C4 - computing expectations via conditioning

similar to how the law of total probability expresses the probability of an event as a sum of conditional probabilities, expectations can be expressed as sum of conditional expectations

theorem

E [X] = \sum_{i = 1}^{\infty} E [X | F_{i}] \cdot P (F_{i})

given a discrete RV, $Y$ , with an event $Y = y :$

E [X] = \sum_{y} E [X | Y = y] \cdot P (Y = y)

theorem

E [g (X)] = \sum_{y} E [g (X) | Y = y] \cdot P (Y = y)