P(x_1, ..., x_n)=0</math>则<math>P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)]</math>定义为0。

其中<math>x_1,...,x_n</math>分别是<math>X_1,...,X_n</math>的特定值，<math>P(x_1, ..., x_n)</math>是这些值产生交集的概率，如果<math>P(x_1, ..., x_n)=0</math>那么<math>P(x_1, ..., x_n) \log_2[P(x_1, ..., x_n)]</math>定义为0。

 

 

== Properties 属性 ==

== Properties 属性 ==