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→Motivation
== Motivation ==
== Motivation ==
Let <math>\Eta(Y|X=x)</math> be the [[Shannon Entropy|entropy]] of the discrete random variable <math>Y</math> conditioned on the discrete random variable <math>X</math> taking a certain value <math>x</math>. Denote the support sets of <math>X</math> and <math>Y</math> by <math>\mathcal X</math> and <math>\mathcal Y</math>. Let <math>Y</math> have [[probability mass function]] <math>p_Y{(y)}</math>. The unconditional entropy of <math>Y</math> is calculated as <math>\Eta(Y) := \mathbb{E}[\operatorname{I}(Y)]</math>, i.e.
Let <math>\Eta(Y|X=x)</math> be the [[Shannon Entropy|entropy]] of the discrete random variable <math>Y</math> conditioned on the discrete random variable <math>X</math> taking a certain value <math>x</math>. Denote the support sets of <math>X</math> and <math>Y</math> by <math>\mathcal X</math> and <math>\mathcal Y</math>. Let <math>Y</math> have [[probability mass function]] <math>p_Y{(y)}</math>. The unconditional entropy of <math>Y</math> is calculated as <math>\Eta(Y) := \mathbb{E}[\operatorname{I}(Y)]</math>, i.e.
设H（Y ǀ X = x）为离散随机变量<math>Y</math>的熵，条件是离散随机变量<math>X</math>取一定值<math>x</math>。
用<math>\mathcal X</math>和<math>\mathcal Y</math>表示<math>X</math>和<math>Y</math>的支撑集。令<math>Y</math>具有概率质量函数<math>p_Y{(y)}</math>。
<math>Y</math>的无条件熵计算为<math>\Eta(Y) := \mathbb{E}[\operatorname{I}(Y)]</math>。
:<math>\Eta(Y) = \sum_{y\in\mathcal Y} {\mathrm{Pr}(Y=y)\,\mathrm{I}(y)}
:<math>\Eta(Y) = \sum_{y\in\mathcal Y} {\mathrm{Pr}(Y=y)\,\mathrm{I}(y)}