Conditioned on a discrete random variable, the conditional expectation is given by the formula : $$E(X|Y=y)=\sum xp(x|Y=y)$$ However I've found another formula in Wikipedia that given an event H: $$E(X|H)=\frac{E(X 1_H)}{p(H)}$$ Can anyone provide the derivation from one formula to the other ? (H is the event Y=y) Thank you for help
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$$\sum_x xP(x|Y=y) = \sum_x \cfrac{xP(x, Y=y)}{P(Y=y)} = \cfrac{1}{P(Y=y)}\sum_xxP(x \cap Y=y)$$
$$X1_{Y=y} = \begin{cases} X, &Y=y \\ 0, &\text{else} \end{cases} \\ \therefore P(X1_{Y=y} = x) = P(X=x, Y=y) \therefore \sum_x xP(x \cap Y=y) = \mathsf E[X1_{Y=y}]$$
