Law of Total Expectation (Partition of Events)

$$ \text{Let } {A_1, A_2, \dots, A_n} \text{ be a partition of } \Omega \text{ with } P(A_i) > 0. $$ $$\large \mathbb{E}[X] = \sum_{i=1}^{n} \mathbb{E}[X \mid A_i] . P(A_i) $$

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Law of Total Expectation (Conditioning on a Discrete Random Variable)

$$\large \mathbb{E}[X] = \sum_{y} \mathbb{E}[X \mid Y = y]. P(Y = y) $$

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Law of Iterated Expectation (General / Tower Property)

$$ \large \mathbb{E}[X] = \mathbb{E}\left[\mathbb{E}[X \mid Y] \right] $$

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Law of Total Expectation (Continuous Case)

$$ \large \mathbb{E}[X] = \int_{-\infty}^{\infty} \mathbb{E}[X \mid Y = y] , f_Y(y), dy $$

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Conditional Law of Total Expectation

$$ \large \mathbb{E}[X \mid Z] = \mathbb{E}\left[ \mathbb{E}[X \mid Y, Z] \mid Z \right] $$

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Conditional Event-Based Version

Let (A) be an event with $\mathbb{P}(A) > 0$ and let ${E_1, E_2, \dots, E_n}$ be a partition of (A).

$$
\mathbb{E}[X \mid A]

\sum_{i=1}^{n}
\mathbb{P}(E_i \mid A).
\mathbb{E}[X \mid A \cap E_i]
$$

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