Show the law of iterated expectation: E[E[X | Y = y]] = E[X] Show the projection property of conditional expectation: E[?(Y)X | Y] = ?(Y)E[X | Y]
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This can be written as: $E[X|Y=y] = \sum_x x \cdot p(x|y)$ Now, we want to find the expectation of this conditional expectation, which is: $E[E[X|Y]] = \sum_y E[X|Y=y] \cdot p(y)$ Substitute the expression for $E[X|Y=y]$: $E[E[X|Y]] = \sum_y \left(\sum_x x Show more…
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