Law of Total Expectation - Proof in The Discrete Case

Proof in The Discrete Case

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

Read more about this topic: Law Of Total Expectation

Famous quotes containing the words proof, discrete and/or case:

“If we view our children as stupid, naughty, disturbed, or guilty of their misdeeds, they will learn to behold themselves as foolish, faulty, or shameful specimens of humanity. They will regard us as judges from whom they wish to hide, and they will interpret everything we say as further proof of their unworthiness. If we view them as innocent, or at least merely ignorant, they will gain understanding from their experiences, and they will continue to regard us as wise partners.”
—Polly Berrien Berends (20th century)

“The mastery of one’s phonemes may be compared to the violinist’s mastery of fingering. The violin string lends itself to a continuous gradation of tones, but the musician learns the discrete intervals at which to stop the string in order to play the conventional notes. We sound our phonemes like poor violinists, approximating each time to a fancied norm, and we receive our neighbor’s renderings indulgently, mentally rectifying the more glaring inaccuracies.”
—W.V. Quine (b. 1908)

“In case you’ve never done it, this is known as washing your hands.”
—Robert Pirosh, U.S. screenwriter, George Seaton, George Oppenheimer, and Sam Wood. Dr. Hugo Z. Hackenbush (Groucho Marx)