\begin{definition}[Definition:Conditional Expectation]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $B$ be an event in $\struct {\Omega, \Sigma, \Pr}$ such that $\map \Pr B > 0$.
The '''conditional expectation of $X$ given $B$''' is written $\expect {X \mid B}$ and defined as:
:$\expect {X \mid B} = \ds \sum_{x \mathop \in \image X} x \condprob {X = x} B$
where:
:$\condprob {X = x} B$ denotes the conditional probability that $X = x$ given $B$
whenever this sum converges absolutely.
\end{definition}