Total Expectation Theorem

Theorem
Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a discrete random variable on $\EE$.

Let $\set {B_1 \mid B_2 \mid \cdots}$ be a partition of $\Omega$ such that $\map \Pr {B_i} > 0$ for each $i$.

Then:
 * $\ds \expect X = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}$

whenever this sum converges absolutely.

In the above:
 * $\expect X$ denotes the expectation of $X$
 * $\expect {X \mid B_i}$ denotes the conditional expectation of $X$ given $B_i$.