Total Expectation Theorem

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

Let $x$ be a discrete random variable on $\mathcal E$.

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

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 $B_i$.

Also known as
Some sources refer to this as the partition theorem, which causes ambiguity, as that name is used for other things as well.