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 any proof were needed of the progress of the cause for which I have worked, it is here tonight. The presence on the stage of these college women, and in the audience of all those college girls who will some day be the nation’s greatest strength, will tell their own story to the world.”
    —Susan B. Anthony (1820–1906)

    “We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the child’s life, these words are discrete and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”
    —Selma H. Fraiberg (20th century)

    “Never in any case say I have lost such a thing, but I have returned it. Is your child dead? It is a return. Is your wife dead? It is a return. Are you deprived of your estate? Is not this also a return?”
    —Epictetus (c. 55–135)