Definition:Conditional Expectation/Discrete Case
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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.
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 2.5$: Conditional expectation and the partition theorem
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conditional expectation