The Law of Total Expectation for Discrete Random Variables

The law of total expectation states that the expected value of a discrete random variable $X$ can be computed by averaging its conditional expectations, weighted by the probabilities of the conditioning events.

If $\{B_1, B_2, \ldots, B_k\}$ is a partition of the sample space with $P(B_i) > 0$ for each $i$, then

$$E[X] = \sum_{i=1}^{k} E[X \mid B_i] \cdot P(B_i).$$

Equivalently, if $Y$ is a discrete random variable taking values $y_1, y_2, \ldots, y_k$, then

$$E[X] = \sum_{i=1}^{k} E[X \mid Y = y_i] \cdot P(Y = y_i).$$

For example, suppose

• $E[X \mid Y = 0] = 4$ with $P(Y = 0) = 0.3,$
• $E[X \mid Y = 1] = 10$ with $P(Y = 1) = 0.7.$

Then

$$E[X] = 4 \cdot 0.3 + 10 \cdot 0.7 = 1.2 + 7 = 8.2.$$

This formula breaks a complicated expectation into a simple weighted average over branches.