Definition:Conditional Expectation

Definition
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.

Also see

 * Compare with expectation.