Law of Total Expectation - Proof in The Discrete Case

Proof in The Discrete Case

\begin{align} \operatorname{E}_Y \left( \operatorname{E}_X(X|Y) \right) &{} = \operatorname{E}_y \Bigg \\ &{}=\sum_y \Bigg \cdot \operatorname{P}(Y=y) \\ &{}=\sum_y \sum_x x \cdot \operatorname{P}(X=x|Y=y) \cdot \operatorname{P}(Y=y) \\ &{}=\sum_x x \sum_y \operatorname{P}(X=x|Y=y) \cdot \operatorname{P}(Y=y) \\ &{}=\sum_x x \sum_y \operatorname{P}(X=x, Y=y) \\ &{}=\sum_x x \cdot \operatorname{P}(X=x) \\ &{}=\operatorname{E}(X). \end{align}

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